Probabilistic Forecasting of Temporal Trajectories of Regional Power Production - Part 1: Wind
PProbabilistic Forecasting of Temporal Trajectories of Regional PowerProduction—Part 1: Wind
Thordis L. Thorarinsdottir, Anders Løland, and Alex Lenkoski ∗ March 5, 2019
Abstract
Renewable energy sources provide a constantly increasing contribution to the total energy pro-duction worldwide. However, the power generation from these sources is highly variable due totheir dependence on meteorological conditions. Accurate forecasts for the production at varioustemporal and spatial scales are thus needed for an efficiently operating electricity market. In thisarticle – part 1 – we propose fully probabilistic prediction models for spatially aggregated windpower production at an hourly time scale with lead times up to several days using weather forecastsfrom numerical weather prediction systems as covariates. After an appropriate cubic transformationof the power production, we build up a multivariate Gaussian prediction model under a Bayesianinference framework which incorporates the temporal error correlation. In an application to predictwind production in Germany, the method provides calibrated and skillful forecasts. Comparison ismade between several formulations of the correlation structure.
Recent years have seen a worldwide proliferation in energy production from renewable energy sources.In Germany, for instance, renewable energy accounted for 36 .
0% of the total national energy productionin 2017 compared to 6 .
6% in 2000 according to the Arbeitsgemeinschaft Energiebilanzen, a workinggroup founded by energy related associations in Germany. This increase is to a large extent due toexpansion in wind and photovoltaic (PV) solar power production. (Wind power production accountedfor 1 .
6% in 2000 and as much as 17 .
6% in 2017.) However, as these energy sources rely on theprevailing wind and solar irradiance conditions, as well as other weather variables, the resulting powergeneration is highly variable and uncertain. Simultaneously, accurate production forecasts are neededfor the management of electricity grids, for scheduling of the production at conventional power plantsas well as for general decision making on the energy market e.g. [20, 5]. These different contexts implyvarying loss functions which, together with the need to control the trade-off between risk and return,calls for a probabilistic forecasting framework [9]. Probabilistic forecasts are becoming increasinglyfrequent for wind power forecasting [1, 31, 17, 29, 30, 16, 4].Time series approaches e.g. [11, 30, 4] usually outperform other methods for lead times up to 3-6 hafter which they may be improved upon by statistical methods that relate the expected production toweather forecasts from numerical weather prediction (NWP) models. The usual approach is to modela single unit or a farm. [36] utilize the local wind speed observations to calibrate wind speed densityforecasts which are subsequently transformed to wind power while [1] and [32] directly model therelationship between the wind speed forecasts and the power production. Alternatively, [24] employan inverse power curve transformation in a regression framework and [17] consider a stochastic powercurve model. A recent comprehensive review of available wind power prediction models at varioustime scales is [8]. The Global Energy Forecasting Competitions (GEFCom2012 [15] and GEFCom2014[16]) have attracted hundreds of participants worldwide, who contributed many novel ideas to theenergy forecasting field, and day ahead wind power forecasting in particular. A clear majority of the ∗ Norwegian Computing Center, Oslo, Norway (e-mail: [email protected]). a r X i v : . [ s t a t . A P ] M a r ontestants applied machine learning methods, like gradient boosting regression and quantile regressionforest [27] or K-nearest neighbors [38].Many end-users require forecasts of aggregated power production over a market region or for aregional transmission organization. Regional forecasts are often formed by an upscaling of a set ofindividual sites [20, 35]. This requires an up-to-date account of the overall installed capacity, hourlyproduction data for the region as a whole and production data from a representative set of sites. Incountries such as Germany with continued expansion of renewable energy production, this can be acumbersome task. Instead, we propose to directly predict the aggregate country-wide production usingspatially averaged NWP forecasts of the relevant weather variables as inputs. Similarly, applications insystem operation and planning call for forecasts over multiple lead times returning calibrated forecasttrajectories. Several studies have applied copula approaches to account for the error correlationstructure across lead times [33] or the correlation between different locations [13, 23, 28]. However,the marginal predictive distributions are often modeled independently in a non-parametric fashion e.g.[33]. We specify the probabilistic prediction model as a Bayesian hierarchical model, which allows usto incorporate a correlation structure in both the model parameters associated with each lead timeas well as the error structure across lead times. A recent review [6] notes that renewable energyforecasting systems that focus on long lead-times, regional level data and use the combination ofmeteorological and production data is largely unexplored in the literature, making this system one ofthe first to combine these aspects.The NWP forecasts and the German power production data are introduced in the next Section 2.The prediction models and the statistical inference methods are derived in Section 3. The forecastverification methods we employ are described in Section 4, and the results are presented in Section 5.We conclude with a discussion in Section 6. We employ the NWP forecast ensemble issued by the European Centre for Medium-Range WeatherForecasts (ECMWF) which has been shown to perform well in this setting [7]. The 50-memberECMWF ensemble system operates at a global horizontal resolution of 0 . × .
