arXiv subjects

Carlos Ruiz

Publications and source records attributed to Carlos Ruiz.

14 recordsLinked to original sources

arXiv2025

Optimal placement of wind farms via quantile constraint learning

Wind farm placement arranges the size and the location of multiple wind farms within a given region. The power output is highly related to the wind speed on spatial and temporal levels, which can be modeled by advanced data-driven approaches. To this end, we use a probabilistic neural network as a surrogate that accounts for the spatiotemporal correlations of wind speed. This neural network uses ReLU activation functions so that it can be reformulated as mixed-integer linear set of constraints (constraint learning). We embed these constraints into the placement decision problem, formulated as a two-stage stochastic optimization problem. Specifically, conditional quantiles of the total electricity production are regarded as recursive decisions in the second stage. We use real high-resolution regional data from a northern region in Spain. We validate that the constraint learning approach outperforms the classical bilinear interpolation method. Numerical experiments are implemented on risk-averse investors. The results indicate that risk-averse investors concentrate on dominant sites with strong wind, while exhibiting spatial diversification and sensitive capacity spread in non-dominant sites. Furthermore, we show that if we introduce transmission line costs in the problem, risk-averse investors favor locations closer to the substations. On the contrary, risk-neutral investors are willing to move to further locations to achieve higher expected profits. Our results conclude that the proposed novel approach is able to tackle a portfolio of regional wind farm placements and further provide guidance for risk-averse investors.

stat.ML↗

arXiv2024

Leveraging Neural Networks to Optimize Heliostat Field Aiming Strategies in Concentrating Solar Power Tower Plants

Concentrating Solar Power Tower (CSPT) plants rely on heliostat fields to focus sunlight onto a central receiver. Although simple aiming strategies, such as directing all heliostats to the receivers equator, can maximize energy collection, they often result in uneven flux distributions that lead to hotspots, thermal stresses, and reduced receiver lifetimes. This paper presents a novel, data-driven approach that integrates constraint learning, neural network-based surrogates, and mathematical optimization to overcome these challenges. The methodology learns complex heliostat-to-receiver flux interactions from simulation data, constructing a surrogate model that is embedded into a tractable optimization framework. By maximizing a tailored quality score that balances energy collection and flux uniformity, the approach yields smoothly distributed flux profiles and mitigates excessive thermal peaks. An iterative refinement process, guided by the trust region and progressive data sampling, ensures the surrogate model improves the obtained solution by exploring new spaces during the iterations. Results from a real CSPT case study demonstrate that the proposed approach surpasses conventional heuristic methods, offering flatter flux distributions and safer thermal conditions without a substantial loss in overall energy capture.

eess.SY↗

arXiv2024

A Quantile Neural Network Framework for Two-stage Stochastic Optimization

Two-stage stochastic programming is a popular framework for optimization under uncertainty, where decision variables are split between first-stage decisions, and second-stage (or recourse) decisions, with the latter being adjusted after uncertainty is realized. These problems are often formulated using Sample Average Approximation (SAA), where uncertainty is modeled as a finite set of scenarios, resulting in a large "monolithic" problem, i.e., where the model is repeated for each scenario. The resulting models can be challenging to solve, and several problem-specific decomposition approaches have been proposed. An alternative approach is to approximate the expected second-stage objective value using a surrogate model, which can then be embedded in the first-stage problem to produce good heuristic solutions. In this work, we propose to instead model the distribution of the second-stage objective, specifically using a quantile neural network. Embedding this distributional approximation enables capturing uncertainty and is not limited to expected-value optimization, e.g., the proposed approach enables optimization of the Conditional Value at Risk (CVaR). We discuss optimization formulations for embedding the quantile neural network and demonstrate the effectiveness of the proposed framework using several computational case studies including a set of mixed-integer optimization problems.

math.OC↗

arXiv2022

A Neural Network-Based Distributional Constraint Learning Methodology for Mixed-Integer Stochastic Optimization

