# Finding Stable Price Zones in European Electricity Markets: Aiming to Square
the Circle?

Teodora Dobos<EMAIL_ADDRESS>Martin Bichler<EMAIL_ADDRESS>Johannes
Knörr<EMAIL_ADDRESS>
###### Abstract

The European day-ahead electricity market is split into multiple bidding zones
with a uniform price. The increase in renewables leads to a growing number of
interventions in the generation of energy sources and increasing redispatch
costs. To ensure efficient congestion management, the EU Commission mandated a
Bidding Zone Review (BZR) to reevaluate the configuration of European bidding
zones. An integral part of this process was a locational marginal pricing
study. Based on this study, alternative bidding zone configurations were
proposed. These bidding zones shall be stable and robust over time. For
Germany, four configurations were suggested. We analyzed the proposed
configurations considering different clustering algorithms and periods based
on the publicly released data set. We found that the configurations do not
reduce the price standard deviations within zones much, and the average prices
across zones are similar. Other configurations identified based on clustering
the prices lead to lower price variance but they are not geographically
coherent. Independent of the clustering features and algorithms used, the
resulting clusters are not stable over time. While the costs of a bidding zone
split in Germany are high, the effect on prices would be low based on an
analysis of the new BZR data.

###### keywords:

Energy pricing , Price zones , Bidding Zone Review , Electricity markets ,
Energy policy

††journal: Energy Policy

[1]organization=School of Computation, Information and Technology, Technical
University of Munich,addressline=Boltzmannstrasse 3, city=Garching,
postcode=85748, country=Germany

##  1 Introduction

The European electricity market is split into multiple bidding zones (BZs),
aka. price zones. Within these zones, market participants can submit bids and
offers, which are subsequently matched without considering the capacities of
the transmission network.111We will use the terms price zone and bidding zone
interchangeably. BZs are generally defined depending on national borders, such
that a country represents a single bidding zone (e.g., France and Poland).
However, there are also countries that are divided into several BZs, such as
Norway, Sweden and Italy, or BZs that consist of more than one country, like
the German-Luxembourgish bidding zone.

An important characteristic of BZs is that in the day-ahead market, they
implement a uniform pricing mechanism, meaning that a single market-clearing
electricity price applies to all participants belonging to a BZ. A uniform
price in a BZ promotes transparency due to a simple single price signal in a
large region, and it limits the market power that individual parties can
exercise. However, a uniform electricity price also leads to significant
negative ramifications. There are no local incentives to adapt to changes in
the volatile supply of renewables, there are significant welfare losses due to
the uniform price constraint, and large redispatch costs. The latter is
because transmission constraints are ignored within a BZ when supply is
matched with demand (Energywende, 2018), and the market-based allocation is
not physically feasible. In addition, uniform prices do not set appropriate
investment incentives.

As an example, in the German-Luxembourgish BZ strong winds along the North Sea
and Baltic Coast can contribute to elevated power production from wind farms
in northern Germany. The energy produced in the north must be transported to
the demand centers in the south of the country. However, the transmission
lines connecting these regions might become congested and thus electricity
cannot be delivered as determined in the day-ahead market. To stabilize the
grid, redispatch by the Transmission System Operators (TSOs) is required. TSOs
often have to throttle the generation of northern conventional and wind power
plants. At the same time, they instruct power plants in the south to increase
energy production. Such redispatch actions incur substantial costs. In 2023,
the redispatch costs in Germany were 3.1 billion
Euros.222https://www.bundesnetzagentur.de/

###  1.1 The Bidding Zone Review

Given these considerations, in August 2022 the European Union Agency for the
Cooperation of Energy Regulators (ACER) decided that alternative bidding zone
configurations should be analyzed as part of the Bidding Zone Review (BZR),
which is organized by the European Network of Transmission System Operators
for Electricity (ENTSO-E, 2023). The goals of the BZR are described on a high
level in different documents. According to Commission Regulation (EU)
2015/1222 and 2019/943, bidding zones “should be defined in a manner to ensure
efficient congestion management and overall market efficiency” (Commission,
2015) and “shall be based on long-term, structural congestions in the
transmission network” (Commission, 2019).

Annex I of ACER’s decision on alternative bidding zone configurations to be
considered in the bidding zone review process (ACER, 2022a ) describes that
different configurations were identified by solving a clustering problem
“based on nodal prices as clustering feature”. The document argues that nodal
prices “can be used as proxies for economic efficiency, in line with the
objectives of the Electricity Regulation”, and that the objective of
clustering algorithms is to “minimize price dispersion within each bidding
zone.” The document discusses three clustering algorithms: constrained K-Means
clustering, Spectral Clustering with constrained K-Means, and Spectral
Clustering with Constrained Deterministic Iterative Refinement Clustering
(CDIRC). Moreover, a constraint related to the minimum size of each zone was
included.333The threshold depends on the number of price zones under analysis
and is between 7% and 10% of the nodes.

Based on this ACER methodology (ACER, 2020), TSOs were tasked to calculate
locational marginal prices (essentially, a nodal price for each substation in
the transmission grid444https://www.tennet.eu/news/bidding-zone-review-tsos-
investigate-alternative-bidding-zone-configurations) for the target year 2025,
for which generation and demand scenarios were created. These scenarios
originate from ENTSO-E’s Data & Models working group. Weather data from 1989,
1995, and 2009 was used to create scenarios for representative reference
climate years. The grid model, the demand, the generation capacities, and
prices for gas, coal and CO2 were based on predictions for 2025. ENTSO-E
computed nodal prices for the respective weeks in a report published in June
2022 (ENTSO-E, 2022). As outlined above (ACER, 2022b ), the time series of
nodal prices represented the clustering features that were used to partition
the set of nodes in the transmission network into a predefined number of
clusters or BZs. A BZ configuration or a clustering thus represents the
assignment of every node to a specific cluster or price zone. Finally, ACER
suggested four alternative bidding zone configurations for Germany ACER, 2022a
based on the results from the Locational Marginal Pricing (LMP) Study
(ENTSO-E, 2022, 2023). The evaluation of these configurations proposed by ACER
in the BZR is ongoing at the time of writing. In this process, the TSOs are
evaluating 22 indicators related to market efficiency, network security, and
the stability and robustness of the proposed configurations over time (ACER,
2020).

###  1.2 Contributions

In this paper, we leverage the data from the LMP study provided by ENTSO-E and
study how stable and robust the configurations of price zones (or,
clusterings) proposed by ACER are with respect to different time frames,
different inputs (i.e., nodal prices with or without geographic coordinates),
and different clustering algorithms. We define three criteria for analyzing
clustering stability and robustness and conduct a thorough empirical
evaluation of the suggested configurations using the data published by
ENTSO-E. Following ACER’s methodology, we implement K-Means and Spectral
Clustering and evaluate the clusterings computed with these methods. We want
to understand if the clustering algorithms compute the same clusters when
nodal prices of different time periods are considered. How different are
clusters resulting from different clustering algorithms? Also, how similar are
the clusters of a particular configuration with respect to prices, and how
large are the price standard deviations within clusters?

We find that the configurations proposed by ACER are not stable across time
and algorithm used. Moreover, the configurations do not reduce the price
standard deviations within zones considerably, and the average prices between
zones are similar. If we recompute the clusters and take only prices into
account555This means that we do not include constraints in the clustering
computation such that we obtain balanced clusters in terms of either the
number of nodes or geographical coverage., we can reduce the price standard
deviation within clusters further. However, the resulting clusters are not of
equal size and are not geographically coherent. Importantly, neither the
configurations proposed by ACER nor the clusters that we computed are stable
across time. Depending on the time frame taken into account, different
clusterings would result.

Note that our analysis is performed based on the same target year 2025 as in
the BZR. Given the significant changes in supply and demand and in the weather
over time, we can expect substantial changes in the supply in future years. In
particular, Germany has ambitious goals concerning the share of renewable
energies and aims for 80% in
2030.666https://www.energiewechsel.de/KAENEF/Navigation/DE/Energiewechsel/Erneuerbare-
Energien/erneuerbare-energien.html There is a danger that even if a robust
configuration could be selected for 2025, it will be outdated soon after. The
data from the LMP study of the BZR can be considered the most comprehensive
data set based on up-to-date information from European TSOs. Based on this
data set, the effect of a zonal split on prices and redispatch costs is low,
while the expected costs of a zonal split are high.

This paper is structured as follows. In Section 2 we discuss the literature
related to our work, while in Section 3 we describe the algorithms we
implement to compute clusterings and introduce the metrics considered to
evaluate stability. The data we use in our experiments is discussed in Section
4, while the experimental results are presented in Section 5. Finally, we
summarize our main findings and their policy implications in Section 6.

