Building safe, reliable, fully automated energy smart grid systems requires a trustworthy electric load forecasting system. Recent work has shown the efficacy of Long Short-Term Memory neural networks in energy load forecasting. However, such predictions do not come with an estimate of uncertainty, which can be dangerous when critical decisions are being made autonomously in energy production and distribution. In this paper, we present methods for evaluating the uncertainty in short-term electrical load predictions for both deep learning and gradient tree boosting. We train Bayesian deep learning and gradient boosting models with real electric load data and show that an uncertainty estimate may be obtained alongside the prediction itself with minimal loss of accuracy. We find that the uncertainty estimates obtained are robust to changes in the input features. This result is an important step in building reliable autonomous smart grids.

Many statistical and machine learning models for prediction make use of historical data as an input and produce single or small numbers of output values. To forecast over many timesteps, it is necessary to run the program recursively. This leads to a compounding of errors, which has adverse effects on accuracy for long forecast periods. In this paper, we show this can be mitigated through the addition of generating features which can have an “anchoring” effect on recurrent forecasts, limiting the amount of compounded error in the long term. This is studied experimentally on a benchmark energy dataset using two machine learning models LSTM and XGBoost. Prediction accuracy over differing forecast lengths is compared using the forecasting MAPE. It is found that for LSTM model the accuracy of short term energy forecasting by using a past energy consumption value as a feature is higher than the accuracy when not using past values as a feature. The opposite behavior takes place for the long term energy forecasting. For the XGBoost model, the accuracy for both short and long term energy forecasting is higher when not using past values as a feature.

In this paper, we prove the existence of fixed points for nonlinear and semilinear operators on order intervals. The abstract results unified some methods in studying the existence of positive solutions for boundary and initial value problems of nonlinear difference and differential equations. Applications are shown by examples.

This paper studies a class of integral boundary value problem of fractional q-difference equations. We first give an explicit expression for the associated Green’s function and obtain an important property of the function. The new property allows us to prove sufficient conditions for the existence of positive solutions based on the associated parameter. The results are derived from the application of a fixed point theorem on order intervals.