Wind power forecasting errors modelling approach considering temporal and spatial dependence
 2.3k Downloads
 9 Citations
Abstract
The uncertainty of wind power forecasting significantly influences power systems with high percentage of wind power generation. Despite the wind power forecasting error causation, the temporal and spatial dependence of prediction errors has done great influence in specific applications, such as multistage scheduling and aggregated wind power integration. In this paper, PairCopula theory has been introduced to construct a multivariate model which can fully considers the margin distribution and stochastic dependence characteristics of wind power forecasting errors. The characteristics of temporal and spatial dependence have been modelled, and their influences on wind power integrations have been analyzed. Model comparisons indicate that the proposed model can reveal the essential relationships of wind power forecasting uncertainty, and describe the various dependences more accurately.
Keywords
PairCopula Wind power forecasting Temporal dependence Spatial dependence Wind power integrations1 Introduction
With mounting concerns over global warming and fossil fuel depletion, in recent years wind power generation has been greatly increased in the worldwide. However, due to the stochastic behaviors of wind resources cannot be fully predicted, the induced uncertainties significantly influence the power system operations, such as reserve deployment, unit commitment, power dispatch, power system’s security and reliability assessments, etc [1, 2, 3, 4, 5, 6]. Therefore, a thorough uncertainty model for wind power forecasting error is imperative for power system operators and participants, in terms of evaluating the benefits and costs of wind power integration and thus supporting tradeoff decisions.
The shortterm uncertainty of wind power generation has been typically modelled by the distribution of the wind power forecasting error. Numerous distributions and modelling methods have been proposed, such as the Gaussian distribution [7], Beta distribution [8] and nonparametric approach [9]. These studies mainly focused on the error distribution of a single wind farm based on a perprediction horizon. However, due to the similar meteorological conditions, the errors of close prediction horizons and nearby farms influence with each other. The temporal dependence of forecast series is crucial for multistage decisionmaking problems, such as energy storage planning and scheduling. In parallel, the spatial dependence of multiple farms has been particular concerned to cluster operators in terms of transmission congestion and reserve deployment. Thus more detailed models which consider the temporal or spatial dependence of wind power forecasting errors have drawn wide attentions in wind power integration applications.
Under this background, researchers are inspired to develop feasible models for the correlation structure of wind power forecasting [10, 11, 12, 13, 14, 15, 16]. General methods can be roughly classified into three main kinds: time series model, Markov model, and highdimensional joint probability model. Time series model is able to discover change rules of historical series, the most widelyused of which is auto regression moving average (ARMA) model. ARMA model has advantages of simple structure and easily fitting [10]. However, ARMA model faces difficulty in dealing with nonstationary stochastic process. Additionally, since there exists essential difference in characteristics between temporal dependence and spatial dependence, simply using the same noise signal for temporal and spatial modelling lacks theoretical support [11]. Markov model depicts the state transition distribution of random variables, with advantages of low requirement to sample size and high fitting effect for shortterm characteristics [12]. Yet, due to the restriction of model order, it may be difficult to apply Markov model in uncertainty modelling for multiple wind farms [13]. By regarding forecasting errors of different wind farms and different times as dependent multivariate random variables, highdimensional joint probability model performs mathematical and statistical description for uncertainty characteristics through joint probability distribution function. Copula theory is a typical method in this area, which has been widely applied in financial market analyses, portfolio investments, and risk assessments [14, 15].
Recently copula theory has been applied in power system uncertainty analyses. The spatial dependence of regional wind power outputs using Gaussian Copula was investigated in [17]. Similar methods were used to model the temporal relevance of forecast errors [18, 19, 20, 21]. Although Gaussian distribution has been the default Copula choice in numerous studies for practical reasons, its appropriateness for wind power forecasting dependent structure has not been rigorously investigated. Reference [22] evaluated the goodnessoffit of several bivariate elliptical and Archimedean Copulas for modelling the wind power spatial dependence and concluded that Gumbel Copula was preferred over Gaussian Copula. However, when extending the bivariate model to higher dimensions, Díaz [23] noted that the multivariate Gumbel Copula did not outperform Gaussian Copula due to its monotonous structure and fewer parameters [24]. Thus, although Archimedean Copula family provides various dependent structures, its applicability in high dimensions is significantly restricted due to a lack of feasible extension methods.
In response to these limitations, this paper introduces PairCopula theory to model the temporal and spatial dependence of wind power forecasting errors. The paper is organized as follows. Section 2 introduces the PairCopula theory and proposes the detailed procedures of modelling wind power forecasting dependence and sampling wind power uncertainty scenarios. Based on the regional wind power dataset from the National Renewable Energy Laboratory (NREL), the stochastic characteristics of temporal and spatial dependence have been analyzed and modelled in Sect. 3. In Sect. 4, different dependent models have been compared to validate the effectiveness of the proposed method, and the effects of temporal and spatial dependence on power system operations are addressed. Finally, concluding remarks have been presented in Sect. 5.
2 PairCopula theory based uncertainty modelling algorithm
2.1 Review of Copula and PairCopula theory
2.1.1 Copula theory
The Copula modelling approach has several attractive properties [26]: 1) it allows the marginal distributions and dependent structure to be modelled separately; 2) it does not require the margins to be normally or uniformly distributed; 3) it contains various Copula functions, which makes it flexible in modelling complex dependences, including nonlinear and asymmetric relationships.
2.1.2 PairCopula theory
In practical applications, highdimensional dependence requires feasible multivariate Copulas and even flexible combinations of different Copula functions. To resolve this problem, hierarchical and nested approaches were proposed, which used submodels of lower dimensions to construct the joint Copula function [27, 28]. However, these techniques faced the compatibility problem. Strict limitations were imposed on the selected types and parameters to satisfy the nested condition [28]. On this basis, the PairCopula construction method was proposed to build flexible multivariate distributions.
PairCopula theory maintains the merits of Copula theory, and further provides a flexible and intuitive way to build the multivariate Copula using bivariate Copula blocks. Compared with the other multivariate Copula extension methods, PairCopula imposes no restrictions on the selected types or parameters and is thus more flexible and practical for modelling and analysing complex dependences.
2.2 Model construction for wind power forecasting errors
To specify the PairCopula model for the wind power forecasting uncertainty, the temporal dependence model are derived as an example, and the spatial dependence can be constructed in the same manner. Suppose that {e _{1}, e _{2}, …, e _{ n }} are the forecast errors of prediction horizons. According to (5), the model construction includes two parts: 1) margin distribution estimation and 2) dependent model construction, i.e., the estimation of every bivariate pair density c _{ j,i+jj+1,…,i+j−1} evaluated at conditional CDF F(e _{ j } e _{ j+1},…,e _{ i+j−1}) and F(e _{ i+j } e _{ j+1},…,e _{ i+j−1}). As the modelling methods for the marginal distribution have been widely investigated, the estimation of dependent model is proposed in the following.
2.2.1 Parameter estimation and Copula selection
2.2.2 Dependent structure estimation
Through the construction process, it can be seen that the PairCopula method allows mixing different Copula types and sets no limitations on types or parameters, thus it can select the bestfitted distributions and gain a higher accuracy. When the Gaussian Copula is nominated for all pairs, the PairCopula model is proven to be equal to the multivariate Gaussian Copula. Therefore, the PairCopula model can replace the conventional multivariate models without sacrificing the accuracy of the model.
2.3 Wind power uncertainty scenario simulation
The most important application of the wind power uncertainty model is to evaluate the benefits and costs of wind power integration. For endusers, Monte Carlo simulated scenarios provide uncertainty information in a discrete form, which are practical for risk assessments and optimization programming. The simulation process is also based on conditional probability theory.

