=Paper=
{{Paper
|id=Vol-3741/paper30
|storemode=property
|title=Nowcasting of the energy production of wind power plants through spatially-aware model trees
|pdfUrl=https://ceur-ws.org/Vol-3741/paper30.pdf
|volume=Vol-3741
|authors=Annunziata D’Aversa,Gianvito Pio
|dblpUrl=https://dblp.org/rec/conf/sebd/DAversaP24
}}
==Nowcasting of the energy production of wind power plants through spatially-aware model trees==
https://ceur-ws.org/Vol-3741/paper30.pdf
Nowcasting of the energy production of wind power
plants through spatially-aware model trees
(Discussion Paper)
Annunziata D’Aversa1,2,* , Gianvito Pio1,2
1
Dept. of Computer Science, University of Bari "Aldo Moro", Via E. Orabona, 4, 70125 Bari, Italy
2
Data Science Lab, National Interuniversity Consortium for Informatics (CINI), Via Volturno, 58, 00185 Roma, Italy
Abstract
The accurate prediction of the energy production from renewable power plants in short-term intervals is
of paramount importance in smart grids, to ensure an efficient distribution of energy within the network.
Existing predictive approaches are mainly based on autoregressive models, machine learning methods
and, more recently, on neural network architectures that also exploit spatio-temporal information.
However, most of them are not able to capture spatial information at different degrees of locality, and
tend to impose the presence of linear (or non-linear) dependencies among data.
In this paper, we discuss a novel approach that is based on linear model trees, to simultaneously
model linear and non-linear dependencies, properly extended to capture the spatial dimension at different
degrees of locality. The proposed approach is able to work in the multi-step predictive setting, that
means that it can simultaneously provide predictions for multiple time intervals in the future.
Our experiments on a real dataset about the energy produced by wind power plants demonstrate the
effectiveness of our method also in comparison with state-of-the-art neural network architectures.
Keywords
Time series nowcasting, Spatio-temporal autocorrelation, Multi-step prediction
1. Introduction
Smart grids are networks that distribute electricity with the support of sensors, advanced
communication technologies, and predictive components. Within the latter, models able to
forecast the energy consumption and production play a fundamental role. Indeed, in long-
term scenarios, they can support planning interventions on the network, aiming not only to
decrease production costs but also to contribute to the reduction of greenhouse gas emissions.
On the other hand, in short-term scenarios, the forecasting (usually called nowcasting, in the
case of very short-term timeframes) of energy production and consumption can be useful for
performing real-time load balancing actions, that may include powering on backup plants or
drawing energy from customers’ accumulators.
In general, predictive models can be built by relying on machine learning methods by exploit-
ing historical data and spatial information of nodes. Indeed, the spatial dimension may introduce
SEBD 2024: 32nd Symposium on Advanced Database Systems, June 23-26, 2024, Villasimius, Sardinia, Italy
*
Corresponding author.
$ annunziata.daversa@uniba.it (A. D’Aversa); gianvito.pio@uniba.it (G. Pio)
0000-0003-1791-5998 (A. D’Aversa); 0000-0003-2520-3616 (G. Pio)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
spatial autocorrelation phenomena, which refer to dependencies that may exist among obser-
vations at nearby geographical locations. In this context, the spatial proximity among power
plants or among customers can influence measurements due to similar climatic conditions.
Another important aspect is that real-world time-series coming from sensor measurements
often exhibit a combination of linear and non-linear trends. This is very common when mea-
surements depend on weather conditions, which may easily show non-linear phenomena, e.g.,
possibly due to storms or other extreme events. Non-linear phenomena may also emerge in the
case of power grid failures. Therefore, capturing both linear and non-linear trends and relation-
ships, along with the exploitation of historical data and spatial information, could improve the
model performance and lead to more accurate predictions.
In the literature, several nowcasting approaches have been proposed leveraging on autore-
gressive models[1, 2], machine learning models [3, 4] and hybrid models [5]. However, in
the literature, we can find only a few works that also take into account the spatial dimension
[6, 7, 8, 9]. For instance, in [6] the authors propose a method for 5-minute ahead wind power
forecasting. The authors capture spatio-temporal dependencies using a method based on sparse
parametrization of VAR models, which selects coefficients that link sites with a spatial co-
dependence, discarding those exhibiting weak dependencies. Another relevant example is [8],
where the authors proposed a spatio-temporal graph convolution neural network for the short-
term prediction of the energy produced by wind power plants. The authors consider a multi-step
setting, where 16 future values (at a 15-minutes interval) are predicted simultaneously.
