RESGen: Renewable Energy Scenario Generation Platform

Transcription

1 1 RESGen: Renewable Energy Scenario Generation Platform Emil B. Iversen, Pierre Pinson, Senior Member, IEEE, and Igor Ardin Abstract Space-time scenarios of renewable power generation are increasingly sed as inpt to decision-making in operational problems. They may also be sed in planning stdies to accont for the inherent ncertainty in operations. Similarly sing scenarios to derive chance-constraints or robst optimization sets for corresponding optimization problems is sefl in a power system context. Generating and evalating sch spacetime scenarios is difficlt. While qite a nmber of proposals have appeared in the literatre, a gap between methodological proposals and actal sage in operational and planning stdies remains. Conseqently, or aim here is to propose an open-sorce platform for space-time probabilistic forecasting of renewable energy generation (wind and solar power). This docment covers both methodological and implementation aspects, to be seen as a companion docment for the open-sorce scenario generation platform. It can generate predictive densities, trajectories and space-time interdependencies for renewable energy generation. The nderlying model works as a post-processing of point forecasts. For illstration, two setps are considered: the case of day-ahead forecasts to be issed once a day, and for rolling windows with reglar pdates, with application to the western part of the United States, with both wind and solar power generation. Index Terms Spatio-temporal forecasting, Probabilistic forecasting, Scenario generation, Renewable energy, Qantile regression, Copla I. INTRODUCTION Energy generated from renewable sorces has attracted attention in recent years as a sstainable soltion to the world s growing energy demand. While hydro is easily dispatched, provided the reservoirs are fll, wind and solar power are intermittent and ncertain and ths provides challenges for their sccessfl integration into the energy system. To partly offset or mitigate these isses, accrate forecasts of ftre wind and solar power generation are reqired. For an efficient tilization of wind and solar power, in terms of economic costs as well as grid stability, forecasts providing the fll predictive density of wind and solar power generation are to be preferred ([1], [2]). These types of forecasts are referred to as probabilistic forecasts. They stand in contrast to point forecasts, which are single-valed, typically informing abot the expected vale of power generation per location, energy type and lead time. Energy systems are distribted in natre. Conventionally, on the demand side, power is consmed at nmeros distinct locations ranging from hoseholds to factories, typically distribted over a large geographical area. Renewable energy generation is also distribted, ranging from rooftop photovoltaic installations to wind trbines spread ot across the contryside or at sea. Frther, renewable generation in adjacent locations may be highly correlated as it is governed by the same weather system [3]. Ths, in view of the distribted natre of spply and demand, combined with network constraints, probabilistic spatio-temporal forecasts comprise optimal inpt to decisionmaking problems. While probabilistic spatio-temporal forecasting of renewable energy generation is novel, qite a bit has been done already on the basics of wind power forecasting. The propagation of wind power forecast errors has been stdied in [10] and [11], where it is sggested that a spatio-temporal model wold allow prodcing better forecasts. In [12] an ellipsoidal ncertainty set is sed to captre correlation between wind farms at different locations. Very-short-term forecasting of wind power generation has been considered in [13] sing a sparse vector ato regressive model to forecast the power prodction at 22 different sites. A vector ato regressive model is also considered in [14], however, the focs is on providing better point forecasts. Copla models were employed in [15], forecasting densities of weather variables, in [16], where a machine learning and copla based approach is sed to predict wind power, and in [17] where a copla approach is sed to model wind power prediction errors while redcing the parameters needed throgh a precision matrix formlation. The relevant challenges with probabilistic renewable energy forecasting inclde (i) the dynamic and time-varying natre of the weather, and (ii) the nonlinear and bonded natre of the power conversion process. Spatio-temporal probabilistic forecasting is frther complicated by the large data sets and the dimension of the models, which in trn can lead to large parameter spaces and high comptational costs. In this work we describe a generic tool for obtaining probabilistic spatio-temporal forecasts for renewable energy generation, within a modlar framework. The tool makes it possible to extract predictive qantiles, moments, prediction intervals, predictive densities and to simlate trajectories. The prediction tool is coded in python as many non-statisticians are more familiar with this langage as opposed statistical software sch as R or SAS. This is intended to allow for people with high programming skills to tweak and improve pon the code withot having an in-depth knowledge of statistics. The code provided is open sorce and free to sed nder an open sorce BSD 3-clase. The remainder of the docment is strctred as follows. In section 2 the data sed for this stdy is detailed. Next, section 3 gives a brief introdction to the mathematics nderlying the prediction tool. Section 4 highlights how to rn the code in python. In section 5 the otpts of the model are considered. Finally section 6 concldes the paper. II. DATA The data sed in this stdy covers wind and solar power generation for the Western Interconnection in the United States, which is shown in Figre 1.

