arXiv subjects

Luis Badesa

Publications and source records attributed to Luis Badesa.

At least 19 recordsLinked to original sources

arXiv2026

A Primal-Dual Formulation for Pricing Static Voltage Stability Services within a Unit Commitment Model

In modern power systems with high penetration of Inverter-Based Resources (IBR), most converters operate in Grid-Following (GFL) mode. Some buses exhibit inherently low Short-Circuit Ratios (SCRs), a property majorly shaped by network topology. The integration of GFL-IBR onto such weak buses thus demands attention to static voltage stability. To address this issue, market mechanisms have been proposed to incentivize generators to provide voltage stability services, such as commitment of synchronous generators for reducing the equivalent impedance at low SCR buses and adaptive reactive power support from GFL-IBR. To compute shadow prices for these services, previously proposed methods such as the `restricted' and `dispatchable' approaches may fail to guarantee operating cost recovery for voltage-stability service providers. As the resulting prices are determined purely from a social surplus maximization objective, the profitability of units is entirely overlooked. This suggests that new pricing methodologies are needed to satisfy cost-recovery requirements. Therefore, this paper proposes a pricing method based on a primal-dual formulation. Case studies demonstrate that the proposed method can consistently produce revenue-adequate shadow prices, enabling all participating units to recover their costs without supplementary uplift payments.

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arXiv2026

Shadow Pricing of Static Voltage Stability Services within Unit Commitment for Inverter-Dominated Power Systems

Modern power systems are increasingly dominated by Inverter-Based Resources (IBR), most of which work in Grid-following (GFL) mode. This implies that they do not directly control their terminal voltage, so the static voltage stability at these buses may be compromised, especially under constant-power-factor operation that lacks voltage-adaptive reactive support. In addition, weather-driven IBR are often installed in electrically remote areas with low Short-Circuit Ratio (SCR), further exacerbating voltage issues. To address this challenge, grid-forming control can be utilized to enhance low-SCR buses, while GFL-IBR could be explicitly required to provide voltage support through grid codes. As an alternative, a market mechanism could be devised that incentivizes relevant generators to proactively adjust their operating points as a service to maintain voltage stability, while the theoretical framework for such a market has not been developed. To fill this gap, this work adopts a second-order cone-based static voltage stability constraint for GFL-IBR buses within a unit commitment problem, and proposes a mechanism to assign shadow prices to this ancillary service. To determine appropriate price values under non-convex conditions, different pricing schemes are assessed. Using a modified IEEE 30-bus system, we demonstrate that both the dispatchable and restricted pricing methods can yield revenue-adequate service prices, though the former may deliver less efficient price signals and the latter may require well-defined uplift payments. This implies that, given differentiated pricing mechanisms and price signals, operators need to select a suitable pricing method in accordance with actual system conditions and market rules.

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arXiv2026

Fair Allocation of Operating Envelopes for Distribution Networks Considering Voltage Unbalance

Operating envelopes (OEs) are increasingly used to allocate limits to distributed energy resources (DERs) while maintaining secure distribution network operation. In unbalanced low-voltage feeders, OE calculation based only on voltage magnitude and thermal constraints can yield overly optimistic limits because power quality constraints such as voltage unbalance are neglected. This paper proposes a three-phase unbalanced AC optimal power flow framework for computing coupled P--Q OEs with explicit voltage unbalance factor (VUF) constraints. In addition, two fairness mechanisms for allocating the available P--Q flexibility across multiple PV units are embedded and compared: (i) network-weighted proportional fairness and (ii) lexicographic max--min fairness. Case studies on unbalanced test feeders illustrate how VUF constraints reshape the P--Q feasible region and the impact of power quality-constrained operation. The comparison highlights the trade-off between the efficiency, equity, and practicality of fairness allocation methods.

