Power ramps are sudden changes in turbine power and must be accurately predicted to minimize costly imbalances in the electrical grid. Doing so requires reliable wind speed forecasts, which can be obtained from ensembles of physical numerical weather prediction (NWP) models through statistical postprocessing. Since the probability of a ramp event depends jointly on the wind speed distributions forecasted at multiple future times, these postprocessing methods must not only correct each individual forecast but also estimate the temporal dependencies among them. Typically though, crucial dependencies are adopted directly from the raw ensemble, and the postprocessed forecast is limited to the tens of members computationally feasible for an NWP model.

We extend statistical postprocessing to include temporal dependencies using novel multivariate Gaussian regression models that forecast 24-dimensional distributions of next-day hourly wind speeds at three offshore wind farms. The continuous joint distribution forecast is postprocessed from an NWP ensemble using flexible generalized additive models for the components of its mean vector

Wind power is an environmentally friendly energy source but difficult to
integrate with the electrical grid because of its high temporal variability

There is no single definition for a power ramp – they form a broad class of
events with different durations, magnitudes, or types. In all cases though, ramps are joint events which depend on the power – and thus the underlying
wind speed – at multiple times. Since the wind speed forecasts for these
individual times cannot be assumed to be independent, the true temporal
dependencies among them must be estimated in order to reliably predict power
ramps

Predicting the probability that a power ramp occurs during the next day
requires reliable wind speed forecasts, which are commonly based on physical
numerical weather prediction (NWP) models

Improved probabilistic wind speed forecasts that are calibrated – i.e.,
statistically consistent with observations – and sharp – i.e., with as little uncertainty as possible – can be obtained by postprocessing NWP ensembles using statistical methods

Here we employ novel multivariate Gaussian regression (MGR) models

Ramp probabilities derived from MGR models are compared to ECC and other reference
methods that forecast a joint distribution of wind speeds – relying either on (i) a multivariate Gaussian distribution fit

Observational data and the NWP ensemble are described in Sect.

In order to predict power ramps, numerical weather predictions (NWPs) of wind
speed are first postprocessed using the methods presented in
Sect.

Observations and predictor variables used to model distributional parameters for wind speed. The placeholders

Observations of wind speed are taken from 100 m a.g.l. (above ground level) – the approximate hub height of large offshore wind turbines – at meteorological towers on the three Forschungsplattformen in Nord- und Ostsee (FINO) research platforms. These are located near offshore wind farms in the North Sea (FINO1 and FINO3) and in the Baltic Sea (FINO2). The distributions of wind speeds observed at the three sites are skewed, but since the NWP ensemble (Sect.

Wind speed forecasts in the period 26 November 2016 and 24 May 2021 are taken from the ensemble system of the European Centre for Medium-Range Weather Forecasts (ECMWF). The 50 perturbed ECMWF ensemble members have a spatial resolution of 18 km, with 91 model levels and a temporal resolution of 1 h and are always initialized at 00:00 UTC. Forecasts of horizontal wind components

Predictor variables are derived from the ECMWF forecast (Table

For each of the three FINO towers, a dataset is constructed containing values
for all variables in Table

Accurately predicting next-day power ramps requires reliable probabilistic wind speed forecasts, which we obtain by statistically postprocessing NWP ensembles to improve their skill (Table

Information about the models used for postprocessing multivariate wind speed forecasts including (i) the section in which they are described, (ii) whether or not the postprocessed forecast is a joint probability density function, (iii) whether or not dependencies are flexibly modeled, and (iv) if postprocessing is a one-step procedure or marginal distributions and their dependencies are treated in separate steps.

At the FINO stations, hourly wind speed observations

Power ramps depend jointly on wind speeds at different times, so the goal of
postprocessing must be a multivariate forecast which not only contains the
univariate distributions of Eq. (

For a fully parametric multivariate wind speed forecast, Eq. (

The joint distribution in Eq. (

Multivariate Gaussian regression (MGR, Sect.

Joint wind speed distribution forecasts are subsequently evaluated using the
Dawid–Sebastiani score (Sect.

