Maintaining power system stability is paramount, and a crucial factor in this stability is the system inertia. Inertia, a measure of a power system's ability to resist frequency fluctuations, is traditionally provided by the kinetic energy stored in the rotating masses of synchronous generators. However, the rapid increase in renewable energy sources, many of which connect to the grid through power electronic interfaces (inverters), is significantly altering the dynamics of power systems. Inverters, unlike synchronous generators, do not inherently contribute to system inertia. This shift towards inverter-based resources (IBRs) results in decreased overall system inertia and increased variability, making the system more susceptible to frequency oscillations caused by power imbalances. Accurate and continuous estimation of system inertia is therefore vital for maintaining grid stability and reliability. Existing methods for inertia estimation fall into two main categories: those triggered by significant disturbances and those employing measurements under normal operating conditions. Disturbance-triggered methods analyze frequency and active power measurements after a significant event, but their accuracy depends heavily on accurately identifying the disturbance, and they do not provide continuous estimations. Methods based on normal operating conditions often involve system identification procedures or rely on accurate real-time data, which may not always be available or practical. This research addresses these limitations by proposing a data-driven approach to continuously estimate system inertia using convolutional neural networks (CNNs). CNNs, known for their success in various pattern recognition tasks, offer the potential to learn the complex relationships between electrical variables and system inertia from data collected under normal operating conditions, without the need for explicit system modeling or disturbance-triggered events. This continuous estimation capability is particularly crucial for real-time monitoring and control in increasingly complex and variable power systems.

Numerous methods have been developed for estimating power system inertia, often categorized as either disturbance-triggered or methods using normal operating conditions. Disturbance-triggered approaches analyze the system response to large disturbances, but they suffer from the limitations of requiring a significant event and lacking continuous updates. Methods using normal operating conditions may rely on ambient measurements and system identification or require real-time data. These techniques often involve assumptions about underlying power system models or specific system identification procedures that may affect the accuracy and generality of the estimation. Some methods utilize dynamic mode decomposition or system identification to estimate inertia, but these approaches may still have limitations regarding continuous estimation and data requirements. Several studies have explored the use of artificial neural networks (ANNs) for inertia estimation, demonstrating promising results. However, the mechanisms underlying the networks' functioning often remain unclear, limiting the understanding of the relevant input features and data requirements for optimal performance. This research aims to address the gaps in existing literature by offering a data-driven, continuous estimation method using CNNs, along with a detailed analysis of its underlying mechanisms and data requirements.

This study develops a framework for continuous momentum estimation (momentum being an equivalent measure to inertia) using convolutional neural networks (CNNs). The methodology involves several key steps: 1. **Data Generation:** The researchers utilize a modified IEEE 39-bus benchmark system, incorporating stochastic loads to simulate real-world power fluctuations and introducing a synchronous compensator to add an additional inertia source. The system is simulated with variations in inertia constants of generators and the compensator, creating a diverse dataset for training and testing the CNN. Time-domain simulations, using the PAN simulator, model the stochastic fluctuations of loads using Ornstein-Uhlenbeck (OU) processes. The linearity of the model allows for a relationship to be established between stochastic load variations and system variable fluctuations. 2. **CNN Architecture:** A CNN architecture is designed, consisting of three preprocessing blocks, each with a convolutional layer and a max pooling layer. Multiple preprocessing pipelines process different voltage signals simultaneously. The outputs are then flattened and fed into two fully connected layers for momentum estimation. The Adam optimizer and a cyclical learning rate schedule are used during training. The loss function is the mean absolute error (MAE). 3. **Training and Validation:** The CNN is trained using the generated data. The training process is evaluated by monitoring training and validation losses to detect and prevent overfitting. The performance of the CNN is evaluated using a separate test set, with metrics such as mean absolute percentage error (MAPE) used to assess accuracy. 4. **Spectral Analysis and Correlation Maps:** To understand the CNN's decision-making process, spectral analysis of voltage signals is performed, and correlation maps are computed between convolutional layer outputs and the power of samples in the neurons' receptive fields for different frequency bands. This analysis helps to identify the frequency bands most influential in the CNN's momentum estimations. The goal is to determine which frequency bands of the input signals are most relevant for accurate prediction. 5. **Comparative Analysis:** The performance of the CNN is compared with other machine learning (ML) methods, including multi-layer perceptron (MLP), support vector regression (SVR), kernel ridge regression, K-nearest neighbor, and random forest. This comparison examines the relative accuracy of different ML approaches for this particular problem. 6. **Robustness Analysis:** The robustness of the CNN is examined by assessing its performance under varying conditions such as changes in load damping and the presence of different devices in the system, such as a virtual synchronous generator (VSG). These experiments provide insights into the generalization capabilities of the model.

