# Distributed Reconciliation in Day-Ahead Wind Power Forecasting

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## Abstract

**:**

## 1. Introduction

## 2. Hierarchical Time-Series and Forecast Reconciliation

#### 2.1. The Forecast Reconciliation Problem

#### 2.2. An Overview of the State of the Art

#### 2.3. Generalized Least Squares Reconciliation

#### 2.4. Trace Minimization Reconciliation

#### 2.4.1. Ordinary Least Squares (OLS) Reconciliation

#### 2.4.2. Weighted Least Squares (WLS) Reconciliation

#### 2.4.3. Hierarchical Least Squares (HLS) Reconciliation

#### 2.4.4. Minimum Trace (MinT) Reconciliation and Shrinkage Estimator

## 3. Proposed Distributed Reconciliation Methods

#### 3.1. Game Theoretical Optimal (GTOP) Reconciliation

#### 3.2. Constrained GTOP Solved by ADMM

Algorithm 1 constrained GTOP Algorithm |

1: Require: base forecasts $\widehat{Y}={[{\widehat{y}}_{\mathrm{AGG}},{\widehat{y}}_{1},{\widehat{y}}_{2},\dots ,{\widehat{y}}_{K}]}^{\top}$; $A=\mathrm{diag}(\sqrt{{a}_{\mathrm{AGG}}},\sqrt{{a}_{1}},\dots ,\sqrt{{a}_{K}})$;2: aggregated consistency $\mathcal{A}$; boundary constraint $\mathcal{B}:{B}^{\mathrm{up}},{B}^{\mathrm{low}}$ 3: Input: $\Delta ={\widehat{y}}_{\mathrm{AGG}}-{\sum}_{k=1}^{K}{\widehat{y}}_{k},\Delta {y}_{k}^{i},{\overline{z}}^{i},{u}^{i}$4: Output: reconciled forecasts $\tilde{Y}={[{\tilde{y}}_{\mathrm{AGG}},{\tilde{y}}_{1},{\tilde{y}}_{2},\dots ,{\tilde{y}}_{K}]}^{\top}$5: while stopping criteria do6: $\Delta {y}_{k}^{i+1}=t{h}_{k}({({a}_{k}+\rho )}^{-1}\rho (\Delta {y}_{k}^{i}-\Delta {\overline{y}}^{i}+{\overline{z}}^{i}-{u}_{k}^{i}))$ 7: ${\overline{z}}^{i+1}={(K{a}_{\mathrm{AGG}}+\rho )}^{-1}\left(\right)open="("\; close=")">\rho (\Delta {\overline{y}}^{i+1}+{u}^{i})+{a}_{\mathrm{AGG}}\Delta $ 8: ${u}^{i+1}={u}^{i}+\Delta {\overline{y}}^{i+1}-{\overline{z}}^{i+1}.$ 9: $i=i+1$ 10: end while11: ${\tilde{y}}_{k}={\widehat{y}}_{k}+\Delta {y}_{k}$, ${\tilde{y}}_{\mathrm{AGG}}={\sum}_{k=1}^{K}{\tilde{y}}_{k}$ |

#### 3.3. Online Estimate of Individual Variance

#### 3.4. Boundary Constraint

## 4. Application and Case-Studies

#### 4.1. Framework and Verification

#### 4.2. Reconciliation on the Simulated Dataset

#### 4.3. Reconciliation on the NREL Dataset

#### 4.4. Reconciliation on the Sardinia Dataset

## 5. Conclusions and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Radial Basis Function Based Support Vector Regression

