Bridging an energy system model with an ensemble deep-learning approach for electricity price forecasting

Abstract

This paper combines a techno-economic energy system model with an econometric model to maximise electricity price forecasting accuracy. The proposed combination model is tested on the German day-ahead wholesale electricity market. Our paper also benchmarks the results against several econometric alternatives. Lastly, we demonstrate the economic value of improved price estimators maximising the revenue from an electric storage resource. The results demonstrate that our integrated model improves overall forecasting accuracy by 18 %, compared to available literature benchmarks. Furthermore, our robustness checks reveal that a) the Ensemble Deep Neural Network model performs best in our dataset and b) adding output from the techno-economic energy systems model as econometric model input improves the performance of all econometric models. The empirical relevance of the forecast improvement is confirmed by the results of the exemplary storage optimisation, in which the integration of the techno-economic energy system model leads to a revenue increase of up to 10 %.

Full text

Bridging an energy system model with an
ensemble deep-learning approach for
electricity price forecasting
Souhir Ben Amor∗Thomas Möbius†Felix Müsgens‡
November 8, 2024
Abstract
This paper combines a techno-economic energy system model with an
econometric model to maximise electricity price forecasting accuracy.
The proposed combination model is tested on the German day-ahead
wholesale electricity market. Our paper also benchmarks the results
against several econometric alternatives. Lastly, we demonstrate the
economic value of improved price estimators maximising the revenue
from an electric storage resource.
The results demonstrate that our integrated model improves overall
forecasting accuracy by 18%, compared to available literature bench-
marks. Furthermore, our robustness checks reveal that a) the En-
semble Deep Neural Network model performs best in our dataset and
b) adding output from the techno-economic energy systems model as
econometric model input improves the performance of all econometric
models.
The empirical relevance of the forecast improvement is confirmed by
the results of the exemplary storage optimisation, in which the inte-
gration of the techno-economic energy system model leads to a revenue
increase of up to 10%.
∗Brandenburgische Technische Universität Cottbus-Senftenberg Chair of Energy Eco-
nomics, Email: benamor@b-tu.de
†Brandenburgische Technische Universität Cottbus-Senftenberg Chair of Energy Eco-
nomics, Email: Thomas.Moebius@b-tu.de
‡Brandenburgische Technische Universität Cottbus-Senftenberg Chair of Energy Eco-
nomics, Email: felix.muesgens@b-tu.de
1
arXiv:2411.04880v1 [econ.GN] 7 Nov 2024

Keywords: Electricity price forecasting, Econometric price forecasts, Techno-
economic models, Ensemble Deep Neural Network model, Energy trading,
Storage bidding strategy.
JEL classification: C52, C53, Q40, D44.
1 Introduction and Motivation
Producers and consumers in many electricity systems worldwide are trading
on liquid wholesale markets. Prices are vital in coordinating supply and de-
mand. Market participants need effective price forecasts to optimise their
business strategies.
The forecasting of electricity prices focuses on either the short-term, with
forecasting horizons in the order of hours or days, or the medium to long-
term, in months or years (Weron 2014). Fundamentally different types of
models are used, mostly oriented around these time frames. Short-term price
forecasting is predominantly approached with econometric methods, while
long-term price forecasting is predominantly achieved using techno-economic
models formulated as operations research-type optimisation problems. These
techno-economic bottom-up models are often referred to as ’energy systems
models’ (ESM). Prices in ESM are estimated from dual variables (typically
”shadow prices” of demand constraints, see e.g. Müsgens 2006).
In this paper, we focus on short-term electricity price forecasting (EPF).
Short-term EPF is challenging due to the complexity and the particular eco-
nomic and physical characteristics of the electricity market, which account
for the simultaneous occurrence of spikes, volatility, long memory, mean re-
version, negative prices, and regime switches. Numerous researchers have
devoted their time and expertise to these complexities and developed various
econometric approaches (see Nowotarski and Weron 2018 and Weron 2014
for an overview).
Although state-of-the-art econometric models incorporate explanatory vari-
ables such as electricity demand, fuel prices, and generation from renewable
energy sources, they analyse market prices statistically, i.e. without explic-
itly modelling the underlying techno-economic objectives and constraints on
the demand and supply sides, which in combination determine market prices.
In contrast, ESMs replicate price formation in electricity markets deciding
on an efficient market equilibrium, taking into account the physical and dy-
namic properties of the market on both the supply and demand sides. ESMs
play to their strengths in long-term forecasts but tend to be less effective
when dealing with the formation of high-frequency price patterns, which in
2

turn is a strength of econometric models. Nowadays, a small but growing
number of researchers are interested in hybrid methods that combine ESM
and econometric models for short-term price forecasting (e.g., Watermeyer
et al. 2023,De Marcos et al. 2019c and Maciejowska and Nowotarski 2016).
Despite these first attempts, there exists a research gap in understanding
whether results from a techno-economic energy system model add value to
short-term price forecasts from a state-of-the-art econometric model. Our
first objective in this paper is to close that gap. We combine an ESM and
an econometric model to benefit from the respective strengths of both model
types. For the proposed combination, both individual models should be
among the best in their respective classes to achieve meaningful practical
results. In this context, we select the em.power dispatch model, a techno-
economic dispatch ESM specifically tailored to provide day-ahead price fore-
casts in high temporal resolution. For the econometric model, we select the
state-of-the-art Ensemble Deep Neural Network (Ens-DNN) model.1The link
between the two models is the market clearing price (MCP). In the ESM,
the MCP is the model output calculated as the shadow price of the demand
constraint. In the econometric models, the MCP is an independent variable.
Adding the MCP as an independent variable in the Ens-DNN improves fore-
casting accuracy by nearly 20%, providing very accurate forecasts.
Our second objective is to assess the value of ESM input in econometric
models in general. We provide quantitative results with an analysis of sev-
eral other econometric models (ranging from simple to sophisticated). In
addition to enabling a more general statement on the use of ESM results
as input in econometric model forecasts, this also tests the relative forecast
quality of the Ens-DNN model. We find that adding the ESM’s MCP as an
independent variable in econometric models improves forecast quality for all
econometric models already performing reasonably well on their own. Hence,
this part of our work serves as a robustness check for the inclusion of ESM
results as an independent variable in econometric models.
Our third contribution is to quantify the impact of the number of indepen-
dent variables on forecasting accuracy and test the possibility of relying only
on the ESM’s MCP as a single regressor in econometric models. This also
answers the question of to what extent the MCP computed by an ESM ac-
commodates all the techno-economic variables usually taken into account in
econometric models to capture electricity price variation.
1The choice of the Ens-DNN model is motivated by the results of Lago, Marcjasz, et al.
2021, which demonstrate that the Ens-DNN model provides accurate forecasts. They also
suggest it to benchmark results.
3

Finally, we present a storage bidding strategy for the day-ahead market based
on the forecasting results from the proposed model and the benchmark mod-
els that allow us to measure, in monetary terms, the economic effects of error
prediction.
More accurate electricity price forecasts are relevant for all parties taking
open positions in power markets: access to better day-ahead price fore-
casts than the competition directly increases market revenues. Furthermore,
power generation companies and utilities benefit when making dispatch de-
cisions, for single units as well as for portfolios of power plants and sector
coupling technologies. On the demand side, our improved forecasts can be
used to schedule flexible consumption at low-price hours. These examples
suggest that forecasts constitute techno-economic inputs to energy company
decision-making. The high demand for improved EPF models is therefore
unquestionable.
The remainder of this paper is structured as follows: Section 2 summarises
the related research on EPF and discusses the adopted methodology and the
gaps left to be filled by our investigation. Section 3 outlines the theoretical
background of the ESM, the econometric models, and the combined model
and explains the storage optimisation strategy. Section 5 presents the case
studies in which the proposed forecasting method has been tested. Comple-
mentary to the overall results, in section 6, we discuss the implications of the
forecast results for the storage bidding strategy. Subsequently, section 7 con-
tains the conclusions drawn in this paper, incorporating recommendations
for market participants.
2 Related Literature
Forecasting in the electricity market has attracted academics, policymakers,
and practitioners since the 1990s. Researchers have focused on developing
reliable EPF models, and while statistical and econometric approaches are
generally the dominant approach, long-term contexts also require a more fun-
damental understanding of market behaviour and dynamics (Weron 2014).
Statistical models (Ben Amor et al. 2018a,Ben Amor et al. 2018b,Weron
and Misiorek 2008,Ziel 2016,Lago, De Ridder, et al. 2018, and Conejo et
al. 2005) as well as artificial intelligence (AI) and machine learning (ML)
methods (Abedinia et al. 2015,Pórtoles et al. 2018,Lago, Marcjasz, et al.
2021, and Jiang and Hu 2018) have been widely accepted due to their abil-
ity to capture linear and non-linear trends, respectively. There have been
studies that combine both approaches into a pure econometric hybrid model
(Nikodinoska et al. 2022,Chang et al. 2019) and Watermeyer et al. 2023.
4

In addition, several researchers emphasise the importance of incorporating
fundamental variables in econometric models when forecasting electricity
prices (Karakatsani and D. W. Bunn 2008a,Kristiansen 2012, and Gian-
freda et al. 2020).
Despite econometric models providing information about physical price drivers,
for instance, demand, CO2prices, fuel prices, renewable energy, etc. (Karakat-
sani and D. W. Bunn 2008b), these models analyse market prices statistically
and do not address the underlying demand and supply functions that drive
market prices. Additionally, these models are usually built on historical data,
which means that they can perform under the assumption that history will
repeat itself in the future. However, in today’s complex electricity markets,
this is not always the case. Hence, it is important to realise that although
statistical models may be able to incorporate operations and market dynam-
ics into their EPF, they often struggle to represent regulatory and market
structural changes.
To address these aspects, the ESM takes into consideration the various
technical features of generation units and their production costs. More pre-
cisely, it simulates price formation from the intersection of demand with the
physical and dynamic properties of the generators’ supply functions, reflect-
ing a more techno-economic "bottom-up" approach. As a result, the events
that are difficult to model using statistics or econometrics are reflected in
the estimated MCP of the ESM. Recently, the ESM has received much at-
tention when it comes to deriving electricity price estimators. For example,
several studies have been conducted in which competitive market prices have
been modelled to investigate market power events (Müsgens 2006,Borenstein
et al. 2002). Sensfuß et al. 2008 analyse the impact of renewable electricity
generation on spot market prices, the so-called merit-order effect. Hirth
2013 apply an ESM to derive the market value of variable renewable en-
ergy. Additionally, an advantage of ESMs is that they have the option to
incorporate market participants’ preferences, such as risk preferences (see
e.g., Möbius, Riepin, et al. 2023). However, despite being used widely
in research, industry, and politics, ESMs are not the ideal tool for short-
term price forecasting. This is because their performance when it comes to
capturing short-term price dynamics is poor (Bello, Reneses, et al. 2016),
while time-series econometric models are more accurate when it comes to
providing short-term forecasts (D. W. Bunn 2000). Hence, there is a grow-
ing interest in combining ESMs with econometric methodologies to address
their shortcomings and improve predictive performance. More precisely, this
combination allows us to include fundamental market mechanisms and the
most relevant economic drivers of electricity prices, such as demand, supply,
and technical constraints, into the econometric model. This is in addition to
5

