Continuous estimation of power system inertia using convolutional neural networks

The increasing penetration of inverter-based resources (IBRs) due to growing renewable energy shares changes the dynamics and stability of power systems compared to traditional synchronous generator-dominated grids. System inertia—linked to kinetic energy in rotating masses—helps counteract frequency excursions following power imbalances, but IBRs typically do not provide inertia, leading to reduced and more variable system inertia. This threatens frequency stability and motivates reliable, preferably online, inertia estimation methods. Existing approaches are often event-triggered (requiring disturbance detection and precise timing) or rely on ambient data and/or probing signals and system identification, each with practical limitations. The authors reformulate the characterization via momentum M = Σ(2 H_i S_i / f_n), an additive, incremental measure equivalent in information to inertia but better suited when devices are added. They propose a data-driven framework using convolutional neural networks (CNNs) to estimate, continuously and online, the momentum of power system areas from voltage time series measured at a limited set of buses under normal operating conditions with stochastic load fluctuations. Validation is conducted on a modified IEEE 39-bus system with stochastic loads and additional devices, and spectral analyses with input–output correlation maps are used to interpret how the CNN exploits frequency-domain features relevant to inertia.

Inertia estimation methods are broadly divided into: (i) event-triggered algorithms that analyze frequency and power measurements after significant disturbances; these require accurate event timing and do not provide continuous updates; (ii) methods based on ambient measurements or responses to injected probing signals; the former often require system identification and accurate real-time data, while the latter are impractical for large systems and perturb the system. Comprehensive reviews exist on inertia estimation in power systems. Prior data-driven and modal-identification approaches (e.g., covariance-based, dynamic mode decomposition, ARMAX-based) have been explored. ML methods, including ANNs and CNNs, have been applied to inertia estimation, but typically require probing or lack interpretability. Compared to these, this work is fully data-driven using natural load-induced fluctuations and provides an in-depth mechanistic analysis of CNN operation via spectral correlation mapping, guiding training data selection.

Problem formulation and metric: Instead of inertia H, the study uses area momentum M = Σ_j (2 H_j S_j / f_n) over synchronous generators in an area, an additive measure aligned with network changes. The network is partitioned into areas (e.g., the IEEE 39-bus split into four areas), enabling estimation of area momentum and reducing prediction dimensionality. Data generation and system setup: Simulations are conducted on a modified IEEE 39-bus benchmark system (46 lines, 10 generators), with a synchronous compensator added at bus 8 (S_c = 100 MVA). Loads are stochastic: each load’s active power follows an Ornstein–Uhlenbeck (OU) process (mean equal to original load, σ = 0.5% of mean, mean reversion rate 2 s), introducing continuous ambient perturbations around the power-flow (PF) operating point. The compensator’s reactive power is set to zero at PF via an fsolve-based optimization of its voltage setpoint. A virtual synchronous generator (VSG) model (grid-forming control implementing swing equation and PLL) is used in some experiments to test generalization. Time-domain simulations are performed using the PAN simulator capable of solving stochastic differential-algebraic equations. For training datasets, grids over generator inertia constants (e.g., 6×6 across H_G2 and H_G3 spanning roughly ±1 s around nominal values) and, in extended settings, over compensator inertia (e.g., 0.1, 2.5, 5 s) are sampled. For each combination, long simulations (~300,000 s) are run; input windows are 60 s sampled at 40 Hz (2400 samples). Traces are normalized to zero mean and unit variance (subtracting PF value). Inputs/outputs: Inputs are time series of voltages at a limited number of buses (e.g., direct/quadrature components at buses 3, 14, 17, 39; in simple tests, a single channel V_d at bus 3). Outputs are the area momentum values (single target L=1 per model). CNN architecture and training: Architecture inspired by Deep Filtering: for each input signal, a preprocessing pipeline of three 1D convolutional layers (with 16, 32, 64 filters; kernel size 5; stride 1) each followed by max pooling (pool size 4) extracts features. No nonlinear activation in preprocessing layers. Outputs from all pipelines are concatenated and flattened (e.g., M×N×64×36 = 9216 features) and fed to two dense layers (first with 64 ReLU units; final linear unit for momentum). Loss is mean absolute error (MAE). Optimizer: Adam with fixed lr 5e−5 or cyclical learning rate (base 5e−5, max 2e−3, step size ≈ 10×epochs). Glorot initialization. Hyperparameter sweeps considered kernel sizes/strides/pooling sizes; chosen configuration balances slightly higher validation loss with greater downstream feature dimensionality to aid generalization. Evaluation uses validation/test splits with inertia offsets between train/val/test to probe generalization. Correlation map analysis: Following Schirrmeister et al., input-feature unit-output correlation maps are built by bandpass filtering inputs across 0.1–20 Hz into logarithmically spaced bands, computing squared mean envelopes over each convolutional unit’s receptive field, and correlating with unit outputs to reveal frequency bands the CNN exploits. Experimental designs: (1) Two-class momentum discrimination using averaged low vs high momentum conditions (aggregating nearby H_G2/H_G3 points) with single-bus input; (2) Inclusion of compensator inertia variability to test model robustness and expanded training grids; (3) Continuous regression over full momentum grid using multi-bus inputs; (4) Stepwise momentum changes over 3-hour simulations to assess temporal tracking; (5) Robustness tests to load damping variations D∈[0,4]; (6) Generalization test replacing a synchronous generator with a VSG. Spectral analyses: PSDs and spectrograms of voltage traces are computed to interpret how generator vs compensator inertia modulate spectral peaks: generator inertia predominantly shifts peaks in ~0.5–1.5 Hz; compensator inertia shifts higher-frequency peaks ~5–20 Hz. These insights guide training set augmentation to cover distinct spectral regimes critical for CNN performance.

