The bootstrap resampling method is applied to an ensemble artificial neural network (ANN) approach (which combines machine learning with physical data obtained from a numerical weather prediction model) to provide a multi-ANN model super-ensemble for application to multi-step-ahead forecasting of wind speed and of the associated power generated from a wind turbine. A statistical combination of the individual forecasts from the various ANNs of the super-ensemble is used to construct the best deterministic forecast, as well as the prediction uncertainty interval associated with this forecast. The bootstrapped neural-network methodology is validated using measured wind speed and power data acquired from a wind turbine in an operational wind farm located in northern China.
There has been an increasing emphasis towards a greater use of renewable energy (e.g., solar, wind, geothermal) as a strategy to reduce greenhouse gas emissions and to mitigate climate change. In this context, one of the fastest growing sources of renewable energy for the generation of “green electricity” is the power obtained from wind turbines. The ever increasing use of wind power poses new challenges. One important challenge is how to accommodate the unpredictable fluctuations in wind speed and direction which lead to variability and uncer- tainty in the wind power generation. The latter has significant implications for unit commitment and deter- mination of scheduling and dispatch decisions (economic dispatch) needed for the optimal utilization of wind energy within a mixed power system. In this regard, wind power forecasting has become a critical component in the efficient management of a green electrical power system (required by generation companies and utilities) and in electrical market operations (required by energy market analysts and traders).
The development of wind power forecasting models for improving the efficiency and reliability of mixed electrical power systems and for supporting electrical market operations has been reviewed by Costa et al. [
The second general class of models is physically-based models for wind speed prediction based on numerical weather prediction (NWP) or computational fluid dynamics (CFD). Utilizing equations of physics such as the conservation principles of mass, momentum, and energy in conjunction with various parameterizations for sub- grid scale physical processes that cannot be resolved explicitly by the necessarily finite number of grid points that are used to represent the atmospheric flow, NWP and/or CFD models provide hydrodynamic and thermo- dynamic models of the atmosphere that can be used to furnish a prediction of the flow field in a prescribed region. The prediction of the wind velocity field can be used in conjunction with the power curve for a wind tur- bine to provide a generated power forecast. Numerical weather prediction models have a number of limitations, including limited spatial resolution resulting in a coarse representation of the local terrain [
The third general class of models for wind power forecasting is based on machine learning approaches such as artificial neural networks, fuzzy systems, and support vector machines [
In this paper, we propose to use the bootstrap resampling method in conjunction with an ensemble artificial neural network (ANN) approach for the multi-step-ahead forecasting of wind speed and generated power. The artificial neural network combines machine learning with physical modeling by using NWP wind speed data from a physical model as the exogenous input to the network. The purpose of the bootstrap resampling method is to reduce the bias in prediction of the wind speed and power and to obtain more accurate estimates for the standard deviation (uncertainty) of these predictions. More importantly, the confidence bands in these pre- dictions can be determined, which can be used to provide a more rigorous uncertainty assessment in wind speed and power forecasting.
As discussed in the previous section, a major concern of wind energy management is the uncertainty quan- tification of multi-step-ahead predictions of the wind speed (at the turbine hub height) and the corresponding power generated by the turbine. Instead of choosing a single best ANN for forecasting, we propose instead to use the bootstrap resampling method in the context of an ensemble of ANNs for predictive uncertainty analysis.
Let
integer
We want to first represent (model) the functional relationship between
The nonlinear parameterized mapping
The output of the ANN is a continuous function of the input and
for the hidden layer and
for the output layer. In Equations (2) and (3), the index
Note that each neuron in the network is a unit that combines and processes all the data coming into the layer and then passes the transformed data (output of the activation function) to all the neurons of the successive layer. Specifically, the input of a neuron is a weighted sum of the outputs of all the neurons in the previous layer plus a bias. The weights
and the mean absolute error (MAE)
The minimization of these error functions is achieved using the particle swarm optimization algorithm [
To apply the bootstrap resampling procedure [
For each of these bootstrap samples, we can train an ensemble of ANNs with the same network architecture, but with each member of the ensemble having different numbers of neurons in the hidden layer (recall that the number of neurons in the input and output layers are determined a priori by the dimension of the input and output vectors, respectively). To be more specific, assume that the number of neurons in the hidden layer of the network architecture varies from
The procedure for bootstrapping an ensemble of neural networks is summarized as follows:
1. Assign the nonparametric distribution
2. Draw a (nonparametric) bootstrap sample (with replacement) from the empirical distribution function
and train
3. Repeat Step 2
We use a two-stage weighted averaging method to provide the predictive uncertainty assessment for the wind speed and power. For each bootstrap sample, we calculate the predictions (forecasts) of the multi-step-ahead wind speed and power using the
For each bootstrap sample,
which we denote as
where
Euclidean norm and
Once the weights
denoted by
(in-sample and multi-step-ahead) are given by
where
For a fixed bootstrap sample, the standard deviation vector
step-ahead prediction of
obtained from the ANN ensemble at the fixed time index
where
to obtain an unbiased estimator of the standard deviation.