the behavioural aspects (speculative and strategic behaviour) and statistical
price features (spikes (Christensen et al. 2012), high volatility, seasonality,
etc.) provided by the econometric approach. Therefore, the behaviour and
operation of the power market can be successfully incorporated into electric-
ity price forecasts, which is of significant interest to its participants.
Such a combination has produced favourable outcomes for both point and
probabilistic forecasting, particularly in medium-term scenarios (Watermeyer
et al. 2023,Bello Morales et al. 2017,Bello, D. Bunn, et al. 2016,Gonzalez
et al. 2011, and Karakatsani and D. W. Bunn 2008a).
Despite the importance of short-term price forecasting, short-term applica-
tions are very scarce and limited to the following studies: De Marcos et al.
2019a and De Marcos et al. 2019b propose a short-term hybrid EPF model,
which combines a cost-production optimisation model with an Artificial Neu-
ral Network model to forecast the Iberian electricity market. In the same
vein, Gonzalez et al. 2011 combines a techno-economic model, formulated
with supply stack modelling, with an econometric model (regime-switching
model) using data on price drivers to predict the APX power exchange for
Great Britain.
Considering the rarity of the published studies and the simplicity of the com-
parisons (limited to outdated methods and avoiding state-of-the-art econo-
metric methods), the results cannot be generalised. In the preceding para-
graphs, several deficiencies and scarcities have been highlighted in the context
of short-term EPF, which motivates our proposed methodology.
Price forecasting models that combine techno-economic optimisation mod-
els and econometric modelling approaches have shown promise over the medium
term. Their performance in the short term, however, is rarely tested (see Wa-
termeyer et al. 2023). Therefore, it would be interesting to determine whether
the same advantages can be achieved over the short term.
Moreover, another aspect of great importance is the way in which the techno-
economic and econometric models are combined. The most widely used pro-
cedure is to obtain the MCP from the ESM, which is later included in the
econometric model’s input dataset. In previous research, the selection of the
model components is neither analysed nor justified. In this regard, it is un-
clear to what extent the individual components are relevant or useful. As a
result, it is imperative to carefully choose each model component.
Aside from this, the models used are outdated, including artificial neural net-
works and autoregressive models with exogenous variables (ARX) that have
shown weaknesses in forecasting electricity prices (Lago, De Ridder, et al.
2018,Weron 2014).
Firstly, one of the neural network’s drawbacks is associated with the high
computational training cost (Lago, De Ridder, et al. 2018). With the in-
6

troduction of deep learning, this limitation has been overcome (Goodfellow
et al. 2016). Indeed, using the greedy layer-wise pretraining algorithm Hin-
ton et al. 2006 has demonstrated that deep belief networks can be efficiently
trained, allowing researchers to train complex neural networks with more
than one hidden layer.
To fill this gap in the literature, we propose to combine the ESM with the
state-of-the-art Ensemble Deep Neural Network model, which is positioned
at the forefront of electricity price forecasting models (Lago, Marcjasz, et al.
2021).
Secondly, ARX is a linear model that incorporates an extensive number of
input features. The traditional methods of estimating the ARX model (or-
dinary least square) are not able to deal with redundancy in the input fea-
tures, resulting in unsatisfactory forecasting accuracy (Ziel and Weron 2018,
Uniejewski and Weron 2018, and Ziel 2016). To address this restriction, use
of the least absolute shrinkage and selection operator (LASSO) (Tibshirani
1996) and elastic nets (Zou and Hastie 2005) is proposed. These two regu-
larisation strategies set some of the parameters to zero by jointly minimising
the squared errors and a penalty factor of the model parameters. This results
in the removal of redundant regressors.
In light of the benefits associated with LASSO techniques (Uniejewski and
Maciejowska 2023,Uniejewski, Nowotarski, et al. 2016), we have opted to
employ the Ensemble LASSO Estimated AutoRegressive (Ens-LEAR) tech-
nique for benchmarking purposes.
This leads us to another issue when discussing combined approaches, in gen-
eral, which is their comparison with benchmarks. In most research, advanced
models are avoided (Singh et al. 2018, and Naz et al. 2019) or only outdated
models are adopted for comparisons (Bento et al. 2018, and Peesapati, Ku-
mar, et al. 2017). In addition, a comparison of models that combine techno-
economic and econometric approaches for short-term power price forecasting
is restricted to their individual components and a naïve model. As a result, it
is impossible to determine with precision how accurate the newly suggested
approaches are. To fill this literature gap, a variety of econometric models
commonly used in EPF, besides the Ens-LEAR model, were also adopted for
comparison purposes. Each econometric model is combined separately with
the techno-economic ESM, and their forecasting accuracy is then evaluated.
This enables us to conduct a thorough evaluation, facilitating a robust analy-
sis that effectively compares the performance of various approaches. This, in
turn, enables us to confidently determine the robustness and generalisability
of our methodology.
7

3 Methodology
In this section, we discuss our proposed forecasting methodology, which com-
bines the strengths of both the techno-economic energy system model and the
econometric model to improve electricity price forecasting. The methodology
section starts with a comprehensive overview of the ESM. The second subsec-
tion presents the econometric models, presenting the Ensemble Deep Neural
Network (Ens-DNN) model first (subsection 3.2.2). We chose the Ens-DNN
model as our first-choice econometric model because it is "highly accurate",
based on findings from Lago, Marcjasz, et al. 2021, who also suggest it as an
open-source benchmark model for all "new complex EPF forecasting meth-
ods" (Lago, Marcjasz, et al. 2021).
We also combine the ESM with six other econometric models, which hence-
forth are referred to as ’benchmark models’. These benchmark models are de-
scribed in subsection 3.2. Examining various econometric models in addition
to the Ens-DNN holds significance for two key reasons: first, it benchmarks
the forecast quality of the Ens-DNN model. Second, if including the MCP as
an independent variable increases accuracy in different econometric models,
our results can be considered more robust. The third subsection defines how
the ESM and the econometric models are coupled. In the last subsection, we
present the methodology for the empirical valuation of the additional price
forecast accuracy in an electricity storage optimisation problem.
3.1 Energy System Model
We apply the em.power dispatch model2as presented in Möbius, Watermeyer,
et al. 2023, which is formulated as a linear optimisation problem minimis-
ing overall system costs. The model includes a comprehensive representa-
tion of key techno-economic elements of a liberalised European electricity
sector, including international trade between markets, minimum generation
and start-up restrictions on power plants, electricity generation from variable
renewable energy sources, combined heat and power plants, energy storage,
and control power provision. A demand constraint guarantees that supply
and demand are balanced across all market zones and time periods. Fol-
lowing economic theory, the em.power dispatch model calculates a market
equilibrium in the electricity market, minimising total system costs. Whole-
sale electricity prices are computed during the optimisation as shadow prices
2Model code and input data are available on GitHub:
https://github.com/ProKoMoProject/Enhancing-Energy-System-Models-Using-Better-
Load-Forecasts
8

of the demand constraint.
Our methodology can be applied to many liberalised electricity markets.
We provide an empirical example for the German market. Due to the market
integration of European electricity markets, we include most of the EU’s 27
member states 3as well as Norway, Switzerland, and the United Kingdom
to include cross-border flows. Figure 1 depicts the geographical scope of the
em.power dispatch model including all interconnections between the different
market zones.
Figure 1: Geographical scope of the energy system model
The model is specifically designed to generate hourly day-ahead electricity
price forecasts. In contrast to many other energy system models, it uses a
rolling window approach to clear the market optimally and uses exclusively
input data known to market participants when submitting their bids. Further
details about the proposed model can be found in Möbius, Watermeyer, et al.
2023.
3.2 Econometric Models
This section will first describe the methodology of our state-of-the-art econo-
metric model, the Ensemble Deep Neural Network model, and then introduce
the econometric benchmark models.
3Bulgaria, Cyprus, Greece, Iceland, Ireland, Malta and Romania are omitted.
9

3.2.1 Ensemble Deep Neural Network Model
The ensemble model category is one of the most commonly used forms of
hybrid models aimed at overcoming the limitations of individual models and
achieving highly accurate forecasting results. Theoretical and empirical ev-
idence suggests that combining different models can significantly improve
forecasting accuracy (Ben Amor et al. 2018a,Niu and Wang 2019). Its pri-
mary aim is to leverage the advantages and strengths of individual models
simultaneously. This conclusion is supported by Lago, Marcjasz, et al. 2021,
who compared the forecasting accuracy of state-of-the-art models, specifi-
cally Deep Neural Network (DNN) and the Lasso Estimated AutoRegressive
(LEAR) models, with their ensemble versions. Their findings demonstrated
that the ensemble models, namely Ensemble DNN (Ens-DNN) and Ensem-
ble LEAR (Ens-LEAR), outperformed the individual models (i.e., DNN and
LEAR, respectively).
Building on this finding, and recognising that the Ens-DNN model is the
most accurate model (Lago, Marcjasz, et al. 2021), we will consider it as the
primary econometric component. This model will incorporate the MCP as
an exogenous variable to test whether it benefits from the introduction of
the output of the ESM model, further enhancing its forecasting accuracy.
Following Lago, Marcjasz, et al. 2021, the Ens-DNN model is constructed
by taking the arithmetic average of four distinct DNNs, each derived from
running the hyperparameter and feature selection procedure four times. Hy-
perparameter optimisation, in theory, is a deterministic process over an infi-
nite number of iterations, eventually identifying the global optimum. How-
ever, in practice, with a finite number of iterations and varying initial random
seeds, the process becomes non-deterministic. This leads to different sets of
hyperparameters and features each time it is run. These results, representing
local minima, have nearly identical performance on the validation dataset,
making it difficult to determine which is superior. This phenomenon occurs
because deep neural networks (DNNs) are highly flexible, allowing multiple
network architectures to achieve equally good results. In this context, the
combination of four distinct DNNs, each configured with unique hyperpa-
rameters, significantly enhances predictive accuracy compared to each DNN
operating independently. The individual DNN model is explained in section
3.2.2, where it is also introduced as an additional benchmark model.
3.2.2 Econometric Benchmark Models
We use six additional econometric models to benchmark the results from the
Ens-DNN: a DNN model, an Ensemble LASSO Estimated AutoRegressive
10

(Ens-LEAR) model, the individual LEAR model, a Long-Short-Term Mem-
ory model (LSTM), a LASSO autoregressive with exogenous variables model
(LARX), and a Random Forest model (RF).
The Deep Neural Network Model
The DNN is a deep feedforward neural network with four layers and a multi-
variate framework (one model with 24 outputs). Without expert knowledge,
its inputs and hyperparameters can be optimised for each case study. Figure
2 shows the DNN model architecture. To estimate the hyperparameters, the
Figure 2: The DNN architecture adapted from Lago, De Ridder, et al. 2018
training dataset comprises data from the four years before the testing period.
Throughout the training process, the training datasets are split into a train-
ing and a validation dataset, with the latter being used for two purposes:
processing early stops to avoid overfitting and optimising hyperparameters.
The validation dataset consists of 42 weeks.
Similar to the original DNN paper (Lago, De Ridder, et al. 2018), the tree-
structured Parzen estimator (Bergstra et al. 2011), a Bayesian optimisation
approach based on sequential model-based optimisation, is used to jointly
optimise the hyperparameters and input features.
11