##  2 Related Literature

This paper complements existing literature computing bidding zone
configurations. In particular, we draw on a unique data set with LMP prices
that is provided by ENTSO-E. Bovo et al., (2019) identifies the main network
indicators that should be considered for defining suitable bidding zone
configurations. LMPs and power transfer distribution factors (PTDFs) represent
such indicators. While some papers focus on identifying configurations by
clustering PTDFs (e.g., (Kumar et al., 2004) and (Kłos et al., 2014)), most
works use LMPs as clustering features (Stoft, 1997). Burstedde, (2012)
computes LMPs for the scenario years 2015 and 2020 considering a simplified
72-node representation of the European grid, and evaluates the shapes and
sizes of the zones obtained by clustering these LMPs using hierarchical
clustering. Breuer et al., (2013) implement a genetic algorithm and apply it
on a dataset consisting of 380 kV nodes considering a projected system for
2016 and 2018. Based on six scenarios, Felling & Weber, (2018) compute robust
price zone configurations for Central Western Europe by employing hierarchical
clustering based on the LMPs of approximately 2,200 nodes and assigning
weights to nodes depending on their demand- and supply situation. These
configurations are robust in the sense that LMPs from multiple scenarios are
integrated into their computation. Considering a network model consisting of
around 2,700 nodes, Brouhard et al., (2023) design nine zone delineations for
Continental Europe by implementing K-Means and hierarchical clustering. The
authors calculate LMPs for multiple, independent single-period optimization
problems (i.e., optimal power flow with direct current approximation), and use
geographic coordinates and these prices as clustering features.

Sarfati et al., (2015) introduce five indicators to assess the impact of
bidding zone configurations in zonally-priced electricity markets. The
indicators include commercial exchange and welfare evolution, as well as price
convergence and divergence, and were analyzed on the Nordic 32-bus example
system. Using a five-node example with a single congested line, Bjorndal &
Jornsten, (2001) show that defining zones based on nodal prices is not
guaranteed to result in a zonal configuration with maximum welfare. The
authors formulate a mixed-integer nonlinear program that, for a given number
of zones, computes a zonal configuration with maximum welfare.

In terms of the German-Luxembourgish bidding zone, Zinke, (2023) identifies
north-south splits using hierarchical clustering based on the LMPs associated
to approximately 500 nodes that are computed within a linear market and grid
model for four scenario years. The author considers uncertain factors such as
short-term weather patterns and long-term system changes to evaluate the
potential decrease in redispatch costs resulting from such splits. Moreover,
Ambrosius et al., (2020) consider a scenario for the year 2035 and determine
welfare-maximizing price zones and available transfer capacities for a
simplified representation of the German market area consisting of 28 nodes.
Such zones are identified based on a mixed-integer nonlinear model that
incorporates a graph partitioning problem on the first level (Grimm et al.,
2019, Kleinert & Schmidt, 2019). The prior literature used different data
sources for these analyses.

##  3 Methodology

The process of computing price zones can be reduced to solving a clustering
problem. Formally, given a set of $N$ $D$-dimensional data points
$X=\\{x_{1},x_{2},...,x_{N}\\}$ with $x_{i}\in\mathbb{R}^{D}$ and a number of
clusters $k$, the objective of the clustering problem is to find for each
point $x_{i}$ the corresponding assignment $z_{i}\in\\{1,...,k\\}$ to one of
the $k$ clusters such that (1) points within the same cluster are similar to
each other and (2) points between different clusters are dissimilar from each
other. A clustering is a solution to this problem and is denoted by
$\mathcal{C}=\\{C_{1},\dots,C_{k}\\}$, where $C_{i}=\\{x_{j}:z_{j}=i\\}$ is a
cluster. The centroid of a cluster $C_{i}$ is defined as the mean of the data
points assigned to $C_{i}$ and is depicted by $c_{i}$. In our setting, the
data points are the nodes in the transmission network and the dimensions
represent the features we use to compute clusters (see Section 4.1).

###  3.1 Clustering Algorithms

Following ACER’s approach to identify alternative price zone configurations,
we consider two clustering algorithms in this paper. ACER used three
clustering approaches (ACER, 2022b ), i.e., K-Means, Spectral Clustering with
constrained K-Means and Spectral Clustering with CDIRC, but the implementation
details are not publicly available. Therefore, it is impossible to reproduce
the exact algorithmic configuration ACER implemented. Instead, we consider the
same family of clustering methods to evaluate alternative price zone
configurations. In what follows, we briefly describe the clustering algorithms
we implement.

####  3.1.1 K-Means

K-Means (Lloyd, 1982) is a simple algorithm that clusters data points by
minimizing within-cluster variances, i.e., squared Euclidean distances. The
method consists of three main steps. In the first step, the centroids are
initialized randomly. In the second step, each data point is assigned to its
nearest centroid with respect to the Euclidean distance, specifically to its
corresponding cluster. The third step consists of updating the centroids,
given the new points assigned to the clusters. The last two steps are repeated
until the difference between the old and the new centroids is below a
threshold. Importantly, the final clusters might differ depending on the
initialization of centroids.

A constrained version of K-Means was considered in the BZR analysis (ACER,
2022b ), in which the centroids are initialized such that they are spread out
spatially. We note that this version of K-Means is known as K-Means++ (Arthur
& Vassilvitskii, 2007). We implement the constrained version of K-Means by
Bennett et al., (2000) and incorporate constraints for the minimum size of
points included in each cluster. We update the clusters until the squared
Euclidean distance between each old and new centroid is smaller than 0.1.
Moreover, we standardize the data such that all features have a mean of 0 and
a standard deviation of 1.

####  3.1.2 Spectral Clustering

Spectral clustering is a technique used for clustering network data. Given a
graph $G=(V,E)$, the method clusters the set of vertices $V$ based on a low-
dimensional spectral embedding of $G$ and consists of three main steps. First,
the Laplacian matrix $L$ is constructed. Then, the first $r$ eigenvectors
(i.e., the eigenvectors corresponding to the $r$ smallest eigenvalues of $L$)
are computed. In the last step, a matrix $\tilde{X}\in\mathbb{R}^{n\times r}$
is considered, whose columns are the selected eigenvectors and the $i$-th row
is an embedding of the $i$-th node in $G$. The graph nodes are then clustered
by applying, e.g., K-Means on the node embeddings. We refer to Aggarwal &
Wang, (2010) for a comprehensive overview of Spectral Clustering.

For our setting, we define a complete graph $G$ in which $V$ is the set of
nodes considered in the BZR analysis. We set the weight $w_{e}$ of an edge
$e=(u,v)\in E$ as the similarity between nodes $u$ and $v$ and compute it
using the radial basis function kernel. Thus, with $x_{u},x_{v}$ denoting the
features corresponding to nodes $u$ and $v$, respectively, we define
$w_{e}=\exp(-\frac{\lVert x_{u}-x_{v}\rVert^{2}}{2\sigma^{2}})$, where
$\sigma$ is a parameter that controlls the width of node-neighborhoods. We
normalize the edge weights such that $w_{e}\in[0,1]$ for all $e\in E$ and set
$\sigma=1$. To cluster the nodes using $\tilde{X}\in\mathbb{R}^{n\times r}$,
we apply the constrained version of K-Means mentioned in Section 3.1.1.
Moreover, we set $r=k$ and thus consider the first $k$ eigenvectors as
clustering features.

###  3.2 Stability Evaluation

In what follows, we discuss three criteria that we consider to evaluate the
stability of pricing zones. Given a period $t$ and a node $x$, we denote by
$x^{t}$ the vector containing the time series of prices corresponding to $x$
in period $t$. We define $\overline{x}^{t}$ as the average price of node $x$
in period $t$, which is simply the average over all values in $x^{t}$. We also
denote the distance between two points $x$ and $y$ in period $t$ by
$d^{t}(x,y)$ and define it as
$d^{t}(x,y)=\lvert\overline{x}^{t}-\overline{y}^{t}\rvert$. Whenever the
period we are referring to is evident, we simplify notation by omitting the
reference to $t$ in $d^{t}(x,y)$.

####  3.2.1 Intra- and Inter-Cluster Similarity

In this section, we assume that a clustering
$\mathcal{C}=\\{C_{1},\dots,C_{k}\\}$ is given for a period $t$. We consider
$\mathcal{C}$ to be stable if it is comprised of clusters having points that
are similar with respect to their prices. In other words, $\mathcal{C}$ has
large intra-cluster similarity. To analyze intra-cluster similarity, we
consider the price standard deviation of each cluster $C_{i}$ and denote it by
$\sigma_{i}$. We compute $\sigma_{i}$ based on the vectors $x^{t}$
corresponding to the nodes assigned to $C_{i}$. The average price standard
deviation $\sigma(\mathcal{C})$ of a clustering $\mathcal{C}$ is the average
of all cluster price standard deviations $\sigma_{i},C_{i}\in\mathcal{C}$.
Whenever the clustering we are referring to is clear, we simplify notation by
dropping the reference to $\mathcal{C}$. A low average price standard
deviation indicates large intra-cluster similarity.