The contribution of the temporal and spatial dimensions has also been considered in the
context of the more classical forecasting scenarios to predict the hourly energy production of
photovoltaic power plants 24 hours ahead [10], or to predict the monthly energy consumption
of customers one year ahead [11]. Also these works consider a multi-step setting, where the 24
hourly predictions (in [10]) and the 12 monthly predictions (in [11]) are returned simultaneously
by the model, possibly exploiting dependencies among them. The spatial dimension is considered
by resorting to two well known techniques in spatial statistics: the Local Indicator of Spatial
Association (LISA), that represents a local measure of spatial autocorrelation [12], and the
Principal Coordinates of Neighbour Matrices (PCNM), that represent the spatial structure in
the data [13]. Such indicators are used to augment the feature space of training instances.
Recently, several neural network architectures that consider both temporal and spatial dimen-
sions have been proposed, but they were applied in different application domains. A relevant
example is MTGNN [14], that is a graph convolutional network applied to multiple domains, in-
cluding energy and traffic speed forecasting. MTGNN employs multiple temporal convolutional
networks (TCNs) with various kernel sizes, for learning temporal dependencies at different
scales, and a self-adaptive adjacency matrix to capture spatial correlations.
It is noteworthy that, although some of the mentioned approaches are able to represent and
exploit the spatial information, they cannot capture spatial dependencies at different degrees
of locality. A first attempt to capture local spatial information can be found in [15], where the
authors proposed the method D2 STGNN applied to the traffic speed forecasting. D2 STGNN
identifies both diffusion signals, representing how traffic conditions spread through the network,
and inherent patterns, such as recurring traffic patterns or daily / seasonal variations. The model
adopts a spatio-temporal localized convolution to capture hidden diffusion time series, while a
combination of GRU (for short-term dependencies) and multi-head self-attention mechanism
Figure 1: An example showing the difference between a classical regression tree and a linear model tree.
(for long-term dependencies) is employed to model hidden inherent time series.
In this paper, we discuss an approach to solve nowcasting tasks in the context of the prediction
of the energy produced by wind power plants, in a multi-step setting. Specifically, we aim at
learning a nowcasting model capable of predicting the energy production for 12 time-steps, at a
15-minutes granularity. Methodologically, contrary to most existing approaches, we capture
both linear and non-linear phenomena through linear model trees. Moreover, we extend them
to effectively capture and model the spatial information at different levels of locality.
2. Spatially-aware linear model trees
As introduced in Section 1, we aim at adopting an approach that is able to capture both linear and
non-linear dependencies. In this respect, we argue that linear model trees [16] can represent a
possible solution, since they combine the ability to model non-linear dependencies of regression
trees with that of linear models. Existing methods for the construction of model trees employ a
learning process characterized by a top-down induction procedure that recursively partitions
the training set, which is analogous to that adopted by conventional tree-based algorithms.
In linear model trees, in the leaf nodes we find linear models instead of constant approxima-
tions of classical regression trees. More formally, given a set of independent variables and a
dependent variable 𝑦, a standard regression tree returns, for each leaf node 𝑘, a constant value
𝑐𝑘 , namely, 𝑦 = 𝑐𝑘 for all the instances falling in the leaf node 𝑘. Such constant value is usually
an aggregation (mean, median, etc.) of the value 𝑦 of the training instances falling in the leaf
node. On the other hand, in model trees, each leaf node of the tree contains a linear regression
model that predicts the target variable based on the data points that reach that leaf. An example
to illustrate the difference between a regression tree and a linear model tree is shown in Fig. 1.
The quality of a split is usually measured using a criterion that quantifies how well the split
separates data with respect to the target variable. For example, in CART [17], the quality of a
split is evaluated by the Mean Squared Error (MSE). When a node is split, the MSE is computed
for each resulting child node, and the weighted sum (according to the number of instances) of
these MSE values represents the quality of the split. The best split is defined as the one that
minimizes the MSE. In the case of linear model trees, the behavior is similar: the only difference
is that the MSE on the child nodes is computed after fitting a linear model on them.
