2.1.1. Overall prediction process and problem modeling.

With the advancement of electricity marketization reform, the improvement of load forecasting accuracy is of great significance for reducing power generation costs, optimizing energy dispatch, and supporting the consumption of new energy. Especially with the increasing proportion of new energy, the volatility of power load has increased, making it difficult for traditional forecasting methods to meet actual demand [17]. PLF typically follows a systematic process, as shown in Fig 1.

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Fig 1. Power load forecasting process.

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Fig 1 shows the entire process, which includes requirements analysis, data collection and processing, feature extraction, method selection, parameter optimization, model prediction, and result correction. Among them, feature extraction and method selection are key steps that affect prediction accuracy. Traditional time series analysis methods, including autoregressive moving average models and exponential smoothing, perform well on linear trend data, but are difficult to effectively handle complex nonlinear load changes [18–19].

2.1.2. LSTM and bilstm time series modeling mechanisms.

In addition, in practical applications, a single prediction method often has limitations, such as traditional statistical methods being difficult to capture nonlinear changes in load. Although DL models have strong fitting ability, there is still room for optimization in extracting long-term features [20–21]. Therefore, this study attempts to construct CDL and comprehensively utilize the advantages of multiple DL methods to lift the accuracy and robustness of PLF. The framework of CDL is shown in Fig 2.

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Fig 2. Structural framework of CDL.

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In Fig 2, CDL adopts the TCN-LSTM-BiLSTM structure, which combines TCN, LSTM, and BiLSTM to fully extract the temporal features of power load data, improving prediction accuracy and generalization ability. Firstly, TCN serves as a feature extraction layer, utilizing causal convolution to ensure that predictions do not rely on future data. That is, for input sequence , the output at time only depends on the current and past time steps, as calculated in Equation (1).

(1)

In Equation (1), represents the convolution output at time ; represents the value of the input sequence at time . and are the convolution kernel’s weight and size. By expanding convolution, the Receptive Field (RF) exponentially grows with the number of layers, resulting in the RF of the -th layer as shown in Equation (2).

(2)

In equation (2), represents the receptive field size of the layer; represents the number of network layers. means the expansion rate of layer . In addition, TCN uses residual connections to alleviate the gradient vanishing issue, and the output at layer is shown in Equation (3).

(3)

In Equation (3), represents the output feature of the layer; represents the input feature of the layer; represents the convolution kernel weights of the layer; represents the convolution operation; represents the bias term; represents the nonlinear activation function. Through the operations of Equations (1) to (3), the features generated by TCN will be used for LSTM to further learn the dynamic changes in power load. In LSTM memory units, state updates rely on gating mechanisms, where the Forget Gate (FG), Input Gate (IG), and Output Gate (OG) respectively control the forgetting, storage, and output of information, as defined in Equation (4).

(4)

In Equation (4), is the Sigmoid function. is the Hadamard product. is the cellular state. (,,) and (,,) are the activation values and bias terms of the FG, IG, and OG. corresponds to the Hidden State (HS) value at . (,,) and (,,) are the weight matrices of the FG, IG, and OG in the HS and current state. , , and are parameters corresponding to IGs used to update cell states. is the activation value of the OG. LSTM is mainly used to enhance the temporal dependence of power loads, while BiLSTM further strengthens the bidirectional modeling ability of sequences. BiLSTM includes forward and backward LSTM. For the input sequence , the formulas for its forward HS and backward HS are shown in Equation (5).

(5)

The final HS splicing is shown in Equation (6).

(6)

Finally, the features extracted by BiLSTM are mapped through a fully connected layer to obtain the final load prediction. Compared with traditional RNN-based models, CDL integrates the parallel capability of TCN, the long-term dependency modeling of LSTM, and the bidirectional learning of BiLSTM, improving prediction accuracy and robustness.

2.1.3. Multi-model weighted reconstruction method.

However, multi-scale feature fusion remains suboptimal, as TCN focuses on short-term patterns while LSTM-based modules emphasize long-term dependencies, leading to imbalanced information contributions. To address this issue, WR is introduced to adaptively fuse multi-scale features through learnable weights, enabling dynamic adjustment of each component’s contribution. The formulation is given in Equation (7).

(7)

In Equation (7), means the final predicted value. denotes the amount of time series prediction algorithms used. is the predicted result of the -th model. is the weight of the predictive model. To address potential conflicting predictions among TCN, LSTM, and BiLSTM during load fluctuations—such as one model emphasizing short-term volatility while another prioritizes long-term trends, leading to inconsistent prediction directions or magnitudes—WR dynamically optimizes the weight vector using a regularized least-squares method. During optimization, if outputs from different submodels diverge significantly, the algorithm automatically reduces the weight of the more volatile submodel. This enhances the overall smoothness and stability of the prediction results.

