Probabilistic Forecasting of Temporal Trajectories of Regional Power Production - Part 2: Photovoltaic Solar
PProbabilistic Forecasting of Temporal Trajectories of Regional PowerProduction—Part 2: Photovoltaic Solar
Thordis L. Thorarinsdottir, Anders Løland, and Alex Lenkoski ∗ March 5, 2019
Abstract
We propose a fully probabilistic prediction model for spatially aggregated solar photovoltaic(PV) power production at an hourly time scale with lead times up to several days using weatherforecasts from numerical weather prediction systems as covariates. After an appropriate logarithmictransformation of the power production, we develop a multivariate Gaussian prediction modelunder a Bayesian inference framework. The model incorporates the temporal error correlationyielding physically consistent forecast trajectories. Several formulations of the correlation structureare proposed and investigated. Our method is one of a few approaches that issue full predictivedistributions for PV power production. In a case study of PV power production in Germany, themethod gives calibrated and skillful forecasts.
Part 1 of the paper proposed fully probabilistic prediction models for spatially aggregated wind powerproduction at an hourly time scale with lead times up to several days using weather forecasts fromnumerical weather prediction systems as covariates. This part of the paper (Part 2) is concerned withcorresponding models for photovoltaic (PV) solar power production. The proposed PV models sharethe same framework as the wind power model from Part 1.The increase in energy production from renewable energy sources is driven by wind and PV powerproduction. In Germany, for instance, renewable energy accounted for 36 .
0% of the total nationalenergy production in 2017 compared to 6 .
6% in 2000 according to the Arbeitsgemeinschaft Energiebi-lanzen, a working group founded by energy related associations in Germany. This increase is to alarge extent due to expansion in wind and photovoltaic (PV) solar power production. (PV productionaccounted for 0 .
0% in 2000 and increased to 6 .
6% in 2017.) As discussed in Part 1, management ofelectricity grids, scheduling of the conventional production and general energy market decisions callfor a probabilistic forecasting framework [11]. While probabilistic forecasts are becoming increasinglyfrequent for wind power forecasting [6, 30, 16, 28, 29, 14, 8] they have been less common for PV powerforecasts [20, 5, 2, 33, 13] and only few approaches have issued full predictive distributions. However,the Global Energy Forecasting Competition (GEFCom2014 [14]) has spurred probabilistic forecastsbased on non-parametric approaches [24, 26, 15, 1, 13].While the wind power model we developed in Part 1 relied on wind speed forecasts from numericalweather prediction (NWP) models, the essential input for a NWP-based prediction model for PV solarpower is a forecast of the downward solar radiation at the surface, which is also called global horizontalirradiance (GHI), see e.g. [4] and [23]. An explicit physical model for a single unit may also includeforecasts of cloud cover, temperature and snow cover [22, 27].As for the wind model, we propose to directly predict the aggregate country-wide PV productionusing spatially averaged NWP forecasts of the relevant weather variables as inputs. We specify theprobabilistic prediction model for PV power as a Bayesian hierarchical model, which allows us toincorporate a correlation structure in both the model parameters associated with each lead time aswell as the error structure across lead times. ∗ Norwegian Computing Center, Oslo, Norway (e-mail: [email protected]). a r X i v : . [ s t a t . A P ] M a r he NWP forecasts and the German PV power production data are introduced in Section 2.The prediction models and the statistical inference methods are presented in Section 3. The results,including forecast verifications, are given in Section 4. Section 5 contains a final discussion. We employ the NWP forecast ensemble issued by the European Centre for Medium-Range WeatherForecasts (ECMWF), which has been shown to perform well for solar power [23]. 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 up toten days [19, 25]. We restrict attention to the forecast initialized at 00:00 UTC, corresponding to 2:00am local time in summer and 1:00 am local time in winter, and lead times up to 72 h for accumulatedglobal horizontal irradiance (GHI).The hourly solar 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 trainingperiod in the parameter estimation as well as for determining the prior parameters of the Bayesianmodel. Given these values, we then test our methods on data from 2012. In order to obtain equallylong training periods for all dates, data from the previous year is used for the parameter estimationat the beginning of a year.We reduce the ECMWF ensemble to a single forecast given by the ensemble average. For theoperation and management of electricity grids, power production predictions are needed on an hourlybasis. However, for the first 72 h, the ECMWF forecasts have a temporal resolution of 3 h. Wetherefore derive hourly forecasts through a spline interpolation conditional on the variables being non-negative. A more advanced interpolation approach for GHI performs a temporal interpolation overthe clear sky index, see e.g. [21]. In a third preprocessing step, we aggregate the forecasts in space bytaking the spatial average. For GHI, several studies have found that spatial averaging increases theskill of the forecast due to difficulty in dealing with changing cloud cover [10, 20, 23]. We employ herethe average GHI forecast over all grid locations within Germany, resulting in a GHI forecast which isan average over 724 grid locations.