In our approach, we considered the multi-step (MS) setting proposed in [11] that consists
in predicting multiple future values of the target variable simultaneously. In particular, our
approach falls in the Multi-Input Multi-Output (MIMO) category [18], which goal is learning a
global predictive model that returns the whole vector of predictions, also taking into account
the possible dependencies between future values, that in principle may be beneficial in terms of
forecasting accuracy. More formally, we consider as input features 𝑤 historical values of the
target variable 𝑦𝑡−𝑤 , 𝑦𝑡−𝑤+1 , ..., 𝑦𝑡−1 , in order to predict the value of the target variable for ℎ
future timesteps 𝑦𝑡 , 𝑦𝑡+1 ..., 𝑦𝑡+ℎ , simultaneously. Note that, in this case, the reduction of the
MSE of a split is evaluated as the average reduction of MSE over all the future timesteps.
In the literature, we can find several implementations of linear model trees [16, 19, 20]. In
this work, we consider the simplest implementation, where internal nodes are simple tests
involving descriptive variables, while leaf nodes are linear models, as shown in the right part
of Fig. 1. This choice makes our extension towards the consideration of the spatial dimension
more straightforward. Specifically, as introduced in Section 1, we aim at extending linear model
trees to effectively capture and model the spatial dimensions at different levels of locality.
Methodologically, we introduce the consideration of the spatial dimension as a post-processing
step of the tree construction: we aim at capturing spatial relationships within each subset
implicitly defined by a leaf node of the model tree, potentially capturing spatial relationships at
different levels of locality. Considering to have multiple different positions (e.g., production
plants or consumers), each represented through several 𝑤-dimensional training instances, we
act as follows: for each instance 𝑥𝛼,𝑡 fallen into a leaf node 𝑙, related to the time step 𝑡 and to
the geographic position 𝛼, we compute a set of additional features 𝑆𝛼,𝑡,𝑙 . These features are
computed as the weighted average of the 𝑤-dimensional historical observations at the same
time step 𝑡 from other positions1 in 𝑙, where the weights are determined by the spatial closeness
between 𝛼 and the other positions (see Figure 2). More formally, 𝑆𝛼,𝑡,𝑙 is defined as follows:
1 ∑︁
𝑆𝛼,𝑡,𝑙 = ∑︀ · 𝐶[𝛼, 𝛽] · 𝑥𝛽,𝑡 (1)
𝛽∈𝑃𝑙 ,𝛽̸=𝛼 𝐶[𝛼, 𝛽] 𝛽∈𝑃𝑙 ,𝛽̸=𝛼
where 𝑃𝑙 is the set of distinct positions of the training instances fallen into the leaf node 𝑙; 𝑥𝛽,𝑡
is the vector of 𝑤 historical observations of the location 𝛽 at the time step 𝑡; 𝐶[𝛼, 𝛽] is the
spatial closeness between the positions 𝛼 and 𝛽 computed as follows:
𝐷[𝛼, 𝛽]
𝐶[𝛼, 𝛽] = 1 − (2)
𝑚𝑎𝑥(𝐷)
where 𝐷 is the distance matrix among locations computed according to the geodesic distance.
The additional features are computed and added to all the training instances falling into the leaf
node. Finally, a new linear model is trained and the contribution of the added features is assessed
using a validation set. Therefore, we compare two distinct linear models, as depicted in Figure
3. The first model is exclusively trained on the original features (during the construction of the
tree), while the second model incorporates both the original features and the additional ones
computed according to the spatial closeness. We selectively retain the model that demonstrates
1
Note that, if a given leaf node contains training instances associated with only one position, this step is skipped.
Figure 2: An example of computation of additional features 𝑆𝛼,𝑡,𝑙 for the instance 𝑥𝛼,𝑡 fallen in the leaf
node 𝑙, given the presence of other instances in 𝑙 belonging to the geographic positions 𝛽, 𝛾, and 𝜆.
Figure 3: Comparison of the predictive performance of two linear models on the training instances of a
leaf node: the first model is learned from the original features, while the second is learned from the
original features expanded with the features computed according to the spatial closeness.
the lowest validation error within each leaf node. This selection process ensures that we tailor
our modeling approach to the specific peculiarities of each subset of data falling into leaf nodes.
Consequently, within this tree, some leaf nodes may employ models that incorporate spatial
features, while others may rely only on the original features (i.e., when the additional features
based on spatial closeness appear to provide no advantage).
