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Very short-term load forecasting predicts load over one hour into the future in five minute steps and performs the moving forecast every five minutes. This is essential for area generation control and resource dispatch, and helps operators make good decisions. To quantify prediction accuracy, it is desirable to have a confidence interval for the forecasted load in real time. However, effective prediction is difficult in view of complicated dynamic load features. This paper develops an interacting multiple model approach using Kalman filter-trained neural networks. Because the hourly load input-output relations can be nearly-linear or nonlinear and it is not easy to know which one plays a more important role, it is difficult to accurately capture dynamic load features. Our key idea is to use a neural network trained by an extended Kalman filter to capture nearly-linear input-output load features, and a neural network trained by an unscented Kalman filter for nonlinear features. The overall estimate (together with confidence interval estimation) is then the dynamic mixing of the two model-conditioned results. Numerical testing demonstrates the significant value of the method for load forecasting with good confidence interval estimation.