The downward solar radiation at the surface, also called global horizontal irradiance (GHI), is com-posed of the direct solar radiation at the surface and a sky diffusion component. For an individual PVsystem, the two components of the GHI are used to generate a tilted forecast of irradiance in the planeof the PV arrays, x ∗ s [27]. Given x ∗ s and the local ambient temperature forecast, x a , the power outputof the PV system is then given by x ∗ s f ( x ∗ s , x a ) for a non-linear function f which parameters dependon the type of installment [9, 22, 27]. Here, we model the aggregated power output as a functionof the predicted GHI only, which is similar to the approach of [4]. Including temperature and snowdepth as covariates did not improve the average marginal predictive performance (results not shown).Installing PV modules on northern latitudes can lead to snow losses of up to 20%, but the effect ofsnow on individual PV modules can be predicted fairly well [3].Denote by x st the GHI forecast for lead times t and let ˜ y t denote the most recent available observedpower production of the hour of the day that is being predicted at lead time t . That is, for T = 72 itholds that ˜ y t = ˜ y t +24 = ˜ y t +48 for all t ∈ { , . . . , } . We can now define T + := { t ∈ { , . . . , T } : ˜ y t , x st > } and T := { t ∈ { , . . . , T } : ˜ y t = 0 or x st = 0 } , such that T + and T are disjoint sets with T + ∪ T = { , . . . , T } . The solar power production Y st attime t ∈ { t , . . . , t | T + | } = T + is then given bylog( Y st ) = β t + β t log( x st ) + ε t , (1)2here β t , β t ∈ R and the error vector fulfills ε ∼ N | T + | (0 , K − ) for some precision matrix K . Asshown in Fig. 1, the relationship between the log transformed GHI forecast and the log transformedsolar power production depends somewhat on the time of the day. l l ll ll l ll llll l l llll l llll ll l l ll llll ll l ll lll ll ll lll l l l lll l ll l l lllll lllll lll llll l l llllll ll llll lll ll ll l lll l l lllllllllllll l l llll l llllllll lllll l ll llll lll ll lll lllllll lll ll l ll ll lll l llll llll lll l l lll lll lll l ll l ll ll ll l l llll l l llll ll ll llll ll llll llll ll ll lll ll llll lll l l lll l llll l lll l lllll lllllll lll ll l l l l lllll l llllll lllll llll llllll l lllllll ll llll ll l lll llll lll ll l lll lll l l llllll ll l lll l Log Transformed GHI Forecast
Log T r an s f o r m ed S o l a r P o w e r Figure 1: Relationship between hourly solar power production in Germany in 2011 and the corre-sponding global horizontal irradiation (GHI) forecasts at 10:00 am (9:00 am) local time in summer(winter) ( ◦ ), 4:00 pm (3:00 pm) local time in summer (winter) (+) and 9:00 pm (8:00 pm) local timein summer (winter) ( × ). The plot only shows data points where both the power production and theGHI forecast are positive.We now set X = [ I | T + | Diag(log( x s ))]with x s = ( x st , . . . , x st | T + | ) (cid:62) and, correspondingly, Y = (log( Y st ) , . . . , log( Y st | T + | )) (cid:62) . The likelihoodmodel for Y is then given by Y ∼ N | T + | ( X β , K − ) , (2)where β = ( β (cid:62) , β (cid:62) ) (cid:62) with β i = ( β it , . . . , β it | T + | ) (cid:62) for i = 0 ,
1. Inference is performed under aBayesian paradigm in the same manner as described in Part 1. The regression parameters β are givena normal conjugate prior distribution and the conjugate prior distribution for the precision matrix K is the G-Wishart distribution K ∼ W G (3 , I T ) [32, 18]. The support of W G is the space of allsymmetric positive definite matrices which fulfill the conditional independence structure given by thegraph G = ( V, E ) where V = { , . . . , | T + |} and E ⊂ V × V . That is, K ij = 0 whenever ( i, j ) / ∈ E .For instance, if G is the conditional independence structure of an autoregressive process of order 1,AR(1), then it holds that ( i, j ) ∈ E if and only if | i − j | ≤ | t i − t j | ≤