Furthermore, a stable clustering is characterized by low inter-cluster
similarity or, equivalently, large inter-cluster dissimilarity. With $\mu_{i}$
denoting the mean price of a cluster $C_{i}$777Note that, similar to
$\sigma_{i}$, $\mu_{i}$ is computed based on the vectors containing the time
series of prices corresponding to the nodes assigned to $C_{i}$., we define
the following distance measures to assess the dissimilarity between two
clusters $C_{i},C_{j}\in\mathcal{C}$:

The average distance between cluster mean prices quantifies the
distinctiveness of clusters based on mean prices:

$d_{\mu}(C_{i},C_{j})=d(\mu_{i},\mu_{j}).$

The nearest neighbor cluster distance reflects how well-separated the clusters
are:

$d_{\min}(C_{i},C_{j})=\min_{x\in C_{i},y\in C_{j}}d(x,y).$

The average distance between clusters considers all pairs of points to measure
the dissimilarity between $C_{i}$ and $C_{j}$:

$d_{\text{avg}}(C_{i},C_{j})=\frac{1}{|C_{i}|\cdot|C_{j}|}\sum\limits_{x\in
C_{i}}\sum\limits_{y\in C_{j}}d(x,y).$

The following quantity is used to evaluate the global inter-cluster
dissimilarity of $\mathcal{C}$:

$\Delta_{q}(\mathcal{C})=\frac{1}{\binom{K}{2}}\sum\limits_{\begin{subarray}{c}C_{i},C_{j}\in\mathcal{C}\\\
i<j\end{subarray}}d_{q}(C_{i},C_{j}),$

where $q$ is one of {$\mu$, min, avg}.

We also consider the Davies-Bouldin index to assess both the intra-cluster
similarity and inter-cluster dissimilarity of $\mathcal{C}$. With $s_{i}$
denoting the diameter888The diameter of a cluster is the maximum distance
between any two points assigned to this cluster. of cluster $C_{i}$ and
$d_{ij}$ the distance between $c_{i}$ and $c_{j}$, the Davies-Bouldin Index
[16] is defined as:

$DB=\frac{1}{k}\sum\limits_{i=1}^{K}\max_{i\neq j}R_{ij},$

were $R_{ij}=\frac{s_{i}+s_{j}}{d_{ij}}$. Intuitively, $DB$ quantifies the
average similarity of every cluster with its most similar cluster. Here,
similarity is defined as the ratio of within-cluster distances to between-
cluster distances. A low $DB$ index indicates a good clustering, and the
minimum value is 0. Clusters that are well-separated and less dispersed are
rewarded by the $DB$ index.

####  3.2.2 Temporal Stability

While geographic coordinates are static across time, prices are time-
sensitive. Therefore, we are interested to see if pricing zones are stable
over time. To decide this, we compare two clusterings $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ that are computed for two different time periods with the
same algorithm and considering the same features. For this comparison, we use
the Rand index, which measures the similarity between
$\mathcal{C}_{1},\mathcal{C}_{2}$ considering all pairs of data points and
counting the pairs that are assigned to the same or to different clusters in
both $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$:

$RI(\mathcal{C}_{1},\mathcal{C}_{2})=\frac{\sum\limits_{1\leq l<r\leq
N}\gamma(x_{l},x_{r})}{\binom{N}{2}},$

where

$\gamma(x_{l},x_{s})=\begin{cases}1&\text{if $x_{l}$ and $x_{r}$ are in the
same cluster in $\mathcal{C}_{1}$ and in $\mathcal{C}_{2}$}\\\ 1&\text{if
$x_{l}$ and $x_{r}$ are in different clusters in $\mathcal{C}_{1}$ and in
$\mathcal{C}_{2}$}\\\ 0&\text{otherwise.}\end{cases}$

Rand indices are between 0 and 1, and 1 indicates a perfect match, that is,
the same clustering is obtained in both time periods.

####  3.2.3 Spatial Coherence

We evaluate clustering stability also with respect to whether the clusters are
geographically separated. Geographically separated or spatially cohesive
clusters can be identified on the map as groups of points (nodes) that are
geographically close.

We assess whether clusters are spatially cohesive considering spatial
autocorrelation. Thus, for a clustering $\mathcal{C}$, a positive spatial
autocorrelation indicates that $\mathcal{C}$ has spatially cohesive clusters
since neighboring locations are more likely to share similar prices and belong
to the same cluster. To measure spatial autocorrelation, we use Global Moran’s
I (Moran, 1950), which is defined as:

$I=\frac{N}{W}\frac{\sum\limits_{x\in X}\sum\limits_{y\in
X}w_{xy}(c(x)-\overline{c})(c(y)-\overline{c})}{\sum\limits_{x\in
X}(c(x)-\overline{c})^{2}},$

where $c(x)$ is the centroid of the cluster to which point $x$ is assigned,
$w_{xy}$ denotes the spatial distance between points $x,y$ and
$W=\sum\limits_{x\in X}\sum\limits_{y\in X}w_{xy}$. Also, $\overline{c}$
represents the average of all centroids. $I$ values are between -1 and +1,
where +1 defines perfect clustering of similar values, 0 indicates no
correlation and -1 suggests perfect clustering of dissimilar values.

##  4 Data

The data we use in our experiments was published as part of the locational
pricing study conducted by ENTSO-E999The datasets can be accessed at
https://www.entsoe.eu/network_codes/bzr/ (January 2024). and includes
locational (i.e., nodal) prices. In this study, 24 weeks spanned across 3
years were analyzed: 1989 (weeks 04, 10, 11, 17, 20, 31, 40, 52), 1995 (weeks
02, 12, 16, 21, 27, 36, 38, 49) and 2009 (weeks 04, 08, 11, 15, 16, 21, 31,
48). For each day in these weeks, nodal prices for every 2-hour interval are
available. The file containing the prices for the German-Luxembourgish bidding
zone includes 6,090 unique node IDs and 14,170,464 nodal prices. Moreover, for
1,893,024 ($\approx 13\%$) nodal prices, which correspond to 939 distinct
nodes, the node ID is “nan”. Consequently, these prices cannot be considered
in our experiments, as insufficient data is provided for identifying the nodes
to which these prices correspond.

Note that ENTSO-E did not publish the geographic coordinates of the nodes
analyzed in the locational pricing study. However, these are essential for
identifying and analyzing price zones. Therefore, we implement the following
approach to obtain the geographic location of nodes. We consider the XML files
describing the German grid model that we obtained from the Bidding Zone Review
website (ENTSO-E, 2023). The grid is composed of elements such as substations,
voltage levels and topological nodes, which form a hierarchy as follows:
substations consist of one or multiple voltage levels and voltage levels
consist of one or multiple topological nodes. Altogether, the German grid
encompasses 834 substations, 1,697 voltage levels and 2,898 topological nodes.
The name of each substation is included in the XML files and based on this we
identified the geographic location of each substation using OpenStreetMap and
(JAO) Static Grid Model101010https://www.jao.eu/static-grid-model. Since a
topological node is mapped to a specific substation, in this process we also
obtain the geographic coordinates of the topological nodes.

Next, we identify the prices corresponding to the topological nodes. We do
this based on the IDs of these nodes and the IDs of the 6,090 nodes included
in the file containing LMPs. For 2,200 out of 2,898 topological nodes we could
identify their IDs in this file. These nodes belong to 714 substations across
Germany. Looking at the different LMP prices computed by ENTSO-E ordered by
substation, it turns out that prices for different topological nodes
associated with a substation are the same or very close. So, with the 714
substations covered in our study, we could reconstruct most of the German
substations, their prices and location across Germany.

###  4.1 Clustering Features & Periods

We apply K-Means and Spectral Clustering for two feature configurations. In
the first configuration we use time series of nodal prices as clustering
features, that is, a feature represents the nodal price of a specific point in
time. In the second configuration, we additionally consider latitude and
longitude. We compute one clustering for every year analyzed in the bidding
zone review, i.e., 1989, 1995 and 2009. For each year, 672 prices are reported
per node ($12\text{ hours}\times 7\text{ days}\times 8\text{ weeks}$), which
implies that 672 and 674 features are considered for the first and second
configuration, respectively. Note that the first configuration was also used
by ACER (ACER, 2021).

##  5 Results

In this section, we first analyze the price zone configurations proposed by
ACER. Then, we evaluate the optimal price zone configurations computed with
K-Means and Spectral Clustering for the years 1989, 1995 and 2009. Considering
the metrics defined in Section 3.2, we assess the stability of the resulting
clusterings and compare them to those proposed by ACER.

