Forecasting Hierarchical Time Series in Power Generation
Abstract
:1. Introduction
2. Materials and Methods
2.1. The Bottom-Up (BU) Approach
2.2. The Top-Down (TD) Approach
2.3. The Optimal Reconciliation Approaches
2.4. ARIMA and ETS Formulation
2.5. Evaluating Forecast Accuracy
3. Results and Discussion
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
ARIMA | Autoregressive integrated moving average model |
BU | Bottom-up |
ETS | Error, trend, and seasonality model |
GWh | Gigawatt hours |
MAPE | Mean absolute percentage error |
MinT | Minimum trace reconciliation |
OLS | Ordinary least squares |
ONS | Operator of the National System |
TD | Top-down |
TDFP | Top-down forecast proportions |
TDGSA | Top-down Gross-Sohl method A |
TDGSF | Top-down Gross-Sohl method F |
WLS | Weighted least squares |
Nomenclature
Level of disaggregation | |
Forecast horizon | |
-dimensional vector of -step-ahead forecasts | |
Unknown conditional mean of the most disaggregated series | |
Error for each forecast horizon | |
Covariance matrix | |
Set of proportions in an m-dimensional vector | |
The average of the historical proportions | |
Summing matrix | |
The sum of the -step-ahead forecasts for TD | |
Covariance matrix of the corresponding h-step ahead base forecast errors | |
Total level of power generation | |
an -dimensional vector of -step-ahead forecasts | |
The -step-ahead forecast for TD | |
Reconciled forecasts | |
Shrinkage estimator |
Appendix A
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Subsystem/Source | Wind | Hydro | Thermal | Solar | Nuclear | Total (GWh—Subsystem) | % | |
---|---|---|---|---|---|---|---|---|
North | (A) | 2688 | 125,182 | 31,489 | 0 | 0 | 159,359 | 14.3% |
Northeast | (B) | 85,377 | 37,705 | 36,699 | 4626 | 0 | 164,407 | 14.7% |
Southeast/Midwest | (C) | 0 | 518,714 | 73,555 | 2437 | 31,805 | 626,511 | 56.1% |
South | (D) | 11,326 | 135,914 | 19,472 | 0 | 0 | 166,712 | 14.9% |
Total (GWh—Source) | 99,391 | 817,516 | 161,215 | 7063 | 31,805 | 1,116,989 | 100% | |
% | 8.9% | 73.2% | 14.4% | 0.6% | 2.8% | 100% | - |
TD Gross-Sohl Method A TDGSA | TD Gross-Sohl Method F TDGSF | TD Forecast Proportions TDFP |
---|---|---|
for . Each proportion reflects the average of the historical proportions of the bottom-level series , t over the period relative to the total aggregate . | for . Each proportion takes the average historical value of the bottom-level series related to the average value of the total aggregate . | where , is the -step-ahead forecast and is the sum of the -step-ahead forecasts below the node that is levels above node . |
Procedure | Description |
---|---|
OLS | where . This is the most simplifying premise, and collapses the MinT estimator to the OLS estimator, proposed by Hyndman et al. [19]. This is optimal when the base forecast errors are uncorrelated and equivariant. |
WLSv | where and: , is the unbiased sample covariance estimator of the in-sample one-step-ahead base forecast errors. In this case, we can describe MinT as a WLS estimator applying variance scaling [27]. |
WLSs | where and with being a unit column vector of dimension . We assume that each of the bottom-level base forecast errors has a variance and is uncorrelated between nodes. Consequently, every element of the diagonal matrix receives the number of forecast error variances contributing to that aggregation level [27]. This estimator depends only on the grouping structure of the hierarchy. |
MinT (Sample) | where , the unrestricted sample covariance estimator for [27]. In the results section, we denote this as MinT (Sample). |
MinT (Shrink) | ; , is a shrinkage estimator with diagonal target, , which is a diagonal matrix comprising the diagonal entries of , and is the shrinkage intensity parameter. Thus, off-diagonal elements of are shrunk toward zero and diagonal elements (variances) remain unchanged [27]. Wickramasuriya, Athanasopoulos and Hyndman [27] suggested a scale and location invariant shrinkage estimator by parameterizing the shrinkage in terms of variances and correlations: , where is the th element of , the -step-ahead sample correlation matrix to shrink it toward an identity matrix. |
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Silveira Gontijo, T.; Azevedo Costa, M. Forecasting Hierarchical Time Series in Power Generation. Energies 2020, 13, 3722. https://doi.org/10.3390/en13143722
Silveira Gontijo T, Azevedo Costa M. Forecasting Hierarchical Time Series in Power Generation. Energies. 2020; 13(14):3722. https://doi.org/10.3390/en13143722
Chicago/Turabian StyleSilveira Gontijo, Tiago, and Marcelo Azevedo Costa. 2020. "Forecasting Hierarchical Time Series in Power Generation" Energies 13, no. 14: 3722. https://doi.org/10.3390/en13143722