1, such that correlation isonly possible between consecutive time points.If the training data contains instances in which either the solar power production or the GHIforecast is equal to zero, we treat these as missing data. For lead times t ∈ T , we follow [27] and setthe predicted power production equal to the average observed production for this lead time duringthe training period. Typically, this value will be equal to zero. For instance, approximately 43% ofthe observed hourly solar power production values in 2011 are equal to zero; for five night time hoursno production is recorded throughout the entire year. Here, we present assessments of marginal and multivariate predictive performance. The predictiveperformance is measured in terms of calibration and accuracy. A forecasting model is said to becalibrated if predicted probabilities are observed with the same relative frequency in the observations.This is assessed empirically through probability integral transform (PIT) histograms marginally andthrough band depth rank histograms in higher dimensions [7, 34]. For both cases, a uniform histogramindicates a calibrated forecast, while deviations from uniformity may provide information regarding3he misspecification of the prediction model. Prediction accuracy is assessed by using proper scoringrules where a smaller score indicates a better performance with the errors given in the unit of thepredictand [12]. Specifically, we apply the continuous ranked probability score (CRPS) which assessesthe full predictive distribution as well as the mean absolute error (MAE) and the root mean squarederror (RMSE) which assess the median and the mean of the predictive distribution, respectively.Further details on the forecast verification methods are given in Part 1.
We assess the influence of the amount of training data on the results by comparing the averagemarginal predictive performance under rolling training periods of different lengths. The performanceof the marginal prediction models for solar power production is significantly more sensitive to thelength of the rolling training period than is the case for wind power, see Fig. 2. Results based on otherperformance measures show a similar pattern. Following these results, we employ a rolling trainingperiod of length 20 days for the full model and all marginal models. For the multivariate copulamodels, we observe nearly identical results when predicting daily sums and maxima using rollingtraining periods of length 50 to 150 days. Shorter training periods give somewhat worse performancefor the maxima. We thus use a training period of 100 days for estimating the multivariate correlationstructure in the copula models.
Days In Training Period CR PS ( M W )
20 40 60 80 100
Figure 2: Average continuous ranked probability score (CRPS) for solar power predictions aggregatedover lead times up to 24 hours and the months of January, April, July and October of 2011 as afunction of the number of days in the rolling training period. The plot shows the results for the fullmodel (gray dashed line) as well as for independent marginal models with independence (black solidline), AR(1) (black dashed line), or AR(2) (gray solid line) structure on the regression coefficients.