The use of machine learning methods helps to improve decision making in different fields. In particular, the idea of bridging predictions (machine learning models) and prescriptions (optimization problems) is gaining attention within the scientific community. One of the main ideas to address this trade-off is the so-called Constraint Learning (CL) methodology, where the structures of the machine learning model can be treated as a set of constraints to be embedded within the optimization problem, establishing the relationship between a direct decision variable $x$ and a response variable $y$. However, most CL approaches have focused on making point predictions for a certain variable, not taking into account the statistical and external uncertainty faced in the modeling process. In this paper, we extend the CL methodology to deal with uncertainty in the response variable $y$. The novel Distributional Constraint Learning (DCL) methodology makes use of a piece-wise linearizable neural network-based model to estimate the parameters of the conditional distribution of $y$ (dependent on decisions $x$ and contextual information), which can be embedded within mixed-integer optimization problems. In particular, we formulate a stochastic optimization problem by sampling random values from the estimated distribution by using a linear set of constraints. In this sense, DCL combines both the high predictive performance of the neural network method and the possibility of generating scenarios to account for uncertainty within a tractable optimization model. The behavior of the proposed methodology is tested in a real-world problem in the context of electricity systems, where a Virtual Power Plant seeks to optimize its operation, subject to different forms of uncertainty, and with price-responsive consumers.

math.OC↗

arXiv2022

On data-driven chance constraint learning for mixed-integer optimization problems

When dealing with real-world optimization problems, decision-makers usually face high levels of uncertainty associated with partial information, unknown parameters, or complex relationships between these and the problem decision variables. In this work, we develop a novel Chance Constraint Learning (CCL) methodology with a focus on mixed-integer linear optimization problems which combines ideas from the chance constraint and constraint learning literature. Chance constraints set a probabilistic confidence level for a single or a set of constraints to be fulfilled, whereas the constraint learning methodology aims to model the functional relationship between the problem variables through predictive models. One of the main issues when establishing a learned constraint arises when we need to set further bounds for its response variable: the fulfillment of these is directly related to the accuracy of the predictive model and its probabilistic behaviour. In this sense, CCL makes use of linearizable machine learning models to estimate conditional quantiles of the learned variables, providing a data-driven solution for chance constraints. An open-access software has been developed to be used by practitioners. Furthermore, benefits from CCL have been tested in two real-world case studies, proving how robustness is added to optimal solutions when probabilistic bounds are set for learned constraints.

math.OC↗

arXiv2022

Contract design in electricity markets with high penetration of renewables: A two-stage approach

The interplay between risk aversion and financial derivatives has received increasing attention since the advent of electricity market liberalization. One important challenge in this context is how to develop economically efficient and cost-effective models to integrate renewable energy sources (RES) in the electricity market, which constitutes a relatively new and exciting field of research. This paper proposes a game-theoretical equilibrium model that characterizes the interactions between oligopolistic generators in a two-stage electricity market under the presence of high RES penetration. Given conventional generators with generation cost uncertainty and renewable generators with intermittent and stochastic capacity, we consider a single futures contract market that is cleared prior to a spot market where the energy delivery takes place. We introduce physical and financial contracts to evaluate their performance assess their impact on the electricity market outcomes and examine how these depend on the level of RES penetration. Since market participants are usually risk-averse, a coherent risk measure is introduced to deal with both risk-neutral and risk-averse generators. We derive analytical relationships between contracts, study the implications of uncertainties, test the performance of the proposed equilibrium model and its main properties through numerical examples. Our results show that overall electricity prices, generation costs, profits, and quantities for conventional generators decrease, whereas quantities and profits for RES generators increase with RES penetration. Hence, both physical and financial contracts efficiently mitigate the impact of uncertainties and help the integration of RES into the electricity system.

econ.GN↗

arXiv2022

Optimal day-ahead offering strategy for large producers based on market price response learning

In day-ahead electricity markets based on uniform marginal pricing, small variations in the offering and bidding curves may substantially modify the resulting market outcomes. In this work, we deal with the problem of finding the optimal offering curve for a risk-averse profit-maximizing generating company (GENCO) in a data-driven context. In particular, a large GENCO's market share may imply that her offering strategy can alter the marginal price formation, which can be used to increase profit. We tackle this problem from a novel perspective. First, we propose a optimization-based methodology to summarize each GENCO's step-wise supply curves into a subset of representative price-energy blocks. Then, the relationship between the market price and the resulting energy block offering prices is modeled through a Bayesian linear regression approach, which also allows us to generate stochastic scenarios for the sensibility of the market towards the GENCO strategy, represented by the regression coefficient probabilistic distributions. Finally, this predictive model is embedded in the stochastic optimization model by employing a constraint learning approach. Results show how allowing the GENCO to deviate from her true marginal costs renders significant changes in her profits and the market marginal price. Furthermore, these results have also been tested in an out-of-sample validation setting, showing how this optimal offering strategy is also effective in a real-world market contest.