## References

- Fliedner, G. Hierarchical forecasting: Issues and use guidelines. Ind. Manag. Data Syst.
**2001**, 101, 5–12. [Google Scholar] [CrossRef] - Christoph, W. Essays in Hierarchical Time Series Forecasting and Forecast Combination. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 2018. [Google Scholar]
- Lobo, M.G.; Sanchez, I. Regional wind power forecasting based on smoothing techniques, with application to the spanish peninsular system. IEEE Trans. Power Syst.
**2012**, 27, 1990–1997. [Google Scholar] [CrossRef] - Fabbri, A.; Roman, T.G.S.; Abbad, J.R.; Quezada, V.H.M. Assessment of the cost associated with wind generation prediction errors in a liberalized electricity market. IEEE Trans. Power Syst.
**2005**, 20, 1440–1446. [Google Scholar] [CrossRef] - Lu, Z.; Ye, X.; Qiao, Y.; Min, Y. Initial exploration of wind farm cluster hierarchical coordinated dispatch based on virtual power generator concept. CSEE J. Power Energy Syst.
**2015**, 1, 62–67. [Google Scholar] [CrossRef] - Meng, K.; Zhang, W.; Li, Y.; Dong, Z.Y.; Xu, Z.; Wong, K.P.; Zheng, Y. Hierarchical SCOPF considering wind energy integration through multiterminal VSC-HVDC grids. IEEE Trans. Power Syst.
**2017**, 32, 4211–4221. [Google Scholar] [CrossRef] - Zhang, Y.; Dong, J. Least Squares-based optimal reconciliation method for hierarchical forecasts of wind power generation. IEEE Trans. Power Syst.
**2018**, 1. [Google Scholar] [CrossRef] - Yang, D.; Quan, H.; Disfani, V.R.; Liu, L. Reconciling solar forecasts: Geographical hierarchy. Sol. Energy
**2017**, 146, 276–286. [Google Scholar] [CrossRef] - Hyndman, R.J.; Lee, A.J.; Wang, E. Fast computation of reconciled forecasts for hierarchical and grouped time series. Comput. Data Anal.
**2016**, 97, 16–32. [Google Scholar] [CrossRef] - Athanasopoulos, G.; Hyndman, R.J.; Kourentzes, N.; Petropoulos, F. Forecasting with temporal hierarchies. Eur. J. Oper. Res.
**2017**, 262, 60–74. [Google Scholar] [CrossRef] - van Erven, T.; Cugliari, J. Game-theoretically optimal reconciliation of contemporaneous hierarchical time series forecasts. In Modeling and Stochastic Learning for Forecasting in High Dimensions; Springer International Publishing: Cham, Switzerland, 2015; pp. 297–317. [Google Scholar]
- Lei, M.; Shiyan, L.; Chuanwen, J.; Hongling, L.; Yan, Z. A review on the forecasting of wind speed and generated power. Renew. Sustain. Energy Rev.
**2009**, 13, 915–920. [Google Scholar] [CrossRef] - Giebel, G.; Brownsword, R.; Kariniotakis, G.; Denhard, M.; Draxl, C. The State-Of-The-Art in Short-Term Prediction of Wind Power: A Literature Overview, 2nd ed.; ANEMOS.plus: Paris, France, 2011. [Google Scholar] [CrossRef]
- Hong, T.; Pinson, P.; Fan, S. Global Energy Forecasting Competition 2012. Int. J. Forecast.
**2014**, 30, 357–363. [Google Scholar] [CrossRef] - Siebert, N. Development of Methods for Regional Wind Power Forecasting. Ph.D. Thesis, École Nationale Supérieure des Mines de Paris, Paris, France, 2008. [Google Scholar]
- Yan, J.; Zhang, H.; Liu, Y.; Han, S.; Li, L.; Lu, Z. Forecasting the high penetration of wind power on multiple scales using multi-to-multi mapping. IEEE Trans. Power Syst.
**2018**, 33, 3276–3284. [Google Scholar] [CrossRef] - Hyndman, R.; Athanasopoulos, G. Optimally reconciling forecasts in a hierarchy. Foresight
**2014**, 2014, 42–48. [Google Scholar] - Hyndman, R.J.; Ahmed, R.A.; Athanasopoulos, G.; Shang, H.L. Optimal combination forecasts for hierarchical time series. Comput. Stat. Data Anal.
**2011**, 55, 2579–2589. [Google Scholar] [CrossRef] - Wickramasuriya, S.L.; Athanasopoulos, G.; Hyndman, R.J. Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization. J. Am. Stat. Assoc.
**2018**, 1–16. [Google Scholar] [CrossRef] - Jooyoung, J.; Anastasios, P.; Fotios, P. Reconciliation of probabilistic forecasts with an application to wind power. arXiv, 2018; arXiv:1808.02635. [Google Scholar]
- Yang, D.; Quan, H.; Disfani, V.R.; Rodríguez-Gallegos, C.D. Reconciling solar forecasts: Temporal hierarchy. Sol. Energy
**2017**, 158, 332–346. [Google Scholar] [CrossRef] - Campbell, S.; Meyer, C. Generalized Inverses of Linear Transformations; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2009. [Google Scholar] [CrossRef]
- Schäfer, J.; Strimmer, K. A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Stat. Appl. Genet. Mol. Biol.
**2005**, 4, 32. [Google Scholar] [CrossRef] [PubMed] - Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; Eckstein, J. Distributed optimization and statistical learning via the Alternating Direction Method of Multipliers. Found. Trends Mach. Learn.
**2011**, 3, 1–122. [Google Scholar] [CrossRef] - Goldstein, T.; O’Donoghue, B.; Setzer, S.; Baraniuk, R. Fast alternating direction optimization methods. SIAM J. Imaging Sci.
**2014**, 7, 1588–1623. [Google Scholar] [CrossRef] - Harris, R.I.; Cook, N.J. The parent wind speed distribution: Why Weibull? J. Wind Eng. Ind. Aerodyn.
**2014**, 131, 72–87. [Google Scholar] [CrossRef] - Lydia, M.; Kumar, S.S.; Selvakumar, A.I.; Kumar, G.E.P. A comprehensive review on wind turbine power curve modeling techniques. Renew. Sustain. Energy Rev.
**2014**, 30, 452–460. [Google Scholar] [CrossRef] - Jowder, F.A. Wind power analysis and site matching of wind turbine generators in Kingdom of Bahrain. Appl. Energy
**2009**, 86, 538–545. [Google Scholar] [CrossRef] - Draxl, C.; Clifton, A.; Hodge, B.M.; McCaa, J. The Wind Integration National Dataset (WIND) Toolkit. Appl. Energy
**2015**, 151, 355–366. [Google Scholar] [CrossRef] - Pennock, K. Updated Eastern Interconnect Wind Power Output and Forecasts for ERGIS: July 2012. Available online: https://www.nrel.gov/docs/fy13osti/56616.pdf (accessed on 1 October 2012).
- Zheng, L.; Hu, W.; Min, Y. Raw wind data preprocessing: A data-mining approach. IEEE Trans. Sustain. Energy
**2015**, 6, 11–19. [Google Scholar] [CrossRef] - Kramer, O.; Gieseke, F. Short-term wind energy forecasting using Support Vector Regression. In Proceedings of the 6th International Conference SOCO—2011 Soft Computing Models in Industrial and Environmental Applications, Salamanca, Spain, 6–8 April 2011; Corchado, E., Snášel, V., Sedano, J., Hassanien, A.E., Calvo, J.L., Ślȩzak, D., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 271–280. [Google Scholar]
- Zeng, J.; Qiao, W. Support vector machine-based short-term wind power forecasting. In Proceedings of the 2011 IEEE/PES Power Systems Conference and Exposition, Phoenix, AZ, USA, 20–23 March 2011; pp. 1–8. [Google Scholar] [CrossRef]

