A General Probabilistic Forecasting Framework for Offshore Wind Power Fluctuations
Abstract
:1. Introduction
 the modeling of a nonlinear and nonstationary stochastic process for which we propose a model that allows to capture up to three different time series effects: autocorrelation, heteroscedasticity and regime switching (the generic name of our model is MSARGARCH),
 the numerous issues linked to the practical implementation of such model as it requires an advanced estimation method based on a Markov Chain Monte Carlo (MCMC) algorithm,
 the gap between applying such model to synthetic data and real world observations.
2. Motivations Based on the StateoftheArt
3. Data from Large Offshore Wind Farms
4. Model Specifications
4.1. Wind Power Predictive Density
4.2. GARCH Models in Meteorology
4.3. Existing Markov Switching Models with GARCH Errors
 the robustness of MCMC samplers to starting values can be evaluated by running several Markov chains with different starting values and tested for differences in their outputs,
 inequality constraints can be handled through the definition of prior distributions (Gibbs sampler) or through a rejection step when the constraint is violated (Metropolis–Hastings sampler),
 theoretically, local minima pitfalls are avoided by simulating the Markov chain over a sufficiently large number of iterations (law of large numbers),
 misspecification of the number of states of the Markov chain can be assessed by a visual inspection of the parameter posterior distributions (check for multiple modes).
4.4. The Model Definition
5. MCMC Implementation
 sample the regime sequence by data augmentation,
 sample the transition probabilities from a Dirichlet distribution,
 sample the AR and GARCH coefficients with the GriddyGibbs sampler.
5.1. Sampling the Regime Sequence
 the filtered probabilities$P({S}_{t}=k{y}_{[1,t]},\Theta )$ which infer the state variable ${S}_{t}$ conditioning upon the vector of parameters and all past and present information ${y}_{[1,t]}$,
 the smoothed probabilities$P({S}_{t}=ky,\Theta )$ which are the outputs of the inference of ${S}_{t}$ using the past, present and future information $y={y}_{[1,T]}$,
 the predicted probabilities$P({S}_{t+1}=k{y}_{[1,t]},\Theta )$ which correspond to the onestep ahead inference ${S}_{t+1}$ at time t and only use past information $y={y}_{[1,t]}$.
5.2. Transition Probability Matrix Sampling
5.3. AR and GARCH Coefficient Sampling
5.4. Implementation Details
5.4.1. Prior Distributions
 a burnin phase whose draws are discarded until the Markov chain reaches its stationary distribution,
 a second phase at the end of which posterior density estimates are computed and prior bounds are refined (the draws generated during this second phase are also discarded),
 a last phase with adjusted prior bounds at the end of which the final posterior densities are computed.
5.4.2. Label Switching
5.4.3. Grid Shape
5.4.4. Mixing of the MCMC Chain
5.4.5. Implementation Summary
Algorithm 1 MCMC procedure for the estimation of MSARGARCH models 

