Robust DC optimal power flow with modeling of solar power supply uncertainty via R-vine copulas

Abstract

We present a robust approximation of joint chance constrained DC optimal power flow in combination with a model-based prediction of uncertain power supply via R-vine copulas. It is applied to optimize the discrete curtailment of solar feed-in in an electrical distribution network and guarantees network stability under fluctuating feed-in. This is modeled by a two-stage mixed-integer stochastic optimization problem proposed by Aigner et al. (Eur J Oper Res (2022) https://doi.org/10.1016/j.ejor.2021.10.051). The solution approach is based on the approximation of chance constraints via robust constraints using suitable uncertainty sets. The resulting robust optimization problem has a known equivalent tractable reformulation. To compute uncertainty sets that lead to an inner approximation of the stochastic problem, an R-vine copula model is fitted to the distribution of the multi-dimensional power forecast error, i.e., the difference between the forecasted solar power and the measured feed-in at several network nodes. The uncertainty sets are determined by encompassing a sufficient number of samples drawn from the R-vine copula model. Furthermore, an enhanced algorithm is proposed to fit R-vine copulas which can be used to draw conditional samples for given solar radiation forecasts. The experimental results obtained for real-world weather and network data demonstrate the effectiveness of the combination of stochastic programming and model-based prediction of uncertainty via copulas. We improve the outcomes of previous work by showing that the resulting uncertainty sets are much smaller and lead to less conservative solutions while maintaining the same probabilistic guarantees.

Generating synthetic rainfall fields by R‐vine copulas applied to seamless probabilistic predictions

“…For a more detailed description regarding the fitting procedure of R‐vine copulas to data, we refer to Joe and Kurowicka (2011), and also see Aigner et al . (2023).…”

“…Once fitted to data, these bivariate copulas can generate samples from the fitted copula or obtain functions such as its density. Note that this example was previously presented in Aigner et al (2023). multivariate samples of the multivariate random vector considered with the help of common approaches such as parametric probability distributions, kernel density estimators, or empirical CDFs.…”

“…Although it is outside of the scope of this article to discuss vine copulas in more detail, it should be mentioned that the vine structure is specifically built in such a way that, once the bivariate copulas within the vine structure are fitted, they can be utilized to obtain the vine copula's CDF or PDF or to generate samples. For a more detailed description regarding the fitting procedure of R-vine copulas to data, we refer to Joe and Kurowicka (2011), and also see Aigner et al (2023).…”

“…Vine copulas have recently been used in the field of energy meteorology to model the spatial interrelation of errors in probabilistic forecasts to assess the uncertainties in the power generation of photovoltaic systems (Aigner et al ., 2023; Schinke‐Nendza et al ., 2021), or for the inclusion of spatial dependencies between multiple wind farms in wind energy scenarios (Li et al ., 2022; Tu et al ., 2023). Another field of application is within so‐called weather generators.…”

Optimal management of coupled shared autonomous electric vehicles and power grids: Potential of renewable energy integration

“…Due to the computational diculty of the OPF problem, some approximation methods exist in the literature. One of the most frequently used approximation methods is the DC-OPF model (Aigner et al, 2022), where the constraints are as follows: Eq. ( 15) limits the electricity generation of each node.…”