On autoregressive deep learning models for day-ahead wind power forecasts with irregular shutdowns due to redispatching

3.1.1 DeepAR

The forecasting method [7] leverages recurrent LSTM layers [45] to capture temporal dependencies in the data. Recurrent layers allow – unlike uni-directional feed-forward layers – the output from neurons to affect the subsequent input to the same neuron. Such a neuron maintains a hidden state that captures information about the past observations in the time series, i.e., the output depends on the prior elements within the input sequence. Consequently, DeepAR is autoregressive in the sense that it receives the last observation as input, and the previous output of the network is fed back as input for the next time step.

3.1.2 N-HiTS

The forecasting method [8] is based on uni-directional feed-forward layers with ReLU activation functions. The layers are structured as blocks, connected using the double residual stacking principle. Each block contains two residual branches, one makes a backcast and the other makes a forecast. The backcast is subtracted from the input sequence before entering the next block, effectively removing the portion of the signal that is already approximated well by the previous block. This approach simplifies the forecasting task for downstream blocks. The final forecast is produced by hierarchically aggregating the forecasts from all blocks.

3.1.3 TFT

The forecasting method [9] combines recurrent layers with the transformer architecture [46], which is built on attention units. An attention unit computes importance scores for each element in the input sequence, allowing the model to focus on those elements in a sequence that significantly influence the forecast. While the attention units are used to capture long-term dependencies, short-term dependencies are captured using recurrent layers based on the LSTM.

3.2.1 Wind power curve

A WP curve y k = f v eff k describes the turbine’s power generation y k as a function of the wind speed v eff k at a reference height above ground level. The curve is separated into four sections by the cut/in, nominal, and cut/out wind speed, see Figure 1a. The reference height of WP curves provided by turbine OEMs is commonly the hub height. Using wind speed forecasts as inputs for OEM WP curves necessitates a height correction,^[4] as wind speed forecasts are commonly available at 10 m and 100 m above ground level. Differences in wind speed at different heights arise from the interaction between the wind and the earth’s surface (so/called terrain roughness) and the wind (so/called atmospheric boundary layer) [47]. A common and straightforward modeling approach for this phenomenon is the wind power law [48]

where the empirically/determined exponent α _h depends on the terrain [49] and is used to establish the relation between the wind speeds v _a and v _b at the heights h _a and h _b above ground level. The impact of different values of α _h is exemplified in Figure 1b. For the height correction of the wind speed forecast based on (2), we perform two steps [50]: First, we take α _h = 1/9 for offshore and α _h = 1/7 for onshore WP turbines [51], and estimate the turbine’s effective hub height

with the averages of the wind speed forecast at 100 m v ̂ ̄ 100 and the wind speed measurement at hub height v ̄ eff on the training data sub/set. Second, each value of the wind speed forecast v ̂ 100 [ k ] can now be corrected to hub height using

Figure 1:

Using the WP curve to make forecasts requires the wind speed forecast to be corrected to the curve’s reference height [10]. (a) The WP curve is the power generation as a function of the wind speed at a reference height and consists of four sections separated by the cut/in, nominal, and cut/out wind speed. (b) Height correction using the wind profile power law (2) with the known wind speed v _b at height h _b above ground level outlined for different values of the exponent α _h.

The advantage of this two/step approach is that knowledge about the actual hub height is not required and extrapolations from the 100 m reference are near the hub height of today’s WP turbines with a typical hub height between 80 m and 140 m above ground level [48]. Finally, the height-corrected wind speed forecast v ̂ eff [ k ] is used as the WP curve’s input to forecast the turbine’s power generation:

3.2.2 AutoWP

AutoWP is based on the underlying idea of representing a new WP turbine by the optimally weighted sum of WP curves a sufficiently diverse ensemble.^[5] The method includes three steps [10]: (i) creation of the ensemble with normalized WP curves, (ii) computation of the normalized ensemble WP curve using the optimally weighted sum of considered normalized WP curves, and (iii) re/scaling of the ensemble WP curve based on the peak power rating of the new WP turbine. These three steps are exemplified in Figure 2 and detailed in the following.