To do this, the features are modelled as a set of hyperparameters, each of
which is a binary variable that determines whether or not a particular feature
is included in the model (Lago, De Ridder, et al. 2018). More specifically,
the method uses 11 decision variables, or 11 hyperparameters, to determine
which of the 241 possible input attributes are relevant:
•Four binary hyperparameters that determine whether or not the his-
torical day-ahead prices pd−1,pd−2,pd−3, and pd−7should be taken into
account
•Two binary hyperparameters that specify whether or not to take into
account the x1
dand x2
dday-ahead forecasts
•Four binary hyperparameters that determine whether or not the his-
torical day-ahead forecasts x1
d−1,x1
d−7,x2
d−1, and x2
d−7should be taken
into account
•One binary hyperparameter that determines whether or not the vari-
able zd, which stands for the day of the week, should be used
In other words, there are 10 binary hyperparameters that each determine
whether or not to include 24 inputs. In addition, there is another binary
hyperparameter that determines whether or not to include a dummy variable.
The technique additionally optimises eight more hyperparameters in addition
to choosing the features: the number of neurons in each layer, the activation
function, the learning rate, the dropout rate, the type of data preprocessing
method, the use of batch normalisation, the initialisation weights of the DNN,
and the coefficient for L1 regularisation that is applied to the kernel of each
layer.
The hyperparameters and features are adjusted once using four years of
data prior to the testing period, unlike the DNN weights, which are calibrated
daily. It is crucial to remember that the algorithm runs for T iterations (T
is set to 1500), during which time it infers a putative optimal subset of hy-
perparameters/features and assesses this subset on the validation dataset.
The Ensemble LASSO Estimated AutoRegressive Model
As for the Ens-DNN, the Ensemble LEAR model (Ens-LEAR) is con-
structed by taking the arithmetic average of individual forecasts derived from
four different calibration window lengths. These lengths are specifically 1.5
years (78 weeks), 2 years, 3 years, and 4 years. This approach allows LEAR
to incorporate a range of historical data perspectives, from short-term to
12

long-term trends and cycles. With Ens-LEAR, forecasts are averaged over
these varied time horizons, achieving a balance between recent data and more
enduring historical patterning, and thus enhance prediction accuracy and ro-
bustness. Detailed information about the individual LEAR is provided in
section 3.2.2. It is also considered a benchmark.
The LASSO Estimated AutoRegressive Model
LEAR is a parameter-rich Autoregressive ARX model estimated using LASSO
as an implicit feature selection approach (Uniejewski, Nowotarski, et al.
2016). The LEAR model is briefly outlined in this paragraph, with more
comprehensive details provided in Lago, Marcjasz, et al. 2021.
The model to predict electricity price pd,h on day dand hour his given by:
pd,h =fpd−1,pd−2,pd−3,pd−7,xi
d,xi
d−1,xi
d−7,θh+εd,h
=
24
X
i=1
θh,i ·pd−1,i +
24
X
i=1
θh,24+i·pd−2,i
+
24
X
i=1
θh,48+i·pd−3,i +
24
X
i=1
θh,72+i·pd−7,i
+
24
X
i=1
θh,96+i·x1
d,i +
24
X
i=1
θh,120+i·x2
d,i
+
24
X
i=1
θh,144+i·x1
d−1,i +
24
X
i=1
θh,168+i·x2
d−1,i
+
24
X
i=1
θh,192+i·x1
d−7,i +
24
X
i=1
θh,216+i·x2
d−7,i
+
7
X
i=1
θh,240+i·zd,i +εd,h
(1)
where θh= [θh,1, . . . , θh,247]⊤contains the LEAR model parameters for hour
haccounts of 247 parameters. As Equation 1 is estimated using LASSO,
many of these parameters become zero:
ˆ
θh= argmin
θh
{RSS +λ∥θh∥1}
= argmin
θh(RSS +λ
247
X
i=1
|θh,i|)(2)
where RSS =PNd
d=8 (pd,h −ˆpd,h)2is the sum of squared residuals. Ndis the
13

number of days in the training sample, ˆpd,h is the price forecast, and λ≥0
is the regularisation hyperparameter of LASSO.
A hyperparameter that regulates the L1penalty is optimised upon recalibra-
tion every day. This can be accomplished using an ex-ante cross-validation
procedure.
Long-Short-Term Memory Model
One of the most crucial components of both current deep learning models
and power market price forecasting is understanding long-term dependencies,
which is the starting point for choosing and creating a forecasting model. Like
typical recurrent neural network models, the LSTM model has emerged as
a crucial instrument for analysing complex feature analyses and sequence
dependencies. The memory capacity of LSTM was greatly enhanced when
compared to recurrent neural networks. More specifically, LSTM can handle
long-term dependencies because it has built-in mechanisms that regulate how
information is retained or forgotten over time. Due to their structure, these
models are more effective at handling continuous sequences; as a result, they
have a variety of applications in the field of EPF right now (Xiong and Qing
2022, and W. Li and Becker 2021).
Hochreiter and Schmidhuber 1997 first proposed the LSTM architecture,
which has since been improved by other researchers to yield superior perfor-
mance (Shao, Zheng, et al. 2021, and Shao, Yang, et al. 2022).
The LSTM introduces the concept of "gates", specifically a forget gate, input
gate, and output gate. As shown in Figure 3, in LSTMs, the forget gate con-
trols how much information from the previous cell is retained, the input gate
controls the input and update of the cell state, and the output gate computes
the output. The forget gate ftis calculated at the ttime step based on the
hidden state htand the current input xtand then outputs data to multiply by
the previous memory cell Ct−1. By using the forget gate ft, data are output
from 0to 1, where 0represents completely forgotten data and 1represents
completely reserved data. Thereafter, the input gate determines how much
information can be transferred to the memory cell Ct. As part of the input
gate process, the first step is to determine what information can be updated
via sigmoid layers and the second step is to generate new information C∗
t
through the tanh layers. Following that, each output from the forget gate
and the input gate is used to calculate the current memory cell Ct. Finally,
the output gate determines which information will be transmitted.
The following are the equations used in the LSTM model:
•Input gate, itat time t, and candidate cell state, C∗
t:
14

Figure 3: LSTM model architecture, adapted from Kawakami 2008
.
it=σ(Wxixt+Whi ht−1+bi)(3)
C∗
t= tanh (Wxcxt+Whcht−1+bc)(4)
•Activation of the memory cells’ forget gate, ftat time t, and new state
Ct:
ft=σ(Wxf xt+Whf ht−1+bf)(5)
Ct=ftCt−1+itC∗
t(6)
•Activation of the cells’ output gate, ot, at time t, and their final outputs,
ht:
ot=σ(Wxoxt+Who ht−1+bo)(7)
ht=ottanh (Ct)(8)
where:
it,ft, and otare the input gate, the forget gate, and the output gate, respec-
tively, Wxi,Wxc,Wxf , and Wxo stand for the weight coefficients, bi,bc,bf,
and borepresent biases, C∗
trepresents the output in the input gate process,
and σand tanh are activation functions.
The LASSO Autoregressive Model With Exogenous Variables
The autoregressive LASSO (Least Absolute Shrinkage and Selection Op-
erator) technique suggested by P. Li et al. 2017 is also adopted as a powerful
statistical model. Statistical approaches have the advantage that their model
complexity is much lower than that of ML methods, which means that they
15

require fewer computational resources.
The use of autoregressive models in the prediction and inference of time se-
ries data is widespread. In light of this fact, we opted to adopt a linear
autoregressive model with exogenous variables based on LASSO estimation.
Throughout the manuscript, LARX is used to refer to this model.
The autoregressive model with exogenous variables is given by the following
equation:
yd,h =β0+
I
X
i=1
βh,ixh,i +εd,h (9)
where yd,h stands for the electricity prices of day dand hour h,xirefers to
the regressors stated in section 4.2.2, βiare their corresponding coefficients,
and εd,h are Gaussian variables, where εd,h ∼N(0,1).
The following notation can be used to write the compact matrix form of the
model in Equation 9:
y=Xβ+ε(10)
In shrinkage (or regularisation), the coefficients of irrelevant explanatory
variables are lowered towards zero so that the most relevant regressors are
retained. In this way, the algorithm avoids overfitting and takes into account
the most relevant regressors (Gareth et al. 2013).
A LASSO is a generalisation of linear regression that minimises the residual
sum of squares (RSS) and a linear penalty function of the βs instead of just
minimising the RSS:
ˆ
βL
h= min
β{RSS + λ∥βh∥1}= min
β(RSS + λ
n
X
i=1
|βh,i|)(11)
where λ≥0is the parameter that controls the level of sparsity in the
model and is known as the tuning parameter (or regularisation parameter). It
controls how many regressors are used to predict electricity prices (i.e. spar-
sity). The bigger λis, the sparser the solution ˆ
βLASSO will be, which means
that fewer data points from the input regressors will be used to predict the
future. For λ= 0, the model is reduced to a traditional AR model (Equation
9). While in the case λ→ ∞ all coefficients βh,i →0tend towards zero, for
intermediate values of λa balance must be struck between minimising RSS
and shrinking coefficients towards zero. Practically, we estimate the value of
λusing k-fold cross validation (CV) (Hastie et al. 2009).
In recent years, LASSO has become increasingly popular in EPF (Ludwig
et al. 2015,Ziel, Steinert, et al. 2015). As part of our paper, we analyse
LASSO’s ability to predict day-ahead electricity prices and provide a com-
parison with some other well-known prediction methods, i.e. LSTM model.
16

Compared to the LEAR model, LARX represents the foundational version
from which Lago, Marcjasz, et al. 2021 has developed the state-of-the-art
LEAR model. In our research, the use of both LARX and LEAR models
offers insights into their distinct reactions under diverse input data. This
comparative analysis serves as a valuable guide for market participants, al-
lowing them to choose between the simpler LARX and the more complex
LEAR models based on the characteristics of their data.
The Random Forest RF Model
The RF algorithm presented in this paper is based on bagging or boot-
strap aggregation to tree learners. In this case, bagging involves repeatedly
selecting Bbootstrap samples from the training set and fitting trees Tbto
these samples where Bis the number of trees and Tbis a random forest tree.
For the model’s parameter estimation, inputs x= [x1, . . . , xN]T∈Xand
responses y∈Yare introduced. Following training, each tree’s prediction
is given. The predictors from all regression trees are averaged in the final
prediction using the following formula:
ˆ
YB
rf (x) = 1
B
B
X
b=1
ˆ
Yb(x)(12)
where ˆ
Yb(x),b= 1, ...B is the prediction of the bth RF tree.
This model is often used for short-term EPF and is shown to provide accurate
forecasts (Ludwig et al. 2015,Pórtoles et al. 2018, and Mei et al. 2014).
3.3 Model Combination
The model combination is straightforward: we use the key output from the
ESM, i.e. the market clearing price MCP, as an independent variable in the
selected econometric model. To assess the value of this particular variable, we
compare different models (ESM, Ens-DNN, and the six econometric bench-
mark models) as well as different data inputs in the econometric models:
market fundamentals only, MCP only, and market fundamentals plus MCP.
We thus have two dimensions of the analysis: first, the type of model, and
second, the independent data entering the econometric models.
3.4 Storage Optimisation
After combining the ESM with the econometric models, we test the effec-
tiveness of our price forecasts simulating the profit contribution of a storage
17

application. Transporting energy over time, storage systems exploit price
spreads and are therefore dependent on good price forecasts. To maximise
their profits, they must first identify the time with the lowest electricity
prices in order to charge their storage and then, at the time with the highest
electricity prices, generate electricity and sell it back to the market.
To evaluate the effectiveness of a price forecast model, we generate a
linear optimisation problem that maximises the profit contribution (Π) of
a storage system, which results from purchases and sales of electricity at
different hours (h) with different prices for a given number of days (d).
The objective function of this problem is represented in Equation 13.
Using price forecasts (ˆpd,h), optimal quantities and times for charging (Cd,h)
and electricity generation (Gd,h) are determined.
max Π = X
d,h
ˆpd,h ·Gd,h −X
d,h
ˆpd,h ·Cd,h (13)
The actual profit contribution (Πact) of the storage system then results
from the combination of the storage operations with the actual electricity
prices pd,h (Equation 14).
Πact =X
d,h
pd,h ·Gd,h −X
d,h
pd,h ·Cd,h (14)
Comparing this to the highest possible profit contribution, which is cal-
culated using the actual electricity prices for determining optimal storage
operations, we can derive the effectiveness of a price forecast.
Note that we focus on day-ahead price forecasts. Hence, we assume the
storage only operates on the day-ahead market. Other streams of income
such as intraday markets and control power markets are not taken into ac-
count. Additionally, we assume that the bidding strategy is based on the
price forecasts for the following day only. To isolate the impact of daily price
forecasts, the possibility of storing electricity for several days is neglected.
Hence, the storage dispatch is determined daily before the day-ahead market
closes at noon exclusively for the 24 hours of the following day.
Both electricity generation (Gd,h) and consumption (Cd,h) are limited by
the generation capacity (cap) of a storage system (Equation 15). When
analysing the relative effectiveness of a price forecast (i.e., the ratio of the
realised profit contribution to the optimal profit contribution), the generation
capacity can be set to one.
Gd,h +Cd,h ≤cap ∀d∈D, h ∈H(15)
18