To ensure principled optimization, WR follows two guidelines: multi-scale complementarity and risk minimization. The weight vector is constrained within a probability simplex to guarantee normalized contributions, while the objective function balances prediction error and stability penalty to achieve a Pareto-optimal trade-off. The prediction risk is measured by weighted mean squared error within each time window, and the stability term constrains weight fluctuations via L2 regularization or adjacent weight differences, thereby reducing variance and oscillation. The optimal weights are obtained by minimizing the objective function in Equation (8).

(8)

In Equation (8), is the weighted error loss function. is the actual power load value. is the time step. From this, it can be known that the entire formula consists of the first term, the prediction error, the second term, the regularization penalty term, and the third term, the constraint . These three elements jointly ensure that the multi-objective optimization problem can be transformed into a single-objective convex problem and naturally converge to the Pareto optimal equilibrium solution. is a regularization term used to prevent overfitting. The regularization parameter not only prevents overfitting but also directly determines the system’s ability to maintain prediction stability under varying load dynamics. Its mathematical relationship can be expressed as follows: when is large, the regularization term suppresses extreme deviations in the weight distribution, promoting more balanced contributions from multiple submodels.

2.1.4. Analysis of multi-scale feature fusion mechanism.

Consequently, during periods of intense load fluctuations, the upper bound of prediction variance is suppressed, resulting in more stable overall outputs. When is small, the model relies more heavily on a single submodel. While this may improve local accuracy, stability decreases. Therefore, the magnitude of positively correlates with the system’s predictive stability under complex load conditions. Selecting an appropriate value for achieves a balance between prediction accuracy and stability.

In terms of convergence, Equation (8) is optimized via mini-batch training, where CLPO performs global search and local refinement of the weight vector under fixed TCN/LSTM/BiLSTM parameters. Convergence is reached when the validation loss no longer decreases or the maximum iteration is met. In online prediction, Equation (8) is applied within a sliding window, where new samples update errors via sliding or exponential decay, enabling lightweight updates without retraining. Moreover, WR dynamically adjusts submodel contributions based on residuals and real-time conditions, assigning higher weights to TCN for short-term fluctuations, LSTM for long-term trends, and BiLSTM for bidirectional corrections. The resulting CDL-WR framework is illustrated in Fig 3.

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Fig 3. Multi-scale feature fusion and weight adaptation mechanism in the CDL framework based on WR.

https://doi.org/10.1371/journal.pone.0351428.g003

As shown in Fig 3, the entire model consists of an input data layer, a DL prediction model layer, a weight adjustment layer, and a final prediction output layer. The workflow begins with data input, incorporating various influencing factors such as historical electricity load data, weather characteristics, and temporal features, followed by the DL modeling phase. Specifically, the Time-Correlation Network (TCN) handles local pattern learning at short time scales, the LSTM model captures long-term dependency relationships, while the BiLSTM leverages bidirectional propagation advantages to enhance periodic feature extraction. After modeling completion, the three models generate prediction outputs that feed into the WR layer. This architecture not only demonstrates the feature extraction pathways of TCN, LSTM, and BiLSTM across different time scales but also achieves dynamic multi-scale information fusion through the WR layer. Unlike simple weighting methods, WR dynamically adjusts output weights of sub-models by integrating historical residuals with current input features during computation, maintaining dynamic equilibrium between short-term fluctuations and long-term trends across various load scenarios. For instance, during load spikes or peak periods, TCN pathway weights are significantly increased to strengthen local response capabilities, while in cyclical or stable phases, LSTM and BiLSTM weights dominate, thereby improving overall prediction stability and accuracy.

2.2.1. Basic principles of po algorithm.

After constructing a new type of CDL model, this study finds that although WR can optimize the fusion effect of different time scale features to some extent, improve the stability of load forecasting, there are still some problems. For example, the optimal value of the weight matrix depends on the distribution of the training data, making it difficult to maintain generalization across different datasets. In addition, the optimization process of the weight matrix usually relies on gradient descent methods, but when faced with complex nonlinear time series data, gradient descent methods may fall into local optima, leading to a decrease in the model’s performance [22–23]. Therefore, to further optimize the adjustment strategy of the weight matrix and improve the global optimization ability, this study introduces the Parrot Optimizer (PO) algorithm for model optimization. Fig 4 shows the flowchart of PO [24–26].

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Fig 4. Multi-strategy search mechanism of the PO algorithm including foraging, residence, and communication behaviors.

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As shown in Fig 4, PO initializes search agents and evaluates fitness, followed by ranking and strategy selection. Each agent randomly adopts one of three update behaviors. Specifically, foraging enables global exploration via high-variance perturbations, residence performs local refinement through contraction toward the current optimum, and communication enhances global perception by sharing differential information among individuals, forming implicit gradient directions in the weight space [27]. These mechanisms jointly balance global exploration, local exploitation, and directional search within a unified iteration, enabling stable and diverse optimization in high-dimensional non-convex spaces. Compared with PSO, GA, and SA, CLPO maintains solution diversity and convergence without gradient dependence, reducing sensitivity to initialization and improving the ability to escape local optima with smoother convergence [28–29]. The foraging behavior explores high-fitness regions and updates positions based on fitness gradients, as described in Equation (9).

