We address day-ahead transmission topology planning and congestion management as a sequential, multi-objective optimization problem and develop two complementary algorithms for it: an exact enumeration method and a tailored evolutionary heuristic. The problem is formulated with four operational objectives reflecting real TSO decision criteria: worst-case line loading under N-1 security, topological depth, number of switching actions, and time spent in non-reference topologies, over a 24-hour horizon. We introduce the block algorithm, an exact method that exploits the temporal block structure of feasible strategies to enumerate the complete Pareto front; for fixed operational bounds on depth and switch count, its evaluation count grows polynomially with the planning horizon. We complement it with a multi-objective evolutionary algorithm based on NSGA-III, with structure-guided initialization and problem-specific variation operators tailored to the topology-planning structure. Using real operational data from the Dutch high-voltage grid operated by TenneT TSO, we show that the block algorithm computes the full Pareto front for a highly congested day in under three minutes, and that the evolutionary algorithm converges toward but does not recover the exact front. The block algorithm thus provides both a practical decision-support tool and a ground-truth benchmark for future heuristic and learning-based methods on this problem class.

Network design under uncertainty arises in countless real-world settings and can be captured by the Stochastic Steiner Tree Problem (SSTP). Although there are a few approaches specifically tailored to this stochastic optimization problem, there are considerably more state-of-the-art heuristics for its deterministic variant, the Steiner Tree Problem (STP). In this work, we show how to leverage an existing STP heuristic in building a novel method for solving its stochastic variant, the SSTP. This approach is a powerful, yet simple and easy-to-implement way of solving this complex problem. We test our method using benchmark instances from the literature. Numerical results show considerably faster computation times compared to the state-of-the-art, with a gap of approximately 5

One of the main responsibilities of a Transmission System Operator (TSO) operating an electric grid is to maintain a designated frequency (e.g., 50 Hz in Europe). To achieve this, TSOs have created several products called frequency-supporting ancillary services. The Frequency Containment Reserve (FCR) is one of these ancillary service products. This article focuses on the TSO problem of determining the volume procured for FCR. Specifically, we investigate the potential benefits and impact on grid security when transitioning from a traditionally static procurement method to a dynamic strategy for FCR volume. We take the Nordic synchronous area in Europe as a case study and use a diffusion model to capture its frequency development. We introduce a controlled mean-reversal parameter to assess changes in FCR obligations, in particular for the Nordic FCR-N ancillary service product. We establish closed-form expressions for exceedance probabilities and use historical frequency data as input to calibrate the model. We show that dynamic dimensioning approaches for FCR have the potential to significantly reduce exceedance probabilities (up to 37%) while maintaining the total yearly procured FCR volume equal to that of the current static approach. Alternatively, a dynamic dimensioning approach could significantly increase security at limited extra cost.

Solving large two-stage stochastic mixed-integer programs is computationally challenging. We propose LOTUS, a subset-based warm-start framework that enhances Dual Decomposition under fixed time budgets. By initializing the dual search with informed multipliers, LOTUS accelerates primal convergence and partially alleviates the impact of weak LP relaxations. Through an extensive computational study on production planning instances, we show that, within two hours, LOTUS yields significantly better primal solutions in 45.83

Graphs are commonly used to model various complex systems, including social networks, power grids, transportation networks, and biological systems. In many applications, the connectivity of these networks can be expressed through the Mean First Passage Times (MFPTs) of a Markov chain modeling a random walker on the graph. In this paper, we generalize the network metrics based on Markov chains' MFPTs and extend them to networks affected by uncertainty, in which edges may fail and hence not be present according to a pre-determined stochastic model. To find optimally connected Markov chains, we present a parameterization-free method for optimizing the MFPTs of the Markov chain. More specifically, we present an efficient Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm in the context of Markov chain optimization. The proposed algorithm is suitable for both fixed and random networks. Using various numerical experiments, we demonstrate scalability compared to established benchmarks. Importantly, our algorithm finds an optimal solution without requiring prior knowledge of edge failure probabilities, allowing for an online optimization approach.