25 degrees, a resolutionof approximately 32 ×
32 km over Germany, and a temporal resolution of 3–6 h with lead times upto ten days [22, 26]. We restrict attention to the forecast initialized at 00:00 UTC, corresponding to2:00 am local time in summer and 1:00 am local time in winter, and lead times up to 72 h for 100 mwind speed.The hourly wind power production data for Germany are obtained from the European Energy Ex-change (EEX) where they are available to all members that trade on the EEX, see . We use data from the calendar year 2011 to assess the optimal length of the training pe-riod for parameter estimation as well as for determining the prior parameters of the Bayesian model.Given these values, we then test our methods on data from 2012. In order to obtain equally longtraining periods for all dates, data from the previous year is used for the parameter estimation at thebeginning of a year.The differences between the individual members in an ECMWF ensemble stem from randomperturbations in initial conditions and stochastic physics parameterizations in the numerical model.The ensemble members are thus statistically indistinguishable, or exchangeable, and should be givenequal weights in a regression framework. We therefore reduce the ensemble to a single forecast givenby the ensemble average. For the operation and management of electricity grids, power productionpredictions are needed on an hourly basis. However, for the first 72 h, the ECMWF forecasts have atemporal resolution of 3 h. We derive hourly forecasts through a spline interpolation conditional onthe variables being non-negative.In a third preprocessing step, we aggregate the forecasts in space by taking the spatial average.The wind power production is largely concentrated in the northern half of the country. Rather thanemploying the aggregated forecasts over the entire country, we thus focus on the northern half only(latitudes greater than 51) for the wind speed, which results in a stronger relationship between theforecasts and the power production. As a result, the wind speed forecast is an average over 371 grid2ocations.
The nonlinear relationship between wind speed and the power output from an individual turbine isdescribed by the power curve. The turbine blades begin to rotate at the cut-in speed and the maximumpower output of the turbine is generated from the rated speed until the cut-out speed, at which speedthe blades stop rotating to prevent damage. These parameters may vary between different turbinesand, in practice, the power curve is not deterministic [17]. Our production data is the aggregatedpower output from thousands of wind turbines spread over a large geographic area. It is thus highlyunlikely that the wind speed is below the cut-in speed or above the cut-out speed simultaneously atall the turbines. This is confirmed by Figure 1 which shows that the data does not appear heavilycensored. llll l llllll ll l lll lllll ll llllllll l l l lll ll l lll l lllll llll ll llll lll l ll l ll ll ll l lllllll llllllll l ll ll l l ll lll l llllll lll l llllll l lll lll ll lllll ll l l lllll l l ll l ll ll l ll ll llll llll l lll ll l lll lllll ll ll l lll ll lllll l l ll llll l l llllll l llll llll ll llllll llll lll llll l l ll llll l ll l llll l l llll ll lll lllllll llll l ll ll ll l lllll ll lllll l l lll ll ll ll l lll llll llll lllllllll l l l lll ll ll ll ll l llll l lll llllll l l l lll lll llll l lllll l ll l lll lllll ll llllllll l l l lllll l lll l lllll llll ll llll lll l ll l ll llll l lllllll llllllll l ll ll l l ll lll l llllll lll l lllll l l lll lll ll lllll ll l l lllll ll ll l ll ll l ll ll llll llll llll ll l lll lllll ll ll l lll ll lllll l l ll llll l l llllll l llll llll ll ll llll ll ll lll llll l l ll llll l ll l llll l l llll ll lll lllllll llll l ll ll ll l lllll ll 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Cubed Wind Speed Forecast W i nd P o w e r ( M W ) llll l lllll l ll l lll lllll ll lllllll l l l l lll ll l lll l lllll llll ll llll lll l ll l ll ll ll l lllll l l lll lllll l ll ll l l ll lll l llllll lll l lllll l l lll lll l l lllll ll l l llll l l l ll l ll ll l ll ll lll l ll ll l ll l ll l ll l lllll ll ll l lll ll lllll l l ll llll l l lllll l l lll l llll ll llllll ll ll lll llll l l ll llll l ll l llll l l llll ll ll l lll llll llll l ll ll ll l llll l ll lllll l l lll ll ll ll l lll l lll llll l lllll lll l l l lll ll ll ll ll l llll l lll llllll l l l lll lll llll l lllll l ll l lll lllll ll lllllll l l l l lllll l lll l llll l llll ll lll l lll l ll l ll ll ll l lllll ll lll lllll l ll ll l l ll lll l llllll lll l lllll l l lll lll l l lllll ll l l llll l ll ll l ll ll l ll ll lll l ll ll l ll l ll l ll l lllll ll ll l lll ll lllll l l ll llll l l lllll l l lll l llll ll ll lll l ll ll lll llll l l ll llll l ll l llll l l llll ll lll lllllll llll l ll ll ll l lllll ll lllll l l lll ll ll ll l lll lll l llll l lllll lll l l l lll ll ll ll ll l llll l lll llllll l l l lll lll llll l lllll l ll l lll lllll ll ll lllll l l l l lllll l lll l llll l llll ll lll l lll l