2 2 is sed to estimate marginal predictive densities and a copla is sbseqently applied to model the interdependence in time and space. The implementation is not dependent on specific data sets, time resoltion, locations or forecast horizons and the approach can be readily adapted to other data sets. In this section a brief overview of the mathematics behind the implementation is presented. A. Qantile regression density estimation Sppose is a real valed random variable with cmlative distribtion fnction F (x) = P ( x). The τ th qantile of is given as 1 Q (τ ) = F (τ ) = inf {x : F (x) τ } Fig. 1. Interconnections, United States Within the Western Interconnection a total of 39 load regions exist. In Figre 2 the approximate location of these different load regions are shown [18]. Only a sbset of the load regions are present in the wind or solar data sets. (1) where τ [0, 1]. Next define the loss fnction as ρτ (x) = x(τ I(x<0) ), where I is an indicator variable, that is one when the condition is met and zero otherwise. A distinct qantile can be obtained by minimizing the expected loss of with regards to : mine(ρτ ( )) Z = min(τ 1) (x )df (x) Z +τ (x )df (x). (2) Set the derivative of the expected loss fnction to 0 and let qτ be the soltion of the following eqation, Z qτ Z 0 = (1 τ ) df (x) τ df (x). (3) qτ This eqation redces to F (qτ ) = τ. Fig. 2. Load regions, Western Interconnection The data contains aggregate power generation from wind and solar. There are 22 load regions for wind and 26 load regions for solar present in the data. Frther the data also contains varios forecasts for power generation. These inclde day-ahead forecasts with an horly time resoltion spanning the next day and intra-day forecast spanning 15-min ahead to 5-hor ahead with a 15 minte time resoltion. Each of the data sets contain one year of data. The data sets sed in this stdy were of high qality. While the qality of the data is high, the same conclsion cannot be reached for the qality of the point forecasts sed in this stdy. This may to some degree be explain by the forecast being compted before a market gate closre. It is clear that the point forecasts do not make se of past observations of generated power to predict the following day. Ths, there is a clear avene for improvements to yield a better predictive performance. III. M ATHEMATICAL F OUNDATIONS The forecasting tool provides a non-parametric approach to spatio-temporal probabilistic forecasting. Qantile regression (4) Ths qτ is τ th qantile of the stochastic variable. The τ sample qantile can be fond by solving the minimization problem: q τ = arg min q R n ρτ (xi q) (5) = arg min (1 τ ) q R (xi q) + τ xi <q (xi q). xi q Solving this minimization problem for a sitable nmber of qantiles the marginal cmlative density fnction and its inverse can now be estimated. Notice here, however, that this formlation does not allow for the density to depend on external regressors. This is remedied by introdcing the conditional qantile regression. B. Conditional qantile regression density estimation Let the τ th conditional qantile fnction be given by Q Z (τ ) = Zβτ. Given the distribtion fnction of, βτ can be obtained by solving βτ = arg mine(ρτ ( Zβ)). β Rk (6)

3 3 The sample analog can be solved, which yields the estimator of β τ, i.e., ˆβ τ = arg min β R k n (ρ τ ( i Zβ)). (7) Solving this minimization problem for a sitable nmber of qantiles we obtain estimates of parameter vales in the model for each specific qantile. Again, collecting these qantiles the marginal cmlative density fnctions can be estimated. While in this specific case Q Z (τ) is linear this is in general not a reqirement. C. Copla estimation Consider a random vector ( 1, 2,..., d ). Sppose its margins are continos, i.e. the marginal CDFs F i (x) = P[ i x] are continos fnctions. By applying the probability integral transform to each component, the random vector (U 1, U 2,..., U d ) = (F 1 ( 1 ), F 2 ( 2 ),..., F d ( d )) (8) has niformly distribted marginals. The copla of ( 1, 2,..., d ) is defined as the joint cmlative distribtion fnction