(9)

In Equation (9), denotes the current location of the -th search agent in the -th generation. is the step size factor. is the fitness gradient of the current position. and are both perturbation terms. The stay behavior maintains the current solution to preserve local optima and triggers fine-tuning when fitness fluctuations are small, as calculated by Equation (10).

(10)

In Equation (10), is the location of the best individual in the current population. is the convergence factor. is the adjustment range. Interaction behavior adjusts positions based on neighborhood information, enabling information sharing among individuals and enhancing global search capabilities, as calculated by Equation (11).

(11)

In Equation (11), is the neighborhood of the -th individual. is the coefficient of influence between individuals. After completing the position update, the algorithm determines whether the maximum iteration count has been reached. If the condition is met, the optimal solution is output; otherwise, the process enters the boundary control phase to ensure the agent remains within the feasible solution space and continues iteration.

2.2.2. CLPO improvement strategy.

To enhance the GSA and local search capabilities of PO, the study makes multiple improvements to PO and proposes CLPO. The process is shown in Fig 5.

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Fig 5. Improved CLPO optimization process with chaotic initialization and adaptive parameter update strategy.

https://doi.org/10.1371/journal.pone.0351428.g005

As shown in Fig 5, CLPO enhances traditional PO algorithms by incorporating chaotic initialization and dynamic weight adjustment mechanisms, significantly improving search efficiency and stability. The chaotic mapping ensures more uniform initial population distribution, strengthening global search capabilities. Meanwhile, the dynamically decreasing inertial weights and velocity adjustment factors progressively reduce search step sizes during iterations, enabling a smooth transition from exploration to convergence phases. This approach facilitates the acquisition of more stable optimal solutions in complex high-dimensional parameter spaces. The optimization of CLPO can be summarized in four steps: ① Initialization via chaotic search generates a uniform initial population in the high-dimensional hyperparameter space encompassing prediction weights, learning rate, error correction coefficient, etc., with fitness calculated using objective function A (Equation 15); ② Dynamic classification of individuals into explorers and exploiters based on fitness, where explorers perform large-step jumps while exploiters fine-tune within the optimal neighborhood; ③ Continuous learning records historical optima, balancing global search and local convergence through decreasing inertia weights and velocity adjustment factors; ④ Iteration terminates when the upper limit is reached or the objective function’s decline over B consecutive generations falls below threshold C, outputting optimal parameters.

It should be noted that the “foraging–resting–communicating” behaviors in CLPO are heuristic abstractions of search operators, aiming to balance global exploration and local exploitation in non-convex, multi-modal spaces without relying on gradients. For the highly nonlinear objective composed of WR weights, network hyperparameters, and ECM terms, CLPO mainly enhances the ability to escape local optima and reduces sensitivity to initialization. When integrated with the combinatorial DL framework, CLPO encodes all key parameters into a unified high-dimensional search vector for joint optimization. By incorporating chaotic perturbation and dual convergence criteria, it effectively avoids local minima and ensures stable convergence. The CSI formulation is given in equation (12) [30–32].

(12)

In Equation (12), means the initial location of the -th search agent. and are the upper and lower bounds of the search space. is a chaotic variable generated by logistic mapping. In the position update stage, the PO algorithm adopts a fixed step size strategy for search. CLPO introduces the DDIW control strategy, gradually reducing the inertia weight during the search process to balance global and local search capabilities, as shown in Equation (13).

(13)

In Equation (13), is the inertia weight of generation . and refer to the maximum and minimum inertia weights. In addition, to enhance the convergence speed of CLPO and avoid oscillation problems during the search process, CLPO further introduces SAF during position updates, dynamically adjusting the movement speed of the search agent to make it more adaptable. The mathematical expression is shown in Equation (14) [33,34].

(14)

In Equation (14), means the speed of the -th individual in the -th generation. is the individual’s historical optimal position. denotes the global optimal position. and are learning factors. and are random numbers. is the velocity decay factor. This method enables the search agent to have a higher movement speed in the early stages to enhance GSA, and gradually converges to a local optimal solution in the later stages to improve the stability of optimization. To further enhance standardization, the optimization process of CLPO is formalized into the following objective function as shown in equation (15).

(15)

In equation (15), represents the actual load, while denotes the prediction result generated by the CDL framework under parameter and weight . The core optimization task involves dynamically updating in high-dimensional space through CLPO to minimize the objective function . Regarding convergence criteria, CLPO employs a dual-criterion approach: first, reaching the maximum iteration count; second, early termination when the fitness decline falls below a threshold for consecutive generations, ensuring computational efficiency.

2.2.3. ECM error correction mechanism.

In addition, this study introduces ECM to optimize BiLSTM in the third layer CDL framework, as shown in Fig 6.

