Stochastic Model Predictive Control based on Mixed Random Variables for Economic Energy Management

Abstract

Optimal scheduling of batteries has significant potential to reduce electricity costs and to enhance grid resilience. However, effective battery scheduling must account for both physical constraints as well as uncertainties in consumption and generation of renewable energy sources. Instead of optimizing fixed battery power setpoints, we propose an approach that optimizes battery power intervals, allowing the optimization to explicitly account for uncertain consumption and generation as well as how the battery system should respond to them within its physical limits. Our method is based on mixed random variables, represented as mixtures of discrete and continuous probability distributions. Building on this representation, we develop an analytical stochastic formulation for minimizing electricity costs in a residential setting with load, photovoltaics, and battery storage. We demonstrate its effectiveness across real-world data from 15 residential buildings over five consecutive months. Compared with deterministic and probabilistic benchmark controllers, the proposed interval-based optimization achieves the lowest costs. These results show that mixed random variables are a practical and promising tool for decision-making under uncertainty.

Figures (5)

• Figure 1: General setting. Net-load $p_L(k)$, battery power $p_B(k)$ and grid power $p_G(k)$ are in balance pinter_probabilistic_2025.
• Figure 2: Separation of the continuous net-load PDF (a) in two mixed random variable PDFs featuring discrete and continuous properties (b) and (c). The battery power PDF (b) is restricted to a closed interval. The grid power PDF (c) contains a discrete event at $p_G^{des}$. The bigger the interval $[\underaccent{\bar{}}{p}_B, \Bar{p}_B]$, the more likely it is that the battery can be controlled such that the desired grid exchange $p_G^{des}$ can be followed, thus the bigger the discrete green event in (c). Adapted from pinter_probabilistic_2025.
• Figure 3: Probabilistic net-load forecast. The lower-right panel visualizes the forecasted PDF at 13:00, including the respective quantiles, the expected value, and the ground truth.
• Figure 4: Exemplary two-week period of selected import and export tariffs based on 2025 wholesale electricity prices.
• Figure 5: Operation of one building over three summer days. Under the deterministic MPC-FG, the fully discharged battery leads to grid imports at several time steps, causing unnecessary additional costs (black circles).