We compare three models, (i) “Full Model” which has a AR(1) structure on the precision matrix forboth the regression coefficients and residuals, (ii) “Fully Independent” which has fully independentresiduals and regressions coefficients and “Independent Residuals” which models dependent regressioncoefficients but indepdent errors. We start by assessing the marginal predictive performance of thesethree models. The PIT histograms (Fig. 3) all indicate that the full model is over-dispersive, asthe majority of observations fall in the middle quantiles of the distribution. However, both the FullyIndependent and Independent Residual models show some signs of slight upward bias but otherwisegood calibration.We further measure calibration and sharpness of marginal predictions for sun power by the widthand coverage of 80% prediction intervals. The results are aggregated over lead times of 1-24h (Day4 ull Model Day 1
PIT
Fully Ind. Day 1
PIT
Ind. Residuals Day 1
PIT
Figure 3: Probility integral transform (PIT) histograms for marginal sun 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 for which a probabilistic forecast was issued,a total of 9741 forecast cases. The dashed lines indicate the level of a perfectly flat histogram.1), 25-48h (Day 2) and 49-72h (Day 3) in Table 1. The width of the prediction intervals is roughlyconstant with respect to lead time, possibly due to the systematic componenent in the GHI forecasts.The Independent Residuals model gives the narrowest prediction intervals, while the Full Model hasthe best coverage, close to the desired 80%. However, this improved coverage comes at the expense ofprediction intervals that are roughly 37% larger than the Fully Independent or Independent Residualmodels.Table 1: Calibration and sharpness of marginal predictions for sun power as measured by the widthand coverage 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 italics.Width (MW) Coverage (%)Day 1 Day 2 Day 3 Day 1 Day 2 Day 3Full Model 5100 5099 5099
Fully Independent 3721 3719 3717 0.761 0.761 0.761Independent Residuals
455 448 487 974 1007 1056 344 354 370
Independent Residuals 461 454 498 987 1021 1067 346 357 3745 .3 Multivariate calibration
Figure 4 shows the band depth rank histograms for the joint predictive distribution over hours 1-24.For each of the three approaches, the independent model is shown on the top row while the bottomrow shows the results after copula post-processing.Figure 4 shows two interesting features. First, the Full Model shows substantial underdispersionand copula post processing does not help this matter. Secondly, in the case of the Fully Independentand Independent Residuals models the univariate fits show a lack of multivariate calibration whichoccurs when the predictive distribution is either too focused or too dispersed relative to the truetrajectory. In both cases, copula post-processing nearly eliminates this feature. This corroborratesthe result in Part 1 that marginal modeling with copula post-processing is an effective way of achievinga sharp and calibrated multivariate predictive distribution.
Full Model Univariate Day 1
Band Depth
Fully Ind. Univariate Day 1
Band Depth
Ind. Residuals Univariate Day 1
Band Depth
Full Model Copula Day 1
Band Depth
Fully Ind. Copula Day 1
Band Depth
Ind. Residuals Copula Day 1
Band Depth
Figure 4: Band depth rank histograms for hours 1-24 by model type under either the univariate (toprow) or copula (bottom row) approach.
We now consider the total PV solar production and maximum hourly PV production over the 72 hours.Since these two quantities are affected by the joint behavior of the underlying forecast, assessments oftheir distributional 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 PV solar power by method.MAE (MW) RMSE (MW) CRPS (MW)Full Model Univariate 24650 36975 18587Fully Independent Univariate 23829 32514 18787Independent Residuals Univariate 24702 33615 19536Full Model Copula 24341 37064 18962Fully Independent Copula
Independent Residuals Copula 24382 33598 17531Table 3 shows the scores for each method for the sum of wind power across all 72-hours. We seethat acording to all methodologies the fully independent model with a copula post-processing shows6he best performance. Inn all cases, copula post processing improved the final model score.Table 4: Scores for predicting the maximum of 72-hours ahead production of PV solar power bymethod. MAE (MW) RMSE (MW) CRPS (MW)Full Model Univariate 5210 8270 3734Fully Independent Univariate 4312 6245 3326Independent Residuals Univariate 4296 6279 3308Full Model Copula 3543 6495 2507Fully Independent Copula
Independent Residuals Copula 2209 3699 1616Table 4 shows similar scores for the maximum. The conclusions here are broadly in-line with thosefrom Table 3. We see that the fully independent errors model with copula performs best. In the case ofthe maximum this improvement is dramatic, with a roughly 50% reduction in each score. This speaksthe importance of modeling the joint error behavior, but also shows that copula post-processing issufficient to capture these dependencies.