5.5. Simulation on Synthetic Time Series
Param.  True Value  Intial prior  50 sample  1 sample  

Posterior mean  Posterior std. dev.  CP  Refined prior support  Posterior mean  Posterior std. dev.  
${\theta}_{0}^{\left(1\right)}$  0.5  [−0.2 ; 1.2]  0.500  0.072  96%  [0.20 ; 0.78]  0.488  0.050 
${\theta}_{1}^{\left(1\right)}$  0.5  [−0.2 ; 1.2]  0.502  0.054  98%  [0.26 ; 0.72]  0.495  0.037 
${\theta}_{2}^{\left(1\right)}$  0.2  [−0.2 ; 1.2]  0.197  0.051  98%  [−0.01 ; 0.43]  0.212  0.035 
${\alpha}_{0}^{\left(1\right)}$  0.1  [0 ; 0.5]  0.109  0.041  94%  [0 ; 0.17]  0.084  0.020 
${\alpha}_{1}^{\left(1\right)}$  0.2  [0 ; 0.5]  0.195  0.068  94%  [0 ; 0.38]  0.175  0.046 
${\beta}_{1}^{\left(1\right)}$  0.6  [0 ; 1]  0.593  0.101  94%  [0.36 ; 0.88]  0.621  0.059 
${\theta}_{0}^{\left(2\right)}$  0  [−0.7 ; 0.7]  0.015  0.041  94%  [−0.44 ; 0.36]  −0.038  0.100 
${\theta}_{1}^{\left(2\right)}$  0.7  [0 ; 1.4]  0.689  0.081  98%  [0.55 ; 0.99]  0.764  0.051 
${\theta}_{2}^{\left(2\right)}$  −0.3  [−1 ; 0.2]  −0.308  0.081  98%  [−0.59 ; −0.17]  −0.381  0.052 
${\alpha}_{0}^{\left(2\right)}$  0.4  [0.1 ; 0.8]  0.512  0.189  98%  [0 ; 0.82]  0.373  0.105 
${\alpha}_{1}^{\left(2\right)}$  0.1  [0 ; 0.5]  0.114  0.073  92%  [0 ; 0.33]  0.135  0.041 
${\beta}_{1}^{\left(2\right)}$  0.85  [0 ; 1]  0.813  0.087  96%  [0.62 ; 1]  0.831  0.044 
${p}_{11}$  0.98  [0 ; 1]  0.977  0.009  90%  [0 ; 1]  0.983  0.005 
${p}_{22}$  0.96  [0 ; 1]  0.950  0.023  92%  [0 ; 1]  0.961  0.012 
Rate of successful  Probability of  

regime inference  regime retrieval  
$P({\widehat{S}}_{t}=1{S}_{t}=1)=0.96$  $P({S}_{t}=1{\widehat{S}}_{t}=1)=0.95$  
$P({\widehat{S}}_{t}=2{S}_{t}=2)=0.90$  $P({S}_{t}=2{\widehat{S}}_{t}=2)=0.91$ 
5.6. Study on an Empirical Time Series of Wind Power
1 Regime: AR(3)GARCH(1,1)  2 Regimes: MS(2)AR(3)GARCH(1,1)  

Initial prior support  Refined prior support  Posterior mean  Posterior std. dev.  Initial prior support  Refined prior support  Posterior mean  Posterior std. dev.  
${\theta}_{0}^{\left(1\right)}$  [−0.01 ; 0.01]  [−0.007 ; 0.006]  −2 × 10${}^{4}$  0.002  [−0.04 ; 0.04]  [−0.004 ; 0.004]  −3 × 10 ${}^{5}$  6 × 10${}^{4}$ 
${\theta}_{1}^{\left(1\right)}$  [1 ; 1.7]  [0.68 ; 2.11]  1.358  0.232  [1 ; 1.8]  [0.64 ; 2.18]  1.417  0.273 
${\theta}_{2}^{\left(1\right)}$  [−085 ; −0.05]  [−1.33 ; 0.34]  −0.460  0.284  [−0.95 ; −0.15]  [−1.36 ; 0.21]  −0.574  0.304 
${\theta}_{3}^{\left(1\right)}$  [−0.15 ; 0.35]  [−0.52 ; 0.72]  0.107  0.206  [−0.35 ; 0.55]  [−0.67 ; 0.99]  0.156  0.300 
${\alpha}_{0}^{\left(1\right)}$  [0 ; 3 × 10${}^{4}$]  [0 ; 3 × 10${}^{4}$]  7 × 10${}^{5}$  6 × 10${}^{5}$  [5 × 10${}^{6}$ ; 10${}^{4}$]  [2 × 10${}^{6}$ ; 10${}^{5}$]  3 × 10 ${}^{6}$  2 × 10${}^{7}$ 
${\alpha}_{1}^{\left(1\right)}$  [0.2 ; 1]  [0.03 ; 1]  0.513  0.161  [0 ; 1]  [0.23 ; 0.74]  0.499  0.077 
${\beta}_{1}^{\left(1\right)}$  [0 ; 0.7]  [0 ; 0.95]  0.467  0.161  [0 ; 1]  [0.25 ; 0.74]  0.489  0.074 
${\theta}_{0}^{\left(2\right)}$          [0.06 ; 0.10]  [−0.04 ; 0.09]  0.011  0.013 
${\theta}_{1}^{\left(2\right)}$          [0.7 ; 1.7]  [0.27 ; 2.02]  1.178  0.285 
${\theta}_{2}^{\left(2\right)}$          [−0.7 ; 0.3]  [−1.22 ; 0.58]  −0.323  0.341 
${\theta}_{3}^{\left(2\right)}$          [−0.4 ; 0.6]  [−0.76 ; 1.01]  0.126  0.284 
${\alpha}_{0}^{\left(2\right)}$          [1 × 10${}^{3}$ ; 8 × 10${}^{3}$]  [0 ; 4 × 10${}^{3}$]  5 × 10 ${}^{4}$  3 × 10${}^{4}$ 
${\alpha}_{1}^{\left(2\right)}$          [0 ; 1]  [0 ; 0.54]  0.079  0.080 
${\beta}_{1}^{\left(2\right)}$          [0 ; 1]  [0 ; 1]  0.892  0.088 
${p}_{11}$          [0 ; 1]  [0 ; 1]  0.913  0.029 
${p}_{22}$          [0 ; 1]  [0 ; 1]  0.783  0.114 
6. Wind Power Forecast Evaluation
6.1. Approximating the Conditional Variance for Prediction Applications
6.2. Evaluation of Point Forecasts
Model  Oct.  Nov.  Dec.  Jan.  Total 

