# Applications of Probabilistic Forecasting in Demand Response
## 1 Introduction
Researchers have investigated a broad range of methodologies, including time series forecasting techniques such as linear regression, exponential smoothing or ARIMA [5], as well as more advanced approaches like artificial neural networks [6], support vector regression [7], random forests [8], and fuzzy logic [9]. The advancement of deep learning techniques has enabled the development of STLF models that significantly outperform traditional methods. These models have demonstrated their efficiency in identifying and modeling nonlinear patterns within historical load data, leading to enhanced predictive accuracy [10]. For a comprehensive description of the various techniques used in STL, we refer the reader to the exhaustive reviews published in the international literature [11; 12]. In recent years, the development of probabilistic models for STLF has gained considerable attention from the scientific community, though the field is still in its early stages of development [13]. Probabilistic forecasting models provide predictive information that quantifies uncertainty by offering probability distributions over future quantities [14]. A widely adopted approach for obtaining probabilistic forecasts involves generating prediction intervals. A prediction interval defines the range within which the actual value of the dependent variable is expected to lie, based on a specified probability level [15]. This type of prediction is particularly valuable in decision-making processes, as it facilitates the minimization of risks. On the one hand, researchers are trying to develop increasingly accurate STLF models that offer greater insight into the uncertainty of predictions. Among the factors influencing this accuracy, DR programs and actions have emerged as a significant source of uncertainty. On the other, DR programs fundamentally alter the STLF process [16], necessitating a combination of back-casting and two types of forecasting beyond the traditional single forecast based on historical load and weather data. The back-cast, using data from regular hours unaffected by these programs, helps utilities assess the impact of demand response initiatives on load reduction. The first forecast follows traditional methodologies, predicting the load without DR activation. The second forecast, more intricate, utilizes back-cast insights to estimate load curtailment under DR conditions, informing utility decisions on program activation. The irregular activation of DR programs makes accurate estimation challenging, thus highlighting the importance of probabilistic forecasting in this domain as an emerging research focus. The number of published works on probabilistic STLF models constitute only a small fraction compared to those focused on point (deterministic) forecasting models. An example of the application of a short-term probabilistic load forecasting model is provided in [17]. The authors propose a model based on a Quantile Regression Neural Network with a Triangle Kernel Function (QRNNT), enabling the non-parametric estimation of hourly probability density functions and the generation of prediction intervals. The proposed model is applied to three major urban areas in Canada, Singapore, and China, demonstrating a good probabilistic forecasting performance and surpassing traditional models in point prediction accuracy. [PERSON] et al. [18] propose a model based on deep learning techniques to generate prediction intervals. The authors apply the model to five datasets to evaluate its predictive performance, including a dataset of electricity consumption from 370 customers at 15 min intervals. The results demonstrate the superiority of these techniques over other approaches in the international literature, both for point predictions and prediction intervals. [PERSON] et al. [13] propose an STLF model based on deep learning techniques, incorporating the novel inclusion of DR factors.