**Figure 1.**A two-level hierarchy for wind farms and related wind power forecasts in a portfolio or region of interest.

**Figure 2.**Iterations of convergence versus $\rho $ of Alternating Direction Method of Multipliers (ADMM) and fast ADMM (regarding three datasets).

**Figure 3.**boxplots of normalized root mean square errors (NRMSEs) on the simulated dataset: (

**a**) NRMSE of Node “AGG”, (

**b**) NRMSE at bottom level.

**Figure 4.**boxplots of normalized mean absolute errors (NMAEs) on the simulated dataset: (

**a**) NMAE of Node “AGG”, (

**b**) NMAE at bottom level.

**Figure 5.**Improvement of RMSEs (IRMSEs) on the NREL dataset: (

**a**) IRMSE of Node “AGG”, (

**b**) IRMSE at bottom level.

Estimator | Covariance Matrix | Matrix Property |
---|---|---|

OLS | Identity matrix | diagonal matrix |

WLS | $\mathrm{diag}({W}_{t+h|t}^{e})$ | diagonal matrix |

HLS | $SI$ | full matrix |

MinT | ${W}_{t+h|t}^{e}$ | full matrix |

MinT_srk | $\lambda \mathrm{diag}({W}_{t+h|t}^{e})+(1-\lambda ){W}_{t+h|t}^{e}$ | full matrix |