ll l ll llll l lllll l l l llll lll l ll ll l l ll lll l lll lll lll l lllll l l llllll l l lllll ll l l llll l ll ll l ll ll l ll ll lll l ll ll l ll l ll lll l lllll ll ll l lll ll lll ll l l ll llll l l lllll l l lll l llll ll ll lll l ll ll lll llll l l ll llll l ll l llll l l llll ll lll llll lll llll l ll ll ll l lllll ll lllll l l lll ll ll ll l lll lll l llll l lllll lll l l l lll ll ll ll ll l lll l llll lll lll l l l lll lll llll l ll lll l ll l lll lllll ll ll lllll l l l l lllll l lll l llll l llll ll lll l lll l ll l ll llll l lllll l l l llll lll l ll ll l l lllll l lll lll lll l lllll l l llllll l l lllll ll l l llll l ll ll l ll ll l ll ll lll l ll ll lll l ll lll l lllll ll ll l lll ll lll ll l l ll llll l l lllll l l llll ll ll ll ll lll l ll ll lll llll l l llllll l ll l llll l l llll lll ll llll lll llll l ll ll ll l llll l ll lllll l l lll ll ll ll l lll lll l llll l lllll lll l l l lll ll ll ll ll l lll l llll lll lll l l l lll lll lll l l ll lll l ll l lll lllll ll ll lllll l l l l lll ll llll lllll l llll ll lll l lll l ll l ll llll l lllll l l l llll lll l ll l l l l llll l l lll lll lll l lllll l l llllll l l lllll ll l l llll l ll ll l ll ll l ll ll lll l ll ll lll l ll lll l lllll ll ll l lll ll lll ll l l ll llll l l lllll l l llll ll ll ll ll lll l ll ll ll ll lll l l l lllll l ll l llll l l llll lll ll lll l lll l ll l l ll ll ll l llll l ll lllll l l lllll ll ll l lll lll l llll l lllll lll l l l lll l l ll llll l lll l llll lll lll l ll lll lll lll l l ll lll l ll l lll lllll ll ll lllll l l l l lll ll llll lllll l llll ll lll l lll l ll l ll llll l lllll l ll llll lll l lll l l l llll l l lll lll lll l lllll l l llllll l l lllll ll l l llll l ll ll l ll ll l ll ll lll l ll ll lll l ll lll l lllll ll ll l lll ll lll ll l l ll lll l l l l llll l l lll l ll ll ll ll lll l ll ll ll ll lll l l l lllll l ll l llll l l llll lll ll lll l lll l ll l l ll ll ll l llll l ll lllll l l lllll ll ll l lll lll l llll l lllll lll l l l lll l ll l lll l l lll l llll lll lll l ll lll lll lll l l ll lll l ll l llllllll ll ll lllll l l l l lll ll llll lllll l llll ll lll l lll lll l llllll l lllll l ll ll ll lll l lll l l l llll l l lll lll lll l lllll l l llllll l l lllll ll l l llll l ll ll l ll ll l ll ll lll l ll ll l ll l l l lll l lllll ll ll l lll ll lll ll l l ll l ll l l l l llll l l lll l ll ll ll ll lll l ll ll ll ll lll l l l lllll l ll l llll l l l lll lll ll lll l ll l l ll l l l l ll ll l llll l ll lllll l l lllll ll ll l lll lll l llll l lllll lll l l l lll l ll llll l l lll l llll lll lll l ll lll lll lll l l ll lll l ll l llllllll ll ll lllll l l l l lll ll llll lllll l llll ll lll l lll lll l l lllll l lllll l ll ll ll lll l lll l l l llll l l llllll lll l lllll l l ll llll l l lllll ll l l llll l ll ll l ll ll l ll ll lll l ll ll l ll l l l llll lllll ll ll llll ll lll ll l l ll l ll l l l lllll l l lll l ll ll ll ll lll lll ll ll ll lll l l l lllll l l l l llll l l l lll lll ll llll ll ll ll l l l l ll ll l llll l ll lllll l l lllll ll ll l lll lll l llll l lllll lll l ll ll l l ll llll ll lll l lll l lll lll l lllll lll lll l l ll lll l ll l llll llll ll ll llll l l l l l lll ll l lll lllll l llll ll lll l lll lll l l lllll l ll lll l ll llll lll l lll l l l llll l l llllll ll l l lllll l l ll llll l l lllll ll l llllll ll ll l ll ll l ll ll lll l ll ll lll l l l llll lllll ll ll llll ll lll ll l lll l ll l l l lllll l l lll l ll ll l l ll lll lll ll llll lll l l l lllll l l l l llll l l l lll lll ll llll ll ll ll l l l l ll ll l llll l ll lllll l l lllll ll ll l lll lll l llll l lllll lll l ll ll l l ll llll ll lll l lll llll lll l lllll lll lll l l ll lll l ll l llll llll ll ll llll l l l l l lll ll l lll lllll l lll l ll lll l lll lll l l lllll l ll lll l ll llll lll l lll l l l llll l lllllll ll l l lllll l l ll ll ll l l lllll ll l llllll ll ll l ll lll ll ll lll l ll ll lll l l l llll ll lll ll ll llll ll lll ll l lll l ll l l l llll l l l lll l ll ll l l ll lll lll ll ll ll lll l l l lllll l l l l llll l l l lll lll ll llll ll ll ll l l l l ll ll l llll l ll lllll l l lllll ll ll llll lll l llll l lllll lll l l l ll l l ll llllll lll l lll lllllll l llll l lll lll l l ll lll l ll l lll l llll ll ll llll l l l l l lll ll l lll lll ll l lll l ll lll l lll lll l l lllll l ll lll l ll l lll lll l lll l l l llll l lllll ll ll l l lllll l l ll ll ll l l lll ll ll l llllll ll ll l ll lll ll ll lll l ll ll lll l l l llll ll lll ll ll llll ll lll ll l lll l ll l l l llll l l l lll l ll ll l l ll lll lll ll ll ll lll ll l lllll l l l l llll l l l lll lll ll llll ll ll ll l l l l ll ll l llll l ll lllll l l lllll ll ll llll lll l llll l lllll lll l l l ll l l ll ll llll lll l lll lllll ll l llll l lll lll ll ll lll l ll llll l llll l l ll lllll l l l l lll ll l lll lll ll l lll l ll lll l lll lll l l lllll l ll lll l ll l lll lll l lll l l l l lll l llll l ll ll l l ll lll l l ll ll ll l l lll ll ll l llllll ll ll l ll lll ll ll lll l ll ll lll l l l llll ll lll ll ll llll ll l ll ll l lll l ll l l lllll l l l lll l ll ll l l ll lll lll ll ll ll lll ll