The study's key findings demonstrate the effectiveness of the CNN-based approach for continuous momentum estimation: 1. **Accurate Momentum Estimation:** The CNN successfully estimates area momentum, achieving a low mean absolute percentage error (MAPE) on the test set, even with complex, heterogeneous networks. The MAPE value is improved by increasing the training data to include variable compensator inertia. The network learns to accurately predict momentum across a wide range of values. 2. **Frequency Band Significance:** Spectral analysis reveals that the CNN focuses on specific frequency bands (0.5-3 Hz and above 7 Hz for compensator analysis) of the voltage signals for accurate predictions. This indicates that these frequency ranges contain crucial information about the system's momentum. 3. **Superior Performance of CNN:** The CNN outperforms alternative ML methods (MLP, SVR, etc.) in momentum estimation accuracy, showcasing its efficacy in handling time-series data and learning complex relationships in power system dynamics. CNNs are more effective at handling time-series data which contributes to the improved accuracy. 4. **Robustness:** The CNN demonstrates robustness to load damping variability, indicating that the method is not heavily impacted by uncertainty in this system parameter. The methodology is also relatively unaffected by variations in load damping values. 5. **Impact of Device Addition:** The addition of new devices (such as a VSG) to the power system can affect the accuracy of the CNN's predictions unless the training data includes those devices. The CNN may struggle to make accurate predictions when encountering unexpected system configurations. 6. **Continuous Prediction Capability:** The CNN provides continuous momentum estimations, unlike many existing methods that require large disturbances. The ability to provide continuous predictions demonstrates a significant advantage over previous methods which often require large disturbances or specific events to function correctly. The ability to provide continuous predictions is superior to previously available methods. 7. **Data-Driven Approach:** The method is entirely data-driven, requiring only voltage measurements during normal operation, unlike some methods which require system identification or probing signals, simplifying the implementation and data requirements for effective momentum estimation. This data driven approach demonstrates a significant advantage in simplifying the implementation and data requirements for effective momentum estimation.

The results of this study demonstrate that a CNN can effectively learn the complex relationships between voltage signals and power system momentum, providing a data-driven and continuous estimation approach. The identification of significant frequency bands highlights the importance of considering spectral characteristics in inertia estimation. The superior performance of the CNN compared to other ML methods underscores its suitability for this specific problem. The robustness of the model to load damping variations demonstrates the applicability to real-world scenarios where these parameters are difficult to precisely quantify. However, the limitations surrounding the addition of new devices highlight the necessity of incorporating such scenarios into the training process for broader applicability. The findings address the need for accurate and continuous inertia estimations in modern power systems, providing a valuable tool for grid operators to manage stability in increasingly complex and variable power networks.

This research presents a novel CNN-based framework for continuous power system momentum estimation. The results showcase the accuracy, robustness, and data-driven nature of the approach, offering a valuable tool for grid operators. Future work should focus on further enhancing the model's robustness to various operating conditions, including seasonal variations in load, and investigating the optimal selection of voltage measurement locations for improved performance. The integration of this approach into real-time monitoring and control systems holds significant potential for improving power system stability and reliability.

While the proposed method demonstrates promising results, certain limitations should be acknowledged. The accuracy of the CNN depends on the quality and diversity of the training data. Adding new devices to the network not present during training may significantly reduce predictive accuracy. Although the CNN showed robustness to load damping variations within a realistic range, very large deviations might affect prediction accuracy. Future studies should investigate the impact of more complex system dynamics and more realistic load models on the CNN's performance. The current methodology focuses on a specific network topology, and further research is needed to assess its generalizability across various power system configurations.