The possibility to generate or consume electricity depends on the current
state of the storage level (SLd,h) at the end of each hour. Equation 16 shows
that withdrawing electricity from the market when charging increases the
storage level, while generating electricity and selling it back to the market
reduces it. Efficiency losses of the entire storage cycle are considered by the
factor η, which is between zero and one. The energy storage must therefore
charge more energy than it can generate later. Equation 17 states that the
electricity generation must not be higher than the storage level at the end of
the previous hour.
SLd,h ≤S Ld,h−1+Cd,h ·η−Gd,h ∀d∈D, h ∈H(16)
Gd,h ≤SLd,h−1∀d∈D, h ∈H(17)
A fully charged storage system can generate electricity for a certain num-
ber of full-load hours. This value is assumed to be exogenous in our analysis
and is calculated as an energy-to-capacity ratio (ecr) restricting the storage
level in Equation 18.
SLd,h ≤cap ·ecr ∀d∈D, h ∈H(18)
As mentioned above, the storage operation is determined separately for
each day. To avoid arbitrary effects, the storage status must be the same at
the beginning and end of each day. We assume the storage to be empty at
both times, which is controlled by Equations 19 and 20.
SLd,h=1 =Cd,h ·η∀d∈D(19)
SLd,h=24 = 0 ∀d∈D(20)
Finally, the non-negativity constraints are presented in Equation 21
Gd,h, Cd,h , SLd,h ≥0∀d∈D, h ∈H(21)
4 Data and Model Preparation
This section provides a structured overview of the dataset and input data
used in developing the techno-economic energy system model and the econo-
metric model for forecasting electricity prices in the German market.
19

4.1 Input Data for the Energy System Model
We apply the European electricity market model em.power dispatch with
data from January 1st, 2015, until December 31st, 2020. In general, the use
of ESMs requires extensive input data. In this section, we have listed the
input data used to compute the MCP for the German electricity market.
On the demand side, original load forecasts are taken from the ENTSOE
transparency platform (ENTSO-E Transparency Platform 2021f). However,
recent studies show that these data have structural errors that should be
corrected (see, e.g. Maciejowska, Nitka, et al. 2021). Therefore, we run an
error correction model for the day-ahead load forecasts according to (Möbius,
Watermeyer, et al. 2023).
On the supply side, a set of technologies is generating and storing electric-
ity. Data on the technical-economic properties used in the model are listed
in Table 1
The data provided for download on GitHub.4
4.2 Input Data for the Econometric Models
We start this section with a description of the dependent variable, before
turning to the independent variables entering the econometric model.
4.2.1 Dependent Variable
In this paper, we choose the hourly day-ahead wholesale electricity price
for Germany as our target variable (dependent variable). This market is
operated by EPEX SPOT. Our sample covers the period between January
1st, 2015, and December 31st 2020.
Day-ahead wholesale electricity trading is settled at 12 noon for each
hour of the following day, meaning that all 24 hours of one day are traded at
once. The forecasting process is thus divided into 24 separate daily prediction
batches, with each batch corresponding to a specific hour of the day. This
segmentation allows for the independent estimation of electricity prices for
each hour, enabling a focused analysis of the unique factors influencing prices
at different times of the day.
The electricity price data are sourced from the ENTSO-E Transparency
Platform.
4https://github.com/ProKoMoProject/Enhancing-Energy-System-Models-Using-
Better-Load-Forecasts
20

Table 1: Data used in the energy system model
Parameter Source
CO2prices Sandbag 2021
Control power procurement Regelleistung.net 2018
Curtailment costs for RES own assumption: 20 EUR/MWh
Efficiency of generation capacities Schröder et al. 2013,
Open Power System Data 2020a
Efficiency losses at partial load Schröder et al. 2013
Electricity demand
(original day-ahead forecast) ENTSO-E Transparency Platform 2021f
Energy-power factor (for storages) own assumption: 9
Fuel prices Destatis Statistisches Bundesamt 2021,
(Lignite, nuclear, coal, gas, oil) EEX 2021,ENTSO-E 2018
Generation and storage capacity BNetzA 2021,UBA 2020,EBC 2021,
ENTSO-E Transparency Platform 2021e,
Open Power System Data 2020a
Generation by CHP units European Commission 2021
Historic electricity generation ENTSO-E Transparency Platform 2021a
Load-shedding costs own assumption: 3,000 EUR/MWh
Minimum output levels Schröder et al. 2013
NTCs ENTSO-E Transparency Platform 2021c,
JAO Joint Allocation Office 2021
Variable O&M costs Schröder et al. 2013
Power plant outages ENTSO-E Transparency Platform 2021g
RES generation ENTSO-E Transparency Platform 2021d
Start-up costs Schröder et al. 2013
Seasonal availability of hydropower ENTSO-E Transparency Platform 2021a
Temperature (daily mean) Open Power System Data 2020b
Water value ENTSO-E Transparency Platform 2021a,
ENTSO-E Transparency Platform 2021b
21

4.2.2 Independent Variables
A group of exogenous variables has been selected to forecast the wholesale
electricity price. Our selection of independent variables is motivated by the-
oretical considerations (Kanamura and ¯
Ohashi 2007,Mount et al. 2006,No-
gales et al. 2002) and considers publicly available information. Six variables
are considered: load forecasts, CO2emission prices, fuel prices (both gas and
coal prices), as well as renewable generation (production forecasts for both
wind and photovoltaic).
Hourly load forecasts are included as independent variables in most arti-
cles on EPF (Lago, Marcjasz, et al. 2021). As this paper focuses on day-ahead
price forecasting, we use day-ahead load forecasts, improved with an error
correction model (see Möbius, Watermeyer, et al. 2023).
Prices for CO2emission certificates, hard coal, and natural gas are long-term
variables that affect production costs, especially for the marginal plants de-
termining prices with their variable costs. We use daily settlement prices for
the 24 hours of the following day.
We include wind and solar generation forecasts as major renewable energy
technologies in Germany.
Finally, the estimated market price from the techno-economic energy sys-
tem model described in section 3.1, the MCP, is considered as an additional
independent variable, which links the ESM and the econometric model.
The explanatory variables’ sources are referred to in section 4.1.
One point to notice is that the ESM has also taken into account the load
forecast, the wind and solar generation, the gas and coal prices, and the CO2
price. Through the demand balance constraint, these input data influence
the values of other generation units. Econometric models, however, consider
statistical patterns, such as volatility, long memory, and linear and non-
linear trends, between electricity prices and the variables mentioned above.
Although the same data were used in both models, they were not processed
or treated the same way (De Marcos et al. 2019a).
4.2.3 Training and Testing Periods
The training and testing samples were split up to evaluate the out-of-
sample forecasting performance of our model. We use the first 4 years (the
years between 2015 and 2018) as the training period, resulting in 34,680
hourly observations, and the following two years (2019 and 2020) are the
testing period, resulting in 17,640 hourly observations. It is important to
22

note that the length of both the testing and training periods was selected
based on Lago, Marcjasz, et al. 2021, who emphasised the importance of
choosing two years to ensure effective research in EPF.
4.2.4 Preliminary Statistics
According to preliminary statistics, the spot price series exhibits typical
characteristics of electricity prices, such as seasonality, high volatility, excess
kurtosis, negative skewness, and spikes (see Table 5 and Figure 8 in Ap-
pendix).
The test of stationarity (Augmented Dickey-Fuller ADF) rejected the unit
root hypothesis at the 5% significance level, indicating that the electricity
price time series is stationary.
4.2.5 Feature Selection Algorithm
To achieve the highest model performance, it is crucial to choose the
optimum set of features from the original dataset. To avoid the curse of
dimensionality, feature selection is usually an effective method. This process
takes place when developing a predictive model by identifying a subset of
attributes from the original dataset. It can reduce computation time, improve
model predictions, and help us to get a better understanding of the dataset
(Chandrashekar and Sahin 2014,W. Li and Becker 2021). Many feature
selection methods have been proposed (Naz et al. 2019,Gholipour Khajeh
et al. 2018,Gao et al. 2018). In this paper, we present a Recursive Feature
Elimination (RFE) algorithm-based Random Forest (RF) model for selecting
the most relevant features to develop an effective and reliable forecasting
model (Chen et al. 2018).
Figure 4 depicts the RF-RFE approach’s flowchart. First, we train our
model using the RF algorithm with the training set. We then evaluate the sig-
nificance of each feature based on its classification. Afterwards, the features
are ranked by importance from the most important to the least important
(see Figure 5). Once the least relevant feature has been removed from the
feature set, we retrain the RF model with the modified features and obtain
the classification performance from the new feature set. As indicated by the
name, RFE involves iterating until all crucial features are chosen. Several
studies have demonstrated the effectiveness of this method (Liu and Motoda
2007,Liu and Yu 2005).
Based on the feature selection results in Figure 5, demand is the most im-
portant factor, followed by wind generation forecasts and gas prices. In the
rest of the analysis, we keep all six exogenous variables since the algorithm
23

Figure 4: RF-RFE flowchart
does not exclude any of them.
4.3 Evaluation Metrics
The most widely used metrics to evaluate the accuracy of point forecasts in
EPF are mean absolute error (MAE) and root mean square error (RMSE):
MAE = 1
24Nd
Nd
X
d=1
24
X
h=1
|pd,h −ˆpd,h|,(22)
RMSE = v
u
u
t
1
24Nd
Nd
X
d=1
24
X
h=1
(pd,h −ˆpd,h)2,(23)
where pd,h and ˆpd,h denote the real and forecasted prices on day dand hour
h, respectively, while the total number of days in the test period is denoted
by Nd.
We also consider the symmetric mean absolute percentage error (sMAPE) as
24