l lllll l l l l llll l l l lll lll l l llll llll ll l l l l ll ll l llll l ll lllll l l lllll ll ll llll lll l llll l lllll lll l l l ll l l ll ll llll lll l lll lllll ll l llll l lll lll l l ll lll l ll llll l llll l l ll lllll l l l l lllll l lll lll ll l lll l ll lll l lll lll l l lllll l ll lll l ll l lll lll llll l l l l lll l llll l ll ll l l ll l ll l lll ll ll l l lll ll l l l llllll ll ll lll lll ll ll lll l ll ll lll l l l llll lllll ll ll llll ll l ll ll l lll l ll l l lllll l l l lll l ll l l l l ll lll lll ll ll ll lll ll l lllll l l l lllll l ll lll lll l l llll llll ll l l l l ll ll lllll l ll llll l l l ll ll l ll ll ll ll lll l llll l lllll lll l l l ll ll ll ll ll ll lll l lll lllll ll l llll l lll lll l l ll lll l ll lllll llll l l ll lllll l l l l lllll l lll lll ll l lll l ll lll l lll lll l l lllll l ll lll l ll l lll lll llll l l l l lll l llll l ll ll l l ll lll l lll ll ll l l lll ll l l l llllll ll ll lll lll ll ll lll lll ll lll ll l llll lllll ll ll llll ll lll ll l lll l ll l l lllll l l l lll l ll l l l l ll lll lll ll ll ll lll ll l lllll l l l lllll l ll lll lll l l llll lll l ll l l l l ll ll lllll l ll llll l l lll ll l ll ll ll ll lll l lll l l lllll lll l l l ll ll ll ll ll ll lll l lll lllllll l llll l lll lll l l ll lll l ll lllll l lll l l ll lllll l l l l lllll l lll lll ll l lll l ll lll l lll lll l l lllll l ll lll l ll l lll lll llll l l l l lll l llll lll ll l l ll lll l lll ll ll l l lll l l l l l llll ll ll ll lll ll l ll ll lll lll ll lll ll l ll ll lllll ll ll llll ll lll ll l lll l ll l l lllll l l l lll l ll l l l l ll lll lll ll ll ll lll ll l lllll l l l lllll l ll lll l ll l l llll lll l ll l l l l ll ll lllll l ll llll l l lll ll l ll ll ll ll lll l lll l l llll l lll l l l ll ll ll ll ll ll lll l lll llll lll l llll l lll lll l l ll lll l ll lllll l lll l l ll lllll l l l llllll llll lll ll l lll l l l lll l lll lll l l lllll l ll lll l ll l lll lll llll l ll l lll l llll lll ll l l ll lll l lll ll ll l l lll l l l l l llll ll ll ll lll ll lll lllll lll ll lll ll l ll ll llll l ll ll llll ll lll l l l lll l ll l l lllll l l l lll l ll l l l l ll lll lll ll ll ll lll ll l lllll l l l lllll l lllll l ll l l llll lll l ll l l l l ll ll lllll l ll llll l l lll ll l ll ll ll ll lll l lll l l llll l lll l l l ll ll ll ll ll ll lll l lll llll lll l llll l lll lll l lll lll l ll lllll l lll l l ll lllll l ll lll lll llll lllll l lll ll l lll l lll lll l l lllll l ll lll l ll l lll lll llll l ll l lll l llll lll ll l l lllll l lll ll ll l l lll l ll l l llllll ll ll lll ll llllllll lll ll lll ll l ll ll llll l l l ll llll ll l ll l l l lll lll l l lllll l l l lll l ll l l l l ll lll lll l l ll ll ll l ll l lllll l l l llll l l lllll l ll l l llll lll l ll l l l l ll ll lllll l ll llll l l lll ll lll ll llll lll l lll l l llll l lll l l l ll ll ll ll ll lllll l lll llll lll l l lll l lll lll l lll lll ll l lllll lllll l ll lllll l ll lll lll llll lllll l lll ll l lll l lll ll l l l llll l l ll lll l ll l lll lll ll ll l ll l lll l llll ll l ll l l lllll l lll ll ll l l lll l ll l l lllll l ll ll lll ll lll lllll lll ll lll ll l ll l lllll l l l l l llll ll l ll l l l lll lll l l llll l l l l lll l lll l l l ll lll lll l l ll ll ll l l l l lllll ll l llll l l lllll l ll l l llll ll l l ll l l l l lll l lllll l l l llll l l lll ll lll ll llll lll l lll l l llll l lll l l l ll ll ll ll ll lllll l lll lllllll l l lll l lll lll l lll lll ll l lllll lllll l ll lllll l ll lll lll llll lllll l lll ll l lll llll ll l l l ll ll l l ll lll l ll l lll ll l ll ll l ll l lll l llll ll l ll l l lllll l lll ll ll l llll l ll l l lllll l ll ll lllll lll lllll lll ll lll ll l ll l lllll l l l l l llll ll l lll ll ll l lll l l llll l l l llll l lll l l l ll lll lll l l ll ll ll l l l l lllll ll l llll ll llllll ll l lll ll ll l l ll l l l l lll l lllll ll l llll l l lll ll lll l l llll lll l lll l l llll l ll l l l l ll ll ll ll ll lllll l lll llllll l l l lll l lll lll l lll lll ll l lll ll lll ll llllllll l ll lll lll llll lllll l lll ll l lll llll ll l l l ll ll l l ll lll l lll lll ll l ll ll l ll l lll l llll ll l ll l l lllll l lll llll l llll l ll l l lllll l ll ll llll l lll lllll lll ll lll ll l ll l lllll l l ll l llll ll l lll ll ll l lll l l llll l l l llll l lll ll llllll lll l l ll ll ll l ll l llll l ll l llll ll llll ll ll l lll ll ll l l ll l l l llll l lllll ll l llll l l lll ll lll l l llll lll l lll l l llll l ll l l l l ll ll ll ll ll lllll l lll llllll l l l lll l lll lll l lll lll ll l lll lllll ll llllllll l ll lll lll llll lllll l lll ll l lll llll ll l ll ll ll l l ll lll l lll lll ll l ll ll l l l l lll l llll ll l ll l llllll l lll lll l l llll l ll l l lllll l ll ll ll ll l lll lllll lllll lll ll l ll l lllll l l ll l lllllll lll ll ll llll l l llll l l l llll l lll ll llllll lll l l ll ll ll l ll l llll l ll l llll ll llll ll ll l lll ll ll l l ll l l l llll l lllll ll l llll l l lll ll lll l l lll l lll l lll l l llll l ll l l l l ll ll ll ll ll l llll l lll llllll l l l lll l lll lll l lll lll ll l lll lllll ll llllllll l ll lll