Step 1 Form the ndimensional dataset E = {e _{1}, e _{2}, …, e _{ n }} of prediction errors for multiple forecast horizons or wind farms using the historical data.

Step 2 Develop the marginal distribution F(e _{ i }) (i = 1, 2,.., n) and transform the data from the actual domain to the uniform domain.

Step 3 Construct the multivariate dependent model based on the uniform data.

Step 4 Sample correlated uniform error {z _{1}, z _{2},…, z _{ n }} using (9) based on the model estimated in Step 3 .

Step 5 Transform samples back to the actual domain using the inverse margin e _{ i } = F ^{−1}(z _{ i }).

Step 6 Obtain wind power uncertainty scenarios by adding the given point forecasts and simulated errors.
Further extension to spatialtemporal dependence modelling also follows a similar principle, but the model dimension is higher.
3 Modelling temporal and spatial dependence of wind power forecasting errors
3.1 Dataset
In this section, the temporal and spatial dependences of wind power forecasting errors are modelled based on the Wind Integration datasets of the NREL [32]. The wind speeds in the dataset are generated by MASS v.6.8 mesoscale model, which is initialized with input from Atmospheric Research Global Reanalysis and assimilates both surface and rawindsonde data. The wind power outputs are computed by composite turbine power curves, and the forecasts are produced by a statistical forecast tool named SynForecast. Though the data are simulated, they are verified to contain similar stochastic features of actual wind farms and reflect the geographic diversity of wind power outputs. Thus they are suitable for researches of wind power temporal and spatial relevance [22, 23, 33].
Detailed spatial locations of concerning 17 wind farms
Index  Latitude  Longitude  Altitude 