Figure 5: RFE-RF results
an alternative to the mean absolute percentage error (MAPE) 5. The latter is
not very informative, since it becomes very large at prices close to zero and
is dominated by low prices. Taking into consideration that the day-ahead
electricity price sample used in this study includes zero and close-to-zero
prices, we adopt the sMAPE. The sMAPE is given by:
sMAPE = 1
24Nd
Nd
X
d=1
24
X
h=1
2|pd,h −ˆpd,h|
|pd,h|+|ˆpd,h |(24)
The relative MAE (rMAE) is also considered, as a reliable evaluation met-
ric that can overcome some limitations related to the mean absolute scaled
error MASE, widely used in the literature. In particular, the MASE is de-
pendent on the in-sample dataset, which implies that forecasting methods
with different calibration windows will have to take into consideration differ-
ent in-sample datasets. This results in the MASE of each model being based
on a different scaling factor, so comparisons across models are not possible.
To overcome these limitations, the rMAE normalises MAE against a naïve
5MAPE = 1
24NdPNd
d=1 P24
h=1
|pd,h−ˆpd,h|
|pd,h|,
25

forecast based on the out-of-sample dataset, ensuring consistent evaluation
across models (Lago, Marcjasz, et al. 2021). The rMAE is similar to MASE
in the sense that rMAE normalises the MAE by the MAE of a naïve forecast.
The rMAE is described as follows:
rMAE =
1
24NdPNd
d=1 P24
h=1 |pd,h −ˆpd,h|
1
24NdPNd
d=1 P24
h=1 pd,h −ˆpnaive
d,h 
(25)
where ˆpnaive
d,h is the naïve forecast.
5 Achieved Price Forecasting Accuracy
This section evaluates the previously specified models for the forecasting of
electricity prices. For the sake of clarity, we divide the section into two subsec-
tions. The first subsection evaluates the performance of the techno-economic
ESM, the Ens-DNN model, and their combinations, using the evaluation
metrics defined in section 4.3. The second subsection presents the com-
parative analysis to gauge the forecasting accuracy of the ESM–Ens-DNN
model against alternative benchmark models incorporated within the com-
bined framework instead of the Ens-DNN model. This serves two purposes:
(i) it evaluates the effectiveness of the chosen Ens-DNN model as a candi-
date for the econometric model, and (ii) assesses how various statistical and
machine learning models respond when integrated with the ESM, enabling
us to evaluate the robustness and generalisability of improving econometric
short-term price forecasts with results from the techno-economic ESM.
5.1 Performance of the ESM–Ens-DNN Model
The forecasting results in terms of evaluation metrics, the MAE, RMSE,
sMAPE, and rMAE (described in section 4.3), are presented in Table 2, with
bold values representing the most favourable results.
All combinations of Ens-DNN and ESM are considered: first, the ESM is
presented as a stand-alone forecasting model.
Second, the Ens-DNN model is used without incorporating the estimated
price from the ESM. Six exogenous variables are considered in the Ens-DNN:
a day-ahead forecast of demand, a wind generation forecast, and a PV gen-
eration forecast, as well as gas, coal, and CO2prices.
Third, the ESM is incorporated into the Ens-DNN through the addition
of the MCP to the list of exogenous variables, denoted ESM–Ens-DNN+.
Consequently, through a comparison between the stand-alone and combined
26

Table 2: Forecasting results of the ESM–Ens-DNN model
Model MAE RMSE sMAPE% rMAE
ESM 6.116 9.374 23.976 0.607
Ens-DNN 4.272 7.222 19.180 0.457
ESM–Ens-DNN + 3.496 5.907 16.660 0.374
ESM–Ens-DNN 3.857 6.272 18.030 0.413
models with exogenous variables, we can effectively highlight the pivotal role
of including techno-economic information in enhancing the forecasting accu-
racy of the Ens-DNN model.
Fourth, we eliminate all exogenous variables and keep only the MCP, to test
whether the ESM already contains all information about those variables. The
model thus formed is denoted as ESM–Ens-DNN. This helps us to determine
whether we can rely exclusively on the MCP derived from the ESM as a
unique predictor of electricity prices. We can therefore test whether it is
possible to reduce the number of features and thus reduce calculation time
and costs without compromising prediction accuracy.
Looking at the table, several key take-aways can be derived.
27

Result (Three key take-aways on the Ens-DNN)
1. The individual Ens-DNN outperforms the ESM across all evalu-
ation metrics. This reconfirms the result that a (good) statistical
model outperforms a (good) ESM in day-ahead forecasting.
2. Model combination achieves a huge improvement in forecasting ac-
curacy. The arithmetic average improvement from all evaluation
metrics is approximately 18% when comparing evaluation metrics
for Ens-DNN and ESM–Ens-DNN+.
3. Significant improvement can also be achieved with the ESM’s MCP
used as the only independent variable in the Ens-DNN. At the same
time, the accuracy when the other six independent variables are
added to the regression is even higher. The latter confirms the
hypothesis stated in section 4.2.2 that the ESM and the econometric
model treat the same inputs differently (De Marcos et al. 2019b).a
aIn the Ens-DNN, inputs are assigned a weight based on their relative importance.
In the ESM, inputs affect the market clearing equilibrium by shaping the supply and
demand side. Hence, these inputs influence the market price estimator based on
economic theory in the ESM.
28

Figure 6: Comparison of real prices and the ESM, Ens-DNN,ESM–Ens-
DNN+, and ESM–Ens-DNN.
Figure 6 plots the four different models we consider, i.e. the ESM and
Ens-DNN plus the combined models ESM–Ens-DNN+ and ESM–Ens-DNN.
The plots indicate that the forecasting of the Ens-DNN model is closer to
the real data when compared to the ESM. Furthermore, the accuracy of the
Ens-DNN model is further enhanced when combined with the ESM, as seen
in the ESM–Ens-DNN+ and ESM–Ens-DNN models.
Overall, the results suggest that the combined model (in both versions:
ESM–Ens-DNN+ and ESM–Ens-DNN) leverages the strengths of both the
ESM and Ens-DNN models. On the one hand, the ESM simulates price
formation from the intersection of demand and the physical and dynamic
properties of the generators’ supply functions. On the other hand, the Ens-
DNN model excels at capturing complex patterns and nonlinear relationships
within data, making it particularly effective for short-term price forecasting
tasks. By integrating these strengths, the combined model emerges as an
excellent forecasting tool for short-term price predictions.
Our key results can be confirmed when compared with the existing lit-
erature, where various models for electricity price forecasting have been de-
29

veloped. Ziel and Weron 2018, for instance, forecast wholesale electricity
prices with an overall MAE of 5.01 e/MWh for the years 2012 to 2016.
Maciejowska, Nitka, et al. 2021 achieve an RMSE of 8.43 e/MWh and a
MAE of 5.92 e/MWh within the time period from 2016 to 2020 using an
autoregressive model with exogenous variables. Marcjasz et al. 2023 report
a MAE of 3.542 and an RMSE of 6.146 for the German electricity market
for the time horizon from June 26th, 2019, to December 31st, 2020, in the
context of probabilistic forecasting. Applying a techno-economic agent-based
power market-simulation model, Qussous et al. 2022 obtain an RMSE of 11.21
e/MWh and a MAE of 7.89 e/MWh for the years 2016 to 2019. Comparing
our results to these studies, we can re-confirm that techno-economic models
have relatively low forecasting accuracy. While our combination of ESM and
Ens-DNN models outperforms the most accurate forecast by Marcjasz et al.
2023, it needs to be pointed out that the results of EPF models are difficult
to compare in terms of input data and time horizons.
5.2 Comparison with Benchmark Models and Robust-
ness Check
The proposed ESM–Ens-DNN model provides significant improvements in
forecasting accuracy, both compared to the Ens-DNN alone as well as com-
pared to the literature. However, to ensure the reliability and robustness of
our findings, the methodology of using ESM results as input in regressions
is tested on various econometric benchmark models, encompassing both sta-
tistical and machine learning approaches. The results are interpreted in the
following section. The framework of the analysis (time horizon, independent
variables, etc.) is identical to the previous subsection; the single variation is
the chosen econometric model.
The forecasting results of the evaluation metrics of the proposed ESM–
Ens-DNN model against all the other model-data combinations are presented
in Table 3.
30

Table 3: Benchmark models’ forecasting results
Model MAE RMSE sMAPE rMAE
ESM 6.117 9.375 23.980 0.654
Ens-DNN 4.272 7.222 19.180 0.457
DNN 5.092 8.301 21.340 0.545
Ens-LEAR 4.090 6.852 19.860 0.438
LEAR 4.137 7.010 19.420 0.443
LSTM 6.073 8.655 24.840 0.650
LARX 8.576 10.977 29.880 0.917
RF 10.529 14.580 36.220 1.126
ESM–Ens-DNN + 3.496 5.907 16.660 0.374
ESM–DNN+ 3.779 6.375 17.720 0.404
ESM–Ens-LEAR+ 3.688 6.117 17.760 0.395
ESM–LEAR+ 3.859 6.311 18.020 0.413
ESM–LSTM+ 5.724 8.382 23.650 0.612
ESM–LARX+ 10.743 12.683 34.300 1.149
ESM–RF+ 11.808 16.628 38.330 1.263
ESM–Ens-DNN 3.857 6.272 18.030 0.413
ESM–DNN 3.986 6.300 18.400 0.426
ESM–Ens-LEAR 3.896 6.282 18.170 0.417
ESM–LEAR 4.160 6.627 19.000 0.445
ESM–LSTM 5.597 8.259 23.220 0.599
ESM–LARX 6.061 9.000 24.410 0.648
ESM–RF 8.129 12.677 29.210 0.869
31

Result (Key take-aways for all models and robustness)
1. Ens-LEAR outperforms the other models as stand-alone models.
The difference is relatively small compared to Ens-DNN, LEAR,
and DNN, considerable compared to LSTM and ESM, and large
compared to LARX and especially RF.
2. The best overall performance is achieved with the ESM–Ens-DNN+
model: it has minimal errors for all evaluation metrics. Hence,
for all stakeholders exclusively interested in the most accurate day-
ahead electricity price forecast, the ESM–Ens-DNN+ model is an
excellent first candidate to implement.
3. Use of the ESM’s estimated market clearing price (MCP) as the
only independent variable improves the forecasting quality of all
econometric models. This confirms the robustness of our method-
ological approach, i.e. using the ESM’s MCP as a single model in-
put in an econometric model improves forecasting accuracy (lower
third of the table).
All in all, our result emphasises the strategic importance of the MCP
as an exogenous variable in sophisticated econometric models. When the
MCP is included alongside other independent variables, advanced models
such as Ens-LEAR, Ens-DNN, LEAR, and DNN not only retain but also
enhance their accuracy, demonstrating their robustness and compatibility
with a complex variable set. In contrast, less sophisticated models like LARX
and RF exhibit diminished accuracy when additional variables are included,
indicating that their simpler structures might not effectively handle multiple
regressors. Crucially, the Ens-LEAR, Ens-DNN, LEAR, and DNN models
consistently outperform simpler models in all tested configurations. This
disparity underscores the significant advantage of employing more advanced
models for integrating the MCP with other exogenous variables, aligning
with our goal of optimising econometric model performance across various
settings.
In detail, it is interesting to compare the Ens-DNN and the ESM-DNN,
which indicates that the forecasting accuracy of the individual DNN, when
introducing the MCP as a unique regressor, outperforms the ensemble DNN
with the six exogenous variables. This highlights the importance of the MCP
in enhancing the forecasting accuracy of the DNN model, making it outper-
form the four combined DNNs in its ensemble version with an average en-
hancement of 8% across all evaluation metrics. Hence, investors can opt for
32