lll llll lllll llll ll l lll lll l ll l ll ll ll l l ll lll l l ll lll ll l ll ll l l l l lll l llll ll l ll l llllll l lll lll l l llll l ll l l lllll l llll ll ll l lll l llll lllll ll l ll l ll l lllll ll ll l lll ll ll lll ll ll llll l l llll l l l llll l lll ll llllll lll l l ll ll ll l lll llll l ll l llll ll llll ll ll l lll ll ll l l ll l ll llll l lllll ll l llll l l lll ll lll l l lll l lll l lll l lllll l ll l l l l ll ll ll ll ll l llll l lll llllll l l l lll l lll lll l lll lll ll llll lllll ll llllllll l ll ll l lll llll lllll llll ll llll lll l ll l ll ll ll l llllll l l ll lllll l ll ll l l l l lll l llll ll l ll l llllll l lll lll l l llll l ll l l llll l l llll ll ll lllll llll lllll ll l ll l ll l lllll ll ll l lll ll ll lll ll ll llll l l llll l l l llll llll ll llllll llll l ll ll ll l l ll llll l ll l llll ll llll ll ll l lll ll ll llll l ll llll l lllll ll l llll l l lll ll lll l l lll l lll l lll l lllll l ll l l l lll ll ll ll ll l llll l lll llllll l l l lll l lll lll l lll lll ll llll lllll ll lllllll l l ll ll l lll llll lllll llll ll llll lll l ll l ll ll ll l llllll l l ll lllll l ll ll l l l l lll l llll ll l ll l lllll l l lll lll l l lllll ll l l llll l l llll ll ll l llll llll llll l ll l ll l ll l lllll ll ll l lll ll ll lll ll ll llll l l lllll l l llll llll ll llllll ll ll lll ll ll l l ll llll l ll l llll ll llll ll lll lll llll llll l ll llll l lllll ll l llll l l lll ll ll ll l lll l lll llll l lllll l ll l l l lll ll ll ll ll l llll l lll llllll l l l lll llll (b) Wind Speed Forecast (m/s) C ube R oo t o f W i nd P o w e r Figure 1: Relationship between hourly wind power production in Germany in 2011 and the corre-sponding wind speed forecasts: (a) wind power production against 1-24 h cubed wind speed forecasts,and (b) cube root of the wind power production against 1-24 h wind speed forecasts.For wind speed values between the cut-in speed and the rated speed, the power output from anindividual turbine is generally proportional to the cubed wind speed [14]. As shown in Figure 1(a),the relationship between the cubed average wind speed forecasts and the aggregated wind powerproduction is highly heteroskedastic with a larger spread for higher wind speeds. We thus follow [24]and, in our prediction model, we model the relationship between the average wind speed forecasts andthe cube root of the resulting wind power, see Figure 1(b). Denote by x w = ( x w , . . . , x wT ) (cid:62) the windspeed forecast for lead times 1 to T . The wind power production Y wt at time t ∈ { , . . . , T } is thengiven by Y / wt = β t + β t x wt + β t x wt + ε t , (1)where β it ∈ R for i = 0 , , ε = ( ε , . . . , ε T ) (cid:62) are assumed correlated in time, ε ∼ N T (0 , K − ) , (2)for some precision matrix K = { K ij } Ti,j =1 . The particular form of the regression equation in (1) wasselected based on the average marginal predictive performance for 1-24 h forecasts in 2011 (resultsnot shown). Alternatives included regression equations with one to three covariates from the set { x w , x w , x w } . The power production is inherently nonnegative and the normal assumption in (2)might therefore be physically unrealistic. However, as the predictand in (1) never takes values closeto zero, see Figure 1(b), we find that, in practice, the predicted probability of negative production isnegligible. Further model validation criteria are discussed in Section 5 below.3enote by X = [ I T Diag( x w ) Diag( x w )] the T × T joint covariate matrix for Y / w , . . . , Y / wT basedon the model in (1). Here, I T is the identity matrix of size T and Diag( x ) denotes a diagonal matrixwith x on the diagonal. The likelihood model for Y = ( Y / w , . . . , Y / wT ) (cid:62) is then given by Y ∼ N T ( X β , K − ) , (3)where β = ( β (cid:62) , β (cid:62) , β (cid:62) ) (cid:62) with β i = ( β i , . . . , β iT ) (cid:62) for i = 0 , ,
2. We estimate the parameters undera Bayesian inference framework with conjugate prior distributions of the form β | K , n ∼ N T (cid:16) , (cid:2) Diag( n ) ⊗ K (cid:3) − (cid:17) , (4) K ∼ W G (3 , I T ) , (5) K ∼ W G (3 , I T ) , (6) n i ∼ Γ(1 , . , i = 0 , , , (7)where ⊗ denotes the Kronecker product, n = ( n , n , n ) (cid:62) ∈ R and the gamma distributionis parameterized in terms of shape and rate. The three vectors β , β and β are thus assumedindependent under the prior and the inflation factors n account for the potential variation in thescale of the covariates.The conjugate prior distribution for the precision matrix K is the G-Wishart distribution W G [34], where we use the notation of [21]. The support of W G is the space of all symmetric positivedefinite matrices which fulfill the conditional independence structure given by the graph G = ( V, E )where V = { , . . . , T } and E ⊂ V × V . That is, K ij = 0 whenever ( i, j ) / ∈ E . For instance, if G isthe conditional independence structure of an autoregressive process of order 1, AR(1), then it holdsthat ( i, j ) ∈ E if and only if | i − j | ≤