stat.AP↗

arXiv2022

Optimal pricing for electricity retailers based on data-driven consumers' price-response

In the present work we tackle the problem of finding the optimal price tariff to be set by a risk-averse electric retailer participating in the pool and whose customers are price-sensitive. We assume that the retailer has access to a sufficiently large smart-meter dataset from which it can statistically characterize the relationship between the tariff price and the demand load of its clients. Three different models are analyzed to predict the aggregated load as a function of the electricity prices and other parameters, as humidity or temperature. More specifically, we train linear regression (predictive) models to forecast the resulting demand load as a function of the retail price. Then we will insert this model in a quadratic optimization problem which evaluates the optimal price to be offered. This optimization problem accounts for different sources of uncertainty including consumer's response, pool prices and renewable source availability, and relies on a stochastic and risk-averse formulation. In particular, one important contribution of this work is to base the scenario generation and reduction procedure on the statistical properties of the resulting predictive model. This allows us to properly quantify (data-driven) not only the expected value but the level of uncertainty associated with the main problem parameters. Moreover, we consider both standard forward based contracts and the recently introduced power purchase agreement contracts as risk-hedging tools for the retailer. The results are promising as profits are found for the retailer with highly competitive prices and some possible improvements are shown if richer datasets could be available in the future. A realistic case study and multiple sensitivity analyses have been performed to characterize the risk-aversion behavior of the retailer considering price-sensitive consumers.

math.OC↗

arXiv2021

Joint optimization of sales-mix and generation plan for a large electricity producer

The paper develops a typical management problem of a large power producer (i.e., he can partly influence the market price). In particular, he routinely needs to decide how much of his generation it is preferable to commit to fixed price bilateral contracts (e.g., futures) or to the spot market. However, he also needs to plan how to distribute the production across the different plants under his control. The two decisions, namely the sales-mix and the generation plan, naturally interact, since the opportunity to influence the spot price depends, among other things, by the amount of the energy that the producer directs on the spot market. We develop a risk management problem, since we consider an optimization problem combining a trade-off between expectation and conditional value at risk of the profit function of the producer. The sources of uncertainty are relatively large and encompass demand, renewables generation and the fuel costs of conventional plants. We also model endogenously the price of futures in a way reflecting an information advantage of a large power producer. In particular, it is assumed that the market forecast the price of futures in a naive way, namely not anticipating the impact of the large producer on the spot market. The paper provides a MILP formulation of the problem, and it analyzes the solution through a simulation based on Spanish power market data.

math.OC↗

arXiv2021

Dynamic tariff-based demand response in retail electricity market under uncertainty

Demand response (DR) programs play a crucial role in improving system reliability and mitigating price volatility by altering the core profile of electricity consumption. This paper proposes a game-theoretical model that captures the dynamic interplay between retailers (leaders) and consumers (followers) in a tariffs-based electricity market under uncertainty. The proposed procedure offers theoretical and economic insights by analyzing demand flexibility within a hierarchical decision-making framework. In particular, two main market configurations are examined under uncertainty: i) there exists a retailer that exercises market power over consumers, and ii) the retailer and the consumers participate in a perfect competitive game. The former case is formulated as a mathematical program with equilibrium constraints (MPEC), whereas the latter case is recast as a mixed-integer linear program (MILP). These problems are solved by deriving equivalent tractable reformulations based on the Karush-Kuhn-Tucker (KKT) optimality conditions of each agent's problem. Numerical simulations based on real data from the European Energy Exchange platform are used to illustrate the performance of the proposed methodology. The results indicate that the proposed model effectively characterizes the interactions between retailers and flexible consumers in both perfect and imperfect market structures. Under perfect competition, the economic benefits extend not only to consumers but also to overall social welfare. Conversely, in an imperfect market, retailers leverage consumer flexibility to enhance their expected profits, transferring the risk of uncertainty to end-users. Additionally, the degree of consumer flexibility and their valuation of electricity consumption play significant roles in shaping market outcomes.

