Metering-Error Prediction for Electronic Current Transformers Using Improved VMD and Two-Stage RA-BiGRU

Forecasting electronic current transformer (ECT) metering errors is challenging because long-term operation produces nonstationary and multiscale variations under changing load and environmental conditions. This study proposes an improved variational mode decomposition (VMD) and two-stage residual-attention bidirectional gated recurrent unit (RA-BiGRU) framework. A dual-constraint objective evaluates reconstruction fidelity, intermode independence, permutation entropy, and decomposition complexity, while hard constraints prevent low-mode under-decomposition. An improved whale optimization algorithm (IWOA) adaptively determines the VMD parameters. Stage 1 predicts the future increments of the final two trend-dominant modes and reconstructs preliminary ratio-error and phase-displacement forecasts. Stage 2 estimates the remaining forecast residuals, and horizon- and error-specific weights selected on the validation set limit over-correction. Experiments using 15,812 field samples at horizons of 1, 3, 6, and 12 consistently outperform the Persistence baseline. At h = 3, the proposed method reduces the ratio-error and phase-displacement RMSE values by 54.27% and 50.37%, respectively. These results demonstrate the complementary effects of trend-increment prediction and controlled residual correction.