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arXiv2026

Voltage Unbalance-Aware AC Optimal Power Flow in Distribution Networks

The increasing penetration of single-phase loads and distributed generation exacerbates voltage unbalance (VU) in distribution grids, raising concerns about power quality and complicating network operation. However, most market-clearing models and price-based coordination frameworks do not enforce VU limits within a three-phase AC representation, so the implications for grid-code compliance, numerical scalability, and economic signals remain unclear. This paper embeds VU in a three-phase AC optimal power flow market-clearing model and benchmarks two treatments: strict VU limit enforcement and objective function penalization. Building on these insights, an Improved Hybrid Limits (IHL) formulation is proposed that preserves compliance while using a smooth unbalance proxy in the objective to guide the optimization solver. Case studies on a European low-voltage feeder show that IHL maintains feasible operating points, yields price and curtailment signals consistent with conventional hybrid formulations, and converges substantially faster and more reliably than a penalization based on the exact unbalance metric. These results support IHL as a practical and scalable mechanism for VU mitigation in market-based operation of unbalanced distribution systems.

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arXiv2026

Least-Cost Overvoltage Control in PV-Rich Distribution Networks via Unbalanced Optimal Power Flow

The increasing penetration of photovoltaic (PV) generation in low-voltage distribution networks presents operational challenges, with overvoltages being among the most critical. This study introduces a tool based on Unbalanced Optimal Power Flow (UBOPF) to assess cost-effective local inverter control strategies specifically aimed at mitigating overvoltage issues. Two approaches are examined: dynamic active power curtailment and combined active and reactive power control. These strategies are tested on a residential low-voltage network with high PV penetration, where the UBOPF model with voltage-magnitude constraints was implemented in Julia using the JuMP optimization package. The results demonstrate that both methods are effective in maintaining voltage levels within regulatory limits, with the latter leading to lower PV curtailment. The analysis highlights the need to consider these control actions as ancillary services to the grid, which should be properly compensated given their effect on generator revenues.

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arXiv2025

On the Impact of Voltage Unbalance on Distribution Locational Marginal Prices

Finding clear economic signals for distribution-network operation and expansion is increasingly important as single-phase loads and distributed energy resources escalate. These devices create phase-to-phase imbalances that manifest as voltage unbalance, a power quality issue that accelerates insulation aging in machines and increases network losses, thereby raising costs for operators and consumers. Traditional grid codes address unbalance via disparate hard limits on various indices thresholds that differ across standards, offer no dynamic economic incentive and undermine optimality. This paper proposes instead to treat voltage unbalance as a `soft limit' by adding penalty terms to grid operation costs within a three-phase optimal power flow to reflect the cost of the decrease in lifetime of assets due to being subject to voltage unbalance. This unified approach yields dynamic economic signals unbalance-aware Distribution Locational Marginal Prices (DLMP) that reflect the cost of power quality deviations. A novel mathematical decomposition of DLMP is developed, isolating the energy, loss, congestion, and unbalance components. Case studies conducted on two benchmark networks demonstrate the effectiveness and practical value of the proposed method. The results indicate that unbalance penalties reshape nodal prices, produce unexpected phase-level effects, and even allow scenarios where added load reduces unbalance and lowers costs, while providing planners and market designers with actionable insights to balance investment, operation, and power quality in modern distribution systems.

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arXiv2025

Evaluating the Impact of a Load Admittance Approximation in Transient Stability-Constrained Optimal Power Flow

The Transient Stability-Constrained Optimal Power Flow (TSC-OPF) incorporates dynamic stability constraints into the OPF formulation to ensure secure and economical operation under disturbances. While discretizing system dynamics enables the use of nonlinear programming techniques, it significantly increases computational burden. To enhance scalability, many studies simplify the network by representing loads as constant admittances, allowing the use of Kron reduction. However, computing the Kron reduction outside the optimization requires a voltage-based assumption to convert loads from constant power to constant admittance. This paper proposes a practical voltage-based load admittance approximation and evaluates the errors it may introduce in rotor angle and speed deviation trajectories. Case studies on the WECC 9-bus system show that the proposed approach reproduces rotor dynamics consistent with time-domain simulations during the first few seconds while considerably reducing implementation effort and mitigating convergence issues. The proposed framework thus offers a simple and effective strategy for scalable TSC-OPF implementations.