After performing this process on all the leaf nodes, we apply a pruning step to prevent
overfitting and possibly capture more global (i.e., less local) spatial dependencies. In particular,
we propose an extended version of the Reduced Error Pruning (REP) algorithm [21]: starting
from the bottom of the tree and working backward, for each internal node, it compares the
error made by the unpruned tree with that made simulating that the subtree rooted on the
node is pruned. The subtree is actually pruned only if the resulting tree performs no worse
than the unpruned one over the validation set. In our extended version, we also consider the
possible contribution coming from the features based on the spatial closeness. In particular, we
compare the unpruned tree with the pruned tree and with the pruned tree that also considers
the features based on the spatial closeness. Considering the example reported in Figure 4, given
Figure 4: Extended version of the Reduced Error Pruning strategy that also takes into account the
contribution of additional features that consider the spatial closeness.
the internal node 𝑛4 , we compare the errors made on the validation set by three models: i)
the model represented by its two children nodes 𝑙4 and 𝑙5 (see the left part of Figure 4); ii) the
model obtained after pruning the subtree rooted in 𝑛4 and learning a new linear model from the
instances falling into it (see the middle part of Figure 4); iii) the model obtained after pruning
the subtree rooted in 𝑛4 and learning a new linear model from the instances falling into it,
expanded with the features considering the spatial closeness (see the right part of Figure 4).
If the model ii) or the model iii) leads to an improvement on the validation set, the tree is pruned
accordingly. This process continues in a bottom-up fashion until no improvement is obtained.
3. Experiments
In order to assess the effectiveness of the proposed approach, we performed our experiments on a
real-world wind power plants dataset, provided by a lead company in the energy distribution field.
The dataset consists of measurements of the energy production of 60 wind plants, collected every
15 minutes for a period of 1 year. Together with the geographic position (latitude and longitude),
the plants are described by some technical characteristics, namely, avg_wind_turbine_height,
rotor_diameter, and number_of_wind_turbines.
Following a cross-validation setting for time series, we consider a sliding window approach
where the training set consists of 4 months of data, the validation set corresponds to the
last month of the training set, and the test set is the subsequent month. We performed the
experiments considering a multi-step setting, where the goal is to predict the energy production
of 12 target time-steps ahead simultaneously. As historical measurements associated with each
instance, we consider 12 previous values of energy production, i.e., 𝑤 = 12. It is noteworthy
that, in real-world production scenarios, actual measurements are often made available after a
certain amount of time. Therefore, we evaluated the performance of all the models considering
different delays from the last observed measurement and the first target time-step to predict.
The considered delays are 0 hours, 2 hours and 4 hours.
To learn the initial model tree, we considered the implementation available in the linear-tree
python library2 . For all the experiments, we investigated two different configurations of its
parameters, namely: min_samples_leaf = 0.1, max_depth = 5 and min_samples_leaf = 0.05,
max_depth = 20. The original version of this system (henceforth denoted with LT), that ignores
the spatial information, has been considered as the closest competitor to our approach. As
additional competitor systems, we considered three different regressors that are able to work in
the multi-step setting, namely, Linear Regression (henceforth denoted with LR), Random Forests
(henceforth denoted with RF) and XGBoost Regressor (henceforth denoted with XGB). For all
these competitors, we also assessed the performance achieved when the spatial information
is considered by injecting PCNM variables [13]. This allows us to specifically evaluate the
contribution of the novel strategy that we proposed to model the spatial dimension. Finally, we
considered two state-of-the-art neural network architectures that can work in the multi-step
setting and capture spatio-temporal phenomena, i.e., MTGNN [14] and D2 STGNN [15].
As evaluation measure, we collected the Relative Squared Error (RSE) for LT, and the per-
centage of improvement with respect to the best configuration of such a model for the pro-
posed method and for all the considered competitor systems. The RSE is formally defined as
𝑦 )2
∑︀
(𝑦 −̃︀
𝑅𝑆𝐸 = ∑︀ (𝑦𝑡 −𝑦)𝑡 2 , where 𝑦𝑡 and 𝑦̃︀𝑡 are the true and the predicted values, respectively, for the
𝑡 𝑡
𝑡
𝑡-th time-step, while 𝑦 is the average value of a given target time-step in the training set.