1. The autoregressive structure is, however, completely flexibleand may vary over time with the prior parameters in (5) providing a slight shrinkage towards noautocorrelation to prevent potential overfitting. If G is the independence graph with ( i, j ) ∈ E if andonly if i = j , the prior distribution in (5) is equivalent to a Γ(3 / , /
2) prior distribution on eachmarginal precision.
Under the full model, we simultaneously estimate the marginal predictive distributions and the errorcorrelation. Here, we set K = K , implying a weakly informative prior, with G = G the conditionalindependence structure of an AR(1) process. Let us assume that N forecast-observation pairs areavailable. In order to obtain samples from the joint posterior distribution of β and K given the data,we iteratively sample from the full conditional distributions β | K , n , { y n } Nn =1 , { X n } Nn =1 ∼ N T (cid:0) ˜ β , ˜ K − (cid:1) , (8) K | β , n , { y n } Nn =1 , { X n } Nn =1 ∼ W G (6 + N, I T + S ) , (9) n i | β i , K ∼ Γ (cid:16) T + 22 , β (cid:62) i K β i (cid:17) , i = 0 , , , (10)where ˜ K = (cid:2) Diag( n ) ⊗ K (cid:3) + N (cid:88) n =1 X (cid:62) n KX n , ˜ β = ˜ K − N (cid:88) n =1 X (cid:62) n Ky n , S = N (cid:88) n =1 ( y n − X n β )( y n − X n β ) (cid:62) + (cid:88) i =0 n i β i β (cid:62) i . While it is straight forward to sample from the distributions in (8) and (10), we employ the directsampler of [21] to obtain samples from the G-Wishart distribution in (9). Given the posterior parame-ter samples and the current wind speed forecast, we then obtain samples from the posterior predictive4istribution for the wind power production by sampling a value from the likelihood model in (3) foreach posterior parameter sample and transforming these to wind power.
An alternative model construction is a two-stage Gaussian copula model which builds on the work of[3] and [25]. In the first stage, we perform joint estimation of the marginal predictive distributionsfollowing the set up above with G equal to the independence graph. If G is equal to the independencegraph, the marginal predictive distributions are estimated independently, while an AR(1) structurein the graph G imposes an autogressive structure on each of β i for i = 0 , ,
2. We consider both ofthese options.To estimate the error correlation, we proceed as follows. The estimation of the marginal predictivedistributions yields forcast-observation pairs { F tn , y tn } for n = 1 , . . . , N and t = 1 , . . . , T , where F denotes the predictive distribution. We may then infer N latent Gaussian observations { z n } Nn =1 bysetting z tn = Φ − ( F tn ( y tn )), where we denote the standard Gaussian cumulative distribution functionby Φ. The latent Gaussian data has likelihood p ( { z n } Nn =1 | K Z ) = (2 π ) T N/ | K z | N/ exp (cid:16) −
12 tr( K z , U ) (cid:17) , where U = (cid:80) Nn =1 z n z (cid:62) n and K z is an N × N precision matrix. Under a prior distribution of the form(5), the posterior distribution for K z is thus given by K z | { z n } Nn =1 ∼ W G (3 + M, I T + U ) . (11)Finally, a sample ˆ y from the posterior predictive distribution for the wind power production isobtained in three steps:1. Sample ˆ K z from (11).2. Sample z ∗ from N T (0 , ˆ K − z ) and set ˆ z t = z ∗ t / (cid:113) ( ˆ K − z ) tt for t = 1 , . . . , T .3. Set ˆ y t = F − t (Φ(ˆ z t )) for t = 1 , . . . , T , where F − t ( u ) := max { y : F t ( y ) ≤ u } .Here, F t denotes the marginal predictive distribution at time t . Note that the latent Gaussian vectorin step 2 is normalized as the inverse of ˆ K z which may be a covariance matrix rather than a correlationmatrix. We apply various forecast verification methods for probabilistic predictions with the aim of assessingwhich method provides the sharpest predictive distributions subject to calibration [10]. A forecastingmethod is calibrated if events predicted to happen with probability p ∈ [0 ,
1] are also realized withempirical relative frequency p . Calibration of univariate forecasts may be assessed empirically byplotting histograms of the probability integral transform (PIT) F ( y ) for a predictive distribution F and the corresponding realized obervation y over a large set of forecast cases. For a calibrated forecast,the PIT histogram will have a uniform (flat) shape [2]. Alternatively, calibration and sharpness canbe assessed directly for a fixed p by calculating the average coverage and width of the correspondingprediction interval.For assessing multivariate calibration, we calculate the multivariate rank of an observed temporaltrajectory y = ( y , · · · , y T ) in an ensemble with y and m − y , · · · , ˆ y m − from the multi-variate predictive distribution F . Here, we use the band depth ranking of [37]. That is, we first applya pre-rank function ρ : R T → R + given by ρ ( y ) = 1 T T (cid:88) t =1 (cid:2) m − rank( y t ) (cid:3)(cid:2) rank( y t ) − (cid:3) + ( m − , (12)5here rank( y t ) denotes the standard univariate rank of y t in (ˆ y t , · · · , ˆ y ( m − t , y t ). The multivariaterank of y is then given by the univariate rank of ρ ( y ) in ( ρ (ˆ y ) , · · · , ρ (ˆ y m − ) , ρ ( y )). The calibrationmay now be assessed empirically by plotting the histogram of the ranks of ρ ( y ) over multiple forecastcases with a uniform shape indicating a calibrated forecast. Note that the definition in (12) only holdsif, with probability one, no two trajectories in the ensemble are equal, see the discussion in [37].In addition, we calculate multiple proper scores which assess various different aspects of the pre-dictive distribution [12]. The absolute error | median( F ) − y | compares the median of the univariatepredictive distribution F against the observation y , while under the squared error (mean( F ) − y ) , themean of F is the optimal point forecast [9]. These scores are then averaged over multiple forecast casesresulting in the mean absolute error (MAE) and the root mean squared error (RMSE). Similarly, wecalculate the mean continuous ranked probability score (CRPS), which compares the full distribution F against the empirical distribution function of the observation y ,CRPS( F, y ) = (cid:90) + ∞−∞ ( F ( z ) − { x ≥ y } ) d z, (13)where denotes the indicator function. To estimate the integral in (13) we employ the approximationmethods described in [19] as implemented in the R package scoringRules [18]. All three scores arenegatively oriented such that a smaller score indicates a better predictive performance. Furthermore,the score units are equal to that of y or MWs in our case. We assess the influence of the amount of training data on the results by comparing the average marginalpredictive performance under rolling training periods of different lengths. For wind power predictions,this is performed for the full model described in Section 3.1 as well as independent marginals witheither an independent or AR(1) structure on the regression coefficients. Aggregated results for leadtimes up to 24 hours and the months of January, April, July and October of 2011 indicate that theprediction models are very robust against the amount of training data. For rolling training periods oflength 50 to 150 days, the performance of all methods changes by less than 3% when measured by theCRPS. Results for the MAE and the RMSE are similar. In the following, we use a training period of100 days for both the marginal and the multivariate models.
Full model
PIT0.0 0.2 0.4 0.6 0.8 1.0
Ind. marginals
PIT0.0 0.2 0.4 0.6 0.8 1.0
AR(1) marginals
PIT0.0 0.2 0.4 0.6 0.8 1.0
Figure 2: Probility integral transform (PIT) histograms for marginal wind power predictions underthree different marginal models. The PIT values are aggregated over lead times of 1-24h during thetest period from January 1, 2011 to December 31, 2012, a total of 17544 forecast cases. The dashedlines indicate the level of a perfectly flat histogram.We start by assessing the marginal predictive performance of the various models. The probilityintegral transform (PIT) historgrams (Fig. 2) indicate minor deviations from the ideal uniform pre-6ictive distribution, most notably a bias to the right, but not a clear under- or overdispersion. Theresults for the three different marginal models are very similar.Table 1: Calibration and sharpness of marginal predictions for wind power as measured by the coverageand width of 80% prediction intervals. The results are aggregated over lead times of 1-24h (Day 1),25-48h (Day 2) and 49-72h (Day 3), and the test period from January 1, 2011 to December 31, 2012.The best results in each category are indicated in bold.Coverage (%) Width (MW)Day 1 Day 2 Day 3 Day 1 Day 2 Day 3Full model
Fully Ind 76.9 77.5 78.6 2388 2944 3692Table 1 shows the average width and coverage of 80% prediction intervals aggregated over leadtimes of 1-24h (Day 1), 25-48h (Day 2) and 49-72h (Day 3). While the coverage is similar for differentlead times, the width of the prediction intervals expectedly increases with lead time. The full modelhas somewhat wider prediction intervals and slightly better coverage than the other two models.Table 2: Marginal predictive performance of models for wind power production as measured by meanabsolute error (MAE), root mean squared error (RMSE) and mean continuous ranked probabilityscore (CRPS). The results are aggregated over lead times of 1-24h (Day 1), 25-48h (Day 2) and 49-72h(Day 3), and the test period from January 1, 2011 to December 31, 2012. The best results in eachcategory are indicated in bold.MAE (MW) RMSE (MW) CRPS (MW)Day 1 Day 2 Day 3 Day 1 Day 2 Day 3 Day 1 Day 2 Day 3Full model 977 1117 1281 1390 1586 1801 676 778 902Ind Errors
988 1185
699 847Fully Ind
792 987 1184 1176 1428 1707 564 698 846
When the marginal predictive performance is assessed by proper scores, the marginal models per-form considerably better than the full model across all scores, see Table 2. In addition to independentand AR(1) marginals, we have also tested using higher order AR structures which yielded reducedpredictive performance (results not shown). For the best model, the performance is reduced 14-24%from day 1 to day 2 and 20-21% from day 2 to day 3, depending on the score.