At this point, we have
this purpose, we define another weight vector
evaluated similarly by minimizing an objective function similar to Equation (8), except now the error matrix is
constructed from the residuals between
calculated by using Equation (9). Once the weight vector
and the corresponding bootstrapped standard deviations from
where
computed in accordance to Equation (12).
Confidence intervals for the forecast of the output variable
with
The two data sets that we analyse were collected from a specific wind turbine, referred to as WT24 hereafter, located in a wind farm in northern China. One of these data sets corresponds to the hourly-averaged wind speeds measured at the turbine hub height and the other corresponds to the associated hourly-averaged power generated by the turbine. The wind speed and generated power time series, consisting of 432 observations each, were measured over a period of 18 days. The measurements collected over the first 15 days (corresponding to 83% of the entire length) of the time series were used as the training set and the remaining 3 days were reserved for the forecast assessment and validation. In addition to these measured time series, wind speed data at turbine hub height obtained from a numerical weather prediction model was available and this information was used as the exogenous input for artificial neural network training. In particular, the modeled wind speed and direction data over the region occupied by the wind farm were obtained from a NWP model executed with a temporal resolution of 1 h on a computational grid with a 3-km spatial resolution centered on the location of the wind farm. We applied a simple bilinear interpolation (BI) on this coarse-resolution NWP wind speed data to obtain the wind speed at the location of the WT24 wind turbine, which was then subsequently used as an exogenous input for ANN training.
As described in the previous section, we bootstrapped (resampled with replacement) the training data set to generate N = 200 “phantom” (bootstrap) data sets. We used each of these bootstrap data sets to train three-layer ANNs with a variable number of nodes in the hidden layer ranging from 5 to 30 nodes inclusive (so,
Next, we consider the forecast performance for the wind power using the bootstrapped neural-network methodology. The forecast for the generated power is more complicated than that for the wind speed owing to
Criterion | Persistence | Best |
---|---|---|
RMSE | 2.3554 | 1.6394 |
MAE | 1.8693 | 1.2145 |
the fact that the wind power is censored from above. More specifically, wind turbines are designed so that when the wind speed exceeds a certain value (referred to as the rated output wind speed), a limit to the power generation is imposed implying that the power generated is censored from above. To account for this maximum limit in the wind power generation by a wind turbine, the bilinear interpolation of the NWP wind speeds to the location of WT24 were censored (to the rated output wind speed of the turbine) before they were used as the exogenous input for the neural network training. Furthermore, the measured wind power used in the training of the ANNs were already censored from above by the maximum limit for power generation by the turbine. Indeed, for the current example, if the modeled wind speeds exceeded 11.5 m·s–1 (rated output wind speed for WT24), then the generated wind power associated with this range of wind speeds was limited above to 1550 W (rated output power for WT24).
In this paper, we proposed a novel bootstrapped artificial neural-network approach for wind speed and generated power forecasting. The approach provides a multi-ANN model super-ensemble that can be used to provide a best deterministic forecasting for these quantities, as well as to provide a quantitative assessment of the related
Criterion | Persistence | Best |
---|---|---|
RMSE | 811.652 | 292.7335 |
MAE | 653.252 | 210.8805 |
prediction uncertainty. In this approach, the individual ANNs that comprise the super-ensemble are first trained using a data set of available wind speed and power measurements and wind speed predictions (obtained from a numerical weather prediction model). The training consists of fitting various artificial neural network architectures (model structures) against the observation and exogenous input to determine the optimal statistical weights for each model.
The advantage of this methodology is that the biases in the forecast can be reduced and good predictions (forecasts) can be obtained through a statistical combination of the individual forecasts from the super-ensemble to give the best deterministic forecast. Applications to a wind turbine in northern China show that our proposed method works quite well. Because the method also provides prediction uncertainty bounds in the forecasts, it is anticipated that this approach would be very useful for green electrical system power management. Indeed, with the rapid pace of increases in computational power, it will become easier in the near future for power system managers and energy system traders/analysts to take advantage of the super-ensemble approach, for providing optimal forecasts and quantitative assessments of the uncertainty associated with these forecasts (allowing this information to be used in a more accurate and reliable manner for various applications).