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arXiv2025

Analyzing the Impact of Demand Response on Short-Circuit Current via a Unit Commitment Model

In low-carbon grids, system flexibility can be enhanced through mechanisms such as Demand Response (DR), enabling the efficient utilization of renewable energy. However, as Synchronous Generators (SGs) are being replaced by renewable energy sources characterized by Inverter-Based Resources (IBR), system stability is severely affected. Due to the limited overload capability of IBRs, their Short-Circuit Current (SCC) contribution is much smaller than that of SGs. As a result, protection devices may fail to trip during faults. Consequently, the remaining SGs play a key role in providing sufficient SCC. Since the commitment of SGs is closely related to system loading conditions, DR can indirectly affect their SCC provision, a relationship that has not yet been investigated in the literature. Therefore, this paper incorporates both DR and SCC constraints into a unit commitment problem and conducts case studies on an IEEE 30-bus system. The results show that although DR can reduce total costs by adjusting power demand, it may also lead to inadequate SCC levels. Nevertheless, when flexible loads are properly coordinated with SCC requirements, the total cost increases by only 0.3%, which is significantly lower than the cost of system dispatch without DR. This demonstrates that DR can facilitate stable system operation in a cost-effective manner.

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arXiv2025

Pricing Short-Circuit Current via a Primal-Dual Formulation for Preserving Integrality Constraints

Synchronous Generators (SGs) currently provide important levels of Short-Circuit Current (SCC), a critical ancillary service that ensures line protections trip during short-circuit faults. Given the ongoing replacement of SGs by power-electronics-based generation, which has a hard limit on current injection, it has become relevant to optimize the procurement of SCC services provided by remaining SGs. Pricing this service is, however, challenging due to the integrality constraints in Unit Commitment (UC). Existing methods, e.g., dispatchable pricing and restricted pricing, attempt to address this issue but exhibit limitations in handling non-convexities, resulting in SCC prices that either fail to cover the operating costs of units or lack interpretability. To overcome these pitfalls, we adopt a primal-dual formulation of the SCC-constrained dispatch that preserves the binary UC for effectively computing shadow prices of SCC services. Using a modified IEEE 30-bus system, the proposed method is compared with the previously developed pricing schemes. It is demonstrated that, under the proposed pricing method, revenue-adequate and explicit service prices can be assigned without the need for uplift payments, an advantage that cannot be achieved by other pricing approaches.

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arXiv2025

Imperfect Competition in Markets for Short-Circuit Current Services

An important limitation of Inverter-Based Resources (IBR) is their reduced contribution to Short-Circuit Current (SCC), as compared to that of Synchronous Generators (SGs). With increasing penetration of IBR in most power systems, the reducing SCC poses challenges to a secure system operation, as line protections may not trip when required. In order to address this issue, the SCC ancillary service could be procured via an economic mechanism, aiming at securing adequate SCC on all buses. However, the suitability of markets for SCC services is not well understood, given that these could be prone to market power issues: since the SCC contributions from various SGs to a certain bus are determined by the electrical topology of the grid, this is a highly local service. It is necessary to understand if SGs at advantageous electrical locations could exert market power and, if so, how it could be mitigated. In order to fill this gap, this paper, for the first time, adopts an SCC-constrained bilevel model to investigate strategic behaviors of SGs. To address the non-convexity due to unit commitment variables, the model is restructured through a primal-dual formulation. Based on a modified IEEE 30-bus system, cases with strategic SGs placed at different buses are analyzed. These studies demonstrate that strategic agents exerting market power by manipulating service prices and extending operating periods could achieve up to triple revenues from SCC provision, which reduces market efficiency and would increase the financial burden on consumers. These findings highlight the need for careful market design, for which potential measures to mitigate these market power issues are also discussed.

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arXiv2025

Evaluation of Voltage Unbalance Metrics in Distribution Networks with High DER Penetration

Voltage unbalance, caused by variations in voltage magnitude and phase angle, is a significant power quality issue in three-phase systems, leading to equipment inefficiencies and increased system losses. The integration of distributed energy resources (DER) into the grid adds complexity, as DER can either reduce or worsen voltage unbalance, depending on factors such as grid configuration and the distribution of loads and DER themselves. This study explores the effects of DER penetration on voltage unbalance levels and the accuracy of the different indices most commonly used to quantify this unbalance. The results highlight the varying impacts of DER on unbalance and index performance, emphasizing the need for effective strategies to assess voltage unbalance in modern distribution systems.