the simpler ESM-DNN without sacrificing forecasting accuracy. For those
seeking even higher precision, ESM-DNN+ offers superior results. If achiev-
ing the best forecasting outcome is the primary goal, investors should consider
ESM–Ens-DNN+.
This range of models provides investors with the flexibility to select the
most suitable one based on their cost-benefit trade-off, depending on their
specific trading objectives.
Besides evaluation metrics, the error measures are statistically tested
based on the Giacomini-White (GW) test. The results, presented in Ap-
pendix 8.2, confirm all our key take-aways from both sections 5.1 and 5.2.
Hence, from a statistical perspective, the techno-economic ESM results in-
deed improve the econometric models’ performance.
6 Achieved Improvements for Storage Bidding
In this section, we evaluate the practical benefit of our price forecasts. For
this purpose, we assume a storage operator who buys electricity on the day-
ahead market to fill a storage unit in order to withdraw and sell the electricity
at a later time. The storage operator uses our price forecasts to select the time
of storage at the lowest possible prices and the time of electricity withdrawal
at the highest possible prices. Hence, the assumption is that storage dispatch
decisions are made daily based on the available day-ahead price forecasts.
In the second step, price realisations are revealed and storage revenues are
calculated.
For the matter of generalisability, we apply three different storage types that
differ in their energy-to-capacity ratio and storage cycle efficiency:
•Storage 1 has a rather high energy-to-capacity ratio of seven 6and an
efficiency of 75 %.
•Storage 2 has a medium energy-to-capacity ratio of three and an effi-
ciency of 80 %.
•Storage 3 has a low energy-to-capacity ratio of one and an efficiency of
90 %.
We present the optimisation model that determines the optimal storage
dispatch in order to yield the highest possible profit contribution already in
6An energy-to-capacity ratio of seven means that a full storage unit is able to generate
electricity for seven straight hours at full load without being recharged
33

section 3.4. The model is fed with daily price forecasts made using the dif-
ferent forecasting models for 2019 and 2020. The day-ahead storage dispatch
is decided based on these forecasts. Finally, the revenues of the storage unit
are based on the revealed prices on the German wholesale electricity market.
The code for this empirical application is provided on GitHub7.
Note that we focus exclusively on day-ahead price forecasts in our study.
Therefore, the storage units are optimised separately for each upcoming day
and a cross-day storage deployment is not considered. Hence, the storage is
given a fixed charge level at the beginning of each day.
Figure 7 shows the annual profit contribution per installed Megawatt of power
generation capacity of the three storage units using all the forecast models
presented in section 3. A storage operator can achieve the highest possible
return if a perfect forecast is available when planning the storage dispatch,
i.e. the storage operator knows the actual price before the market clearing.
The results of this hypothetical case are shown on the left side of the figure
where the profit contribution is computed based on the actual day-ahead
price (real price). This serves as a benchmark for the considered forecast
models.
Per Megawatt installed capacity, storage with a high energy-to-capacity ratio
generally achieves a higher profit contribution compared to storage with a
lower energy-to-capacity ratio. This is because storage facilities with a high
energy-to-capacity ratio can store electricity for several hours at a time and
later generate electricity for several hours. For such units, a good forecast
must identify not only the lowest and highest prices of a day but also the
third-, fourth-, and fifth-lowest and highest prices.
Some of the applied models seem unable to meet these requirements, namely
the RF and ESM–RF, as they do not capitalise on the value of a high energy-
to-capacity ratio. The results also show that the ESM–RF+ model is gen-
erally not suitable for determining a storage dispatch. With prices ranging
only between 29 and 34 EUR/MWh, the model does not identify sufficiently
high price spreads to make storage worthwhile.
Table 4 presents the profit contribution of storage as a factor to the per-
fect forecast, where the value of one would mean that the forecasting model
achieves the results of a perfect forecast.
For all three storage types, we see a group of model types that achieve very
good results that are relatively close to the optimal solution of a perfect
forecast. These are the ESM, Ens-DNN, ESM–Ens-DNN+, ESM–Ens-DNN,
Ens-LEAR, ESM–Ens-LEAR+, ESM–Ens-LEAR, LEAR, ESM–LEAR+, ESM–
7https://github.com/BTU-EnerEcon/Bridging-ESM-and-Deep-Learning-Models
34

LEAR, ESM–DNN+, ESM–DNN, and ESM–LARX+.
Result (Key take-aways for storage bidding)
•The ESM as a stand-alone model already achieves very good re-
sults. The ESM model is thus better - in comparison to economet-
ric models - in identifying the optimal dispatch of storage than in
forecasting day-ahead wholesale electricity prices. This may result
from the ESM endogenously dispatching all capacities in the system
(including storage) to determine the cost-minimal market clearing
solution. Optimal storage dispatch is thus already part of ESM op-
timisation. Thus, the model is stronger in identifying "scarcity"
hours when storage should produce and "excess" hours when stor-
age should charge than in deriving the absolute level of prices.
•Revenues increase for all econometric models when the ESM’s MCP
is added as a regressor. Eventually, the ESM–DNN+ model, which
combines a deep neuronal network with an energy system model,
yields the highest profit contribution for Storage 1 and Storage 2,
closely followed by the ESM and ESM–LEAR models. For Storage
3, the stand-alone ESM is the most profitable forecasting model,
closely followed by the Ens-DNN+, the ESM–LARX+ and ESM–
DNN+, and the ESM–Ens-LEAR models.
•These two results in combination, i.e. first that the ESM alone
provides near-optimal storage revenues, and second that the econo-
metric models benefit from including the MCP as an independent
variable, re-confirm the high practical value of the techno-economic
ESM in short-term price forecasting. On the econometric side, the
ESM–Ens-DNN+ model combination is found to be an excellent
model to generate accurate price forecasts as well as optimise stor-
age operation (and maximise storage revenues) for all considered
storage types.
35

Figure 7: Annual profit contribution of storage plants in Euros per installed
MW generation capacity
36

Table 4: Profit contribution of storage plants as a factor to a perfect forecast
Storage 1 Storage 2 Storage 3
ESM 0.907 0.916 0.888
Ens–DNN 0.875 0.891 0.858
Ens–LEAR 0.896 0.909 0.874
DNN 0.831 0.852 0.817
LEAR 0.893 0.909 0.870
LARX 0.853 0.888 0.868
RF 0.555 0.643 0.665
LSTM 0.549 0.528 0.411
ESM–Ens-DNN+ 0.911 0.918 0.887
ESM–Ens-LEAR+ 0.911 0.917 0.878
ESM–DNN+ 0.910 0.919 0.880
ESM–LEAR+ 0.902 0.913 0.876
ESM–LARX+ 0.892 0.917 0.881
ESM–RF+ 0.000 0.000 0.201
ESM–LSTM+ 0.542 0.521 0.403
ESM–Ens-DNN 0.899 0.911 0.878
ESM–Ens-LEAR 0.899 0.909 0.880
ESM–DNN 0.894 0.906 0.873
ESM–LEAR 0.889 0.900 0.868
ESM–LARX 0.842 0.866 0.812
ESM–RF 0.626 0.751 0.770
ESM–LSTM 0.553 0.534 0.416
37

7 Conclusion
Our study combines two fundamentally different state-of-the-art price fore-
casting models for electricity prices: a techno-economic energy systems model
and an econometric Ens-DNN model. The purpose is, first, to predict day-
ahead electricity prices in Germany with high accuracy and, second, to assess
the value of ESMs in short-term price forecasting. Our approach is robustified
with a strategic selection of additional econometric models, encompassing a
spectrum of statistical and machine learning models with distinct character-
istics. We included linear and nonlinear models, ensemble learning (decision
trees) models, and statistical state-of-the-art models. This deliberate diver-
sity allows us to thoroughly evaluate how these models respond to the com-
bination with an ESM, thereby assessing the robustness and generalisability
of our proposed methodology. Our research thus compares forecast accu-
racy caused by changes in two dimensions: the type of econometric model
adopted, and the set of the independent variables used in the economet-
ric models (fundamental regressors without the ESM’s market clearing price
estimator, fundamental regressors plus the ESM’s market clearing price esti-
mator, and the ESM’s market clearing price estimator as the only regressor).
Our analysis provides numerous insights. First, we re-confirm that techno-
economic ESMs alone are better suited for medium- to long-term forecasts
and cannot play to their strengths when directly forecasting high-frequency
day-ahead prices. Standing alone, most econometric models are more accu-
rate when parameterised with six fundamental independent variables. Sec-
ond, we find that the picture pivots by 180°when ESM results are used as an
alternative input in econometric models: all seven econometric models con-
sidered become more accurate when the fundamental regressors are replaced
by the ESM’s market clearing price. This result proves that ESM results
can improve electricity price forecasting accuracy considerably when used in
combination with econometric models.
Third, the highest overall EPF accuracy is achieved in the ESM–Ens-
DNN+ model, i.e. when the Ens–DNN model is run with the six fundamental
independent variables plus the ESM’s MCP. In fact, the five most accurate
econometric models achieve the highest forecasting in this combination of
regressors.8In the case of the ESM–Ens-DNN+, the accuracy outperforms
8Exceptions are the LARX and RF models, which show better performance when the
MCP is used as the sole regressor. This result, as well as the result that no further im-
provement is achieved in these two models when the MCP is added to the other regressors,
can be due to their limited feature selection capabilities, which suggests that these mod-
els benefit from a simplified input structure where a strong single predictor dominates,
38

the literature cited in this paper, but we have pointed out that results of
EPF models are difficult to compare as they vary in terms of input data and
time horizons.
From a practical standpoint, we have demonstrated the economic impact
of our forecasting models through the optimisation of an electricity stor-
age resource. Based on our forecasts, the operator optimised storage and
withdrawal timings to maximise the profit contributions. Overall, the most
accurate price forecasts also deliver the highest profit contributions, in par-
ticular the ESM–Ens-DNN+. A notable exception is the individual ESM,
which is stronger in deciding the optimal dispatch of the storage than it is in
forecasting exact prices. Our analysis provides investors with a diverse set
of models to choose from. In particular, we allow them to balance forecast
accuracy with profit objectives and model complexity, effectively balancing
the cost-benefit trade-offs of each forecasting model. With this flexibility,
investors can align their financial goals with market conditions and make
informed decisions.
Although we employ state-of-the-art models and other models with unique
strengths and characteristics, there remain a variety of powerful econometric
models that are worth testing further by applying our methodology. This is
especially pertinent given the rapidly growing body of literature on electric-
ity price forecasting including probabilistic methods (Marcjasz et al. 2023)
and hybrid models, which typically combine data decomposition algorithms,
statistical and machine learning models, and optimisation methods (Ehsani
et al. 2024). One potential future research direction could involve testing our
methodology across various econometric models and multiple datasets.
Acknowledgements
Souhir Ben Amor gratefully acknowledges the support of the German Fed-
eral Ministry for Economic Affairs and Climate Action (BMWK), which
funded this research through the FOCCSI 2 project (Forecast Optimiza-
tion by Correction and Combination Methods for System Integration; Award
No. 03EI1061) as part of the 7th Energy Research Program. Thomas Möbius
thanks the German Federal Ministry of Economic Affairs and Climate Action
through the research project ProKoMo, Award No. 03ET4067A within the
Systems Analysis Research Network of the 6th energy research program. Fe-
lix Müsgens gratefully acknowledges financial support from the Federal Min-
istry of Education and Research, Award No. 19FS2032C, as well as the Ger-
avoiding potential noise from weaker variables.
39

