Virtual Smart Metering in District Heating Networks via Heterogeneous Spatial-Temporal Graph Neural Networks

Intelligent operation of thermal energy networks aims to improve energy efficiency, reliability, and operational flexibility through data-driven control, predictive optimization, and early fault detection. Achieving these goals relies on sufficient observability, requiring continuous and well-distributed monitoring of thermal and hydraulic states. However, district heating systems are typically sparsely instrumented and frequently affected by sensor faults, limiting monitoring. Virtual sensing offers a cost-effective means to enhance observability, yet its development and validation remain limited in practice. Existing data-driven methods generally assume dense synchronized data, while analytical models rely on simplified hydraulic and thermal assumptions that may not adequately capture the behavior of heterogeneous network topologies. Consequently, modeling the coupled nonlinear dependencies between pressure, flow, and temperature under realistic operating conditions remains challenging. In addition, the lack of publicly available benchmark datasets hinders systematic comparison of virtual sensing approaches. To address these challenges, we propose a heterogeneous spatial-temporal graph neural network (HSTGNN) for constructing virtual smart heat meters. The model incorporates the functional relationships inherent in district heating networks and employs dedicated branches to learn graph structures and temporal dynamics for flow, temperature, and pressure measurements, thereby enabling the joint modeling of cross-variable and spatial correlations. To support further research, we introduce a controlled laboratory dataset collected at the Aalborg Smart Water Infrastructure Laboratory, providing synchronized high-resolution measurements representative of real operating conditions. Extensive experiments demonstrate that the proposed approach significantly outperforms existing baselines.

Robust Correlated Equilibrium: Definition and Computation

We study N-player finite games with costs perturbed due to time-varying disturbances in the underlying system and to that end we propose the concept of Robust Correlated Equilibrium that generalizes the definition of Correlated Equilibrium. Conditions under which the Robust Correlated Equilibrium exists are specified and a decentralized algorithm for learning strategies that are optimal in the sense of Robust Correlated Equilibrium is proposed. The primary contribution of the paper is the convergence analysis of the algorithm and to that end, we propose an extension of the celebrated Blackwell's Approachability theorem to games with costs that are not just time-average as in the original Blackwell's Approachability Theorem but also include time-average of previous algorithm iterates. The designed algorithm is applied to a practical water distribution network with pumps being the controllers and their costs being perturbed by uncertain consumption by consumers. Simulation results show that each controller achieves no regret and empirical distributions converge to the Robust Correlated Equilibrium.

On Bellman's principle of optimality and Reinforcement learning for safety-constrained Markov decision process

We study optimality for the safety-constrained Markov decision process which is the underlying framework for safe reinforcement learning. Specifically, we consider a constrained Markov decision process (with finite states and finite actions) where the goal of the decision maker is to reach a target set while avoiding an unsafe set(s) with certain probabilistic guarantees. Therefore the underlying Markov chain for any control policy will be multichain since by definition there exists a target set and an unsafe set. The decision maker also has to be optimal (with respect to a cost function) while navigating to the target set. This gives rise to a multi-objective optimization problem. We highlight the fact that Bellman's principle of optimality may not hold for constrained Markov decision problems with an underlying multichain structure (as shown by the counterexample). We resolve the counterexample by formulating the aforementioned multi-objective optimization problem as a zero-sum game and thereafter construct an asynchronous value iteration scheme for the Lagrangian (similar to Shapley's algorithm. Finally, we consider the reinforcement learning problem for the same and construct a modified Q-learning algorithm for learning the Lagrangian from data. We also provide a lower bound on the number of iterations required for learning the Lagrangian and corresponding error bounds.