###  5.1 Analysis of the price zone configurations proposed by ACER

The configurations proposed by ACER for Germany are the following: two
configurations with two price zones (DE2 (k-means) and DE (spectral)), one
with three price zones (DE3 (spectral)) and one with four price zones (DE4
(spectral)). Appendix A contains approximations of these configurations given
the nodes we consider in our experiments and based on the maps provided by
ACER (ACER, 2022a ). Since it is difficult to manually obtain an accurate
geographic separation of the zones within the proposed configurations, the
configurations from Appendix A are just approximations. In what follows, we
analyze these configurations.

First, we want to understand the extent to which the geographic coordinates of
the nodes contribute to the determination of price zones. To do this, we apply
K-Means with latitude and longitude as features111111We also imposed a
threshold related to the minimum number of nodes to be included in a zone. to
cluster the nodes, but do not consider prices. The computed clusters are shown
in Figure 1.

Figure 1: Price zones computed with K-Means for $k\in\\{2,3,4\\}$ based on
latitude and longitude

###### Result 1

Clustering nodes based solely on latitude and longitude leads to price zones
that closely resemble the configurations proposed by ACER.

We notice that DE2 (k-means), DE3 (spectral) and DE4 (spectral) are similar to
the clusterings computed by K-Means. In fact, the configurations proposed by
ACER seem to be adjustments of the zones illustrated in Figure 1 such that
their borders do not cross any federal state, with few exceptions at the
borders between North Rhein-Westphalia & Lower Saxony (for DE2 (spectral)) and
Hesse & North Rhein-Westphalia & Lower Saxony (for DE3 (spectral) and DE4
(spectral)). Further below, we discuss clusterings that take prices into
account and compare them to those of ACER.

Before this, we analyze the nodal prices for 1989, 1995 and 2009 in the
configurations proposed by ACER. For each year and week, we compute the
cluster price standard deviations and average prices corresponding to
configurations DE2 (k-means), DE2 (spectral), DE3 (spectral) and DE4
(spectral), respectively. The purpose of this analysis is to see what benefit
multiple price zones have on the reduction of the price standard deviation
inside zones. Furthermore, we study whether the clusters are well-defined,
i.e., whether, for a given configuration, the cluster average prices are
different.

The evolution of cluster price standard deviations across the weeks in 1989,
1995 and 2009 is illustrated in Appendix B.1. We also observe the price
standard deviations corresponding to a single price zone or cluster (Figure
5). Clearly, the average cluster price standard deviations decrease as we
consider configurations with 2, 3 or 4 clusters. This is in fact an implicit
objective of clustering nodes based on prices. However, this reduction is not
substantial and varies between 3% to 9% depending on the configuration. For
instance, for 2009, the percentage decreases in average price standard
deviations compared to a single zone range from 1.92 to 5.69. Moreover, the
decrease in average cluster price standard deviation is not monotonic with
respect to the number of clusters. That is, a larger number of clusters does
not necessarily imply smaller average cluster price standard deviations. For
example, for 2009 and configuration DE2 (k-means), the average price standard
deviation is 10.85, while DE3 (spectral) leads to an average price standard
deviation of around 11.26. An explanation for this behavior is that the
average cluster prices are heavily influenced by outliers, which are nodes
having very large or low prices compared to the majority of nodes. For smaller
clusters, that is, for configurations with a larger number of clusters, the
influence of outliers is stronger. Moreover, if we compare the configuration
with one price zone to any configuration with multiple zones (clusters), we
notice that a reduction in average price standard deviation does not
necessarily imply that all cluster standard deviations are lower.

Next, we analyze how different the average cluster prices are for each
configuration. These values are shown in Appendix B.2.

###### Result 2

The difference in the average prices of clusters in the proposed
configurations is low. For all configurations, the yearly differences between
the average cluster prices are less than 6 EUR/MWh and 4.16 EUR/MWh on
average.

The differences between the average prices in Germany and the average prices
in the neighboring countries are generally larger than 6
EUR/MWh.121212https://www.energy-
charts.info/charts/price_average_map/chart.htm?l=en&c=DE&year=2018&interval=year
The German-Austrian price zone was split in 2018. The average electricity spot
market price in 2018 was 41.74 EUR/MWh in Germany and 59.92 EUR/MWh in
Austria, which implies a price difference of 18.18 EUR/MWh between these
zones. Our analysis indicates that the configurations proposed by ACER do not
lead to well-defined clusters in terms of average prices. As the tables in
Appendix B.2 showcase, the cluster average prices are generally similar,
regardless of the configuration, week or year.

Let us now compare the configurations proposed by ACER to the clusterings
computed with our implementation of K-Means and Spectral Clustering. Given the
nodal prices for 1989, 1995 and 2009, respectively, we apply both clustering
algorithms to identify clusterings (i.e., price zones) for each year. The goal
of this comparison is to understand whether the proposed configurations by
ACER are stable over time. We say that the ACER configurations are stable if
they are similar to the optimal clusterings identified for different years.

As described in Section 4.1, we compute clusterings for two scenarios. The
first scenario uses, as clustering features, the time series of nodal prices,
while the second scenario additionally considers the location coordinates.
Table 1 shows Rand indices measuring the similarity between the clusterings
proposed by ACER and the clusterings computed with each algorithm for 1989,
1995 and 2009.

|  | Without loc. | With loc.
---|---|---|---
|  | 1989 | 1995 | 2009 | 1989 | 1995 | 2009
K-Means | DE2 (k-means) | 0.5 | 0.5 | 0.51 | 0.5 | 0.5 | 0.51
DE2 (spectral) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5
DE3 (spectral) | 0.51 | 0.53 | 0.52 | 0.52 | 0.52 | 0.52
DE4 (spectral) | 0.6 | 0.5 | 0.56 | 0.6 | 0.58 | 0.57
Spectral | DE2 (k-means) | 0.51 | 0.51 | 0.51 | 0.51 | 0.51 | 0.51
DE2 (spectral) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5
DE3 (spectral) | 0.52 | 0.44 | 0.52 | 0.52 | 0.44 | 0.52
DE4 (spectral) | 0.55 | 0.45 | 0.57 | 0.56 | 0.46 | 0.56
Table 1: Comparison of alternative price configurations to the price zones
proposed by ACER. The table shows Rand indices, which are in $[0,1]$, 1
indicating perfect similarity. The clusterings analyzed in this table were
computed with K-Means or Spectral Clustering using time series of nodal prices
(columns “Without loc.”) or time series of nodal prices and the location
coordinates of nodes (columns “With loc.”).

###### Result 3

The price zone configurations proposed by ACER are not stable over time. Given
the nodal prices corresponding to 1989, 1995 and 2009, respectively, K-Means
and Spectral Clustering compute price zones that are considerably different
from the proposed configurations.

We recall from Section 3.2 that the Rand index measures the similarity between
two clusterings by considering all pairs of nodes and counting the pairs that
are assigned to the same or to different clusters in both clusterings. As
Table 1 indicates, Rand indices vary considerably when different years are
analyzed. For instance, for DE4 (spectral) and the clustering computed with
Spectral Clustering with price and location features we get $RI=0.56$ for
1989, $RI=0.46$ for 1995 and $RI=0.56$ for 2009. Thus, our analysis suggests
that the configurations proposed by ACER are not stable over time, regardless
of the number of price zones. An explanation is that nodal prices vary both
temporally and spatially. As a result, the spatial distribution of the optimal
clusters (i.e., obtained based on the prices corresponding to a specific time
frame) is different each year. The plots included in Appendix D show that the
clusterings indeed differ across time.

###  5.2 Analysis of the optimal price zone configurations for 1989, 1995 and
2009

In what follows, we analyze alternative price zones computed with K-Means and
Spectral Clustering for the years 1989, 1995 and 2009. Previously we showed
that the configurations proposed by ACER are not stable over these years. Now
we are interested to see whether other price zone configurations exist that
are more stable with regards to the criteria defined in Section 3.2. We report
the results of K-Means and Spectral Clustering as described in Section 3.1. We
evaluate the clusterings computed for each year using (1) time series of nodal
prices as features, and (2) time series of nodal prices and the location
coordinates of nodes.

Based on the results provided in Appendix C, we first evaluate the intra-
cluster similarity and the inter-cluster dissimilarity of the clusterings
computed for all three years. For each year, we want to understand whether the
clusters are well-defined. Well-defined clusters are characterized by low
price standard deviation within clusters (or, large intra-cluster similarity)
and different cluster average prices (or, large inter-cluster dissimilarity).

###### Result 4

The clusterings computed with K-Means for 1989, 1995 and 2009 based only on
prices exhibit lower average price standard deviations compared to the
configurations proposed by ACER.