man Federal Government, the Federal Ministry of Education and Research,
and the State of Brandenburg within the framework of the joint project EIZ:
Energy Innovation Center (project numbers 85056897 and 03SF0693A) with
funds from the Structural Development Act (Strukturstärkungsgesetz) for
coal-mining regions.
References
Abedinia, O, Nima Amjady, Miadreza Shafie-Khah, and Joao PS Catalão
(2015). “Electricity price forecast using combinatorial neural network
trained by a new stochastic search method”. In: Energy Conversion and
Management 105, pp. 642–654. doi:https: // doi. org/ 10. 1016/ j.
enconman.2015.08.025.
Bello, Antonio, Derek Bunn, Javier Reneses, and Antonio Muñoz (2016).
“Parametric density recalibration of a fundamental market model to fore-
cast electricity prices”. In: Energies 9.11, p. 959.
Bello, Antonio, Javier Reneses, Antonio Muñoz, and Andrés Delgadillo (2016).
“Probabilistic forecasting of hourly electricity prices in the medium-term
using spatial interpolation techniques”. In: International Journal of Fore-
casting 32.3, pp. 966–980.
Bello Morales, Antonio, Derek W Bunn, Javier Reneses Guillén, and Anto-
nio Muñoz San Roque (2017). “Medium-term probabilistic forecasting of
electricity prices: a hybrid approach”. In.
Ben Amor, Souhir, Heni Boubaker, and Lotfi Belkacem (2018a). “Forecast-
ing electricity spot price for Nord Pool market with a hybrid k-factor
GARMA–LLWNN model”. In: Journal of Forecasting 37.8, pp. 832–851.
— (2018b). “ Forecasting Electricity Spot Price with Generalized Long Mem-
ory Modeling: Wavelet and Neural Network”. In: International Journal of
Economics and Management Engineering 11.9, pp. 2307–2323.
Bento, PMR, JAN Pombo, MRA Calado, and SJPS Mariano (2018). “A bat
optimized neural network and wavelet transform approach for short-term
price forecasting”. In: Applied energy 210, pp. 88–97.
Bergstra, James, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl (2011).
“Algorithms for hyper-parameter optimization”. In: Advances in neural
information processing systems 24.
BNetzA (2021). Kraftwerksliste der Bundesnetzagentur. Accessed on 15-05-
2021. url:https: / /www .bundesnetzagentur . de/ DE/ Sachgebiete /
ElektrizitaetundGas/Unternehmen_Institutionen/Versorgungssicherheit/
Erzeugungskapazitaeten/Kraftwerksliste/start.html.
40

Borenstein, Severin, James B. Bushnell, and Frank A. Wolak (Dec. 2002).
“Measuring Market Inefficiencies in California’s Restructured Wholesale
Electricity Market”. In: American Economic Review 92.5, pp. 1376–1405.
doi:10.1257/000282802762024557.url:https://www.aeaweb.org/
articles?id=10.1257/000282802762024557.
Bunn, Derek W (2000). “Forecasting loads and prices in competitive power
markets”. In: Proceedings of the IEEE 88.2, pp. 163–169.
Chandrashekar, Girish and Ferat Sahin (2014). “A survey on feature selection
methods”. In: Computers & Electrical Engineering 40.1, pp. 16–28.
Chang, Zihan, Yang Zhang, and Wenbo Chen (2019). “Electricity price pre-
diction based on hybrid model of adam optimized LSTM neural network
and wavelet transform”. In: Energy 187, p. 115804.
Chen, Qi, Zhaopeng Meng, Xinyi Liu, Qianguo Jin, and Ran Su (2018).
“Decision variants for the automatic determination of optimal feature
subset in RF-RFE”. In: Genes 9.6, p. 301.
Christensen, Tim M, A Stan Hurn, and Kenneth A Lindsay (2012). “Forecast-
ing spikes in electricity prices”. In: International Journal of Forecasting
28.2, pp. 400–411.
Conejo, Antonio J, Javier Contreras, Rosa Espínola, and Miguel A Plazas
(2005). “Forecasting electricity prices for a day-ahead pool-based electric
energy market”. In: International journal of forecasting 21.3, pp. 435–462.
De Marcos, Rodrigo A, Antonio Bello, and Javier Reneses (2019a). “Electric-
ity price forecasting in the short term hybridising fundamental and econo-
metric modelling”. In: Electric Power Systems Research 167, pp. 240–251.
— (2019b). “ Short-term electricity price forecasting with a composite fundamental-
econometric hybrid methodology”. In: Energies 12.6, p. 1067.
— (2019c). “ Electricity price forecasting in the short term hybridising funda-
mental and econometric modelling”. In: Electric Power Systems Research
167, pp. 240–251. issn: 0378-7796. doi:10.1016/j.epsr.2018.10.034.
Destatis Statistisches Bundesamt (2021). Erzeugerpreise gewerblicher Pro-
dukte (Inlandsabsatz). Preise für leichtes Heizöl, Motorenbenzin und Diesel.
Accessed on 25-01-2021. url:https://www.destatis.de/DE/Themen/
Wirtschaft / Preise / Erzeugerpreisindex - gewerbliche - Produkte /
_inhalt.html.
EBC (2021). Europe Beyond Coal: European Coal Plant Database, 25 Jan
2021.url:https://beyond-coal.eu/database/.
EEX (2021). European Energy Exchange: Historic gas price data. Accessed
on 03-01-2022.
Ehsani, Behdad, Pierre-Olivier Pineau, and Laurent Charlin (2024). “Price
forecasting in the Ontario electricity market via TriConvGRU hybrid
41

model: Univariate vs. multivariate frameworks”. In: Applied Energy 359,
p. 122649.
ENTSO-E (2018). TYNDP 2018 Scenario Report. Accessed on 23-02-2023.
url:https://tyndp.entsoe.eu/tyndp2018/scenario-report.
ENTSO-E Transparency Platform (2021a). Actual Generation per Production
Type.url:https://transparency.entsoe.eu/.
— (2021b). Day-ahead prices.url:https://transparency.entsoe.eu/.
— (2021c). Forecasted Transfer Capacities - Day Ahead.url:https : / /
transparency.entsoe.eu/.
— (2021d). Generation Forecast - Day ahead.url:https://transparency.
entsoe.eu/.
— (2021e). Installed Capacities per Production Type.url:https://transparency.
entsoe.eu/.
— (2021f). Total Load - Day Ahead / Actual.url:https://transparency.
entsoe.eu/.
— (2021g). Unavailability of Production and Generation Units.url:https:
//transparency.entsoe.eu/.
European Commission (2021). Eurostat Statistics Database. Accessed on 12-
12-2021. url:https://ec.europa.eu/eurostat/data/database.
Gao, Wei, Vali Sarlak, Mohammad Reza Parsaei, and Mohammad Ferdosi
(2018). “Combination of fuzzy based on a meta-heuristic algorithm to pre-
dict electricity price in an electricity markets”. In: Chemical Engineering
Research and Design 131, pp. 333–345.
Gareth, James, Witten Daniela, Hastie Trevor, and Tibshirani Robert (2013).
An introduction to statistical learning: with applications in R. Spinger.
Gholipour Khajeh, Morteza, Akbar Maleki, Marc A Rosen, and Mohammad
H Ahmadi (2018). “Electricity price forecasting using neural networks
with an improved iterative training algorithm”. In: International Journal
of Ambient Energy 39.2, pp. 147–158.
Gianfreda, Angelica, Francesco Ravazzolo, and Luca Rossini (2020). “Com-
paring the forecasting performances of linear models for electricity prices
with high RES penetration”. In: International Journal of Forecasting 36.3,
pp. 974–986.
Gonzalez, Virginia, Javier Contreras, and Derek W Bunn (2011). “Fore-
casting power prices using a hybrid fundamental-econometric model”. In:
IEEE Transactions on Power Systems 27.1, pp. 363–372.
Goodfellow, Ian, Yoshua Bengio, and Aaron Courville (2016). Deep learning.
MIT press.
Hastie, Trevor, Robert Tibshirani, Jerome H Friedman, and Jerome H Fried-
man (2009). The elements of statistical learning: data mining, inference,
and prediction. Vol. 2. Springer.
42

Hinton, Geoffrey E, Simon Osindero, and Yee-Whye Teh (2006). “A fast learn-
ing algorithm for deep belief nets”. In: Neural computation 18.7, pp. 1527–
1554.
Hirth, Lion (2013). “The market value of variable renewables: The effect of
solar wind power variability on their relative price”. In: Energy Economics
38, pp. 218–236.
Hochreiter, Sepp and Jürgen Schmidhuber (1997). “Long short-term mem-
ory”. In: Neural computation 9.8, pp. 1735–1780.
JAO Joint Allocation Office (2021). ATC for Shadow Auction. Accessed on
20-12-2021. url:https://www.jao.eu/implict-allocation.
Jiang, LianLian and Guoqiang Hu (2018). “Day-ahead price forecasting for
electricity market using long-short term memory recurrent neural net-
work”. In: 2018 15th International Conference on Control, Automation,
Robotics and Vision (ICARCV). IEEE, pp. 949–954.
Kanamura, Takashi and Kazuhiko ¯
Ohashi (2007). “A structural model for
electricity prices with spikes: Measurement of spike risk and optimal poli-
cies for hydropower plant operation”. In: Energy economics 29.5, pp. 1010–
1032.
Karakatsani, Nektaria V and Derek W Bunn (2008a). “Forecasting electricity
prices: The impact of fundamentals and time-varying coefficients”. In:
International Journal of Forecasting 24.4, pp. 764–785.
— (2008b). “Intra-day and regime-switching dynamics in electricity price
formation”. In: Energy Economics 30.4, pp. 1776–1797.
Kawakami, Kazuya (2008). “Supervised sequence labelling with recurrent
neural networks”. PhD thesis. Technical University of Munich.
Kristiansen, Tarjei (2012). “Forecasting Nord Pool day-ahead prices with an
autoregressive model”. In: Energy Policy 49, pp. 328–332.
Lago, Jesus, Fjo De Ridder, and Bart De Schutter (2018). “Forecasting spot
electricity prices: Deep learning approaches and empirical comparison of
traditional algorithms”. In: Applied Energy 221, pp. 386–405.
Lago, Jesus, Grzegorz Marcjasz, Bart De Schutter, and Rafał Weron (2021).
“Forecasting day-ahead electricity prices: A review of state-of-the-art al-
gorithms, best practices and an open-access benchmark”. In: Applied En-
ergy 293, p. 116983.
Li, Pan, Baosen Zhang, Yang Weng, and Ram Rajagopal (2017). “A sparse
linear model and significance test for individual consumption prediction”.
In: IEEE Transactions on Power Systems 32.6, pp. 4489–4500.
Li, Wei and Denis Mike Becker (2021). “Day-ahead electricity price predic-
tion applying hybrid models of LSTM-based deep learning methods and
feature selection algorithms under consideration of market coupling”. In:
Energy 237, p. 121543.
43

