Constructing uncertainty sets as unions of multiple subsets has emerged as an effective approach for creating compact and flexible uncertainty representations in data-driven robust optimization (RO). This paper focuses on two separate research questions. The first concerns the computational challenge in applying these uncertainty sets in RO-based predictive control. To address this, a monolithic mixed-integer representation of the uncertainty set is proposed to uniformly describe the union of multiple subsets, enabling the computation of the worst-case uncertainty scenario across all subsets within a single mixed-integer linear programming (MILP) problem. The second research question focuses on mitigating the conservatism of conventional RO formulations by leveraging the structure of the uncertainty set. To achieve this, a novel objective function is proposed to exploit the uncertainty set structure and integrate the existing RO and distributionally robust optimization (DRO) formulations, yielding less conservative solutions than conventional RO formulations, while avoiding the high-dimensional continuous uncertainty distributions and the high computational burden typically associated with existing DRO formulations. Given the proposed formulations, numerically efficient computation methods based on column-and-constraint generation (CCG) are also developed. Extensive simulations across three case studies are performed to demonstrate the effectiveness of the proposed schemes.

Robust Optimal Control (ROC) with adjustable uncertainties has proven to be effective in addressing critical challenges within modern energy networks, especially the reserve and provision problem. However, prior research on ROC with adjustable uncertainties has predominantly focused on the scenario of uncertainties modeled as continuous variables. In this paper, we explore ROC with binary adjustable uncertainties, where the uncertainties are modeled by binary decision variables, marking the first investigation of its kind. To tackle this new challenge, firstly we introduce a metric designed to quantitatively measure the extent of binary adjustable uncertainties. Then, to balance computational tractability and adaptability, we restrict control policies to be affine functions with respect to uncertainties, and propose a general design framework for ROC with binary adjustable uncertainties. To address the inherent computational demands of the original ROC problem, especially in large-scale applications, we employ strong duality (SD) and big-M-based reformulations to create a scalable and computationally efficient Mixed-Integer Linear Programming (MILP) formulation. Numerical simulations are conducted to showcase the performance of our proposed approach, demonstrating its applicability and effectiveness in handling binary adjustable uncertainties within the context of modern energy networks.

The way how the uncertainties are represented by sets plays a vital role in the performance of robust optimization (RO). This paper presents a novel approach leveraging machine learning (ML) techniques to construct data-driven uncertainty sets from historical uncertainty data for RO problems. The proposed method integrates Density-Based Spatial Clustering of Applications with Noise (DBSCAN), Gaussian Mixture Model (GMM), and Principle Component Analysis (PCA) systematically to eliminate the influence of uncertainty scenarios with low occurrence probability and generate a nonconvex uncertainty set that is a union of multiple basic subsets (box or ellipsoid) without sacrificing its computational tractability. In addition to presenting a comprehensive algorithm for uncertainty set development, this paper offers detailed guidelines for parameter tuning and performance analysis. By harnessing the well-established ML packages scikit-learn, a Python-based toolkit for implementing the proposed approach is also provided. Furthermore, a computationally efficient solution for a two-stage linear RO problem with the proposed data-driven uncertainty set is derived, alongside establishing a probabilistic guarantee of constraint satisfaction for out-of-sample uncertainties. Extensive numerical experiments, conducted on both synthetic and real-world datasets as well as an optimization-based control problem, are performed to demonstrate the efficacy of the proposed methodology.

Heat pump and thermal energy storage (HPTES) systems, which are widely utilized in modern buildings for providing domestic hot water, contribute to a large share of household electricity consumption. With the increasing integration of renewable energy sources (RES) into modern power grids, demand-side management (DSM) becomes crucial for balancing power generation and consumption by adjusting end users' power consumption. This paper explores an energy flexible Model Predictive Control (MPC) design for a class of HPTES systems to facilitate demand-side management. The proposed DSM strategy comprises two key components: i) flexibility assessment, and ii) flexibility exploitation. Firstly, for flexibility assessment, a tailored MPC formulation, supplemented by a set of auxiliary linear constraints, is developed to quantitatively assess the flexibility potential inherent in HPTES systems. Subsequently, in flexibility exploitation, the energy flexibility is effectively harnessed in response to feasible demand response (DR) requests, which can be formulated as a standard mixed-integer MPC problem. Numerical experiments, based on a real-world HPTES installation, are conducted to demonstrate the efficacy of the proposed design.

Towards integrating renewable electricity generation sources into the grid, an important facilitator is the energy flexibility provided by buildings' thermal inertia. Most of the existing research follows a single-step price- or incentive-based scheme for unlocking the flexibility potential of buildings. In contrast, this paper proposes a novel two-step design approach for better harnessing buildings' energy flexibility. In a first step, a robust optimization model is formulated for assessing the energy flexibility of buildings in the presence of uncertain predictions of external conditions, such as ambient temperature, solar irradiation, etc. In a second step, energy flexibility is activated in response to a feasible demand response (DR) request from grid operators without violating indoor temperature constraints, even in the presence of uncertain external conditions. The proposed approach is tested on a high-fidelity Modelica simulator to evaluate its effectiveness. Simulation results show that, compared with price-based demand-side management, the proposed approach achieves greater energy reduction during peak hours.

The efficacy of robust optimal control with adjustable uncertainty sets is verified in several domains under the perfect state information setting. This paper investigates constrained robust optimal control for linear systems with linear cost functions subject to uncertain disturbances and state measurement errors that are both residing in adjustable uncertainty sets. We first show that the class of affine feedback policies of state measurements are equivalent to the class of affine feedback policies of estimated disturbances in terms of their conservativeness. Then, we formulate and solve a robust optimal control problem with adjustable uncertainty sets by considering the disturbance feedback policies. In contrast to the conventional robust optimal control, where uncertainty sets are fixed and known a priori, the uncertainty sets themselves are regarded as decision variables in our design. In particular, given the metrics for evaluating the optimal size/shape of the polyhedral uncertainty sets, a bilinear optimization problem is formulated to decide the optimal size/shape of uncertainty sets and a corresponding optimal control policy to robustly guarantee that the system will respect its constraints for all admissible uncertainties. In addition, we introduce a convex approximation for the proposed scheme to provide a computationally efficient inner approximation of the original problem. The proposed scheme is illustrated by numerical simulation of a building temperature control problem to demonstrate its effectiveness.