A  44.97  −72.52  503 
B  44.87  −72.51  806 
C  44.81  −72.69  687 
D  44.92  −71.82  700 
E  44.71  −71.79  744 
F  44.54  −72.18  681 
G  44.2  −72.35  720 
H  44.29  −72.89  949 
I  44.23  −73  627 
J  43.99  −72.8  646 
K  43.68  −72.66  673 
L  43.56  −72.8  691 
M  43.52  −72.62  623 
N  43.35  −72.76  641 
O  43.42  −73.08  754 
P  43.22  −73.14  853 
Q  42.9  −72.8  578 
3.2 Temporal dependence model
As the dimension is 24 (the forecast errors of 24 h), there are 23 trees in the temporal model. Tree1 has 23 pairs, each pair represents the temporal correlation of horizon j and j + 1. In parallel, Tree 2 has 22 pairs, each pair represents the conditional temporal correlation of horizon j and j + 2. The rest trees can be analysed in the same way. The fitted results show that the independent function dominates in Trees 2–23, which implies that nonadjacent errors are conditionally independent, i.e., the wind power forecasting error has a oneorder Markov property. This result is not surprising, as wind speed and wind power outputs have been proven to be Markov chains in many studies [12, 34]. Using the PairCopula approach in this paper, the dayahead wind power forecasting errors are verified to possess a Markov property with an optimal order of 1.
For the dependence of adjacent errors in Tree 1, the Student t is the most favored Copula function due to its elliptical structure and strong tail dependence, and the Gumbel function is also selected for some pairs. For parameter results, strong correlations are found for adjacent errors, and the variation of parameters in Tree 1 is small, indicating that the correlations are not significantly affected by the prediction horizons 4 figures and tables.
3.3 Spatial dependence model
4 Model analysis and comparisons
To illustrate the effectiveness of the proposed method, the PairCopula, Gaussiandependent and the independent (i.e., temporal or spatial correlation is ignored) models are compared in this section. The goodnessoffit evaluation of multivariate distribution needs huge computation and is timeconsuming, especially for high dimensional cases. Hence, the typical application scenes in which the temporal or spatial dependence of wind power forecasting errors are required are investigated. And according to the effects of the dependences in application scenes, the corresponding criterion is proposed to map the multivariate distribution to a univariate one, hence facilitating the comparisons between different multivariate models.
4.1 Temporal dependence model comparison
Therefore, the distribution of E _{b} is regarded as the criterion to evaluate the accuracy of different temporal models. To develop the univariate distribution, temporal related prediction errors are generated by the Monte Carlo algorithm based on different temporal models, E _{b} can then be calculated by (9), and the empirical distribution can be further obtained using the simulated data.
Figure 8 shows the probability density frequency of the ESS energy based on the measurements and the three models studied, and the data are normalized by the wind farm rated capacity. The measured data exhibit a fattailed characteristic due to the presence of continuous errors with the same sign, which agrees with the analysis in temporal modelling. The independent model does not reflect this property and therefore will greatly underestimate the necessary ESS energy capacity. On the other hand, the PairCopula and Gaussian models are alike because the selected Student t Copula has similar stochastic characteristics with Gaussian Copula, and they both fit the data well.
Goodnessoffit indices of three temporal models
Index  PairCopula model  Gaussian model  Independent model 

CvM index  0.0059  0.0100  0.0626 
KS index  0.0153  0.0191  0.1095 
4.2 Spatial dependence model comparison
Due to the weak spatial correlation, the prediction errors of disperse wind farms is offset thus reserve demand can be reduced. Thus the spatial dependence is desired for the assessment of power system operation reliability and security. The distribution of the aggregated prediction error of regional wind farms is selected as the criterion to evaluate the accuracy of different spatial models.
Goodnessoffit indices of three spatial models
Index  PairCopula model  Gaussian model  Independent model 

CvM index  0.0119  0.0123  0.0769 
KS index  0.0376  0.0478  0.2351 
Load shedding and wind curtailing probabilities
Probability  Measured data (%)  PairCopula model (%)  Gaussian model (%)  Independent model (%) 