Figure 2:

The three steps of AutoWP’s automated design exemplified for the real/world WP turbine no. 1. The figure is based on [10]. (a) The first step creates the ensemble pool using N _m = 10 normalized WP curves y ̂ * , n ∈ N 1 N m of the OEP wind turbine library [52]. The selection reduces redundancy and preserves diversity. (b) The seconds step computes the normalized ensemble WP curve y ̂ * as convex linear combination of the pool’s curves, with the weights w ̂ n , n ∈ N 1 N m adapted to optimally fit the new WP turbine. (c) The third step re/scales the ensemble WP curve y ̂ to the peak power rating P _max,new = 1,500 kWp of the new WP turbine. Additionally, the measurement y used to fit the ensemble weights is shown.

In the first step, the ensemble is created using OEM WP curves from the wind turbine library of the OEP [52]. This process involves loading the database, selecting all entries that include a WP curve, resampling the {P _n , v _eff} value pairs to ensure a uniform wind speed sampling rate, and normalizing the power values P _n [v _eff] based on the peak power rating of each WP curve P _max,n :

Such a normalized WP curve outputs y ̂ * [ k ] = f v eff [ k ] as a function of the wind speed v eff [ k ] , k ∈ N K .^[6] Limiting the size of the ensemble without losing diversity is achieved by sorting all WP curves according to their sum of normalized power values, keeping the first and last WP curve, together with eight WP curves in between at equally spaced intervals, i.e., N _m = 10.

In the second step, we compute the normalized ensemble WP curve using the weighted sum of the considered WP curves

which is a convex linear combination with w ̂ n , n ∈ N 1 N m being the weight and y ̂ n being the output of the n-th curve in the ensemble. In order to determine the optimal weights, the target time series y is normalized analogously to (6)

and the optimization problem

is solved with the least squares algorithm of the Python package SciPy [54], and the weights being normalized to hold the constraints.^[7]

In the third step, the ensemble output (7) is re/scaled with the new WP turbine’s peak power rating P _max,new:

3.2.3 Machine learning for wind power curve modeling

ML methods can be used to directly learn the relationship between the wind speed forecast at 100 m and the WP turbine’s power generation, eliminating the need for a height/correction of the wind speed forecast like OEM WP curves. Given the non/linear relationship between wind speed and power generation (see Figure 1a), only non/linear ML methods are considered in the following. Established regression methods that perform well in a variety of tasks include DT/based methods like XGB, SVR, and the MLP. For considering WP curve modeling as a regression problem

the explanatory variables X ̂ are the same as for the autoregressive DL forecasting methods, namely exogenous forecasts of air temperature and atmospheric pressure, as well as wind speed and direction, and the model’s parameters p are determined by training. That is, the regression model f ⋅ uses the exogenous forecasts X ̂ k to estimate the WP generation y ̂ k . Unlike autoregressive forecasting methods, the regression model (11) is static, meaning it does not consider past WP generation values when making a forecast.

4.1.1 Data

For the evaluation, we use a real/world data set consisting of two years (2019, 2020) quarter/hourly energy metering (kWh) from two WP turbines; i.e., the time resolution is 15 min.^[11] To increase interpretability, we transform the energy metering into the mean power generation time series

with the energy generation ΔE[k] (kWh) metered within the sample period t _k (h). Additionally, wind speed measurements at turbines’ hub heights are available (15-min average). Both WP turbines have a peak power rating of 1,500 kWp, are located in southern Germany, and are subject to many shutdowns, see Figures 5 and 6, which are not explicitly labeled by the data provider.^[12]

Figure 5:

Identified WP turbine shutdowns with rule/based filtering [10]. Black fields represent data points at a measured wind speed greater than the cut/in speed with a power generation lower than the cut/in power. The shutdowns for WP turbine no. 1 amount to 49 % and 20 % for no. 2.