We introduce a new Ising-type framework for opinion dynamics that explicitly separates private preferences from publicly expressed binary opinions and naturally incorporates neutral agents. Each individual is endowed with an immutable hidden preference, while public opinions evolve through Metropolis dynamics on a finite graph. This formulation extends classical sociophysical Ising models by capturing the tension between internal conviction, social conformity, and neutrality. Focusing on highly symmetric grid networks and spatially structured hidden-preference patterns, we analyze the resulting low-temperature dynamics using the pathwise approach to metastability. We provide a complete characterization of stable and metastable configurations, identify the maximal stability level of the energy landscape, and derive sharp asymptotics for hitting and mixing times. A central technical contribution is a new family of isoperimetric inequalities for polyominoes on the torus, which emerge from a geometric representation of opinion clusters and play a key role in determining critical configurations and energy barriers. Our results provide a quantitative understanding of how spatial heterogeneity in hidden preferences qualitatively reshapes collective opinion transitions and illustrate the power of geometric and probabilistic methods in the study of complex interacting systems.

Load-alteringattacks (LAAs) targeting a large number of IoT-based high-wattage devices (e.g., smart electric vehicle charging stations) can lead to serious disruptions of power grid operations. In this work, we aim to uncover spatiotemporal characteristics of LAAs that can lead to serious impact. The problem is challenging since existing protection measures, such as N-1 security designed to make the power system resilient to single component failures, also provide resilience to load changes to a large extent. Thus, strategically injected load perturbations that lead to network failure can be regarded as rare events. To this end, we adopt a rare-event sampling approach to uncover LAAs distributed temporally and spatially across the power network. The key advantage of this sampling method is the ability to sample efficiently from multi-modal conditional distributions with disconnected support. Furthermore, we systematically compare the impacts of static (one-time manipulation of demand) and dynamic (attack over multiple time periods) LAAs. We perform extensive simulations using benchmark IEEE test bus systems. The results show (i) the superiority and the need for rare-event sampling in the context of uncovering LAAs as compared to other sampling methodologies, (ii) statistical analysis of attack characteristics and impacts of static and dynamic LAAs, and (iii) cascade sizes (due to LAA) for different network sizes and load conditions.

We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature $\beta$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $\beta\to\infty$ in the case of periodic boundary conditions. The hard-core constraints and the grid symmetry make the structure of the critical configurations, also known as essential saddles, for this transition very rich and complex. We provide a comprehensive geometrical characterization of the set of critical configurations that are asymptotically visited with probability one. In particular, we develop a novel isoperimetric inequality for hard-core configurations with a fixed number of particles and we show how not only their size but also their shape determines the characterization of the saddles.

In transmission networks, power flows and network topology are deeply intertwined due to power flow physics. Recent literature shows that a specific more hierarchical network structure can effectively inhibit the propagation of line failures across the entire system. In particular, a novel approach named tree partitioning has been proposed, which seeks to bolster the robustness of power networks through strategic alterations in network topology, accomplished via targeted line switching actions. Several tree partitioning problem formulations have been proposed by considering different objectives, among which power flow disruption and network congestion. Furthermore, various heuristic methods based on a two-stage and recursive approach have been proposed. The present work provides a general framework for tree partitioning problems based on mixed-integer linear programming (MILP). In particular, we present a novel MILP formulation to optimally solve tree partitioning problems and also propose a two-stage heuristic based on MILP. We perform extensive numerical experiments to solve two tree partitioning problem variants, demonstrating the excellent performance of our solution methods. Lastly, through exhaustive cascading failure simulations, we compare the effectiveness of various tree partitioning strategies and show that, on average, they can achieve a substantial reduction in lost load compared to the original topologies.

Power grid operation is becoming increasingly complex due to the rising integration of renewable energy sources and the need for more adaptive control strategies. Reinforcement Learning (RL) has emerged as a promising approach to power network control (PNC), offering the potential to enhance decision-making in dynamic and uncertain environments. The Learning To Run a Power Network (L2RPN) competitions have played a key role in accelerating research by providing standardized benchmarks and problem formulations, leading to rapid advancements in RL-based methods. This survey provides a comprehensive and structured overview of RL applications for power grid topology optimization, categorizing existing techniques, highlighting key design choices, and identifying gaps in current research. Additionally, we present a comparative numerical study evaluating the impact of commonly applied RL-based methods, offering insights into their practical effectiveness. By consolidating existing research and outlining open challenges, this survey aims to provide a foundation for future advancements in RL-driven power grid optimization.