- Single-bus two-level momentum estimation: A CNN trained on V_d at bus 3 discriminated between low (≈0.176 GW·s²) and high (≈0.266 GW·s²) momentum with test MAPE = 1.79%. Correlation maps showed sensitivity primarily around ~1.2 Hz and across 0.7–3 Hz bands; the 0.5–1 Hz band shows high correlation in trained and untrained networks due to strong signal power and is less informative for classification. - Frequency bands of importance: R² analyses superimposed on average spectra identified (0.7–1) Hz, (1–1.5) Hz, and (1.5–3) Hz as most predictive, with (0.7–1) Hz most crucial in agreement with correlation maps. - Effect of compensator inertia: A CNN trained without compensator variability failed when momentum changes were induced solely by increasing compensator inertia (predicted 0.184 ± 0.003 vs true 0.197 GW·s²). After augmenting training to include compensator inertia up to ~5–6 s, test MAPE improved to 0.87% and predictions matched true momentum for both generator-driven and compensator-driven cases (e.g., 0.192 ± 0.013 and 0.197 ± 0.002 GW·s² for two scenarios with M = 0.197 GW·s²). Correlation maps then showed additional sensitivity above ~7 Hz, consistent with compensator-induced spectral shifts. - Continuous regression across grid: Using voltages from buses 3, 14, 17, 39, the CNN achieved test MAPE = 2.67% over momentum range ≈(0.17–0.28) GW·s². Slightly reduced accuracy in mid-range (0.21–0.23) GW·s² likely due to many H combinations mapping to similar M values. - Stepwise tracking over hours: The CNN accurately tracked step changes in momentum over 3-hour simulations when changes were due to generator inertia variations. Performance initially degraded when only compensator inertia varied; after augmenting training to include compensator inertia values (0.1, 2.5, 5 s), step responses were accurately tracked (overall test MAPE 2.24%). - Spectral mechanism: Generator inertia increases shift PSD peaks in ~0.5–1.5 Hz to lower frequencies and reduce variance; compensator inertia shifts higher-frequency peaks (~5–20 Hz) monotonically with its inertia, enabling learnable mapping when included in training. - Robustness to load damping: Varying load damping D from 0 to 4 caused only slight increases in MAPE; PSD changes were mainly at very low frequencies (<~0.1 Hz) or in peak amplitudes, whereas the CNN relies mostly on peak locations, preserving accuracy. - Generalization to new devices: Replacing a synchronous generator with a VSG altered PSDs around ~1.5 Hz and ~5 Hz; the CNN remained accurate only when PSDs overlapped with the training spectra below ~1 Hz, demonstrating the need to include new device spectral signatures in training for reliable estimation. - Overall: CNNs provide continuous, event-independent momentum estimation with MAPE rarely exceeding ~4% in tested scenarios, rapidly detecting momentum changes and interpreting underlying spectral features via learned filters.

The study demonstrates that CNNs can continuously estimate area momentum from ambient voltage measurements without requiring disturbance detection or probing signals. The convolutional front-end effectively performs linear filtering that emphasizes informative spectral bands tied to inertia-related electromechanical modes, enabling the dense layers to map these features to momentum. This addresses the challenge of online inertia (momentum) estimation in low-inertia, IBR-rich systems by providing continuous updates and responsiveness to changes in generator and compensator inertia. The interpretability through correlation maps clarifies which frequency bands are leveraged, guiding construction of training datasets that cover relevant spectral regimes (e.g., 0.7–3 Hz for generators; >~7 Hz for compensators). The approach is robust to realistic variations in load damping but sensitive to device classes not represented during training (e.g., VSGs), underscoring the importance of including anticipated device configurations and their spectral effects in training to ensure generalization across operating conditions.

This work introduces a data-driven CNN framework for continuous estimation of power system momentum using ambient voltage measurements at a few buses. The method achieves low errors across classification and regression tasks, detects stepwise momentum changes, and provides mechanistic insight via frequency-domain correlation mapping that informs training data design. The approach’s advantages include continuous, event-independent estimation and learned selection of informative frequency bands, avoiding handcrafted features. Future work should: (i) develop heuristics for selecting buses and electrical variables based on network topology and variable statistics; (ii) model daily and seasonal load variability to broaden operating condition coverage; (iii) incorporate additional device types (e.g., grid-forming converters/VSGs) into training to enhance generalization; and (iv) evaluate on real-world PMU datasets and larger networks to assess scalability and practical deployment.

- Generalization depends on training data coverage of spectral conditions: devices or operating regimes absent from training (e.g., VSG replacing a generator, or compensator-induced high-frequency shifts) can degrade accuracy until included in the dataset. - Performance can be sensitive to ambiguous mappings where multiple inertia combinations yield similar momentum values, slightly reducing accuracy in mid-range M. - Validation uses synthetic data from a benchmark network with stochastic loads; applicability to diverse real-world networks and measurement conditions requires further testing. - The method relies on availability of synchronized voltage measurements at selected buses; bus/variable selection strategy is not optimized in this study. - Daily/seasonal variability of loads was not modeled and may affect spectral characteristics and estimation if not represented during training.