Tables 4, 4 and 4 show the average price standard deviations for different
configurations with 1, 2, 3 or 4 price zones or clusters. The configurations
refer to (1) a single price zone (column ‘Single price zone’), (2) the
configurations proposed by ACER (columns ACER DE2 (k-means), DE2 (spectral),
DE3 (spectral), DE4 (spectral)) and (3) the optimal configurations computed
with K-Means and Spectral Clustering based on nodal prices (columns ‘K-Means’
and ‘Spectral Clustering’).

| Single price zone | ACER DE2 (k-means) | ACER DE2 (spectral) | K-Means | Spectral
---|---|---|---|---|---
1989 | 14.54 | 13.09 | 13.98 | 13.40 | 13.39
1995 | 14.31 | 12.93 | 13.80 | 11.16 | 15.16
2009 | 11.55 | 10.85 | 11.33 | 10.80 | 11.61

Table 2: Average price standard deviations of configurations with 1 or 2 price
zones

| Single price zone | ACER DE3 (spectral) | K-Means | Spectral
---|---|---|---|---
1989 | 14.54 | 13.86 | 12.11 | 14.81
1995 | 14.31 | 13.65 | 12.47 | 15.56
2009 | 11.55 | 11.26 | 10.06 | 11.11

Table 3: Average price standard deviations of configurations with 1 or 3 price
zones

| Single price zone | ACER DE4 (spectral) | K-Means | Spectral
---|---|---|---|---
1989 | 14.54 | 13.14 | 12.86 | 12.23
1995 | 14.31 | 13.02 | 11.12 | 15.09
2009 | 11.55 | 10.87 | 10.05 | 10.73

Table 4: Average price standard deviations of configurations with 1 or 4 price
zones

We observe that the average price standard deviations $\sigma$ differ
depending on the clustering algorithm. For instance, for $k=3$, K-Means
(Spectral Clustering) leads to $\sigma=12.11(14.81)$ for 1989,
$\sigma=12.47(15.56)$ for 1995 and $\sigma=10.06(11.11)$ for 2009. However,
regardless of the year and the number of clusters, the clusterings we compute
with K-Means based on nodal prices lead to average price standard deviations
that are generally lower compared to the configurations proposed by ACER.
Recall that the average price standard deviations for the configurations
proposed by ACER are up to 13.98. Nevertheless, the clusters we compute with
K-Means and Spectral Clustering are not necessarily balanced, that is, the
clusters may not contain a similar number of nodes. This fact can be observed
in the plots provided in Appendix D. Given this observation and the fact that
the configurations proposed by ACER appear to be balanced (see Appendix A), it
explains that the price standard deviations inside the clusters we compute
with K-Means and Spectral Clustering are lower.

###### Result 5

The clusterings computed with K-Means and Spectral Clustering for 1989, 1995
and 2009 are characterized by low inter-cluster dissimilarity and are thus not
well-defined. Regardless of algorithm or clustering features, the yearly
differences between the average cluster prices are less than 6 EUR/MWh and
3.45 EUR/MWh on average.

To evaluate inter-cluster dissimilarity, we analyze the values of
$\Delta_{\mu}$, $\Delta_{\min}$ and $\Delta_{\text{avg}}$ included in Appendix
C. We recall from Section 3.2.1 that $\Delta_{\mu}$ denotes the global average
distance between cluster mean prices, $\Delta_{\text{min}}$ represents the
global nearest neighbor cluster distance, while $\Delta_{\text{avg}}$ encodes
the global average distance between clusters. For a given period, the distance
between two nodes is simply the absolute difference between their average
prices in this period. We notice that, regardless of the year, algorithm or
clustering features, $\Delta_{\mu}$ is smaller than 6 EUR/MWh, which suggests
that, in terms of average prices, clusters are rather similar to each other.
Moreover, all values of $\Delta_{\min}$ are smaller than 1, indicating that
clusters are not well-separated. This means that there exist nodes that have
very similar prices (i.e., their price difference is less than 1) that are
assigned to different clusters. We also observe that the values of
$\Delta_{\text{avg}}$ are in the interval $[2.51,5.20]$ and become slightly
smaller as we consider configurations with multiple clusters. In other words,
the average price distance between any pair of two nodes assigned to different
clusters is below 6 EUR/MWh.

Finally, we analyze the Davies-Bouldin indices to measure the average
similarity of every cluster with its most similar cluster. We notice that
these indices are quite large and fluctuate depending on the clustering
algorithm and clustering features, ranging from 0.45 to 7.75. We recall from
Section 3.2.1 that Davies-Bouldin indices are in $[0,\infty)$, where 0
indicates a good clustering having well-separated and less dispersed clusters.
Interestingly, there is no clear evidence that using location coordinates in
the clustering computation leads to smaller indices.

Given these observations, we conclude that the price zones computed for 1989,
1995 and 2009 with K-Means and Spectral Clustering generally exhibit larger
intra-cluster similarity and similar inter-cluster dissimilarity compared to
the configurations proposed by ACER. Large intra-cluster similarity indicates
strong cohesion inside clusters and is a consequence of unbalanced clusters
relative to the ACER configurations.

Next, we evaluate temporal stability considering the Rand indices included in
Table 5.

|  | 1989 | 1995 | 2009
---|---|---|---|---
|  | $k=2$ | $k=3$ | $k=4$ | $k=2$ | $k=3$ | $k=4$ | $k=2$ | $k=3$ | $k=4$
K-Means | 1989 | - | - | - | 0.57 | 0.69 | 0.7 | 0.87 | 0.88 | 0.83
1995 | 0.57 | 0.69 | 0.7 | - | - | - | 0.61 | 0.78 | 0.67
2009 | 0.87 | 0.88 | 0.83 | 0.61 | 0.78 | 0.67 | - | - | -
Spectral | 1989 | - | - | - | 0.68 | 0.5 | 0.5 | 0.72 | 0.48 | 0.78
1995 | 0.68 | 0.5 | 0.5 | - | - | - | 0.7 | 0.48 | 0.44
2009 | 0.72 | 0.48 | 0.78 | 0.7 | 0.48 | 0.44 | - | - | -

|  | 1989 | 1995 | 2009
---|---|---|---|---
|  | $k=2$ | $k=3$ | $k=4$ | $k=2$ | $k=3$ | $k=4$ | $k=2$ | $k=3$ | $k=4$
K-Means | 1989 | - | - | - | 0.95 | 0.95 | 0.76 | 0.59 | 0.93 | 0.86
1995 | 0.95 | 0.95 | 0.76 | - | - | - | 0.57 | 0.92 | 0.66
2009 | 0.59 | 0.93 | 0.86 | 0.57 | 0.92 | 0.66 | - | - | -
Spectral | 1989 | - | - | - | 0.69 | 0.5 | 0.46 | 0.71 | 0.78 | 0.76
1995 | 0.69 | 0.5 | 0.46 | - | - | - | 0.7 | 0.49 | 0.5
2009 | 0.71 | 0.78 | 0.76 | 0.7 | 0.49 | 0.5 | - | - | -

Table 5: Temporal stability evaluation. Tables show Rand indices, which are
always in $[0,1]$, 1 indicating perfect similarity. In the first table, the
results refer to clusterings computed based on prices. In the second table,
the results relate to clusterings computed based on prices and location
coordinates.

###### Result 6

The clusterings computed with K-Means and Spectral Clustering are not
temporally stable. A signficant number of nodes would be assigned to different
clusters in different years.

We notice that Rand indices fluctuate considerably when different pairs of
years are considered and lie in the intervals [0.57, 0.95] for $k=2$, [0.48,
0.95] for $k=3$ and [0.44, 0.86] for $k=4$. This suggests that clusterings are
not temporally stable. The price zone configuration identified for a specific
year might be far from the optimal configuration corresponding to a different
year. This challenges the goal of finding price zones that reflect stable
long-term patterns in nodal prices.

Last but not least, we analyze how spatially coherent the clusterings computed
with K-Means and Spectral Clustering for 1989, 1995 and 2009 are. Figure 2
illustrates the Global Moran’s I values of the clusterings computed using (1)
nodal weekly prices and (2) nodal weekly prices and the location coordinates
of nodes as features.

Figure 2: Global Moran’s I values of the clusterings computed with K-Means and
Spectral Clustering

For all years we consider, the Global Moran’s I values are between 0.01 and
0.5, indicating that clusterings exhibit moderate positive spatial
autocorrelation. Neighboring nodes are more likely to have similar prices and
to belong to the same cluster. Also, considering the location coordinates in
the K-Means computation leads to better Global Moran’s I scores which are
quite stable across time. However, the plots from Figure 2 show that the
clusterings obtained with Spectral Clustering exhibit scores that vary
considerably across years.

Analyzing the clusters on the map of Germany, we notice that isolated points
assigned to a specific cluster are sometimes located in the geographic area
predominantly occupied by points belonging to a different cluster. This
behavior can also be observed when latitude and longitude are included as
features and is independent of the number of clusters or clustering algorithm.
Therefore, a post-processing step is required to obtain spatially coherent
price zones. This is because prices alone do not define geographically well-
separated or spatially coherent clusters. This phenomenon can be observed in
Figures 12 and 11 from Appendix D.