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Fig 6. Error correction mechanism based on ECM integrated with BiLSTM prediction outputs.

https://doi.org/10.1371/journal.pone.0351428.g006

As shown in Fig 6, the ECM processing workflow sequentially comprises data input, feature extraction, sequence modeling, and error correction. The input layer first receives load data and relevant influencing parameters. Convolutional operations are then applied to extract local temporal features, which are subsequently fed into the BiLSTM network. Through bidirectional propagation, the BiLSTM model learns long-term dependencies, enabling superior performance in capturing short-term fluctuations and cyclical trends. Finally, the ECM dynamically corrects short-term prediction errors based on long-term equilibrium relationships, ensuring output results remain stable and accurate under complex load conditions. In this architecture, the ECM does not operate independently. Its key correction parameters—such as the error correction coefficient and long-term equilibrium term weights—are incorporated alongside the CDL–WR module’s weights into the CLPO’s high-dimensional search vector. The CLPO and main network undergo joint optimization under the same loss function. In other words, during each iteration, CLPO dynamically updates ECM parameters using residual information generated by ECM, enabling adaptive adjustment of correction intensity in response to load fluctuations. This establishes a closed-loop compensation mechanism: “prediction → residual → parameter update.” The specific correction calculation method is shown in equation (16).

(16)

In equation (16), represents the error correction value of at the current moment, and the correction process is , where represents the output of TCN-LSTM-BiLSTM, and is generated from the residuals at the previous moment, the long-term equilibrium term, and the predicted changes. is the correction factor. is the coefficient of long-term equilibrium relationship. is a short-term correction term. is a random perturbation term. It is important to emphasize that the introduction of the ECM does not operate in isolation from the DL framework, but rather forms a joint modeling approach with the BiLSTM output layer. Let the short-term prediction error be defined as shown in equation (17).

(17)

In equation (17), represents the actual load; represents the output of the deep network at time . Based on this, ECM introduces the long-term equilibrium relationship as shown in equation (18).

(18)

In equation (18), denotes the incrementally forecasted value corrected by the ECM, represents the coefficient of the error lag term, signifies the long-term equilibrium constraint variable (such as trend terms or seasonal components), and is the disturbance term. Through this formulation, the ECM integrates prior errors with long-term constraints to influence the current forecast, ensuring the network maintains dynamic equilibrium even under non-stationary load conditions. Furthermore, addressing the non-stationary load patterns common in real-world power systems, ECM does not rely on strict stationarity assumptions. Instead, it dynamically updates parameters and to adaptively adjust correction intensity across different load phases, thereby enhancing the model’s adaptability to abnormal scenarios. For instance, in extreme scenarios such as high penetration of renewable energy or sudden power outages, the model automatically increases the weight of long-term equilibrium terms to suppress abnormal fluctuations. During holiday periods with sudden load surges, it reduces long-term constraints to enhance short-term responsiveness, thereby ensuring the robustness and flexibility of the forecast.

2.2.4. CDL-CLPO-ECM collaborative optimization framework.

Finally, the paper proposes a novel PLF model that fuses CDL, CLPO, and ECM, as shown in Fig 7.

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Fig 7. Integrated CDL-CLPO-ECM framework with joint optimization and closed-loop correction.

https://doi.org/10.1371/journal.pone.0351428.g007

As shown in Fig 7, WR, CLPO, and ECM are not independent modules at the operational level but form a tightly coupled relationship through a unified parameter space and closed-loop optimization mechanism. Specifically, WR serves as an intermediate fusion layer that performs weighted combination of outputs from TCN, LSTM, and BiLSTM, with its weight vector constituting a critical component of overall optimization variables. CLPO jointly optimizes WR weights, network hyperparameters, and ECM correction coefficients within this unified parameter space, achieving cross-module parameter coordination. During each iteration cycle, WR-generated predictions first calculate residuals, which are fed into ECM for error correction. The corrected results then participate in CLPO’s fitness evaluation, forming a closed-loop update process of “prediction-residual-correction-reoptimization.” Under this mechanism, WR manages multi-scale feature contribution allocation, ECM corrects systematic biases, and CLPO dynamically coordinates parameter updates through global search. This enables collaborative convergence under a unified objective function while avoiding suboptimal solutions caused by independent optimization of individual modules. Compared to traditional RNN models, the proposed CDL architecture is not a simple concatenation but is built upon the mathematical foundations of time series decomposition and weighted information fusion. Let the input sequence be denoted as x(t), which can be decomposed into three typical components, as shown in equation (19).

(19)

In equation (19), represents the short-term fluctuating component, which is modeled in parallel within a finite RF via dilated convolutions in the TCN; represents the long-term trend component, which is memorized through the gating mechanism of the LSTM via cell state propagation; represents the bidirectional dependency component, jointly captured by the forward and backward hidden states of the BiLSTM. During the feature output stage, the hidden states generated by the three submodels are further weighted and fused as shown in equation (20).