Nonhomogeneous Gaussian regression

Following

While NGR can be used to obtain sharp and calibrated forecasts for individual
times, dependencies between these forecasts are not considered at all.
Ensemble copula coupling

Having postprocessed individual forecasts with NGR, a cumulative distribution
function of the wind speed

To avoid the limitations on ensemble size that come with ECC, a joint
distribution may be estimated from the postprocessed ensemble for each day using a multivariate Gaussian distribution fit

With the model GDF,

Since the number of ensemble members (50) is not much greater than the
distributional dimension (24), estimates for

Alternatively, joint distributions may be generated without using the temporal dependencies from the ECMWF ensemble at all. Instead, a constant correlation matrix – equivalent to a Gaussian copula

The model COP (Obs) estimates correlations from raw observations as
inspired by the Schaake shuffle. The entries of

The model COP (Err) takes an analogous approach but instead estimates

Multivariate Gaussian regression

Modeling the components of

Different parameterizations exist to guarantee positive definite

The model MGR (AR1) uses a variance–correlation decomposition
(Eq.

To ensure

In order to model the correlation parameter on predictors, the link function

While a variance–correlation parameterization (Sect.

The innovation variances are modeled analogously to the standard deviations
in Eq. (

As noted by

Evaluating multivariate forecasts is not straightforward, and no score can
guarantee optimal forecasts for every application. When evaluating multivariate Gaussian forecasts, it is most natural to employ the Dawid–Sebastiani score

Following

Evaluating the skill of these predictions requires observations, which are again obtained through transformations. Here the ramp observation is Boolean: either the event occurred or it did not. It may happen that no ensemble members predict a ramp on a given day and the estimated ramp probability is zero – especially for the ECC-postprocessed ensemble limited to 50 members. For this reason, ramp forecasts are evaluated using skill scores calculated from the area under the receiver operating characteristic (ROC) curve – the ROCSS – and the Brier score – the BSS – rather than a metric based on the Bernoulli likelihood.

The models outlined in Sect.

Sample postprocessed joint probabilistic wind speed scenarios for 9 December 2019 at FINO1 are visualized in Fig.

Wind speed ensembles at FINO1 forecasted for the next day (

Even without postprocessing, the ECMWF ensemble forecast performs quite well on this day. Observations between 00:00 and 06:00 UTC are contained within the narrow ensemble spread (i.e., sharp forecast). All members correctly predict a gradual decline in wind speed during the morning before a return to higher winds by afternoon. Uncertainty in the ECMWF ensemble increases after 06:00 UTC because not all ensemble members predict ramps at the same time but instead up to 6 h apart.

While the methods of Sect.

To quantify the forecast quality across all days, postprocessed joint distributions are evaluated using differences in the DSS
(Sect.

Differences in Dawid–Sebastiani score (DSS) to COP (Err), aggregated by month and year.

The multivariate Gaussian regression model MGR (AR1) performs best at each station. Errors are adequately described by a first-order autoregressive process, and allowing more flexible dependency structures to be modeled using Cholesky-based parameterization – e.g., MGR (AD1) or MGR (AD2) with assumed first- and second-order antedependencies, respectively – does not improve the scores.

Postprocessed joint distributions of wind speeds are used to simulate
1000-member ensembles that are converted into multivariate power scenarios
using the theoretical turbine curve of Fig.

Theoretical power curve of a large offshore wind turbine.

Sample power scenarios at FINO1 are visualized in Fig.

The next-day hourly forecasts (

Multivariate power scenarios are used to derive probabilities for weak and
strong up ramps, where the normalized power increases by 0.3 and 0.6,
respectively. These probabilities are included as

Probabilistic ramp predictions are obtained in this manner for every day and
each station. All combinations of ramp magnitudes (weak or strong), types (up or down), forecast window widths (3 or 6 h), and window positions are
considered. To quantify the skill of these ramp predictions, probabilities
obtained from each postprocessing method are evaluated using the scores
described in Sect.

According to the ROCSS (Fig.

