Assessing the risk of disruption of wind turbine operations in Saudi Arabia using Bayesian spatial extremes

Abstract

Saudi Arabia has been seeking to reduce its dependence on oil by diversifying its energy portfolio, including the largely underused energy potential from wind. However, extreme winds can possibly disrupt the wind turbine operations, thus preventing the stable and continuous production of wind energy. In this study, we assess the risk of disruptions of wind turbine operations, based on return levels with a hierarchical spatial extreme modeling approach for wind speeds in Saudi Arabia. Using a unique Weather Research and Forecasting dataset, we provide the first high-resolution risk assessment of wind extremes under spatial non-stationarity over the country. We account for the spatial dependence with a multivariate intrinsic autoregressive prior at the latent Gaussian process level. The computational efficiency is greatly improved by parallel computing on subregions from spatial clustering, and the maps are smoothed by fitting the model to cluster neighbors. Under the Bayesian hierarchical framework, we measure the uncertainty of return levels from the posterior Markov chain Monto Carlo samples, and produce probability maps of return levels exceeding the cut-out wind speed of wind turbines within their lifetime. The probability maps show that locations in the South of Saudi Arabia and near the Red Sea and the Persian Gulf are at very high risk of disruption of wind turbine operations.

Appendices

Appendix A: Supplementary figures, tables and discussions

Fig. 8

Cluster means for GMLEs of the shape parameter ξ with different values of k and different assigned weights for longitude, latitude and GMLEs for ξ, respectively

Table 2 Range of cluster sizes with different values of k and different weights. The two “best” configurations are in bold

Fig. 9

Example for a cluster neighbor. Each color represents one single cluster, and all the seven clusters represent the cluster neighbor of the central cluster in blue

Fig. 10

QQ-plots of marginal GPD fitting for 9 randomly selected locations, using 95% quantile as the threshold

1.1 Discussion on CAR and IAR specifications for spatial random effects

There have been discussions on whether the proper or improper CAR (referred to as IAR) specification should be used in practice (e.g., Besag et al. 1995, Besag and Kooperberg 1995, Gelfand and Vounatsou 2003 and Banerjee et al. 2014). As Besag et al. (1995) showed, the marginal maximal bivariate correlation that can be captured with a proper Gaussian field is around 0.6. Besag and Kooperberg (1995) pointed out a common disadvantage of a proper CAR that appreciable correlations between the spatial random effects at neighboring sites require parameter values extremely close to a particular boundary of the parameter space. Gelfand and Vounatsou (2003) demonstrated that IAR is analogous to the nonstationary or random walk case in familiar autoregressive time series models and can be advantageous in accommodating more irregular spatial behaviors. Banerjee et al. (2014) also claimed that the breadth of spatial patterns may be too limited if the proper CAR is used, and the improper IAR choice may actually enable a wider scope for posterior spatial patterns.

Fig. 11

Values of threshold and KL divergence for marginal extremes using 90% (top) and 95% (bottom) quantile, respectively, as a rule of thumb in the GPD model

Here we first implement our hierarchical model with the proper CAR prior used for the spatial random effects ϕ. The multivariate CAR model (Kavanagh et al. 2016) we used is given by:

$$ \boldsymbol{\phi} \sim \mathcal{N}_{3N}\left( \boldsymbol{0},[\boldsymbol{Q}(\boldsymbol{W},\rho)\otimes \boldsymbol{\Sigma}^{-1}]^{-1} \right), $$

where Q(W,ρ) = ρ[diag(W1) −W] + (1 − ρ)I (1 is the N × 1 vector of ones, I is the N × N identity matrix) is the N × N precision matrix for the joint distribution corresponding to the CAR prior proposed by Leroux et al. (2000), while Σ is a 3 × 3 cross-variables covariance matrix. The matrix Q(W,ρ) controls the spatial autocorrelation structure of the random effects, and is based on a non-negative symmetric N × N neighborhood (or adjacency) matrix W, and a spatial dependence parameter ρ. We use the common binary specification for W, where its entry w_ij = 1 if the grid cells i and j are adjacent, and is zero otherwise. The parameter ρ is a spatial autoregressive parameter, with ρ close to one corresponding to strong spatial dependence and ρ = 0 corresponding to independence in space. When ρ = 1, we obtain the multivariate IAR model. The correlation structure is specified via the full conditionals:

$$ (\boldsymbol{\phi}_{i}|\boldsymbol{\phi}_{-i},\boldsymbol{W},\boldsymbol{\Sigma},\rho) \sim \mathcal{N}_{3}\left( \frac{\rho{\sum}_{k=1}^{N}w_{ki}\boldsymbol{\phi}_{i}}{\rho{\sum}_{k=1}^{N}w_{ki}+1-\rho}, \frac{\boldsymbol{\Sigma}}{\rho{\sum}_{k=1}^{N}w_{ki}+1-\rho}\right), $$

where ϕ_−i denotes the set of spatial random effects except those at the i th location. With the choice of the matrix W, the conditional expectation of spatial random effect at one location is a weighted average of the random effects in its adjacent locations, and the covariance is weighted by the number of adjacent locations.

In our Bayesian hierarchical model, a Uniform[0, 1] prior is assigned to the spatial autoregressive parameter ρ, as the negative spatial autocorrelation is rarely seen in practice in spatial areal unit data (Tobler 1970), and ρ ∈ [0, 1) is a sufficient condition for the covariance matrix of the joint distribution to be nonsingular (Banerjee et al. 2014). The parameter ρ is updated with the MH algorithm, where the candidate for ρ is drawn from a truncated normal distribution in the unit interval so as to bound ρ in [0, 1). Other settings for priors and computational details are the same as in the main text. The posterior density from MCMC samples for ρ is peaked near ρ = 1 for all subregions (see Figs. 12 and 13), suggesting that there is more spatial dependence in the data than the model can capture. Therefore, we replace the multivariate CAR with the multivariate IAR in order to capture the irregular and strong spatial dependence in our high-resolution data. Although the IAR is improper, we are only using it as a prior; the posterior will typically still emerge as proper, so Bayesian inference can still proceed. On the other hand, we can impose a sum-to-zero constraint on ϕ as a remedy of impropriety, which is numerically convenient in the MCMC sampling procedure.

Fig. 12

Trace plot of 10,000 MCMC iterations for the parameter ρ in the MCAR model in a random spatial cluster as selected in Fig. 1 in the main text

Fig. 13

Posterior mean of the spatial autoregressive parameter ρ in the MCAR model from MCMC samples with Bayesian hierarchical spatial extremes modeling

Fig. 14

Potential best locations for siting wind farms over Saudi Arabia where the wind speeds exceed 9 m/s for at least half of the time in Summer, and the risk of disruption of wind turbine operations is lower than 1% for the setting of a k = 200, Threshold = 95% quantile and b k = 250, Threshold = 95% quantile. “prob” stands for “probability of the 30-year return levels that exceed 25 m/s”

Fig. 15

Cluster means of shape GMLEs with a all data and b locations with highly uncertain marginal shape estimates (i.e., where the length of temporal cluster maxima, $N_{u_{i}}$, is less than 50) removed

Appendix B: Supplementary R codes

The datasets generated and/or analyzed during the current study were used under license, and so are not publicly available. Data are available however from the corresponding author upon reasonable request. The R codes related to this article can be found online at the github repository: https://github.com/wanruofenfang123/Bayesian-Hierarchical-Spatial-Extremes.