This paper presents a novel market clearing mechanism for manual Frequency Restoration Reserve (mFRR) capacity markets, focusing on the Nordic market setup. This market enhances the reliability of the Nordic system by ensuring availability of mFRR. In contrast to today’s market setup which only accounts for capacity costs, the proposed market clearing mechanism accounts for both expected energy activation costs and mFRR capacity costs. The market clearing problem is formulated as a two-stage stochastic Mixed Integer Linear Program (MILP). To efficiently solve the resulting optimization problem, we introduce a nested decomposition algorithm that combines the Benders Decomposition (BD) and Surrogate Absolute Value Lagrangian Relaxation (SAVLR) methods. For practical implementation, input data scenarios are generated using day-ahead (DA) price data modeled by a Bayesian Neural Network (BNN). A real-world case study in Sweden demonstrates that the proposed mechanism can reduce daily mFRR costs by 600-14,500 €, due to reduced energy activation costs. Simulation results further show that the nested decomposition algorithm converges to a near-optimal solution more quickly than standard BD.

This paper proposes an integrated centralized multiperiod planning framework for local flexibility markets (LFMs) in unbalanced distribution networks (UDNs). It co-optimizes line reinforcement and phase-aware flexibility procurement subject to unbalanced three-phase AC power-flow constraints using a rectangular-coordinate branch current-voltage representation. The resulting MBQCQP captures AC bilinearities and investment-dependent network constraints. For each fixed investment vector, the model is reformulated as a bounded nonconvex QCQP and lifted into a completely-positive-programming-based (CPP-based) representation, from which semidefinite programming (SDP) and SDP-Moment relaxations are derived. Numerical results on a modified IEEE 13-node feeder show that generationside flexibility reduces reinforcement costs by 22.86% relative to the no-flexibility case, while the combined use of local load- and generation-side flexibility fully eliminates reinforcement investment. The SDP-Moment relaxation closes the gap in all tested cases and recovers the exact investment decisions, demonstrating that strengthened lifted relaxations can provide reliable bounds for phase-aware investment planning in UDNs.

Conventional capacity expansion planning (CEP) relies on a perfect-foresight planning horizon and linear investment optimisation, which fail to capture the non-linear dynamics of electricity markets. In the Nordics, hydro-related weather variability plays a critical role in maintaining the robustness of the power system. This paper addresses the intra-year perfect-foresight limitation in current CEP models, focusing on hydro-dominated power systems with substantial hydro reservoir capacity, using Sweden's decarbonisation pathway toward 2050 as a case study. Our approach provides a robust long-term CEP framework by leveraging short-term price forecasts to guide storage dispatch decisions. The proposed CEP model has been historically validated and captures the dynamics of seasonal storage hydro reservoirs, achieving deviations of less than €1/MWh in annual average prices across all Swedish bidding zones. A comparative analysis between the proposed and conventional CEP models (cGrid and GenX), together with the Ten-Year Network Development Plan (TYNDP 2024), reveals a broad alignment in capacity expansion and dispatch under an average weather year. However, in a problematic weather year, with correlated low wind output and reduced hydro inflows, significant divergences emerge, with half-year price averages differing by up to ±€40/MWh. These discrepancies are mainly driven by contrasting approaches to hydro reservoir modelling. Notably, the proposed CEP model recommends a 37.5 % increase in firm nuclear capacity to mitigate supply shortages, whereas the conventional CEP suggests a 4.4 % reduction, thereby increasing reliance on weather-dependent resources. These findings underscore the limitations of perfect-foresight CEP in power systems with substantial seasonal storage resources.

Electric power systems are increasingly turning to energy storage systems to balance supply and demand. But how much storage is required? What is the optimal volume of storage in a power system and on what does it depend? In addition, what form of hedge contracts do storage facilities require? We answer these questions in the special case in which the uncertainty in the power system involves successive draws of an independent, identically-distributed random variable. We characterise the conditions for the optimal operation of, and investment in, storage and show how these conditions can be understood graphically using price-duration curves. We also characterise the optimal hedge contracts for storage units.

Integrating renewable energy sources into a power system imposes uncertainty in the power generation, rendering traditional power flow methods ineffective. By calculating uncertain power flow, we can obtain more realistic and reliable estimates of the system state. Interval methods have been recognized as a powerful tool for analyzing uncertain power systems and increasing their overall reliability. In this paper, an approach is proposed to formulate the uncertain power flow with interval uncertainties, called Interval Power Flow (IPF), as a convex feasibility problem. To attain this goal, the IPF problem is written in the form of Bilinear Matrix Inequalities. Then, the polytopic model of IPF is derived and it is proved that to guarantee the validity of IPF for the whole range of renewable energy changes, it is enough to solve the matrix inequalities in the corner points of the polytopic uncertain space. Then, the Inside-Ellipsoids Outside-Sphere model is applied to the IPF model resulting in a convex feasibility problem, plus a non-convex quadratic constraint which is later relaxed to achieve an LMI problem. The final problem is solved by one of the off-the-shelf solvers and a robust operating point for the IPF problem is obtained. The approach is tested for various case studies and the results prove its efficacy compared to the existing method.