### Sensitivity Analysis regarding Outliers

In what follows, we briefly discuss the influence of outliers on the resulting
price zones. We classify a node as an outlier if it has a price lower than 0
EUR/MWh or larger than 100 EUR/MWh in any of the periods considered in the
clustering computation. Out of 1,478,400 nodal prices reported for a specific
year, 38,378 ($\approx 2.59\%$) lie outside the interval $[0,100]$.
Interestingly, this number is identical in all three years we consider. We
want to understand the influence of outliers on the clustering computation. To
do this, we clip the prices to the interval $[0,100]$ and apply the clustering
algorithms. Table 6 contains the average yearly prices of the clusters
computed with K-Means and Spectral Clustering after clipping the prices.

|  |  | K-Means | Spectral
---|---|---|---|---
Year | $k$ | Loc. | $\Delta_{\mu}$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{3}$ | $\mu_{4}$ | $\Delta_{\mu}$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{3}$ | $\mu_{4}$
1989 | 2 |  | 5.7 | 46.59 | 40.89 | - | - | 3.2 | 43.66 | 46.86 | - | -
1995 | 2 |  | 5.89 | 41.54 | 47.43 | - | - | 3.2 | 43.66 | 46.86 | - | -
2009 | 2 |  | 5.9 | 47.44 | 41.54 | - | - | 3.2 | 43.66 | 46.86 | - | -
1989 | 2 | x | 5.79 | 41.25 | 47.04 | - | - | 3.13 | 43.63 | 46.76 | - | -
1995 | 2 | x | 5.68 | 46.55 | 40.87 | - | - | 3.13 | 43.63 | 46.76 | - | -
2009 | 2 | x | 5.79 | 41.32 | 47.11 | 3.13 | 43.63 | 46.76 | - | - |  |
1989 | 3 |  | 3.95 | 41.52 | 41.47 | 47.39 | - | 2.21 | 43.39 | 46.64 | 46.7 | -
1995 | 3 |  | 4.27 | 41.2 | 47.6 | 46.04 | - | 3.67 | 43.94 | 46.9 | 41.4 | -
2009 | 3 |  | 4.44 | 40.91 | 47.57 | 44.16 | - | 3.76 | 43.96 | 41.26 | 46.9 | -
1989 | 3 | x | 4.75 | 42.43 | 47.38 | 40.26 | - | 3.71 | 43.91 | 46.85 | 41.29 | -
1995 | 3 | x | 5.06 | 41.36 | 46.72 | 48.95 | - | 3.71 | 43.91 | 46.85 | 41.29 | -
2009 | 3 | x | 5.02 | 48.61 | 41.08 | 46.43 | - | 3.71 | 43.91 | 46.85 | 41.29 | -
1989 | 4 |  | 3.96 | 47.1 | 41.02 | 41.76 | 39.43 | 3.04 | 43.74 | 46.28 | 46.57 | 41.33
1995 | 4 |  | 4.7 | 40.72 | 43.74 | 48.92 | 47.32 | 2.86 | 40.94 | 46.27 | 46.58 | 46.51
2009 | 4 |  | 3.94 | 41.34 | 47.48 | 40.24 | 43.25 | 3.07 | 43.72 | 46.53 | 41.33 | 46.54
1989 | 4 | x | 4.2 | 41.21 | 47.48 | 43.73 | 39.92 | 1.85 | 43.16 | 46.64 | 46.61 | 45.94
1995 | 4 | x | 3.82 | 40.62 | 43.89 | 41.62 | 47.5 | 1.79 | 42.99 | 46.21 | 46.25 | 46.56
2009 | 4 | x | 4.5 | 41.51 | 47.69 | 45.65 | 40.06 | 1.85 | 43.16 | 46.64 | 46.61 | 45.94
Table 6: Cluster average prices after clipping nodal prices to $[0,100]$
(EUR/MWh)

We observe that these prices are similar to the average yearly prices shown in
Table 13, in which clusterings were computed without clipping prices. The
global average distances between cluster mean prices (i.e., $\Delta_{\mu}$)
illustrated in Table 6 are also close to the average distances included in
Table 13. In particular, the differences between the values of $\Delta_{\mu}$
shown in these tables are smaller than 6 EUR/MWh. Hence, the impact of
outliers on the resulting clusters or price zones is not considerable.

##  6 Conclusions and Policy Implications

In this paper, we analyzed the alternative price zone configurations proposed
by ACER for Germany, namely DE2 (k-means), DE2 (spectral), DE3 (spectral) and
DE4 (spectral). Similar to ACER’s approach (the algorithms used are publicly
known, but the implementation details are not), we computed alternative price
zone configurations by solving a clustering problem using K-Means and Spectral
Clustering. Clusterings that we obtained with K-Means using only latitude and
longitude as features are similar to the configurations proposed by ACER.
Additionally, we computed alternative configurations by taking nodal prices or
nodal prices and geographic coordinates into account and compared them with
the configurations proposed by ACER. The resulting price zones were different
from the proposed ACER configurations. Moreover, our results show that
clusters with “minimal price dispersion within each bidding zone” (ACER, 2022a
) were not geographically coherent on the map, and they do not have similar
numbers of nodes. Note that ACER might have had access to a more complete
dataset covering all substations and also the clustering algorithms might
differ in details.

We evaluated the stability of the alternative configurations considering three
criteria, namely intra- and inter-cluster similarity, temporal stability and
spatial coherence. Our findings have important policy implications concerning
the ongoing discussion of whether a split of the German bidding zone should be
performed.

Firstly, splitting the German bidding zone into 2, 3 or 4 zones does not
reduce the price dispersion within zones by more than 3-9%. Moreover, the
decrease in average price standard deviation is not monotonic with respect to
the number of zones. Configurations with 3 or 4 zones do not necessarily
reduce the average price standard deviation compared to configurations with
less zones. Furthermore, for all proposed configurations, the average prices
across zones do not differ considerably. The annual differences between the
average cluster prices are less than 6 EUR/MWh for all three years we
analyzed. Given that two of the main goals of a split are to reduce the price
dispersion within individual bidding zones and to reflect long-term
differences in the average prices across resulting zones, the alternative
configurations proposed by ACER appear to be unsatisfactory.

Secondly, independent of how the alternative price zone configurations were
determined and whether we look at those suggested by ACER or our own, none of
them is temporally stable. What appears to be a good configuration in one time
period might turn out suboptimal in another time period. Based on the data set
from the LMP study, it appears that central goals for the Bidding Zone Review,
such as increasing market efficiency and being robust and stable over time,
are elusive.

Note that our analysis is based on data from the Bidding Zone Review with a
target year 2025. With the ambitious goals for renewable energy in Germany
until 2030 and beyond, and the increasing electrification of the traffic and
heat sectors, supply and demand will change substantially and the analysis
might soon be outdated. Splitting price zones incurs high transition costs and
there is a danger that new reconfigurations of the bidding zones might be in
order in a few years again.

## Acknowledgements

We gratefully acknowledge the financial support of the Kopernikus-Project
“SynErgie” by the Federal Ministry of Education and Research of Germany (BMBF)
and the project supervision by the project management organization
Projektträger Jülich (PtJ).

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##  Appendix A Approximations of the alternative configurations proposed by
ACER

Figure 3: Approximations of the alternative configurations proposed by ACER

##  Appendix B Cluster price analysis for the alternative configurations
proposed by ACER

###  B.1 Cluster price standard deviations

Table 7 shows the average price standard deviations of a single price zone
configuration as well as of the configurations proposed by ACER.

| 1989 | 1995 | 2009
---|---|---|---
Single price zone | 14.54 | 14.32 | 11.48
DE2 (k-means) | 13.09 | 12.93 | 10.85
DE2 (spectral) | 13.98 | 13.80 | 11.22
DE3 (spectral) | 13.86 | 13.65 | 11.26
DE4 (spectral) | 13.14 | 13.02 | 10.87
Table 7: Average price standard deviations of configurations with 1, 2, 3 or 4
price zones

The following plots illustrate, for every configuration, the cluster price
standard deviation corresponding to each week in 1989, 1995, and 2009. A point
represents the price standard deviation of a specific cluster in a specific
week. Depending on the configuration (i.e., number of clusters), multiple
points are shown for the same week. Figure 5 highlights the weekly price
standard deviation of a single price zone for each year we consider.