Liu, Huan and Hiroshi Motoda (2007). Computational methods of feature
selection. CRC Press.
Liu, Huan and Lei Yu (2005). “Toward integrating feature selection algo-
rithms for classification and clustering”. In: IEEE Transactions on knowl-
edge and data engineering 17.4, pp. 491–502.
Ludwig, Nicole, Stefan Feuerriegel, and Dirk Neumann (2015). “Putting Big
Data analytics to work: Feature selection for forecasting electricity prices
using the LASSO and random forests”. In: Journal of Decision Systems
24.1, pp. 19–36.
Maciejowska, Katarzyna, Weronika Nitka, and Tomasz Weron (July 2021).
“Enhancing load, wind and solar generation for day-ahead forecasting of
electricity prices”. In: Energy Economics 99, p. 105273. issn: 01409883.
doi:10 . 1016 / j . eneco . 2021 . 105273.url:https : / / linkinghub .
elsevier.com/retrieve/pii/S014098832100178X.
Maciejowska, Katarzyna and Jakub Nowotarski (2016). “A hybrid model for
GEFCom2014 probabilistic electricity price forecasting”. In: International
Journal of Forecasting 32.3, pp. 1051–1056. issn: 0169-2070. doi:10 .
1016/j.ijforecast.2015.11.008.
Marcjasz, Grzegorz, Michał Narajewski, Rafał Weron, and Florian Ziel (2023).
“Distributional neural networks for electricity price forecasting”. In: En-
ergy Economics 125, p. 106843. issn: 0140-9883. doi:https://doi.org/
10.1016/j.eneco.2023.106843.
Mei, Jie, Dawei He, Ronald Harley, Thomas Habetler, and Guannan Qu
(2014). “A random forest method for real-time price forecasting in New
York electricity market”. In: 2014 IEEE PES General Meeting| Conference
& Exposition. IEEE, pp. 1–5.
Möbius, Thomas, Iegor Riepin, Felix Müsgens, and Adriaan H. van der Wei-
jde (2023). “Risk aversion and flexibility options in electricity markets”.
In: Energy Economics 126, p. 106767. issn: 0140-9883. doi:https :
/ / doi . org / 10 . 1016 / j . eneco . 2023 . 106767.url:https : / / www .
sciencedirect.com/science/article/pii/S0140988323002657.
Möbius, Thomas, Mira Watermeyer, Oliver Grothe, and Felix Müsgens (2023).
“Enhancing Energy System Models Using Better Load Forecasts”. In: En-
ergy Systems.doi:10.1007/s12667-023-00590-3.
Mount, Timothy D, Yumei Ning, and Xiaobin Cai (2006). “Predicting price
spikes in electricity markets using a regime-switching model with time-
varying parameters”. In: Energy Economics 28.1, pp. 62–80.
Müsgens, Felix (2006). “Quantifying market power in the German wholesale
electricity market using a dynamic multi-regional dispatch model”. In:
The Journal of Industrial Economics 54.4, pp. 471–498.
44

Naz, Aqdas, Muhammad Umar Javed, Nadeem Javaid, Tanzila Saba, Musaed
Alhussein, and Khursheed Aurangzeb (2019). “Short-term electric load
and price forecasting using enhanced extreme learning machine optimiza-
tion in smart grids”. In: Energies 12.5, p. 866.
Nikodinoska, Dragana, Mathias Käso, and Felix Müsgens (2022). “Solar and
wind power generation forecasts using elastic net in time-varying forecast
combinations”. In: Applied Energy 306, p. 117983.
Niu, Xinsong and Jiyang Wang (2019). “A combined model based on data pre-
processing strategy and multi-objective optimization algorithm for short-
term wind speed forecasting”. In: Applied Energy 241, pp. 519–539. doi:
https://doi.org/10.1016/j.apenergy.2019.03.097.
Nogales, Francisco Javier, Javier Contreras, Antonio J Conejo, and Rosario
Espínola (2002). “Forecasting next-day electricity prices by time series
models”. In: IEEE Transactions on power systems 17.2, pp. 342–348.
Nowotarski, Jakub and Rafał Weron (2018). “Recent advances in electricity
price forecasting: A review of probabilistic forecasting”. In: Renewable and
Sustainable Energy Reviews 81, pp. 1548–1568.
Open Power System Data (2020a). Data Package National Generation Ca-
pacity. Version 2019-12-02. url:https://doi.org/10.25832/national_
generation_capacity/2019-12-02.
— (2020b). Data Package Weather Data. Version 2020-09-16. url:https:
//doi.org/10.25832/weather_data/2020-09-16.
Peesapati, Rajagopal, Niranjan Kumar, et al. (2017). “Electricity price fore-
casting and classification through wavelet–dynamic weighted PSO–FFNN
approach”. In: IEEE Systems Journal 12.4, pp. 3075–3084.
Pórtoles, Javier, Camino González, and Javier M Moguerza (2018). “Electric-
ity price forecasting with dynamic trees: a benchmark against the random
forest approach”. In: Energies 11.6, p. 1588.
Qussous, Ramiz, Nick Harder, and Anke Weidlich (2022). “Understanding
power market dynamics by reflecting market interrelations and flexibility-
oriented bidding strategies”. In: Energies 15.2, p. 494.
Regelleistung.net (2018). List of tenders capacity. Accessed on 12-12-2021.
url:https://www.regelleistung.net/ext/.
Sandbag (2021). CO2 emission allowance.url:https : / / sandbag . be /
index.php/carbon-price-viewer/.
Schröder, Andreas, Friedrich Kunz, Jan Meiss, Roman Mendelevitch, and
Christian Von Hirschhausen (2013). Current and prospective costs of elec-
tricity generation until 2050. Tech. rep. DIW data documentation.
Sensfuß, Frank, Mario Ragwitz, and Massimo Genoese (2008). “The merit-
order effect: A detailed analysis of the price effect of renewable electricity
45

generation on spot market prices in Germany”. In: Energy Policy 36.8,
pp. 3086–3094. issn: 0301-4215. doi:10.1016/j.enpol.2008.03.035.
Shao, Zhen, Yudie Yang, Qingru Zheng, Kaile Zhou, Chen Liu, and Shanlin
Yang (2022). “A pattern classification methodology for interval forecasts
of short-term electricity prices based on hybrid deep neural networks: A
comparative analysis”. In: Applied Energy 327, p. 120115.
Shao, Zhen, Qingru Zheng, Chen Liu, Shuangyan Gao, Gang Wang, and
Yan Chu (2021). “A feature extraction-and ranking-based framework for
electricity spot price forecasting using a hybrid deep neural network”. In:
Electric Power Systems Research 200, p. 107453.
Singh, Nitin, Saddam Hussain, and Shailesh Tiwari (2018). “A PSO-based
ANN model for short-term electricity price forecasting”. In: Ambient Com-
munications and Computer Systems. Springer, pp. 553–563.
Tibshirani, Robert (1996). “Regression shrinkage and selection via the lasso”.
In: Journal of the Royal Statistical Society: Series B (Methodological)
58.1, pp. 267–288.
UBA (2020). Umweltbundesamt: Datenbank “Kraftwerke in Deutschland".
Accessed on 20-02-2020. url:https : / / www . umweltbundesamt . de /
dokument/datenbank-kraftwerke-in-deutschland.
Uniejewski, Bartosz and Katarzyna Maciejowska (2023). “Lasso principal
component averaging: A fully automated approach for point forecast pool-
ing”. In: International Journal of Forecasting 39.4, pp. 1839–1852.
Uniejewski, Bartosz, Jakub Nowotarski, and Rafał Weron (2016). “Auto-
mated variable selection and shrinkage for day-ahead electricity price
forecasting”. In: Energies 9.8, p. 621.
Uniejewski, Bartosz and Rafał Weron (2018). “Efficient forecasting of elec-
tricity spot prices with expert and LASSO models”. In: Energies 11.8,
p. 2039.
Watermeyer, Mira, Thomas Möbius, Oliver Grothe, and Felix Müsgens (2023).
A hybrid model for day-ahead electricity price forecasting: Combining fun-
damental and stochastic modelling. arXiv: 2304.09336 [econ.EM].
Weron, Rafał (2014). “Electricity price forecasting: A review of the state-of-
the-art with a look into the future”. In: International journal of forecasting
30.4, pp. 1030–1081.
Weron, Rafał and Adam Misiorek (2008). “Forecasting spot electricity prices:
A comparison of parametric and semiparametric time series models”. In:
International journal of forecasting 24.4, pp. 744–763.
Xiong, Xiaoping and Guohua Qing (2022). “A hybrid day-ahead electricity
price forecasting framework based on time series”. In: Energy, p. 126099.
46

Ziel, Florian (2016). “Forecasting electricity spot prices using lasso: On cap-
turing the autoregressive intraday structure”. In: IEEE Transactions on
Power Systems 31.6, pp. 4977–4987.
Ziel, Florian, Rick Steinert, and Sven Husmann (2015). “Efficient model-
ing and forecasting of electricity spot prices”. In: Energy Economics 47,
pp. 98–111.
Ziel, Florian and Rafał Weron (2018). “Day-ahead electricity price forecasting
with high-dimensional structures: Univariate vs. multivariate modeling
frameworks”. In: Energy Economics 70, pp. 396–420.
Zou, Hui and Trevor Hastie (2005). “Regularization and variable selection
via the elastic net”. In: Journal of the royal statistical society: series B
(statistical methodology) 67.2, pp. 301–320.
8 Appendix
8.1 Descriptive Statistics
Table 5: Descriptive statistics for the
German electricity price
Electricity price
Observations 52.608
Mean 34.563
std 16.608
min -130.090
25% 25.920
50% 34.020
75% 43.590
Max 200.040
kurtosis 7.808
skewness -0.216
Jarque_bera test 81047.046
(0.000)***
ADF test -15.288
(0.000)***
Figure 8: Density function
Levels of the significance of Jarque-Bera and ADF
tests are indicated between squared brackets.
*** Denotes significance at the 1% level.
47

8.2 Robustness Based on Statistical Testing
Besides evaluation metrics, it is important to determine whether any differ-
ence in accuracy is statistically significant. To determine whether the forecast
accuracy difference is real and not just due to random differences between
forecasts, it is crucial to perform statistical tests. It has been argued that
the Giacomini-White (GW) test is preferable because it can be viewed as
a generalisation of the Diebold-Mariano (DM) test (the reader may refer to
Lago, Marcjasz, et al. 2021 for more details).
We perform the multivariate GW test jointly for all hours by using the mul-
tivariate loss differential series or the daily loss differential series:
∆A,B
d=
εA
d
p−
εB
d
p,(26)
where εA
d,h =pd,h −ˆpd,h is the prediction error of model Afor day dand hour
h.
According to the results of the multivariate GW test using the L1norm in
Equation 26, we have the following loss differential series:
∆A,B
d=
24
X
h=1 εA
d,h−
24
X
h=1 εB
d,h(27)
In Figure 9 we display the results for all models.
To illustrate the range of obtained p−values, we use heat maps arranged
as chessboards. Given that this test is run at a 5% significance level, results
are interpreted based on p−values as follows:
•p−values < 0.05 (green) indicate significant outperformance of a model
on the X-axis (better) compared to the model on the Y-axis (worse).
•p−values > 0.05 indicate significant underperformance of the model
on the X-axis (worse) compared to the model on the Y-axis (better).
The following results were withdrawn:
•The ESM–Ens-DNN+ column is green. This indicates that this model’s
predictions are statistically significantly better than those of all the
other models.
•The forecasts of all econometric models combined with the ESM are
statistically significantly better than their stand-alone versions. This
shows the importance of the ESM’s MCP in increasing the forecasting
accuracy of the econometric models.
48

Figure 9: Results of the GW test
From a statistical standpoint, the MCP contribution of the ESM enhances
the performance of individual econometric models. Moreover, by examining
the results of models both with and without independent variables, in ad-
dition to the MCP (e.g., ESM-DNN+ vs ESM-DNN), we can observe how
different models respond to the inclusion of extra predictors. This interac-
tion allows us to differentiate between models that perform better with a
larger number of features and those that excel with a few, but more potent,
features.
49