of (U 1, U 2,..., U d ): C( 1, 2,..., d ) = P[U 1 1, U 2 2,..., U d d ]. (9) The copla C contains all information on the dependence strctre between the components of ( 1, 2,..., d ) whereas the marginal cmlative distribtion fnctions F i contain all information on the marginal distribtions. The importance of the above is that the reverse of these steps can be sed to generate psedo-random samples from general classes of mltivariate probability distribtions. That is, given a procedre to generate a sample (U 1, U 2,..., U d ) from the copla distribtion, the reqired sample can be constrcted as ( 1, 2,..., d ) = ( F 1 1 (U 1 ), F 1 2 (U 2 ),..., F 1 d (U d) ). (10) The inverses F 1 i are nproblematic as the F i were assmed to be continos. The above formla for the copla fnction can be rewritten to correspond to this as: C( 1, 2,..., d ) = P[ 1 F1 1 ( 1 ), (11) 2 F2 1 ( 2 ),..., d F 1 ( d)]. In this implementation a Gassian copla is sed. This copla is particlarly simple to estimate de to the fact that it is completely characterized by the covariance matrix of the associated mltivariate normal distribtion. We have that ( Φ 1 (U 1 ), Φ 1 (U 2 ),..., Φ 1 (U d ) ) N R d(0, Σ), (12) where Φ 1 is the inverse cmlative density fnction for the standard normal distribtion and Σ describes the covariance matrix that characterizes the Gassian copla. Ths we can estimate Σ with a method for estimating the covariance matrix for a normal distribtion. The sample covariance matrix is given by Σ = 1 n (x i x)(x i x) T, (13) n 1 d where x i is the i-th observation of the d-dimensional random vector, and x 1 x =. = 1 n x i, (14) n x d is the sample mean. IV. PYTHON IMPLEMENTATION The model implementation consists of two models, one day-ahead model, predicting horly power prodction for the coming day, and one intra-day model, predicting 15 minte power prodction p to the following 5 hors. The main difference between the two models are that the day-ahead model constrcts a marginal density for every hor of the comming day whereas the intra-day model de-seasons the data and constrcts the marginal densities for each lead time instead. This section explains how to rn the different models, however frther details of the code fnctionality is inclded in the code as comments and the athors sggest looking throgh these comments for frther details of the actal fnctionality. A. The Sample Data The day-ahead model (either wind or solar power) ses horly observation of generated power along with day-ahead forecasts of power generation. The forecasts consist of one point forecasts for every location for every hor of the day. The intra-day model (either wind or solar power) ses 15- minte observations of generated power along with intra-day forecasts of power generation. The forecasts for the intra-day model overlap sch that every 15 mintes point forecasts are generated for the next 5 hors with a time resoltion of 15 mintes. B. The Day-Ahead Model A short introdction to rnning the day-ahead model is provided. Loading the data is done by the python script ReadingData.py. The model is fitted to the data by the python script ConditionalQantileRegression.py. The predictive qantiles for the generated power are estimated and are then sed to constrct a predictive cmlative density fnction and its inverse. To obtain niform marginals, the generated power observations are transformed by the predictive cmlative density fnctions. These niform marginals are then converted to standard normal marginals and the covariance matrix is estimated as the sample covariance. Ths the model is specified. In Figre 3 the estimated correlation between different locations at different hors of the day are show for the dayahead solar model. The correlation matrix is ordered sch that location 1 hors 1-24 is in the top left. below that is location 2 hors 1-24 and so on. The day ahead model can be simlated by the python script SimlationConditionalQantileRegression.py. This is done by simlating normally distribted variables with the estimated interdependence strctre and reversing the procedre specified above.

