Optimal placement of wind farms via quantile constraint learning

TL;DR

This work tackles the problem of optimally siting and sizing multiple wind farms within a region under uncertain wind production. It introduces a data-driven surrogate, Incremental Quantile Neural Network (IQNN), that learns conditional wind-power quantiles and is embedded as linearizable constraints in a two-stage stochastic MILP to account for spatiotemporal wind dynamics and risk preferences. The method balances transmission-line costs with CVaR-based risk (tail risk) and demonstrates superior quantile estimation over bilinear interpolation using high-resolution ERA5-Land data from Asturias, Spain. Case studies reveal that risk-averse investors favor spatial diversification and near-substation connections, while risk-neutral investors push toward distant, higher-mean wind sites, providing actionable guidance for portfolio wind-farm development under uncertainty.

Abstract

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.

Figures (8)

• Figure 1: Incremental Quantile Neural Network (IQNN) for wind power quantiles. Figure and caption adapted from alcantara2025quantile.
• Figure 2: 0.1° × 0.1° (latitude, longitude) grid covering Asturias
• Figure 3: Wind speed and power curves for wind farms sample, with sitting coordinates of $\bigl\{(43.0 \degree N, -7.1 \degree E),(42.9 \degree N, -5.5 \degree E)\bigr\}$ and sizing of $(20,20)$ turbines
• Figure 4: IQNN prediction and empirical real values of wind generation quantiles for three samples, including Sites 1 of $\bigl\{(42.9 \degree N, -7.1 \degree E),(43.2 \degree N,-6.4 \degree E)\bigr\}$, Sites 2 of $\bigl\{(43.1 \degree N, -6.8 \degree E),(42.9 \degree N, -6.3 \degree E)\bigr\}$. Each wind farm has the same turbine size of $(20,20)$ or $(50,50)$.
• Figure 5: Geometry instance of bilinear interpolation.