Dataset | Wind Speeds | Power Output |
---|---|---|

Simulated dataset | Randomly generated | Simulated |

NREL dataset | Provided | Simulated |

Sardinia dataset | Provided | Measured |

Parameters | Interval |
---|---|

the cut-in speed ${w}_{\mathrm{cin}}$ m/s | [3, 4] |

the rated speed ${w}_{\mathrm{r}}$ m/s | [12, 15] |

the cut-out speed ${w}_{\mathrm{cout}}$ m/s | [24, 25] |

Weibull shape factor C | [1.6, 2] |

Weibull scale factor $\eta $ | [6, 8] |

wind farm capacity MW | [20, 30] |

Node 1 | Node 2 | Node 3 | Node 4 | Node 5 | Node 6 | Bottom Level | Node AGG | |
---|---|---|---|---|---|---|---|---|

Base | 7.76 | 6.62 | 6.96 | 6.88 | 6.75 | 7.44 | 7.07 | 10.90 |

OLS | 16.69 | 17.68 | 17.35 | 8.88 | 9.92 | 8.90 | 13.24 | 9.95 |

WLS | 7.73 | 6.63 | 6.96 | 7.23 | 6.92 | 8.18 | 7.28 | 6.28 |

OLS-CADMM | 8.34 | 7.25 | 7.56 | 7.74 | 7.67 | 8.43 | 7.83 | 6.91 |

WLS-CADMM | 7.73 | 6.63 | 6.96 | 7.19 | 6.92 | 8.02 | 7.24 | 6.23 |

Node 1 | Node 2 | Node 3 | Node 4 | Node 5 | Node 6 | Bottom Level | Node AGG | |
---|---|---|---|---|---|---|---|---|

Base | 5.34 | 4.62 | 4.82 | 4.90 | 4.79 | 5.18 | 4.94 | 6.65 |

OLS | 9.73 | 9.48 | 9.53 | 6.11 | 6.55 | 6.06 | 7.91 | 6.25 |

WLS | 5.34 | 4.64 | 4.83 | 5.18 | 4.91 | 5.68 | 5.09 | 4.49 |

OLS-CADMM | 6.16 | 5.36 | 5.55 | 5.54 | 5.51 | 5.83 | 5.66 | 4.96 |

WLS-CADMM | 5.34 | 4.64 | 4.83 | 5.14 | 4.91 | 5.57 | 5.07 | 4.45 |

Base | OLS | WLS | OLS-CADMM | WLS-CADMM |
---|---|---|---|---|

4.34 | 13.13 | 3.68 | 4.29 | 3.63 |

Node 1 | Node 2 | Node 3 | Node 4 | Bottom Level | Node AGG | |
---|---|---|---|---|---|---|

Base | 11.68 | 9.15 | 12.29 | 12.40 | 11.38 | 7.56 |

OLS | 11.54 | 8.93 | 11.96 | 12.12 | 11.14 | 7.61 |

WLS | 11.55 | 9.05 | 12.21 | 12.29 | 11.28 | 7.76 |

OLS-CADMM | 11.54 | 8.93 | 11.96 | 12.12 | 11.14 | 7.61 |

WLS-CADMM | 11.60 | 9.07 | 12.15 | 12.28 | 11.27 | 7.76 |

Node 1 | Node 2 | Node 3 | Node 4 | Bottom Level | Node AGG | |
---|---|---|---|---|---|---|

Base | 7.11 | 4.90 | 7.69 | 7.71 | 6.85 | 4.94 |

OLS | 7.09 | 4.85 | 7.67 | 7.65 | 6.81 | 4.95 |

WLS | 7.09 | 4.86 | 7.67 | 7.68 | 6.82 | 4.98 |

OLS-CADMM | 7.09 | 4.85 | 7.67 | 7.65 | 6.81 | 4.95 |

WLS-CADMM | 7.14 | 5.12 | 8.02 | 7.90 | 7.04 | 5.08 |

Base | OLS | WLS | OLS-CADMM | WLS-CADMM |
---|---|---|---|---|

5.82 | 5.61 | 5.75 | 5.61 | 5.75 |

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**MDPI and ACS Style**

Bai, L.; Pinson, P.
Distributed Reconciliation in Day-Ahead Wind Power Forecasting. *Energies* **2019**, *12*, 1112.
https://doi.org/10.3390/en12061112

**AMA Style**

Bai L, Pinson P.
Distributed Reconciliation in Day-Ahead Wind Power Forecasting. *Energies*. 2019; 12(6):1112.
https://doi.org/10.3390/en12061112

**Chicago/Turabian Style**

Bai, Li, and Pierre Pinson.
2019. "Distributed Reconciliation in Day-Ahead Wind Power Forecasting" *Energies* 12, no. 6: 1112.
https://doi.org/10.3390/en12061112