The adoption of the RSE, instead of more commonly adopted measures like MAE/MSE/RMSE,
allows us to evaluate the actual usefulness of the predictive models in real scenarios, with
respect to adopting a baseline predictor that always returns the mean of the measurements: an
RSE value close to 0.0 means that the model returns perfect predictions; an RSE value close to
1.0 corresponds to a model that performs analogously to the baseline that always returns the
mean; an RSE value higher than 1.0 means that the model performs worse than such a baseline.
In Table 1, we report the RSE results, averaged over all the target time-steps and over all
the folds of the cross-validation. As expected, all the considered methods perform worse with
higher delays. Nevertheless, all the RSE values remains under 1.0, which means that they can
still provide more useful indications than those provided by the baseline predictor based on the
average. Looking at the results obtained by our approach, it clearly provides advantages over
LT with all the values of delay and in both configurations of its parameters. On the contrary, all
the other competitors perform worse than (or equal to) LT, except for few specific cases, where
the improvement is no more than 0.6%. These results confirm the adequacy of adopting model
trees in this application domain, due to the co-presence of linear and non-linear phenomena.
Looking at the contribution provided by the PCNM variables to the competitors, we can
observe no evident differences with respect to the same methods with no PCNM features, with
some peculiar cases in which the error also increases (see, for example, RF+PCNM vs RF). This
is possibly due to the fact that PCNM variables do not take historical factors into account. On
the other hand, our approach incorporates additional historical features, taking into account
the spatial closeness at different degrees of locality. This approach clearly performs better than
injecting static features dependent on the positions as seen by the approaches relying on PCNM.
In general, we can observe that our approach outperforms all the considered competitors,
including those based on recent neural network architectures. Surprisingly, they obtained the
2
https://github.com/cerlymarco/linear-tree
Model min_samples_leaf max_depth 0 hours delay 2 hours delay 4 hours delay
LT 0.1 5 0.261 0.499 0.645
RSE
LT 0.05 20 0.260 0.498 0.644
LT+PCNM 0.1 5 0.00% 0.60% 0.30%
% of Improvement
LT+PCNM 0.05 20 -2.70% -9.00% -4.00%
RF default default -3.40% -3.60% -4.70%
RF+PCNM default default -3.40% -4.00% -4.80%
XGB default default 0.00% 0.20% 0.00%
XGB+PCNM default default -0.40% 0.20% 0.00%
LR - - -0.80% -0.20% 0.00%
LR+PCNM - - -0.80% -0.20% 0.20%
MTGNN - - -181.89% -63.73% -35.19%
D2 STGNN - - -123.92% -60.26% -30.96%
Our approach 0.1 5 6.10% 4.40% 2.30%
Our approach 0.05 20 6.50% 5.20% 2.80%
Table 1
Average RSE obtained by LT and percentage of improvement with respect to such a method (with
min_samples_leaf = 0.1 and max_depth = 5), obtained by our approach and by the competitor systems.
Positive and negative percentages are emphasized with green and orange backgrounds, respectively.
worst results among the considered systems. This is possibly due to the complexity of their
architecture that requires a huge amount of training data (possibly much higher than those
available in this context) to properly learn an accurate model.
4. Conclusion
In this paper, we presented an approach for nowcasting the energy produced by wind power
plants in a multi-step predictive setting. We enabled linear model trees to capture spatial
phenomena at different degrees of locality. Specifically, we incorporate additional features that
represent historical observations of other plants, taking into account their spatial closeness.
Moreover, we also extended the REP pruning strategy to consider the spatial dimension.
Our experiments, performed on a real-world dataset, proved the effectiveness of the proposed
approach, in comparison with standard linear trees and other state-of-the-art competitors that
are also able to model the spatial dimension.
For future work, we intend to evaluate the effectiveness of the proposed method in other
domains, and to perform a deep evaluation of the difference in terms of (theoretical and empirical)
model complexity with respect to unpruned linear trees and complex neural networks.
Acknowledgments
This work was partially supported by the project FAIR - Future AI Research (PE00000013),
Spoke 6 - Symbiotic AI, under the NRRP MUR program funded by the NextGenerationEU. The
research of Annunziata D’Aversa is funded by a PhD fellowship within the framework of the
Italian "POR Puglia FSE 2014-2020" – Axis X - Action 10.4 "Interventions to promote research
and for university education - PhD Project n. 1004.121 (CUP n. H99J21006620008).
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