Load shedding probability  2.79  2.44  4.45  0 
Wind curtailing probability  5.36  5.69  4.64  0.01 
4.3 Feasibility evaluation of model
5 Conclusion
Because of the highly variable forecast errors of wind power generation, a thorough uncertainty model describing both the predictive distribution and dependent structure of forecast errors is essential. In this paper, a PairCopula modelling approach has been proposed to construct joint distribution and generate scenarios that mimic wind power stochastic behaviours. The feasibility of the PairCopula approach for wind power temporal and spatial dependence is verified, and the advantages of the PairCopula are noted. First, owing to the basic of conditional dependence, the PairCopula model can reflect the essential relationships between random variables, which the other multivariate models fail to reveal, e.g., the oneorder Markov characteristic of the temporal model. Furthermore, the PairCopula approach is more adaptive and practical in various applications, as it allows the combination of different types of Copulas and thus yields the bestfitted model. In the illustrated example, the symmetrical Student t function is selected for the temporal model, while the asymmetrical Gumbel Copula is chosen for the spatial model.
Based on the developed models, the effects of the temporal and spatial dependence of wind power forecasting errors on power system operations are analysed. For the temporal model, the tail dependence indicates that large prediction errors have stronger correlations, resulting in a fattail distribution of the ESS energy. Thus, a large ESS capacity is required to balance the prediction errors. For the spatial model, the upper tail makes the probability of excess generation higher than that of deficits, causing different demands for upward and downward reserves. Furthermore, goodnessoffit indices of different models are calculated and comparison results show that PairCopula model gives greater accuracy in various application scenes.
The main contribution of this paper is to significantly advance the uncertainty model of wind power and to provide a systematic procedure for modelling and generating wind power scenarios, which can then be applied to a range of wind power integration problems. Although only wind power uncertainty modelling is discussed in this paper, the PairCopula approach provides a powerful tool for constructing flexible multivariate distributions that it can also be applied to a wide range of statistical analyses and stochastic decision fields of power systems.
Notes
Acknowledgements
This work was supported by China’s National High Technology Research and Development Program (No. 2012AA050207), China’s National Nature Science Foundation (No. 51190101) and Science and Technology Projects of the State Grid Corporation of China (No. SGHN0000DKJS130022).
References
 [1]Yuan XM (2013) Overview of problems in largescale wind integrations. J Mod Power Syst Clean Energy 1(1):22–25. doi: 10.1007/s4056501300106 CrossRefGoogle Scholar
 [2]Halamay DA, Brekken TKA, Simmons A et al (2011) Reserve requirement impacts of largescale integration of wind, solar, and ocean wave power generation. IEEE Trans Sustain Energy 2(3):321–328CrossRefGoogle Scholar
 [3]Zhou BR, Geng GC, Jiang QY (2016) Hierarchical unit commitment with uncertain wind power generation. IEEE Trans Power Syst 31(1):94–104CrossRefGoogle Scholar
 [4]Li YZ, Wu QH, Jiang L et al (2016) Optimal power system dispatch with wind power integrated using nonlinear interval optimization and evidential reasoning approach. IEEE Trans Power Syst 31(3):2246–2254CrossRefGoogle Scholar
 [5]Ding Y, Singh C, Goel L et al (2014) Shortterm and mediumterm reliability evaluation for power systems with high penetration of wind power. IEEE Trans Sustain Energy 5(3):896–906CrossRefGoogle Scholar
 [6]Ciupuliga AR, Gibescu M, Pelgrum E et al (2012) Roundtheyear security analysis with largescale wind power integration. IEEE Trans Sustain Energy 3(1):85–93CrossRefGoogle Scholar
 [7]Holttinen H (2005) Impact of hourly wind power variations on the system operation in the Nordic countries. Wind Energy 8(2):197–218CrossRefGoogle Scholar
 [8]Bludszuweit H, DominguezNavarro JA, Llombart A (2008) Statistical analysis of wind power forecasting error. IEEE Trans Power Syst 23(3):983–991CrossRefGoogle Scholar