Ship pipe route design is often overlooked in the context of the energy transition, although it is a crucial driver for design time and costs. Motivated by this, we propose a mathematical approach for modeling uncertainty in pipe routing with deterministic optimization, stochastic programming, and robust optimization. The uncertainty entails not knowing which type of fuel will be used in the ship’s future. All three models are based on state-of-the-art integer linear programming models for the Stochastic Steiner Forest Problem and adjusted to the maritime domain using specific constraints for pipe routing. Our results highlight the importance of accounting for uncertainty in ship pipe routing, demonstrating cost reductions of up to 22% based on experiments with artificial and realistic data. Our methods enable engineers to explore different levels of preparedness for the energy transition with minimal effort during the early design phase.

Two-stage stochastic programs become computationally challenging when the number of scenarios representing parameter uncertainties grows. Motivated by this, we propose the TULIP-algorithm ("Two-step warm start method Used for solving Large-scale stochastic mixed-Integer Problems"), a two-step approach for solving two-stage stochastic (mixed) integer linear programs with an exponential number of constraints. In this approach, we first generate a reduced set of representative scenarios and solve the root node of the corresponding integer linear program using a cutting-plane method. The generated constraints are then used to accelerate solving the original problem with the full scenario set in the second phase. We demonstrate the generic effectiveness of TULIP on two benchmark problems: the Stochastic Capacitated Vehicle Routing Problem and the Two-Stage Stochastic Steiner Forest Problem. The results of our extensive numerical experiments show that TULIP yields significant computational gains compared to solving the problem directly with branch-and-cut.

This work proposes a pricing mechanism for multi-stage electricity markets that does not explicitly depend on the choice of dispatch procedure or optimization method. Our approach is applicable to a wide range of methodologies for the economic dispatch of power systems under uncertainty, including multi-interval dispatch, multi-settlement markets, scenario-based dispatch, and chance-constrained dispatch policies. We prove that our pricing scheme provides both ex-ante and expost dispatch-following incentives by simultaneously supporting per-stage and ex-post competitive equilibria. In numerical experiments on a ramp-constrained test system, we demonstrate the benefits of scheduling under uncertainty and show how our price decomposes into components corresponding to energy, intertemporal coupling, and uncertainty.

Large-scale load-altering attacks (LAAs) are known to severely disrupt power grid operations by manipulating several internet-of-things (IoT)-enabled load devices. In this work, we analyze power grid cascading failures induced by such attacks. The inherent security features in power grids such as the N–1 design philosophy dictate LAAs that can trigger cascading failures are rare events. We overcome the challenge of efficiently sampling critical LAAs scenarios for a wide range of attack parameters by using the so-called “skipping sampler” algorithm. We conduct extensive simulations using a three-area IEEE-39 bus system and provide several novel insights into the composition of cascades due to LAAs. Our results highlight the particular risks to modern power systems posed by strategically designed coordinated LAAs that exploit their structural and real-time operating characteristics.

Synthetic power grids enable secure, real-world energy system simulations and are crucial for algorithm testing, resilience assessment, and policy formulation. We propose a novel method for the generation of synthetic transmission power grids using Exponential Random Graph (ERG) models. Our two main contributions are: (1) the formulation of an ERG model tailored specifically for capturing the topological nuances of power grids, and (2) a general procedure for estimating the parameters of such a model conditioned on working with connected graphs. From a modeling perspective, we identify the edge counts per bus type and $k$-triangles as crucial topological characteristics for synthetic power grid generation. From a technical perspective, we develop a rigorous methodology to estimate the parameters of an ERG constrained to the space of connected graphs. The proposed model is flexible, easy to implement, and successfully captures the desired topological properties of power grids.