(20)

In Equation (20), where represents the final composite representation, and , , and denote dynamic weights. This weighting mechanism ensures that contributions from features across different temporal scales adaptively adjust within the error minimization objective (see Equations (7)-(8)), mathematically achieving complementary information flow: TCN provides high-frequency local responses, LSTM ensures long-term trend continuity, while BiLSTM corrects unidirectional biases under periodic dependencies through bidirectional propagation.

3.1.1. Dataset and experimental design.

The study utilizes two publicly available electricity load datasets as test sources. The first is the UK National Grid Electricity Demand Dataset (UK-NGED), accessible at https://connecteddata.nationalgrid.co.uk/?utm_source=chatgpt.com. Provided by the UK National Grid operator, this dataset covers national electricity load information from 2015 to 2023 with 30-minute recording intervals, incorporating multidimensional variables such as temperature, wind speed, humidity, holiday indicators, and day-night labels. This dataset not only exhibits excellent continuity and representativeness but also supports multi-scale modeling and short-to-medium-term forecasting analysis, making it widely used in smart grid dispatch and demand response research [35]. The second dataset is the Electricité de France Smart Grid Dataset (EDF-SGD), available at https://www.neso.energy/data-portal/historic-demand-data?utm_source=chatgpt.com. Published by EDF, this dataset contains detailed load records for over 20,000 residential households and thousands of commercial and industrial users, with hourly and daily granularity spanning 2017–2022. Beyond load time series, it provides additional data including temperature, humidity, light intensity, electricity consumption category labels, and penetration ratios of renewable energy sources such as solar and wind power. These features enable effective modeling of electricity consumption behavior and are suitable for predictive evaluation in high penetration scenarios [36]. Table 1 shows the experimental environment and parameter settings.

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Table 1. Details of experimental environment and parameters.

https://doi.org/10.1371/journal.pone.0351428.t001

To further enhance model reproducibility and engineering feasibility, key implementation details are supplemented as follows: The TCN component in the CDL framework adopts a 3-layer expanded convolutional architecture with 3-dimensional convolution kernels and expansion rates of 1, 2, and 4 respectively. Both LSTM and BiLSTM networks utilize a 2-layer stacked architecture with 128 hidden units per layer. The ReLU activation function is employed for neural networks, while linear activation functions are applied to the output layer to accommodate regression tasks. During model training, batch size is set to 64 with a maximum training cycle limit of 300, accompanied by an Early Stopping strategy that terminates training prematurely when validation set loss fails to show significant improvement for 15 consecutive cycles. The Adam optimizer is selected with an initial learning rate of 0.001, complemented by a learning rate decay mechanism to improve convergence stability.

To address potential data insufficiency, missing values, and sampling imbalance in actual power load datasets, this study implements unified preprocessing on raw data prior to modeling. First, random missing values are filled using a combination of linear interpolation and sliding window mean filling, while continuous missing intervals are segmented through interpolation based on periodic characteristics of adjacent time periods to ensure temporal sequence continuity. Second, outliers are identified via a joint detection strategy combining boxplots and the 3σ principle, followed by local smoothing techniques to mitigate extreme noise interference during model training. Additionally, to alleviate sample distribution unevenness, resampling enhancement is applied to peak load scenarios and extreme weather events, ensuring stable learning performance across diverse load patterns. Finally, all input features undergo normalization processing to eliminate dimensionality discrepancies and improve model convergence efficiency.

3.1.2. Hyperparameter sensitivity analysis.

Based on the data in Table 1, this study first conducts value selection tests on the two types of hyperparameters that greatly affects the model, namely the prediction weight of the combined model and the correction coefficient of the ECM, as shown in Fig 8.

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Fig 8. Hyperparameter selection test result.

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In Fig 8 (a), when the prediction weight is set to 0.6, the loss value decreases the fastest and stabilizes at a lower level after 250 iterations. When the weights are 0.2 and 0.4, the model converges slowly, and the final loss is high, indicating that smaller weights may make it difficult for the model to fully utilize feature information during the optimization process. Therefore, the optimal value for predicting weights is 0.6, which helps to lift the convergence velocity and prediction accuracy. In Fig 8 (b), when the correction coefficient values are 0.1 and 0.3, the loss decreases slowly, and there is significant fluctuation, indicating weak error correction ability. As the correction coefficient increases, the rate of decrease in loss value accelerates, and the optimal convergence effect is achieved when the correction coefficient is 0.5. This indicates that enhancing the ECM error correction capability appropriately can significantly improve the stability and prediction accuracy. Therefore, the optimal value for the ECM correction coefficient is 0.5, and within this range, the model can converge in fewer iterations and maintain a lower level of error.

3.1.3. Ablation experiment analysis.

This study continues to conduct ablation testing on the proposed new model, as shown in Fig 9.