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arXiv2025

Analyzing the Role of the DSO in Electricity Trading of VPPs via a Stackelberg Game Model

The increasing penetration of distributed energy resources has sparked interests in participating in power markets. Here, we consider two settings where Virtual Power Plants (VPPs) with some flexible resources participate in the electricity trading, either directly in the wholesale electricity market, or interfaced by the Distribution System Operator (DSO) who is the transaction organizer. In order to study the role of DSO as a stakeholder, a Stackelberg game is represented via a bi-level model: the DSO maximizes profits at the upper level, while the VPPs minimize operating costs at the lower level. To solve this problem, the Karush-Kuhn-Tucker conditions of lower level is deduced to achieve a single-level problem. The results show that the role of the DSO as an intermediary agent leads to a decrease in operating costs of the VPPs by organizing lower-level trading, while making a profit for itself. However, this result comes with interests loss of the wholesale market, implying that stakeholders in the market need to abide by regulatory constraints.

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arXiv2023

Who should pay for frequency-containment ancillary services? Making responsible units bear the cost to shape investment in generation and loads

While the operating cost of electricity grids based on thermal generation was largely driven by the cost of fuel, as renewable penetration increases, ancillary services represent an increasingly large proportion of the running costs. Electric frequency is an important magnitude in highly renewable grids, as it becomes more volatile and therefore the cost related to maintaining it within safe bounds has significantly increased. So far, costs for frequency-containment ancillary services have been socialised in most countries, but it has become relevant to rethink this regulatory arrangement. In this paper, we discuss the issue of cost allocation for these services, highlighting the need to evolve towards a causation-based regulatory framework. We argue that parties responsible for creating the need for ancillary services should bear these costs. However, this would imply an important change in electricity market policy, therefore it is necessary to understand the impact on current and future investments on generation, as well as on electricity tariffs. Here we provide a mostly qualitative analysis of this issue, defining guidelines for practical implementation and further study.

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arXiv2023

Cost Allocation for Inertia and Frequency Response Ancillary Services

The reduction in system inertia is creating an important market for frequency-containment Ancillary Services (AS) such as enhanced frequency response (e.g.,~provided by battery storage), traditional primary frequency response and inertia itself. This market presents an important difference with the energy-only market: while the need for energy production is driven by the demand from consumers, frequency-containment AS are procured because of the need to deal with the largest generation/demand loss in the system (or smaller losses that could potentially compromise frequency stability). Thus, a question that arises is: who should pay for frequency-containment AS? In this work, we propose a cost-allocation methodology based on the nucleolus concept, in order to distribute the total payments for frequency-containment AS among all generators or loads that create the need for these services. It is shown that this method complies with necessary properties for the AS market, such as avoidance of cross-subsidies and maintaining players in this cooperative game. Finally, we demonstrate its practical applicability through a case study for the Great Britain power system, while comparing its performance with two alternative mechanisms, namely proportional and Shapley value cost allocation.

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arXiv2022

Reinforcement Learning and Mixed-Integer Programming for Power Plant Scheduling in Low Carbon Systems: Comparison and Hybridisation

Decarbonisation is driving dramatic growth in renewable power generation. This increases uncertainty in the load to be served by power plants and makes their efficient scheduling, known as the unit commitment (UC) problem, more difficult. UC is solved in practice by mixed-integer programming (MIP) methods; however, there is growing interest in emerging data-driven methods including reinforcement learning (RL). In this paper, we extensively test two MIP (deterministic and stochastic) and two RL (model-free and with lookahead) scheduling methods over a large set of test days and problem sizes, for the first time comparing the state-of-the-art of these two approaches on a level playing field. We find that deterministic and stochastic MIP consistently produce lower-cost UC schedules than RL, exhibiting better reliability and scalability with problem size. Average operating costs of RL are more than 2 times larger than stochastic MIP for a 50-generator test case, while the cost is 13 times larger in the worst instance. However, the key strength of RL is the ability to produce solutions practically instantly, irrespective of problem size. We leverage this advantage to produce various initial solutions for warm starting concurrent stochastic MIP solves. By producing several near-optimal solutions simultaneously and then evaluating them using Monte Carlo methods, the differences between the true cost function and the discrete approximation required to formulate the MIP are exploited. The resulting hybrid technique outperforms both the RL and MIP methods individually, reducing total operating costs by 0.3% on average.