econ.GN↗

arXiv2021

Contracts in Electricity Markets under EU ETS: A Stochastic Programming Approach

The European Union Emission Trading Scheme (EU ETS) is a cornerstone of the EU's strategy to fight climate change and an important device for plummeting greenhouse gas (GHG) emissions in an economically efficient manner. The power industry has switched to an auction-based allocation system at the onset of Phase III of the EU ETS to bring economic efficiency by negating windfall profits that have been resulted from grandfathered allocation of allowances in the previous phases. In this work, we analyze and simulate the interaction of oligopolistic generators in an electricity market with a game-theoretical framework where the electricity and the emissions markets interact in a two-stage electricity market. For analytical simplicity, we assume a single futures market where the electricity is committed at the futures price, and the emissions allowance is contracted in advance, prior to a spot market where the energy and allowances delivery takes place. Moreover, a coherent risk measure is applied (Conditional Value at Risk) to model both risk averse and risk neutral generators and a two-stage stochastic optimization setting is introduced to deal with the uncertainty of renewable capacity, demand, generation, and emission costs. The performance of the proposed equilibrium model and its main properties are examined through realistic numerical simulations. Our results show that renewable generators are surging and substituting conventional generators without compromising social welfare. Hence, both renewable deployment and emission allowance auctioning are effectively reducing GHG emissions and promoting low-carbon economic path.

econ.TH↗

arXiv2020

A Single Scalable LSTM Model for Short-Term Forecasting of Disaggregated Electricity Loads

Most electricity systems worldwide are deploying advanced metering infrastructures to collect relevant operational data. In particular, smart meters allow tracking electricity load consumption at a very disaggregated level and at high frequency rates. This data opens the possibility of developing new forecasting models with a potential positive impact in electricity systems. We present a general methodology that is able to process and forecast a large number of smart meter time series. Instead of using traditional and univariate approaches, we develop a single but complex recurrent neural-network model with long short-term memory that can capture individual consumption patterns and also consumptions from different households. The resulting model can accurately predict future loads (short-term) of individual consumers, even if these were not included in the original training set. This entails a great potential for large scale applications as once the single network is trained, accurate individual forecast for new consumers can be obtained at almost no computational cost. The proposed model is tested under a large set of numerical experiments by using a real-world dataset with thousands of disaggregated electricity consumption time series. Furthermore, we explore how geo-demographic segmentation of consumers may impact the forecasting accuracy of the model.

stat.ML↗

arXiv2019

Hierarchical Clustering for Smart Meter Electricity Loads based on Quantile Autocovariances

In order to improve the efficiency and sustainability of electricity systems, most countries worldwide are deploying advanced metering infrastructures, and in particular household smart meters, in the residential sector. This technology is able to record electricity load time series at a very high frequency rates, information that can be exploited to develop new clustering models to group individual households by similar consumptions patterns. To this end, in this work we propose three hierarchical clustering methodologies that allow capturing different characteristics of the time series. These are based on a set of "dissimilarity" measures computed over different features: quantile auto-covariances, and simple and partial autocorrelations. The main advantage is that they allow summarizing each time series in a few representative features so that they are computationally efficient, robust against outliers, easy to automatize, and scalable to hundreds of thousands of smart meters series. We evaluate the performance of each clustering model in a real-world smart meter dataset with thousands of half-hourly time series. The results show how the obtained clusters identify relevant consumption behaviors of households and capture part of their geo-demographic segmentation. Moreover, we apply a supervised classification procedure to explore which features are more relevant to define each cluster.

stat.AP↗

arXiv2018

Convex Hull, IP and European Electricity Pricing in a European Power Exchanges setting with efficient computation of Convex Hull Prices

This paper introduces a computationally efficient comparative approach to classical pricing rules for day-ahead electricity markets, namely Convex Hull Pricing, IP Pricing and European-like market rules, in a Power Exchange setting with non-convex demand bids. These demand bids can, for example, be useful to large industrial consumers, and extend demand block orders in use by European Power Exchanges. For this purpose, we show that Convex Hull Prices can be efficiently computed using continuous relaxations for bidding products involving start-up costs, minimum power output levels and ramp constraints, or analogous versions on the demand side. Relying on existing efficient algorithmic approaches to handle European-like market rules for such bidding products, we provide comparative numerical experiments using realistic data, which, together with stylized examples, elucidates the relative merits of each pricing rule from economic and computational perspectives. The motivation for this work is the prospective need for mid-term evolution of day-ahead markets in Europe and in the US, as well as the importance of day-ahead price signals, since these (spot) prices are used as reference prices for many power derivatives. The datasets, models and algorithms programmed in Julia/JuMP are provided in an online Git repository.

math.OC↗