Persistence  2.41  2.58  3.01  2.47  2.55 
AR(3)  2.36  2.64  2.98  2.46  2.53 
AR(3)GARCH(1,1)  2.29  2.60  2.95  2.41  2.49 
MS(2)AR(3)GARCH(1,1)  2.27  2.50  2.89  2.38  2.44 
MSAR(2,3)  2.28  2.49  2.89  2.37  2.44 
MSAR(3,3)  2.26  2.49  2.89  2.36  2.42 
Model  Oct.  Nov.  Dec.  Jan.  Total 

Persistence  4.17  6.22  5.76  4.28  5.02 
AR(3)GARCH(1,1)  4.00  6.18  5.72  4.24  4.93 
AR(3)  3.98  5.99  5.56  4.17  4.83 
MS(2)AR(3)GARCH(1,1)  3.96  6.00  5.55  4.15  4.82 
MSAR(2,3)  3.98  5.95  5.55  4.17  4.81 
MSAR(3,3)  3.96  5.95  5.55  4.17  4.80 
6.3. Evaluation of Interval and Density Forecasts
Model  Oct.  Nov.  Dec.  Jan.  Total 

AR(3)  1.99  2.33  2.48  2.02  2.15 
MSAR(2,3)  1.81  2.01  2.26  1.88  1.94 
MSAR(3,3)  1.78  1.98  2.24  1.85  1.91 
AR(3)GARCH(1,1)  1.76  1.99  2.24  1.85  1.91 
MS(2)AR(3)GARCH(1,1)  1.76  1.95  2.20  1.83  1.88 
Nom. cov.  Emp. cov.  Emp. cov.  Emp. cov. 

AR(3)  MSAR(3,3)  MS(2)AR(3)GARCH(1,1)  
10  13.2  7.1  9.4 
20  42.6  25.8  20.7 
30  55.5  35.2  31.3 
40  64.3  42.9  42.3 
50  71.4  52.4  63.2 
60  77.2  60.3  71.2 
70  81.6  68.8  78.1 
80  89.9  77.7  84.4 
90  90.0  86.9  90.0 
7. Discussion and Concluding Remarks
Acknowledgements
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Trombe, P.J.; Pinson, P.; Madsen, H. A General Probabilistic Forecasting Framework for Offshore Wind Power Fluctuations. Energies 2012, 5, 621657. https://doi.org/10.3390/en5030621
Trombe PJ, Pinson P, Madsen H. A General Probabilistic Forecasting Framework for Offshore Wind Power Fluctuations. Energies. 2012; 5(3):621657. https://doi.org/10.3390/en5030621
Chicago/Turabian StyleTrombe, PierreJulien, Pierre Pinson, and Henrik Madsen. 2012. "A General Probabilistic Forecasting Framework for Offshore Wind Power Fluctuations" Energies 5, no. 3: 621657. https://doi.org/10.3390/en5030621