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arXiv2022

Chance-constrained allocation of UFLS candidate feeders under high penetration of distributed generation

Under-Frequency Load Shedding (UFLS) schemes are the last resort to contain a frequency drop in the grid by disconnecting part of the demand. The allocation methods for selecting feeders that would contribute to the UFLS scheme have traditionally relied on the fact that electric demand followed fairly regular patterns, and could be forecast with high accuracy. However, recent integration of Distributed Generation (DG) increases the uncertainty in net consumption of feeders which, in turn, requires a reformulation of UFLS-allocation methods to account for this uncertainty. In this paper, a chance-constrained methodology for selecting feeders is proposed, with mathematical guarantees for the disconnection of the required amount of load with a certain pre-defined probability. The correlation in net-load forecasts among feeders is explicitly considered, given that uncertainty in DG power output is driven by meteorological conditions with high correlation across the network. Furthermore, this method is applicable either to systems with conventional UFLS schemes (where relays measure local frequency and trip if this magnitude falls below a certain threshold), or adaptive UFLS schemes (where relays are triggered by control signals sent in the few instants following a contingency). Relevant case studies demonstrate the applicability of the proposed method, and the need for explicit consideration of uncertainty in the UFLS-allocation process.

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arXiv2022

Assigning Shadow Prices to Synthetic Inertia and Frequency Response Reserves from Renewable Energy Sources

Modern electricity grids throughout the world, particularly in islands such as Great Britain, face a major problem on the road to decarbonisation: the significantly reduced level of system inertia due to integration of Renewable Energy Sources (RES). Given that most RES such as wind and solar are decoupled from the grid through power electronics converters, they do not naturally contribute to system inertia. However, RES could support grid stability through appropriately controlling the converters, but currently no market incentives exist for RES to provide this support. In this paper we develop a methodology to optimally clear a market of ancillary services for frequency control, while explicitly considering the participation of grid-forming and grid-following inverter-based technologies. We propose a mathematical framework that allows to compute shadow prices for ancillary services offered by a pool of diverse providers: synchronous and synthetic inertia, enhanced frequency response (e.g. from curtailed RES) and traditional primary frequency response (e.g. by thermal generators). Several case studies are run on a simplified Great Britain system, to illustrate the applicability and benefits of this pricing scheme.

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arXiv2022

Value of Optimal Trip and Charging Scheduling of Commercial Electric Vehicle Fleets with Vehicle-to-Grid in Future Low Inertia Systems

The electrification of transport is seen as an important step in the global decarbonisation agenda. With such a large expected load on the power system from electric vehicles (EVs), it is important to coordinate charging in order to balance the supply and demand for electricity. Bidirectional charging, enabled through Vehicle-to-Grid (V2G) technology, will unlock significant storage capacity from stationary EVs that are plugged in. To take this concept a step further, this paper quantifies the potential revenues to be gained by a commercial EV fleet operator from simultaneously scheduling its trips on a day-ahead basis, as well as its charging. This allows the fleet to complete its trips (with user defined trip length and distance), while taking advantage of fluctuating energy and ancillary services prices. A mathematical framework for optimal trip scheduling is proposed, formulated as a mixed-integer linear program, and is applied to several relevant scenarios of the present and future British electricity system. It is demonstrated that an optimal journey start time can increase the revenue of commercial fleets by up to 38% in summer and 12% in winter. This means a single EV from the maintenance fleet can make additional annual revenue of up to {\pounds}729. Flexible trip schedules are more valuable in the summer because keeping EVs plugged in during peak solar output will benefit the grid and the fleet operators the most. It was also found that a fleet of 5,000 EVs would result in the equivalent $\textrm{CO}_2$ of removing one Combined Cycle Gas Turbine from the system. This significant increase in revenue and carbon savings show this approach is worth investigating for potential future application.

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