Recent challenges in operating power networks arise from increasing energy demands and unpredictable renewable sources like wind and solar. While reinforcement learning (RL) shows promise in managing these networks, through topological actions like bus and line switching, efficiently handling large action spaces as networks grow is crucial. This paper presents a hierarchical multi-agent reinforcement learning (MARL) framework tailored for these expansive action spaces, leveraging the power grid's inherent hierarchical nature. Experimental results indicate the MARL framework's competitive performance with single-agent RL methods. We also compare different RL algorithms for lower-level agents alongside different policies for higher-order agents.

Transmission line failures in power systems propagate and cascade non-locally. In this work, we propose an adaptive control strategy that offers strong guarantees in both the mitigation and localization of line failures. Specifically, we leverage the properties of network bridge-block decomposition and a frequency regulation method called the unified control. If the balancing areas over which the unified control operates coincide with the bridge-blocks of the network, the proposed strategy drives the post-contingency system to a steady state where the impact of initial line outages is localized within the areas where they occurred whenever possible, stopping the cascading process. When the initial line outages cannot be localized, the proposed control strategy provides a configurable design that progressively involves and coordinates more balancing areas. We compare the proposed control strategy with the classical Automatic Generation Control (AGC) on the IEEE 118-bus and 2736-bus test networks. Simulation results show that our strategy greatly improves overall reliability in terms of the N-k security standard, and localizes the impact of initial failures in the majority of the simulated contingencies. Moreover, the proposed framework incurs significantly less load loss, if any, compared to AGC, in all our case studies.

We study opinion dynamics on networks with a nontrivial community structure, assum-ing individuals can update their binary opinion as the result of the interactions with an external influence with strength h is an element of [0, 1] and with other individuals in the network. To model such dynamics, we consider the Ising model with an external magnetic field on a family of finite networks with a clustered structure. Assuming a unit strength for the interactions inside each community, we assume that the strength of interaction across different communities is described by a scalar epsilon is an element of [-1, 1], which allows a weaker but possibly antagonistic effect between communities. We are interested in the stochastic evolution of this system described by a Glauber-type dynamics parametrized by the inverse temperature /3. We focus on the low-temperature regime /3 -> infinity, in which homogeneous opinion patterns prevail and, as such, it takes the network a long time to fully change opinion. We investigate the different metastable and stable states of this opinion dynamics model and how they depend on the values of the parameters epsilon and h. More precisely, using tools from statistical physics, we derive rigorous estimates in probability, expectation, and law for the first hitting time between metastable (or stable) states and (other) stable states, together with tight bounds on the mixing time and spectral gap of the Markov chain describing the network dynamics. Lastly, we provide a full characterization of the critical configurations for the dynamics, i.e., those which are visited with high probability along the transitions of interest.(c) 2023 Published by Elsevier B.V.

The RangL project hosted by The Alan Turing Institute aims to encourage the wider uptake of reinforcement learning by supporting competitions relating to real-world dynamic decision problems. This article describes the reusable code repository developed by the RangL team and deployed for the 2022 Pathways to Net Zero Challenge, supported by the UK Net Zero Technology Centre. The winning solutions to this particular Challenge seek to optimize the UK's energy transition policy to net zero carbon emissions by 2050. The RangL repository includes an OpenAI Gym reinforcement learning environment and code that supports both submission to, and evaluation in, a remote instance of the open source EvalAI platform as well as all winning learning agent strategies. The repository is an illustrative example of RangL's capability to provide a reusable structure for future challenges.

We consider a model for a queue in which only a fixed number $N$ of customers can join. Each customer joins the queue independently at an exponentially distributed time. Assuming further that the service times are independent and follow an exponential distribution, this system can be described as a two-dimensional Markov process on a finite triangular region $\mathfrak S$ of the square lattice. We interpret the resulting random walk on $\mathfrak S$ as a Dyck path that is weighted according to some state-dependent transition probabilities that are constant along one axis, but are rather general otherwise. We untangle the resulting intricate combinatorial structure by introducing appropriate generating functions that exploit the recursive structure of the model. This allows us to derive a fully explicit expression for the probability density function of the number of customers served in any busy period (equivalently, of the length of any excursion of the Dyck path above the diagonal) as a weighted sum with alternating sign over a certain subclass of Dyck paths, whose study is of independent interest.