4 4 lot from the prediction already in the initial hor. This is most likely de to the qality of the prediction, in that it does not se past observations of power generation to predict the otpt power. As sch the models cold be sbstantially improved by a better point prediction of the generated wind power. Notice also the ato-correlation in the residals from the predicted power which is an important featre for simlation tool as to provide reliable mlti-horizon trajectories. Fig. 3. Correlation matrix describing the interdependence strctre. Warmer colors eqal a high correlation. C. The Intra-Day Rolling Model A short introdction to rnning the intraday model is provided. The data is loaded by the python script ReadingDataRolling.py. The model is fitted to the data by the python script RollingQantileRegression.py. First the observations and predictions are transformed to a stationary domain. The predictive qantiles for the transformed observations are estimated via qantile regression which in trn are sed to constrct a predictive cmlative density fnction and its inverse. To obtain niform marginals, the transformed observation on the stationary domain are transformed once more by the predictive cmlative density fnctions. These niform marginals are then converted to standard normal marginals and the covariance matrix is estimated as the sample covariance. Ths the model is specified. The day ahead model can be simlated by the python script SimlationRollingQantileRegression.py. First the niforms with the proper interdependence strctre is simlated. These are then converted to de-seasoned domain by sing the inverse predictive cmlative density fnctions estimated via qantile regression. Frther the predictions on the de-seasoned domain are transformed to the original power domain via a transformation. The intra-day model is different from the day-ahead model in the sense that the forecasts are overlapping. The simlated trajectories are frther consider in the model otpt section. V. MODEL OUTPUTS Model otpts that can be extracted from the models inclde qantiles, moments, prediction intervals, predictive densities and simlated trajectories. Here we focs on the simlated trajectories. A. Day-Ahead Trajectories The day-ahead simlated trajectories span the next day with an horly time resoltion. In Figre 4 sch trajectories are shown for wind power generation at the location APS. Along with the 3 simlated trajectories in ble the predicted power is shown in green. The actal realization is shown in red. Notice how both the simlations and the realization deviate qite a Fig. 4. Day-ahead wind power generation is shown on vertical axis and hor on horizontal axis. Prediction in green, scenarios in ble and realization in red. Similarly for solar power, Figre 5 represent simlated trajectories for day-ahead solar power generation for the location APS. Again the simlations are in ble, the prediction is in green and the realization is in red. Notice here how the dirnal effect is captred by the simlated trajectories. The yearly seasonality enters indirectly into the model throgh the point predictions of solar power generation. Fig. 5. Day-ahead solar power generation is shown on vertical axis and hor on horizontal axis. Prediction in green, scenarios in ble and realization in red. B. Intra-Day Trajectories The intra-day simlated trajectories span the following 15 mintes to 5 hors with a time resoltion of 15 mintes. In Figre 6 simlate trajectories are shown for wind power generation at the location APS. Three simlated trajectories are shown in ble, the predicted power in green and the actal realization is shown in red. As for the day-ahead models, the simlations and the realization deviate qite a lot from the prediction already in the first step, that is 15 mintes ahead. Again the likely explanation is the qality of the point prediction, in that it does not se past observations of power generation to predict the otpt power nor is it post-processed

5 5 in a statistical sense to correct for possible bias. As for the dayahead models, the intra-day models predictive performance cold be sbstantially improved by better point predictions of generated power. Notice again the ato-correlation in the residals from the predicted power. Fig. 6. Intraday wind power generation is shown on vertical axis and lead time on horizontal axis. Prediction in green, scenarios in ble and realization in red. Simlations from the intra-day solar power model are shown in Figre 7 for the location APS. Again the simlations are in ble, the prediction is in green and the realization is in red. In Figre 7 the trajectories simlation starts at 7:00 am and spans the next 5 hors. We see that the dirnal variation is captred nicely by the model. Fig. 7. Intraday solar power generation is shown on vertical axis and lead time on horizontal axis. Prediction in green, scenarios in ble and realization in red. VI. CONCLUSION The work presented here provides a generic, accessible and modlar framework for providing probabilistic spatiotemporal forecasts. From this model it is possible to extract predictive qantiles, moments, prediction intervals, predictive densities and to simlate trajectories, all in a setting that preserves the distribtions of and spatio-temporal dependence in the observations. A versatile and state-of-the-art tool for probabilistic spatio-temporal forecasting is provided. The tool provides models for day-ahead as well as intra-day forecasts for both wind and solar power. The model is provided along with a data set pertaining to the Western Interconnection in the United States. However, the implementation is not dependent on specific data sets, time resoltion, locations or forecast horizons and the approach can be readily adapted to other data sets. The code provided is open sorce and free to se nder an open sorce BSD 3-clase. REFERENCES [1] J. M. Morales, A. J. Conejo, H. 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