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Fig 9. Ablation test results.

https://doi.org/10.1371/journal.pone.0351428.g009

Fig 9 (a) and (b) indicate the results from the UK-HGED and EDF-SGD datasets. In Fig 9 (a), the prediction accuracy of the CDL-CLPO-ECM outperforms other models, reaching above 0.85 after 200 iterations and eventually tending towards 0.95. To quantify its computational cost, the study recorded the training time consumption and inference delay: under the same hardware environment (A100 GPU), a single training convergence took approximately 38 minutes, and the average inference delay was 1.76 ms per 1000 points. Further comparison was made with the multi-scale trapezoidal attention network of the lightweight Transformer. It converges faster in the early stage, but the fluctuation in the stable interval is significantly greater than that of this model, and the inference delay is slightly higher. The t-test results based on 10 independent experiments showed that both had statistically significant differences in stability and delay (p < 0.05). Overall, although this method uses approximately 200 rounds of iterations, the training and online inference overheads are both within a controllable range, and its stability is superior to that of lightweight attention structures at a confidence level of p < 0.05, indicating that the model maintains a reasonable balance between accuracy and efficiency. In addition, CDL-CLPO surpasses CDL and CDL-PO after 100 iterations, demonstrating the advantage of the CLPO algorithm in GSA and enabling the model to approach the optimal solution faster. In Fig 9 (b), CDL-CLPO-ECM surpasses 0.8 after 150 rounds and eventually stabilizes at around 0.96, which converges faster and has higher accuracy. This proves that the combination of CLPO and ECM can effectively optimize the model’s adaptability to multivariate load data on more complex datasets. In contrast, CDL-PO still fails to achieve the accuracy of CDL-CLPO after 300 rounds, indicating that the convergence effect of traditional PO optimization methods on complex datasets is not as good as CLPO. To systematically evaluate the impact of TCN, LSTM, BiLSTM, WR, PO, and CLPO components on model performance and computational cost, ablation experiments were designed focusing on computational overhead, number of Pareto optimal solutions, and inference latency. The results are shown in Table 2.

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Table 2. Ablation results of different components in the CDL-CLPO-ECM framework.

https://doi.org/10.1371/journal.pone.0351428.t002

As shown in Table 2, when used individually, the three base models maintain an overall MAPE of approximately 5%, identified only 3–4 Pareto optimal solutions, and exhibit inference delays within the 1.4–1.5 ms range. This indicates that while TCNs perform slightly better during peak periods, LSTMs demonstrate greater stability in periodic segments, and BiLSTMs offer bidirectional correction capabilities, all three models exhibit significant structural bias. After simple averaging ensemble, MAPE decreases to around 4% and the number of Pareto solutions increased to 6, but computational overhead rose simultaneously, indicating that complementarity is activated but the fusion method remained inefficient. After introducing WR, MAPE further decreases to the 3% level, and the number of Pareto optimal solutions expands to 7–9. Notably, after incorporating the regularization term, the solution set distribution becomes significantly more balanced. The improvement in stability metrics exceeds the increase in latency cost, exhibiting the typical characteristics of “high-efficiency Pareto frontier advancement.” Further combining PO with CLPO increases the number of Pareto solutions to 11 and 14, with coverage jumping to 68.4%. while inference latency increased only from 1.6 ms to approximately 1.76 ms. The computational overhead rises by less than 2 GFLOPs, demonstrating that CLPO’s multi-strategy search significantly expands the non-dominated solution space while maintaining an engineering-acceptable online inference load. This enables the final model to achieve an optimal trade-off between accuracy, solution set diversity, and computational efficiency.

3.1.4. Comparative analysis with baseline model.

To validate the normativity and effectiveness of CLPO, Table 3 presents its comparison results with various benchmark methods in terms of convergence and under different load scenarios.

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Table 3. Comparison of CLPO with baseline methods in convergence and load scenarios.