Full Model Sum Univariate
PIT
Fully Ind. Sum Univariate
PIT
Ind. Residuals Sum Univariate
PIT
Full Model Sum Copula
PIT
Fully Ind. Sum Copula
PIT
Ind. Residuals Sum Copula
PIT
Figure 5: 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.Figure 5 shows the PIT histograms for the total PV production under the various model combi-nations. We see again that while the full model is originally rather calibrated (and indeed becomesless calibrated after a subsequent copula post processing), the best performing models are the fullyindependent and independent residuals models after a subsequent copula post processing.Figure 6 shows the PIT histograms for the maximum PV production over 72 hours. In this case,we see that in all models the predictive distributions are heavily upwards biased. This feature isconsiderably improved (though not completely mitigated) via copula post processing.In general, the results in Tables 3 and 4 alongside the calibration results in Figures 5 and 6 showthat our approach to joint distributional modeling works well. Interestingly and in contrast to theresults in Part 1, modeling dependence in the regression coefficients did not appear to be beneficial.However, a post processing step via a Gaussian copula to correlated samples was highly-beneficialand indeed considerably more useful than directly incorporating dependence into a joint model. Thepredictive distributions for the maximum PV production are still slightly prone to upward bias, even7 ull Model Max Univariate
PIT
Fully Ind. Max Univariate
PIT
Ind. Residuals Max Univariate
PIT
Full Model Max Copula
PIT
Fully Ind. Max Copula
PIT
Ind. Residuals Max Copula
PIT
Figure 6: 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.after copula post-processing, an aspect which constitutes further research.
We have outlined a model for probabilistic forecasting of hourly PV production in Germany for leadtimes up to three days ahead. Our approach used summary measures taken from NWP output as aninput feature in a Bayesian regression model. We entertained two sources of dependence, hierarchicaldependence between regression coefficients and residual dependence amongst forecast errors. It wasfound that for PV forecasting, hierarchical dependence was not useful and – in line with the resultsof Part 1 – that it is better to impose residual dependence in a second copula post processing step.These results point to an important aspect related to the training of Bayesian models for use inpractical forecasting problems. In particular, while it is appealing to believe that a single, large jointmodel can handle all potential dependencies in a system, the potential for model misspecificationgrows in tandem with the model complexity. Indeed, as seen here, the Full Model performed far worsethan more parsimonious alternatives. Part of the reason, we believe, that the copula post-processingimproved model performance was because it uses a second data layer (the actual observed residuals)to incorporate model misspecification into the residual process. We feel this is an important practicallesson for all forecasters to keep in mind.The NWP forecast accuracy for GHI is approximately constant for lead times up to 3 days [31], witha significantly better performance under clear skies than for cloudy conditions [17]. As a result, thepredictive performance and the associated forecast uncertainty of the PV power production forecastsis approximately constant for all lead times up to 72 hours. This was not the case for the wind powerproduction forecasts in Part 1, where the forecast uncertainty increased and the predictive performancedecreased with longer lead times.One final challenge faced by our approach relates to modeling the “shoulder” hours of a given daythroughout the year. For example, the PV power production in hour 8 at the start of spring movesrapidly up from 0 (throughout the winter) to a positive number. In these cases, historical trainingdata is not particularly useful, which partially informed the use of a short training period of 20-days.A more direct model which incorporated this seasonality should be able to capture these dynamicsand thereby enable a wider training window to be entertained.Parts 1 and 2 of this considered renewable energy production from wind and solar power seperately.8learly, forecasts of their joint production is of interest to actors in power markets. The methodologieswe have outlined, especially the copula post-processing steps should be capable of accommodatingthese larger forecasting objectives. Production across multiple countries can be entertained in asimilar manner. In both cases, the graphical conditional independence structure that was imposed inthe copula model should prove a useful means of avoiding over-specification.
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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