 [9]Pinson P (2006) Estimation of the uncertainty in wind power forecastinging. Ph.D. thesis. Ecole des Mines de Paris, Paris, FranceGoogle Scholar
 [10]Billinton R, Chen H, Ghajar R (1996) Timeseries models for reliability evaluation of power systems including wind energy. Microelectron Reliab 36(9):1253–1261CrossRefGoogle Scholar
 [11]Miranda MS, Drun RW (2007) Spatially correlated wind speed modelling for generation adequacy studies in the UK. In: Proceedings of the 2007 IEEE Power Engineering Society general meeting, Tampa, FL, USA, 24–28 Jun 2007, 6 ppGoogle Scholar
 [12]Shamshad A, Bawadi MA, Wan Hussin WMA et al (2005) First and second order Markov chain models for synthetic generation of wind speed time series. Energy 30(5):693–708CrossRefGoogle Scholar
 [13]Bouffard F, Galiana FD (2008) Stochastic security for operations planning with significant wind power generation. IEEE Trans Power Syst 23(2):306–316CrossRefGoogle Scholar
 [14]McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques, tools. Princeton University Press, PrincetonzbMATHGoogle Scholar
 [15]Cherubini U, Luciano E, Vecchiato W (2004) Copula methods in finance. Wiley, New YorkCrossRefzbMATHGoogle Scholar
 [16]Wan C, Xu Z, Pinson P et al (2014) Probabilistic forecasting of wind power generation using extreme learning machine. IEEE Trans Power Syst 29(3):1033–1044CrossRefGoogle Scholar
 [17]Papaefthymiou G, Kurowicka D (2009) Using copulas for modeling stochastic dependence in power system uncertainty analysis. IEEE Trans Power Syst 24(1):40–49CrossRefGoogle Scholar
 [18]Pierre P, Henrik M, Aa NH et al (2009) From probabilistic forecasts to statistical scenarios of shortterm wind power production. Wind Energy 12(1):51–62CrossRefGoogle Scholar
 [19]Ma XY, Sun YZ, Fang HL (2013) Scenario generation of wind power based on statistical uncertainty and variability. IEEE Trans Sustain Energy 4(4):894–904CrossRefGoogle Scholar
 [20]Yang M, Lin Y, Zhu SM et al (2015) Multidimensional scenario forecast for generation of multiple wind farms. J Mod Power Syst Clean Energy 3(3):1–10. doi: 10.1007/s4056501501106 Google Scholar
 [21]Haghi HV, Bina MT, Golkar MA (2013) Nonlinear modeling of temporal wind power variations. IEEE Trans Sustain Energy 4(4):838–848CrossRefGoogle Scholar
 [22]Louie H (2014) Valuation of bivariate archimedean and elliptical copulas to model wind power dependence structures. Wind Energy 17(2):225–240CrossRefGoogle Scholar
 [23]Díaz G (2014) A note on the multivariate archimedean dependence structure in small wind generation sites. Wind Energy 17(8):1287–1295CrossRefGoogle Scholar
 [24]Hofert M, Mchler M, McNeil AJ (2011) Likelihood inference for archimedean copulas in high dimensions under known margins. J Multivar Anal 110:133–150MathSciNetCrossRefzbMATHGoogle Scholar
 [25]Nelsen RB (1999) An introduction to copulas. Springer, BerlinCrossRefzbMATHGoogle Scholar
 [26]Drouet MD, Kotz S (2001) Correlation and dependence. Imperial College Press, LondonzbMATHGoogle Scholar
 [27]Savu C, Trede M (2010) Hierarchies of archimedean copulas. Quant Finance 10(3):295–304MathSciNetCrossRefzbMATHGoogle Scholar
 [28]McNeil AJ (2008) Sampling nested archimedean copulas. J Stat Comput Simul 78(6):567–581MathSciNetCrossRefzbMATHGoogle Scholar
 [29]Czado C (2010) Paircopula constructions of multivariate copulas. In: Jaworski P, Durante F, Hardle W et al (eds) Copula theory and its applications. Springer, Berlin, pp 93–109CrossRefGoogle Scholar
 [30]Genest C, Ghoudi K, Rivest LP (1995) A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3):543–553MathSciNetCrossRefzbMATHGoogle Scholar
 [31]Aas K, Czado C, Frigessi A et al (2009) Paircopula constructions of multiple dependence. Insur Math Econ 44(2):182–198CrossRefzbMATHGoogle Scholar
 [32]Pennock K (2012) Updated eastern interconnect wind power output and forecasts for ERGIS. NREL/SR550056616. National Renewable Energy Laboratory, Golden, CO, USAGoogle Scholar
 [33]Zhang N, Kang CQ, Xu QY et al (2013) Modelling and simulating the spatiotemporal correlations of clustered wind power using copula. J Electr Eng Technol 8(6):1615–1625CrossRefGoogle Scholar
 [34]Papaefthymiou G, Klockl B (2008) MCMC for wind power simulation. IEEE Trans Energy Conver 23(1):234–240CrossRefGoogle Scholar
 [35]Genest C, Rémillard B, Beaudoin D (2009) Goodnessoffit tests for copulas: A review and a power study. Insur Math Econ 44(2):199–213CrossRefzbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.