https://doi.org/10.1371/journal.pone.0351428.t003

As shown in Table 3, CLPO converges in approximately 200–250 iterations on both the UK-NGED and EDF-SGD datasets, significantly faster than other methods. In critical load scenarios, CLPO also demonstrates outstanding performance: achieving 92.4% accuracy for peak load prediction, nearly 90% seasonal stability, and an RMSE of only 4.95 kW under extreme weather conditions. More importantly, statistical tests reveal significant performance differences between CLPO and other methods across various scenarios: the comparison with Transformer-DGNN shows marginal statistical significance at p = 0.04; while comparisons with CDL-PO, EEMD-Attention, and VMD-ResNet revealed even more pronounced differences (p < 0.01 or even p < 0.001). These findings demonstrate that CLPO not only outperforms baseline methods in convergence efficiency and prediction accuracy but also exhibits statistically significant advantages, thereby further validating its applicability and robustness in complex load forecasting environments. To verify the comprehensive performance of the research model in multi-load environments, representative mainstream models in the field of PLF in recent years are selected for validation on publicly available datasets (UK-NGED and EDF-SGD). Among them, the Variational Mode Decomposition and Deep Residual Network Model (VMD-ResNet) model has good performance in modeling non-stationary load data. The Ensemble Empirical Mode Decomposition and Attention Mechanism (EEMD-Attention) model uses empirical mode decomposition to extract multi-scale features and introduces an attention mechanism to enhance feature weighting. It is a typical method for multi-scale forecasting research in recent years. The Transformer-DGNN combines the strong temporal modeling capability of the Transformer structure and the Dynamic Graph Neural Network. The transformer structure has strong time series modeling ability and dynamic graph modeling ability, representing the latest trend of graph neural network application in the field of power. These methods have been published in important journals at the intersection of energy systems and artificial intelligence in 2022–2024, with high citation rates and representativeness, making them suitable as baseline models for comparison. To ensure consistency in comparing models under different load conditions, all Precision, Recall, and F1 metrics are calculated based on the 7:1:2 training-validation-test splits of the UK-NGED and EDF-SGD datasets, organized chronologically. Each model is independently run 10 times under the same data partitioning, and the results are averaged. The results presented in Table 4 represent the average of these multiple runs. The precision, recall, F1, and Mean Forecast Time (MFT) are utilized as indicators, as listed in Table 4.

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Table 4. Test results of different prediction models.

https://doi.org/10.1371/journal.pone.0351428.t004

As shown in Table 4, when the prediction horizon extends from 6 to 12 months, all baseline models exhibit performance degradation, with F1 decreasing to 80%−84% and standard deviations generally increasing, indicating reduced stability in long-term forecasting. This is mainly due to accumulated trend drift and nonlinear amplification effects. In contrast, the proposed model achieves F1-scores of 91.52% ± 0.84% and 88.83% ± 0.91%, with lower variance and p-values of 0.008 and 0.004, demonstrating statistically significant improvements and better stability. The reduced performance attenuation benefits from cross-scale feature modeling and ECM-based error compensation. Meanwhile, inference delay increases only slightly from 1.59 s to 1.72 s, maintaining deployment feasibility.

3.2.1. Impact analysis of new energy penetration rate.

To enhance the representativeness of experiments and the generalization capability of data, this study introduces an additional regional load dataset from a provincial power company in China (China Provincial Multi-Region Load Dataset, CPMR-LD) in Section 3.2, alongside the two public datasets UK-NGED and EDF-SGD. This dataset covers residential, commercial, and industrial sub-area loads across multiple cities from 2019 to 2023, with a sampling interval of 15 minutes. It incorporates various engineering-side features such as meteorological data, holiday markers, equipment operational status, and load anomaly labels. This dataset exhibits typical characteristics of “load structures in developing country power grids” and “high real-time monitoring noise,” including common industrial issues such as sensor noise, intermittent missing records, and abnormal peak spikes. This study applies a unified 7:1:2 time-series partitioning and missing value imputation method to CPMR-LD, using it alongside the first two datasets throughout the simulation experiments in Section 3.2. To validate the effectiveness of this new model in practical electricity load forecasting, the study simulates the impact of varying penetration rates of renewable energy sources—such as photovoltaic and wind power—on load prediction, thereby verifying the model’s adaptability under different renewable energy shares. Fig 10 verifies the adaptability of the model under different proportions of new energy.

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Fig 10. Predictive stability test results of different models with PV and wind access.

https://doi.org/10.1371/journal.pone.0351428.g010

Fig 10 (a) and (b) show the predicted stability test results of different models when photovoltaic and wind energy are connected. As shown in Fig 10, within the range where the PV penetration rate increases from 6% to 36%, the proposed model maintains a prediction stability consistently above 85.6%, with significantly lower fluctuation than all comparison methods. This demonstrates that CDL-CLPO-ECM can continuously withstand load uncertainties caused by PV injection. When the penetration rate exceeds 24%, both VMD-ResNet and EEMD-Attention exhibit a noticeable decline in stability, indicating insufficient adaptability. The same advantage holds in wind energy scenarios: even with 36% wind penetration, the proposed model maintains stability above 80.4% with the smoothest curve. In contrast, Transformer-DGNN and VMD-ResNet rapidly deteriorate beyond 21% penetration, struggling to handle the strong random fluctuations induced by wind energy. Overall, CLPO’s global search and ECM’s long-term equilibrium correction jointly reduce prediction variance in high-penetration scenarios. Simultaneously, the regularization parameter λ in WR maintains a Pareto-optimal balance between prediction error and stability penalties under varying load dynamics, enabling the model to exhibit enhanced robustness in high-penetration PV and wind environments.

3.2.2. analysis of different load periods.

This study continues to test the predictive ability of different electricity peak periods, such as morning peak, evening peak, and nighttime testing models, and analyzes the impact of high load fluctuations on model stability, as shown in Fig 11.