Area under the ROC curve skill score (ROCSS) relative to ECC at the three FINOs for weak and strong up ramps (top row panels) and down ramps (bottom row panels) within 3 and 6 h time frames (leftmost and rightmost two columns, respectively). Individual boxplots are composed of scores computed for specific positions (i.e., starting points) of the time window. Three-hour time windows have 22 possible starting points (

The BSS relative to ECC is visualized in Fig.

As in Fig.

We use novel multivariate Gaussian regression (MGR) models to postprocess joint distributions of the next day's hourly wind speeds from NWP ensemble forecasts. Joint distribution forecasts have advantages over traditional ensembles, some of which are addressed in Sect.

Only tens of ensemble members are computationally feasible for a NWP model.
To make the weather forecast more realistic, it is common to generate a continuous distribution from the discrete ensemble members. There are several ways to achieve this, including (i) replacing individual ensemble members with parametric distributions – e.g., ensemble dressing, Bayesian model averaging – or (ii) modeling a parametric distribution on the ensemble with distributional regression. For multivariate forecasts, there are also similar methods – e.g., ensemble kernel dressing

Multivariate forecasts postprocessed from NWP ensembles need to accurately
describe the true dependencies between forecasts at different times. If the
result of postprocessing is an ensemble with tens of members, it is poorly
suited to accomplish this task except for very small dimensions because the
complexity of the dependencies increases quadratically with the dimension. At all three FINO stations, ramp predictions are improved when using 1000-member wind speed ensembles simulated from joint distributions instead of 50-member ensembles generated with ECC (Sect.

NWP ensembles are postprocessed to ensure that forecasts for any future lead time are calibrated (statistically consistent) and sharp. Since individual ensemble members are distinguishable NWP runs, the dependencies between member forecasts at different times have a physical basis and may contain valuable information regarding the dependencies between the postprocessed forecasts. In state-of-the-art postprocessing, this information is not fully utilized in the same way that the ensemble mean and spread are.

Common methods either directly adopt the order statistics of the NWP ensemble for the dependencies of the postprocessed forecast or apply the same principle using observations. Directly adopting the error dependencies of the NWP ensemble is most sensible when the raw forecast is already quite good, as is the case here with a forecast horizon of at most 48 h and a homogeneous ocean surface surrounding the stations. For the same reason, dependencies estimated from observations are much too strong. Presumably these would more accurately reflect the error structure for a very poor NWP prediction, which approximates an hourly wind speed climatology.

With MGR, a major advantage is the ability to model error dependencies on predictors without a reliance on such assumptions. This allows the temporal error structure of the NWP ensemble to be adjusted for the postprocessed forecast as is commonly done for its location and spread at each time with NGR. If no ensemble is available to obtain the dependencies, MGR can still be used to estimate these from the deterministic forecasts together with external predictors, for example characterizing the time of the year or the synoptic-scale weather situation.

Probabilistic power ramp predictions are obtained for three offshore wind farms from joint distributions of hourly wind speeds postprocessed using novel multivariate Gaussian regression (MGR) models. This model employs a
multivariate Gaussian distribution for the 24-hourly wind speeds of the next
day where the mean vector

The code is available upon request by contacting the correspondence author.

The data are not publicly accessible since they contain forecasts from the ECMWF (

TM, GJM, TS, and AZ planned the research. TM, TS, MNL, and AZ developed software. JWM added expertise in wind energy forecasting. TM wrote the original manuscript draft, and all authors subsequently reviewed and revised it.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank the Bundesamt für Seeschifffahrt und Hydrographie for supplying observations from the FINO towers and the European Centre for Medium-Range Weather Forecasts for the numerical weather predictions. Computational results presented here have been achieved (in part) using the LEO HPC infrastructure of Universität Innsbruck.

This research has been supported by the Austrian Science Fund (grant no. P 31836). Thomas Muschinski was also supported through the doctoral scholarship (Doktoratsstipendium) of the Universität Innsbruck.

This paper was edited by Raúl Bayoán Cal and reviewed by two anonymous referees.