Figure 4: Single price zone (cluster)

Figure 5: DE2 (k-means)

Figure 6: DE2 (spectral)

Figure 7: DE3 (spectral)

Figure 8: DE4 (spectral)

###  B.2 Average cluster prices

In the following tables, $\Delta^{\text{year}}_{\mu}$ represents the average
distance between cluster mean prices for a specific year, i.e.,
$\Delta^{\text{year}}_{\mu}=\frac{1}{8}\displaystyle\sum_{i\in\\{1,\dots,8\\}}\Delta^{\text{W}_{i}}_{\mu}$,
where $\Delta^{\text{W}_{i}}_{\mu}$ is the average distance between cluster
mean prices for week $\text{W}_{i}$ (see Section 3.2.1).

|  | W1 | W2 | W3 | W4 | W5 | W6 | W7 | W8 | $\Delta^{\text{year}}$
---|---|---|---|---|---|---|---|---|---|---
1989 | Cluster 1 | 50.49 | 46.72 | 41.99 | 49.88 | 48.52 | 39.52 | 40.22 | 45.50 | -
1995 | Cluster 1 | 39.87 | 28.09 | 45.89 | 46.11 | 48.61 | 38.35 | 51.76 | 54.50 | -
2009 | Cluster 1 | 50.49 | 51.94 | 41.99 | 47.66 | 45.89 | 46.11 | 39.52 | 40.91 | -
Table 8: Cluster average prices (EUR/MWh) - Single price zone (cluster) |  | W1 | W2 | W3 | W4 | W5 | W6 | W7 | W8 | $\Delta^{\text{year}}_{\mu}$
---|---|---|---|---|---|---|---|---|---|---
1989 | Cluster 1 | 49.09 | 45.71 | 39.18 | 49.32 | 47.94 | 35.84 | 37.44 | 44.38 | 4.3
Cluster 2 | 52.55 | 48.21 | 46.10 | 50.69 | 49.39 | 44.91 | 44.28 | 47.15
1995 | Cluster 1 | 35.02 | 21.21 | 44.56 | 45.13 | 47.65 | 35.63 | 51.22 | 54.14 | 5.75
Cluster 2 | 46.98 | 38.19 | 47.84 | 47.55 | 50.03 | 42.35 | 52.56 | 55.04
2009 | Cluster 1 | 49.09 | 51.21 | 39.18 | 47.11 | 44.56 | 45.13 | 35.84 | 37.25 | 4.67
Cluster 2 | 52.55 | 53.01 | 46.10 | 48.46 | 47.84 | 47.55 | 44.91 | 46.28
Table 9: Cluster average prices (EUR/MWh) - DE2 (k-means) |  | W1 | W2 | W3 | W4 | W5 | W6 | W7 | W8 | $\Delta^{\text{year}}_{\mu}$
---|---|---|---|---|---|---|---|---|---|---
1989 | Cluster 1 | 48.87 | 45.52 | 38.77 | 49.13 | 47.73 | 34.89 | 36.80 | 44.29 | 4.17
Cluster 2 | 52.07 | 47.90 | 45.14 | 50.61 | 49.31 | 44.05 | 43.56 | 46.70
1995 | Cluster 1 | 34.06 | 20.40 | 44.22 | 44.91 | 47.36 | 35.03 | 51.08 | 54.08 | 5.46
Cluster 2 | 45.56 | 35.63 | 47.53 | 47.29 | 49.84 | 41.61 | 52.44 | 54.91
2009 | Cluster 1 | 48.87 | 51.06 | 38.77 | 46.91 | 44.22 | 44.91 | 34.89 | 36.81 | 4.47
Cluster 2 | 52.07 | 52.80 | 45.14 | 48.39 | 47.53 | 47.29 | 44.05 | 44.93
Table 10: Cluster average prices (EUR/MWh) - DE2 (spectral)

|  | W1 | W2 | W3 | W4 | W5 | W6 | W7 | W8 | $\Delta^{\text{year}}_{\mu}$
---|---|---|---|---|---|---|---|---|---|---
1989 | Cluster 1 | 52.68 | 48.38 | 46.38 | 50.72 | 49.36 | 44.96 | 44.58 | 47.43 | 3.09
Cluster 2 | 49.17 | 45.94 | 40.29 | 49.85 | 48.51 | 37.41 | 38.86 | 44.68
| Cluster 3 | 49.72 | 45.81 | 39.04 | 48.88 | 47.51 | 35.88 | 36.81 | 44.32
1995 | Cluster 1 | 47.30 | 38.88 | 47.80 | 47.66 | 50.05 | 42.57 | 52.58 | 55.08 | 3.99
Cluster 2 | 36.69 | 23.49 | 45.31 | 45.88 | 48.37 | 36.26 | 51.77 | 54.24
| Cluster 3 | 35.34 | 21.51 | 44.37 | 44.53 | 47.18 | 36.21 | 50.73 | 54.18
2009 | Cluster 1 | 52.68 | 53.05 | 46.38 | 48.42 | 47.8 | 47.66 | 44.96 | 46.74 | 3.37
Cluster 2 | 49.17 | 51.49 | 40.29 | 47.89 | 45.31 | 45.88 | 37.41 | 38.87
| Cluster 3 | 49.72 | 51.21 | 39.04 | 46.38 | 44.37 | 44.53 | 35.88 | 36.69

Table 11: Cluster average prices (EUR/MWh) - DE3 (spectral)

|  | W1 | W2 | W3 | W4 | W5 | W6 | W7 | W8 | $\Delta^{\text{year}}_{\mu}$
---|---|---|---|---|---|---|---|---|---|---
1989 | Cluster 1 | 49.03 | 46.0 | 39.51 | 49.91 | 48.52 | 36.68 | 38.28 | 44.48 | 3.14
Cluster 2 | 52.30 | 47.85 | 45.41 | 50.94 | 49.77 | 44.74 | 43.06 | 46.37
Cluster 3 | 49.00 | 45.41 | 38.73 | 48.84 | 47.44 | 34.78 | 36.55 | 44.25
Cluster 4 | 52.62 | 48.50 | 46.44 | 50.37 | 48.91 | 44.81 | 45.46 | 47.86
1995 | Cluster 1 | 35.24 | 21.24 | 44.78 | 45.75 | 48.32 | 36.11 | 51.63 | 54.13 | 4.19
Cluster 2 | 45.35 | 35.98 | 47.97 | 47.80 | 50.53 | 41.13 | 53.33 | 55.07
Cluster 3 | 34.37 | 20.74 | 44.23 | 44.60 | 47.05 | 34.99 | 50.87 | 54.11
Cluster 4 | 48.32 | 39.51 | 47.58 | 47.13 | 49.35 | 43.53 | 51.58 | 54.94
2009 | Cluster 1 | 49.03 | 51.29 | 39.51 | 47.72 | 44.78 | 45.75 | 36.68 | 37.71 | 3.34
Cluster 2 | 52.30 | 53.05 | 45.41 | 49.11 | 47.97 | 47.80 | 44.74 | 44.95
Cluster 3 | 49.00 | 51.07 | 38.73 | 46.6 | 44.23 | 44.60 | 34.78 | 36.69
Cluster 4 | 52.62 | 52.87 | 46.44 | 47.67 | 47.58 | 47.13 | 44.81 | 47.18

Table 12: Cluster average prices - DE4 (spectral)