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Fig 11. The relative error results of load forecasting in three periods under different load modes.

https://doi.org/10.1371/journal.pone.0351428.g011

Fig 11 (a) to (d) show the relative error results of power load prediction at different time periods under VMD-ResNet, EEMD-Attention, Transformer-DGNN, and the research model. As shown in Fig 11, the proposed model consistently exhibits the lowest relative error across three typical time periods: during nighttime low-fluctuation conditions, the error drops as low as 2.2%, demonstrating high-precision characterization of steady-state loads. During peak-hour high-fluctuation phases, while errors increase across all methods, the maximum error of the proposed approach remains only about 5.3%, significantly lower than other models. In contrast, VMD-ResNet and Transformer-DGNN frequently exceed 7% error during peak segments, with increased volatility under smaller sample sizes, indicating insufficient adaptability to complex transients. These results demonstrate that the synergistic integration of multi-scale feature extraction (TCN/LSTM/BiLSTM) with CLPO-ECM enables more precise capture of peak transitions and short-term disturbances, thereby maintaining superior performance across all time periods.

3.2.3. Analysis of different electricity usage types.

This study continues to test the ability of four methods for predicting residential and industrial electricity loads, as shown in Fig 12.

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Fig 12. Deviation results of load forecasting for residential and industrial electricity consumption.

https://doi.org/10.1371/journal.pone.0351428.g012

Fig 12 (a) and (b) show the comparison of prediction bias between residential and industrial electricity consumption under four different methods. In Fig 12 (a), in the prediction of residential electricity load, the prediction points of the research model are mostly distributed within a deviation range of ±1kW, and the overall error is the smallest. VMD-ResNet and Transformer-DGNN have significant deviations below 30 A current, with some point errors exceeding 3 kW, indicating their weak adaptability to low-power loads. In Fig 12 (b), in the prediction of industrial electricity load, the research model controls the prediction error within ±2 kW in the high-power region, i.e., above 40A, which is highly consistent with the standard value. EEMD-Attention and VMD-ResNet show an increase in deviation above 40 A, with some points having a deviation of 4–6 kW, indicating that their predictions are unstable under high load fluctuations. Overall, the research method combined with CLPO optimization and ECM shows better prediction accuracy and stability under different load modes, making it suitable for multi-scenario load forecasting in smart grids. Finally, this study takes MAPE, RMSE, throughput, and response time as indices to test the power load in sunny, cloudy, and rainstorm days, as exhibited in Table 5. The extreme weather test was constructed using a subset of the UK-NGED and EDF-SGD test datasets labeled with meteorological conditions. Clear, cloudy, and heavy rain days were segmented and selected based on historical weather records. The data partitioning for the training phase remained unchanged; evaluation was conducted separately for each weather subset during the testing phase. Each model was run 10 times, and the metric values were averaged.

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Table 5. Model prediction indexes under different weather conditions.

https://doi.org/10.1371/journal.pone.0351428.t005

3.2.4. Robustness analysis for extreme weather events.

To avoid context-free response times, this study explicitly defines the metric as corresponding to a 1-hour rolling prediction window, an input sequence length of 96 points (15-minute resolution), and a single-sample online inference scenario. When batch-processing 128 input sequences, the per-sample response time maintains linear scalability.

As shown in Table 5, under sunny conditions, the proposed model achieves MAPE of 2.92% ± 0.27% and RMSE of 2.45 ± 0.23 kW, outperforming all baselines with lower variance. Under cloudy and stormy conditions, although all models show increased errors, the proposed method maintains smaller standard deviations (e.g., MAPE std reduced by ~15%–25%), indicating stronger robustness to weather disturbances. In storm scenarios, its MAPE remains at 6.42% ± 0.55%, still lower than Transformer-DGNN. Additionally, throughput and response time remain stable (1.38–1.72 s), demonstrating consistent computational efficiency. These results confirm that the model maintains both accuracy and stability under complex and uncertain load conditions.

3.2.5. Model robustness and distribution generalization analysis.

To validate the model’s robustness under structural changes and data distribution disturbances, structural substitution and data distribution adjustment experiments are conducted, with results presented in Table 6.

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Table 6. Robustness evaluation under model structure replacement and data distribution alignment.

https://doi.org/10.1371/journal.pone.0351428.t006

As shown in Table 6, the model maintains stable performance under various structural replacements and data distribution perturbations, with MAE and RMSE variations kept within 3%−8%—demonstrating that single-module replacements do not compromise overall performance. Notably, removing WR or substituting CLPO with PSO results in significant performance degradation, further validating WR’s weighted robustness and CLPO’s global search advantage. The data distribution adjustment test reveals that ECM effectively absorbs trend deviations and maintains stability metrics (1 − CV) within acceptable ranges under peak load disturbances and distribution shifts, indicating the structure’s strong distribution generalization capability and robustness in practical scheduling scenarios.