Figure 6:

Data pre/processing to identify abnormal operational states [10]. Rule/based filtering identifies turbine shutdowns and LOF/based filtering identifies turbine stop/to/operation transitions and vice versa. The samples for which the wind speed at hub height is below the cut/in speed v _cut-in = 2.5 m s⁻¹ amount to 16.7 %, while the cut/out speed v _cut-out = 25 m s⁻¹ at which the WP turbine would be shut down for safety reasons is not reached.

To perform day/ahead WP forecasting using the methods described in Section 3, we use day/ahead weather forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF) [55]. Importantly, shutdown handling in the autoregressive models’ past horizon H ₁ is performed with the weather measurements available up to the forecast origin.

We split the data into a training (2019) and a test data sub/set (2020). Additionally, 20 % of the training data sub/set is hold/out for the forecasting methods DeepAR, N-HiTS, and TFT to terminate training when the loss on the hold/out data increases (early stopping).

4.1.2 Evaluation strategy

The forecasting error is evaluated under two scenarios. The first scenario concerns day/ahead WP forecasting including future shutdowns. It evaluates whether autoregressive forecasting methods can forecast both WP generation and future shutdowns. The second concerns day/ahead WP forecasts to be used for communicating the WP turbine’s availability for redispatch planning. The forecast of future shutdowns must be excluded since times of planned shutdowns are communicated via non/availability notifications. The metrics used for assessment are the normalized Mean Absolute Error (nMAE)

and the normalized Root Mean Squared Error (nRMSE)

with the forecast y ̂ [ k ] and the realized value y[k] at time point k.

For all methods, the default hyperparameter configuration of the respective implementation is used. Additionally, methods employing stochastic training algorithm (DeepAR, N-HiTS, TFT, and MLP) are run five times.

4.2.1 Shutdown handling for methods based on autoregression

Table 1 shows the impact of different shutdown handling methods on the forecasting error of state/of/the/art autoregressive DL methods (DeepAR, N-HiTS, and TFT) when considering shutdowns. With regard to WP turbine no. 1, the shutdown handling methods imputation and drop/imputation result in significantly higher forecasting errors than the others. The reason is that numerous shutdowns in the data set (see Figure 5) are imputed with OEM WP curve values (forecasting model’s past horizon), resulting in forecasts of WP generation where the turbine is actually shut down. For WP turbine no. 2, showing regular and time/dependent shutdowns (see Figure 5), considering additional explanatory variables only improves over none shutdown handling for N-HiTS, and results in particularly high errors for DeepAR. For both WP turbines, shutdown handling methods highly impact the forecasting error, but no method consistently performs best across DeepAR, N-HiTS, and TFT.

Table 2 shows the impact of shutdown handling methods on the forecasting error of the state/of/the/art autoregressive DL methods (DeepAR, N-HiTS, and TFT) when disregarding shutdowns. With regard to WP turbine no. 1, the shutdown handling method drop results in the lowest forecasting error for DeepAR and TFT, while none shutdown handling performs best for N-HiTS. With regard to WP turbine no. 2, the shutdown handling method imputation achieves the lowest forecasting error for DeepAR and TFT, while drop/imputation achieves the lowest errors for N-HiTS. Interestingly, shutdown imputation in the past horizon with OEM WP curve values only lead to improvement for WP turbine no. 2. This is because the OEM WP curve’s forecasting error is comparably high for WP turbine no. 1, as evident in Table 3.

4.2.2 Comparison of autoregressive and WP curve-based methods

Table 3 compares the forecasting error of the autoregressive DL methods DeepAR, N-HiTS and TFT (using the respective best/performing shutdown handling method) with WP curve modeling-based forecasting methods. Notably, even with the optimal shutdown handling method, the autoregressive DL methods underperform compared to the methods based on WP curve modeling. Within the WP curve modeling methods, AutoWP, MLP, and SVR achieve similar forecasting errors. While AutoWP achieves the lowest nMAE for both turbines, the SVR achieves the lowest nRMSE for WP turbine no. 1, and the MLP for no. 2, see Table 3.^[13]