##  Appendix C Intra- and inter-cluster similarity evaluation

Year | $k$ | Alg. | Loc. | $\sigma$ | $\Delta_{\mu}$ | $\Delta_{\text{min}}$ | $\Delta_{\text{avg}}$ | $DB$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{3}$ | $\mu_{4}$
---|---|---|---|---|---|---|---|---|---|---|---|---
1989 | 2 | K-Means |  | 13.4 | 5.91 | 0.0 | 5.91 | 0.63 | 47.28 | 41.37 | - | -
1995 | 2 | K-Means |  | 11.16 | 4.66 | 0.0 | 4.97 | 0.76 | 48.47 | 43.81 | - | -
2009 | 2 | K-Means |  | 10.8 | 3.68 | 0.0 | 3.68 | 0.71 | 45.29 | 48.97 | - | -
1989 | 2 | Spectral |  | 13.39 | 3.57 | 0.0 | 4.3 | 1.44 | 43.61 | 47.18 | - | -
1995 | 2 | Spectral |  | 15.16 | 2.87 | 0.0 | 3.57 | 1.89 | 44.74 | 41.87 | - | -
2009 | 2 | Spectral |  | 11.61 | 0.31 | 0.0 | 2.19 | 13.76 | 46.68 | 46.37 | - | -
1989 | 2 | K-Means | x | 13.0 | 5.91 | 0.0 | 5.93 | 0.62 | 41.64 | 47.55 | - | -
1995 | 2 | K-Means | x | 13.03 | 5.9 | 0.0 | 5.93 | 0.56 | 47.88 | 41.98 | - | -
2009 | 2 | K-Means | x | 9.97 | 2.44 | 0.0 | 2.91 | 0.84 | 46.36 | 48.8 | - | -
1989 | 2 | Spectral | x | 15.46 | 1.29 | 0.0 | 3.97 | 5.72 | 44.05 | 42.76 | - | -
1995 | 2 | Spectral | x | 14.2 | 3.0 | 0.0 | 4.41 | 2.26 | 44.11 | 47.11 | - | -
2009 | 2 | Spectral | x | 11.68 | 0.39 | 0.0 | 2.25 | 9.3 | 46.64 | 46.25 | - | -
1989 | 3 | K-Means |  | 12.11 | 4.24 | 0.19 | 4.51 | 1.65 | 41.64 | 47.44 | 48.0 | -
1995 | 3 | K-Means |  | 12.47 | 5.15 | 1.29 | 5.23 | 0.96 | 46.35 | 49.11 | 41.38 | -
2009 | 3 | K-Means |  | 10.06 | 2.61 | 0.0 | 2.89 | 0.94 | 48.59 | 45.16 | 49.08 | -
1989 | 3 | Spectral |  | 14.81 | 4.66 | 0.01 | 4.83 | 1.4 | 47.46 | 41.69 | 40.47 | -
1995 | 3 | Spectral |  | 15.56 | 2.24 | 0.0 | 2.91 | 3.86 | 45.04 | 41.82 | 41.68 | -
2009 | 3 | Spectral |  | 11.11 | 2.51 | 0.0 | 2.57 | 1.09 | 48.81 | 45.04 | 46.4 | -
1989 | 3 | K-Means | x | 11.78 | 4.95 | 0.51 | 5.01 | 0.88 | 46.66 | 48.78 | 41.35 | -
1995 | 3 | K-Means | x | 11.99 | 4.45 | 0.0 | 4.83 | 1.08 | 47.42 | 41.86 | 48.54 | -
2009 | 3 | K-Means | x | 10.17 | 2.88 | 0.43 | 3.03 | 1.17 | 48.47 | 45.11 | 49.43 | -
1989 | 3 | Spectral | x | 12.8 | 4.32 | 0.01 | 4.71 | 2.33 | 41.18 | 46.88 | 47.66 | -
1995 | 3 | Spectral | x | 13.93 | 5.11 | 0.0 | 5.42 | 1.48 | 44.16 | 41.73 | 49.39 | -
2009 | 3 | Spectral | x | 10.97 | 2.1 | 0.0 | 2.47 | 1.43 | 45.42 | 46.36 | 48.57 | -
1989 | 4 | K-Means |  | 12.86 | 4.38 | 0.14 | 4.58 | 1.77 | 48.51 | 41.28 | 42.44 | 47.05
1995 | 4 | K-Means |  | 15.09 | 3.38 | 0.0 | 3.73 | 1.39 | 42.62 | 48.68 | 49.06 | 47.71
2009 | 4 | K-Means |  | 10.73 | 2.8 | 0.45 | 2.97 | 1.57 | 47.6 | 44.85 | 50.31 | 48.05
1989 | 4 | Spectral |  | 12.23 | 4.33 | 0.21 | 4.39 | 1.6 | 41.25 | 47.21 | 49.12 | 44.86
1995 | 4 | Spectral |  | 15.09 | 3.78 | 0.0 | 4.32 | 3.54 | 44.62 | 41.73 | 41.73 | 48.32
2009 | 4 | Spectral |  | 10.73 | 2.46 | 0.0 | 2.76 | 1.72 | 48.91 | 45.32 | 45.0 | 48.37
1989 | 4 | K-Means | x | 14.65 | 4.33 | 1.23 | 4.35 | 0.83 | 41.21 | 47.53 | 39.79 | 43.96
1995 | 4 | K-Means | x | 11.93 | 4.14 | 0.77 | 4.37 | 1.75 | 41.41 | 47.73 | 45.97 | 49.11
2009 | 4 | K-Means | x | 12.22 | 2.14 | 0.0 | 2.28 | 0.65 | 45.35 | 48.75 | 44.57 | 45.04
1989 | 4 | Spectral | x | 12.36 | 3.53 | 0.25 | 4.12 | 3.85 | 46.28 | 47.44 | 40.9 | 47.57
1995 | 4 | Spectral | x | 15.73 | 1.95 | 0.0 | 2.52 | 2.81 | 45.3 | 41.74 | 41.55 | 41.47
2009 | 4 | Spectral | x | 11.59 | 2.04 | 0.0 | 2.19 | 1.7 | 48.76 | 44.93 | 46.35 | 45.58
Table 13: $\sigma$ is the mean of all cluster price standard deviations,
$\Delta_{\mu}$ is the average distance between cluster mean prices,
$\Delta_{\text{min}}$ is the mean nearest neighbour cluster distance,
$\Delta_{\text{avg}}$ denotes the average distance between all points situated
in different clusters, and $DB$ is the Davies-Bouldin index. $DB$ has values
in $[0,\infty)$ and measures the average similarity of every cluster with its
most similar cluster. A low $DB$ index indicates a good clustering. $\mu_{i}$
denotes the mean price of cluster $i$. All metrics were computed considering
the yearly average price for each node.

##  Appendix D Clusterings visualization based on Rand indices

Figure 9: RI = 0.44 - left: D3 (spectral); right: Spectral Clustering based on
prices; $k=3$, 2009
Figure 10: RI = 0.69 - K-Means based on prices; $k=3$; left: 1989, right: 1995
Figure 11: RI = 0.83 - K-Means based on prices; $k=4$; left: 1989, right: 2009
Figure 12: RI = 0.93 - K-means based on prices and geographic coordinates;
$k=3$; left: 1989, right: 2009

##  Appendix E Single-day clustering analysis

The time granularity of the prices included in the clustering computation
influences the stability of the resulting clusterings. In what follows, we
analyze the stability of the clusterings identified given the prices
corresponding to a single day. We select the day of January 22, 1989, and
compute clusterings using K-Means and Spectral Clustering. As clustering
feature(s), we consider (1) the time series of prices of a node (12 hours,
i.e., 12 features) and (2) the time series of prices of a node and its
geographic coordinates.

Table 14 shows the values of the inter- and intra-cluster similarity metrics
introduced in Section 3.2.1. We observe the average distances between cluster
mean prices ($\Delta_{\mu}$) and the average distances between all points
situated in different clusters ($\Delta_{\text{avg}}$) are considerably lower
compared to the values shown in Table 13, indicating that the clusters are
similar to each other and thus not well-defined. We recall that the results
illustrated in Table 13 correspond to clusterings obtained considering the
annual time series of nodal prices as clustering features. With time series of
smaller length, clusterings appear to be less stable.

$k$ | Alg. | Loc. | $\sigma$ | $\Delta_{\mu}$ | $\Delta_{\text{min}}$ | $\Delta_{\text{avg}}$ | $DB$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{3}$ | $\mu_{4}$
---|---|---|---|---|---|---|---|---|---|---|---
2 | K-Means |  | 12.78 | 0.81 | 0.0 | 0.8 | 0.94 | 56.31 | 57.12 | - | -
2 | K-Means | x | 16.17 | 0.39 | 0.0 | 0.56 | 2.03 | 56.35 | 56.74 | - | -
2 | Spectral |  | 13.23 | 0.76 | 0.0 | 0.78 | 1.01 | 56.32 | 57.08 | - | -
2 | Spectral | x | 11.28 | 0.33 | 0.0 | 0.39 | 0.73 | 56.43 | 56.1 | - | -
3 | K-Means |  | 11.16 | 0.73 | 0.09 | 0.75 | 0.83 | 56.34 | 56.07 | 57.17 | -
3 | K-Means | x | 10.38 | 0.23 | 0.0 | 0.34 | 1.09 | 56.43 | 56.4 | 56.08 | -
3 | Spectral |  | 12.62 | 0.77 | 0.0 | 0.78 | 1.04 | 56.36 | 57.13 | 55.97 | -
3 | Spectral | x | 13.44 | 0.47 | 0.0 | 0.55 | 1.71 | 56.39 | 56.78 | 56.07 | -
4 | K-Means |  | 12.88 | 0.57 | 0.08 | 0.57 | 1.11 | 56.61 | 56.13 | 57.23 | 56.47
4 | K-Means | x | 12.94 | 0.3 | 0.0 | 0.39 | 5.21 | 56.68 | 56.08 | 56.35 | 56.33
4 | Spectral |  | 13.02 | 0.67 | 0.0 | 0.68 | 0.94 | 56.39 | 57.2 | 55.97 | 56.08
4 | Spectral | x | 14.52 | 0.16 | 0.0 | 0.27 | 3.11 | 56.29 | 56.14 | 56.44 | 56.35
Table 14: $\sigma$ is the mean of all cluster price standard deviations,
$\Delta_{\mu}$ is the average distance between cluster mean prices,
$\Delta_{\text{min}}$ is the mean nearest neighbour cluster distance,
$\Delta_{\text{avg}}$ denotes the average distance between all points situated
in different clusters, and $DB$ is the Davies-Bouldin index. $DB$ has values
in $[0,\infty)$ and measures the average similarity of every cluster with its
most similar cluster. A low $DB$ index indicates a good clustering. $\mu_{i}$
denotes the mean price of cluster $i$. All metrics were computed considering
the yearly average price for each node.