The electrification of heavy road freight is a promising pathway to decarbonise the sector, yet it depends on the electricity supply system, which itself is undergoing a complex transition to low-carbon energy. Achieving net-zero emissions requires understanding the interdependencies between transport and electricity systems. To address this challenge, we developed a system dynamics (SD) model to explore how freight electrification interacts with electricity supply capacity and market-based pricing. Drawing on literature review and expert interviews, the model integrates three modules: demand (electric truck fleet and charging behaviour), supply (capacity expansion with construction delays), and price (merit-order dispatch). Preliminary results indicate that e-truck adoption has limited direct impact on electricity prices under baseline assumptions. However, cross-sectoral competition emerges as critical, where accelerated electrification in other sectors may drive electricity prices upward, reducing electric truck competitiveness. Charging coordination strategies also influence electricity price dynamics. The model provides a holistic perspective and highlights the need for coordinated planning between transport and electricity sectors. However, model calibration is ongoing, and results should be interpreted cautiously.

This paper proposes a new approach for solving Quadratically Constrained Feasibility Problems (QCFPs). We introduce an isomorphic mapping (one-to-one and onto correspondence), which equivalently converts the QCFP to an optimization problem called the Inside-Ellipsoids Outside-Sphere Problem (IEOSP). This mapping preserves the convexity of convex constraints, but it converts all non-convex constraints to convex ones. The QCFP is a feasibility problem with non-convex constraints, while the IEOSP is an optimization problem with a convex feasible region and a non-convex objective function. It is shown that the global optimal solution of IEOSP is a feasible solution of the QCFP. Comparing the structures of QCFP and the proposed IEOSP, the second model only has one extra variable compared to the original QCFP because it employs one slack variable for the mapping. Thus, the problem dimension approximately remains unchanged. Due to the convexity of all constraints in IEOSP, it has a well-defined feasible region. Therefore, it can be solved much easier than the original QCFP. This paper proposes a solution algorithm for IEOSP that iteratively solves a convex optimization problem. The algorithm is mathematically shown to reach either a feasible solution of the QCFP or a local solution of the IEOSP. To illustrate our theoretical developments, a comprehensive numerical experiment is performed, and 500 different QCFPs are studied. All these numerical experiments confirm the promising performance and applicability of our theoretical developments in the current paper.

This paper addresses a charging-station company (Chargco) for electric and hydrogen vehicles. The optimal trading of the Chargco in day-ahead and intraday auction markets for electricity is modeled as a stochastic Mixed-Integer Quadratic Program (MIQP). We propose a series of linearization and reformulation techniques to reformulate the stochastic MIQP as a mixed-integer linear program (MILP). To model stochasticity, we utilize generative adversarial networks to cluster electricity market price scenarios. Additionally, a combination of random forests and linear regression is employed to model the relationship between Chargco electricity and hydrogen loads and their selling prices. Finally, we propose an Improved L-Shaped Decomposition (ILSD) algorithm to solve our stochastic MILP. Our ILSD algorithm not only addresses infeasibilities through an innovative approach but also incorporates warm starts, valid inequalities and multiple generation cuts, thereby reducing computational complexity. Numerical experiments illustrate the Chargco trading using our proposed stochastic MILP and its solution algorithm.

To ensure that battery energy storage systems (BESSs) are used to facilitate the operation of power systems with high shares of variable renewable energy (VRE) sources, new policies for BESSs are currently being designed. This paper presents a new model for the system cost-minimizing scheduling of BESSs providing energy and balancing services, which can be used as a policy-evaluation tool and as a benchmark for evaluating current market performance. The model is based on a stochastic optimization problem, which we suggest solving using an algorithm combining Benders Decomposition (BD) and Stochastic Dual Dynamic Programming (SDDP). In this paper, we apply the model to study how the system cost-minimal scheduling of BESSs in Sweden is impacted by new requirements for Limited Energy Reservoir (LER) resources providing Frequency Containment Reserves (FCR). Case study results show that new requirements related to guaranteeing a minimum full activation time significantly impact the operation of BESSs, reducing the provision of upregulating FCR capacity for disturbances (FCR-D up) by over 60 %. Also, the case study results show that the consideration of BESSs’ degradation costs reduces the provision of energy as well as balancing services from BESSs. The daily discharged energy from BESSs is more than 85 % lower if the degradation costs are considered, compared to if they are not considered.

Investment in liberalized wholesale electricity markets is typically not driven by spot and forward wholesale energy price signals alone. Instead, in most wholesale electricity markets there is a separate mechanism to procure, fund or subsidize investment in generation or demand response capacity. These mechanisms are known as capacity markets and are often highly controversial. There remains substantial debate over whether these mechanisms are necessary and, if so, how they should be designed. This paper provides a theoretical analysis of the capacity market debate. We observe that an intuitive understanding of the effects of capacity mechanisms may be misleading. While a capacity mechanism may offset the effects of a price cap, if the price cap is funded through a levy on consumption the combined effect of the tax and subsidy is to nullify the price cap. We explore possible reasons for under-investment concerns in wholesale electricity markets, focusing on the potential for a lack of liquidity in the hedge market arising from the presence of uninsurable risks. We identify three groups of uninsurable risks and propose policies to address them.