Figure 3 shows the band depth histograms for the joint predictive distribution over hours 1-24. Foreach of the three approaches, the independent model is shown on the top row while the bottom rowshows the results after copula post-processing.As expected, under the univariate approach only the full model shows evidence of near calibra-tion, while the two approaches with completely independent errors show substantial multivariateover-dispersion. This is unsurprising, given the strong degree that errors are correlated across hours.However, the bottom row of Figure 3 shows that copula post-processing is capable of addressingthese issues, and all three aproaches, after copula processing, show roughly the same degree of mul-tivariate calibration. This suggests that it is acceptable to perform marginal inference first and thensubsequently address the multivariate aspects of the forecast distribution.7 ull Model Univariate Day 1
Band Depth
Ind Errors Univariate Day 1
Band Depth
Fully Ind Univariate Day 1
Band Depth
Full Model Copula Day 1
Band Depth
Ind Errors Copula Day 1
Band Depth
Fully Ind Copula Day 1
Band Depth
Figure 3: Band depth histograms for hours 1-24 by model type (full, independent errors and fullyindependent respectively going from left to rigth) and under either the univariate (top row) or copula(bottom row) approach.
We now consider two distributional forecasts derived from the entire multivariate forecast. Namely,we look at the total wind speed and forecasted maximum windspeed over the 72 hours. Since thesetwo quantities are affected by the joint behavior of the underlying forecast, assessments of theirdistributional performance provides an indication of the quality of the overall joint distributionalforecast.Table 3: Scores for predicting the sum of 72-hours ahead production of wind power by methodMAE (MW) RMSE (MW) CRPS (MW)Full Model Univariate 49224 69786 34868Ind Errors Univariate
Fully Ind Copula 44048 65403 31066Table 3 shows the scores for each method for the sum of wind power across all 72-hours. We seethat acording to the MAE, the full model performs best, while the model with independent marginalerrors followed by a copula post-processing shows the best performance.The conclusion from Table 3 is two-fold. First, it is (self-evidently) important to have dependencein the errors in the joint distribution either by explicit direct modeling (Full model) or via subsequentcopula post-processing. Furthermore, including dependence in the regression coefficients is beneficial,which can be seen by the fact that the Fully Independent model underperforms the other methods,even after copula post processing to correlate the sampling distribution. These results speak to theusefulness of a joint Bayesian model with dependence built into the prior.Table 4 shows similar scores for the maximum. The conclusions here are broadly in-line with thosefrom Table 3. We see that the independent errors model with copula performs best according to MAEwhile the Full model with subsequent copula post processing performs best according to RMSE andCRPS. While the ordering has changed slightly the key points hold, namely that dependence in the8ampling distribution and model-imposition of dependence in regression coefficients are beneficial topredictive performance.Table 4: Scores for predicting the maximun of 72-hours ahead production of wind power by methodMAE (MW) RMSE (MW) CRPS (MW)Full Model Univariate 1340 1752 946Ind Errors Univariate 2156 3225 1679Fully Ind Univariate 2383 6974 1871Full Model Copula 1309
Ind Errors Copula
Full Model Sum Univariate
PIT
Ind Errors Sum Univariate
PIT
Fully Ind Sum Univariate
PIT
Full Model Sum Copula
PIT
Ind Errors Sum Copula
PIT
Fully Ind Sum Copula
PIT
Figure 4: PIT for the sum by model type (full, independent errors and fully independent respectivelygoing from left to rigth) and under either the univariate (top row) or copula (bottom row) approach.
We have built a hierarchical Bayesian model for issuing joint distributional forecasts of wind powerproduction in Germany. This system uses the output of a numerical weather predicition model to9 ull Model Max Univariate
PIT
Ind Errors Max Univariate
PIT
Fully Ind Max Univariate
PIT
Full Model Max Copula
PIT
Ind Errors Max Copula
PIT
Fully Ind Max Copula
PIT
Figure 5: PIT for the max by model type (full, independent errors and fully independent respectivelygoing from left to rigth) and under either the univariate (top row) or copula (bottom row) approach.derive a predictive feature set.Our results are clear in the necessity for a joint predictive distribution. However, we have shownthat copula post-processing of marginal forecasts can be a competitive alternative to building a directfull model.Since neighboring hours are likely to translate features into production estimates in a similarmanner, we introduced dependence between regression coefficients via a G-Wishart prior distribution.Our results show that this approach yields estimates which are sharp and calibrated, both on theunivariate and multivariate scales and outperform approaches that use an independent prior on theregression parameters.There are number of technical manners by which the model could be embellished. For instance,at the moment the NWP output in a given hour is used to form the features for that particular hour.There is reason to believe that sharing this information across hours could be beneficial. More flexiblerepresentations than a linear function relating NWP output and wind power production could alsobe entertained, e.g. smoothing splines which still enable the methodology used here on an expandedfeature set.
Acknowledgment
This work was performed within Big Insight – Centre for Research-based Innovation with supportfrom The Research Council of Norway through grant nr. 237718. We thank Stefan Erath from NorskHydro for sharing his expertise and data.
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