# Zonal vs. Nodal Pricing: An Analysis of Different Pricing Rules in the
German Day-Ahead Market

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

The European electricity market is based on large pricing zones with a uniform
day-ahead price. The energy transition leads to shifts in supply and demand
and increasing redispatch costs. In an attempt to ensure efficient market
clearing and congestion management, the EU Commission has mandated the Bidding
Zone Review (BZR) to reevaluate the configuration of European bidding zones.
Based on a unique data set published in the context of the BZR, we compare
various pricing rules for the German power market. We compare market clearing
and pricing for national, zonal, and nodal models, including their generation
costs and associated redispatch costs. Moreover, we investigate different non-
uniform pricing rules and their economic implications for the German
electricity market. Our results indicate that the differences in the average
prices in different zones are small. The total costs across different
configurations are similar and the reduction of standard deviations in prices
is also small based on this data set. A nodal pricing rule leads to the lowest
total costs. We also analyze the quality of different pricing rules and their
differences with respect to the quality of the price signals and the necessary
uplift payments. While the study focuses on Germany, the analysis is relevant
beyond and feeds into the broader discussion about pricing rules.

††journal: TUM Working Paper

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

##  1 Introduction

A fundamental problem in electricity market design is to compute adequate
price signals in the presence of non-convex preferences of market
participants. Prices should reflect the true value of electricity and provide
the right bidding and investment incentives to all relevant market
participants (Cramton,, 2017). In liberalized markets, day-ahead electricity
prices are derived from welfare maximization problems, but pricing is
challenging due to the non-convexity of these optimization problems
(Liberopoulos & Andrianesis,, 2016). A central discussion on electricity
market pricing revolves around zonal and nodal pricing (aka. locational
marginal pricing). Among the electricity wholesale markets implementing nodal
pricing are Argentina, Chile, Mexico, New Zealand, Peru, Russia, Singapore,
and several regions in the United States (U.S.). In contrast, a zonal pricing
approach has been employed in the large and coupled European markets to reduce
the computational burden of market clearing and pricing. Large groups of nodes
are aggregated into zones, and the market clearing problem only considers flow
constraints between zones. With some exceptions (e.g., Italy, Norway, Sweden),
there is a single price zone within a country, and the day-ahead market
provides a uniform national electricity price. Ignoring transmission
constraints can lead to dispatch decisions that are not physically feasible.
Thus, after the market clearing, transmission operators conduct redispatch to
ensure that the final allocation aligns with physically feasible power flows.

###  1.1 Zonal vs. Nodal Pricing

Due to climate change and the ongoing energy transition, the zonal pricing
paradigm has come under scrutiny (Eicke & Schittekatte,, 2022; Bertsch et
al.,, 2016; Trepper et al.,, 2015). Volatile renewable energy sources such as
wind and solar lead to congestion in different parts of the electricity grid
(Neuhoff et al.,, 2013). In addition, the energy transition can lead to
structural problems. We will focus on Germany as a case in point. Wind farms
have grown significantly in Northern Germany, while much of the industrial
demand is in the south. Energy is cleared as if there were no transmission
constraints, but the traded energy can often not be delivered. As a result,
redispatch costs have increased drastically in recent years to 4.2 billion in
2022 (Bundesnetzagentur,, 2022). The redispatch costs in 2022 were also
influenced by the war in the Ukraine, but have been increasing consistently
over several years. In an attempt to ensure efficient market clearing and
congestion management, the EU Commission – as part of the Clean Energy Package
– has mandated a Bidding Zone Review (BZR) process to reevaluate the
configuration of European bidding zones. An integral part of this process is a
locational marginal pricing (LMP) study conducted by the European Network of
Transmission System Operators for Electricity (ENTSO-E) as a basis to identify
structural congestion and modified bidding zones. Based on this study, in
August 2022, the European Union Agency for the Cooperation of Energy
Regulators (ACER) decided on a set of alternative bidding zone configurations
based on proposals by ENTSO-E (ACER, 2022a, ). Among others, the BZR considers
a split of the German price zone.

In their LMP study, ENTSO-E solved linearized unit commitment models with
marginal pricing to obtain nodal prices (ENTSO-E,, 2022). The data is the main
input for the Bidding Zone Review and constitutes the results of a multi-year
effort of the European TSOs. The discussion of nodal and zonal prices has a
long history, but this unique data set allows for a unique opportunity to
compare different pricing rules on realistic data and estimate the
differences. Note that the allocation considers assets individually at each
node and abstracts from block orders or portfolio bids that are common
practice in Europe today. The computed prices served as input for clustering
approaches to yield alternative bidding zone configurations. Following Article
16 of the BZR methodology, ENTSO-E has published non-confidential data related
to the LMP study (ENTSO-E,, 2023). We leverage this dataset to expand on the
results of the LMP study, focusing on the German bidding zone. Beyond the
choice between four alternative zonal configurations of the German market, our
goal is to understand how these configurations compare to nodal prices and the
current market with a uniform national price.

Specifically, we compare market clearing and pricing for national, zonal, and
nodal models, the generation costs and associated redispatch costs. We
consider a redispatch model that minimizes cost-based compensations provided
to generators for any changes in their dispatched quantities during
redispatch. Moreover, we investigate different non-uniform pricing rules and
their economic implications for the German electricity market. We consider
Integer Programming (IP) pricing (O’Neill et al.,, 2005) and Convex Hull (CH)
pricing (Hogan & Ring,, 2003; Gribik et al.,, 2007) as two established pricing
rules in U.S. markets, and the recently suggested Join pricing rule (Ahunbay
et al.,, 2022). We also implemented a version of the Euphemia algorithm
currently used in European day-ahead markets. In contrast to IP, CH, and Join
pricing, Euphemia sacrifices welfare maximization to achieve linear and
anonymous prices and budget balance. We compare the outcomes of clearing and
pricing rules based on the total costs, average prices, make-whole payments
(to compensate paradoxically accepted bids), and lost opportunity costs. For
Euphemia, we also compute the welfare loss incurred by the algorithm. We
report aggregate results using the generation and demand scenarios for 2025
and climate year data of 2009 provided by the ENTSO-E LMP study. We also focus
on individual days to illustrate our findings and their economic implications.

###  1.2 Contributions

Our study suggests that a nodal pricing rule lowers system costs as costly
redispatch can be avoided. Conversely, by disregarding transmission
constraints, the zonal configurations lead to an inefficient dispatch. Even if
redispatch is conducted at minimal costs, this inefficiency leads to higher
overall system costs. The average price differences across configurations
between 1 and 4 zones for Germany are less than 3 EUR/MWh. This could partly
be attributed to the capacities assumed by ENTSO-E for 2025, e.g., new solar
capacities in the southern region. There are also individual days where the
price difference between zones is more than 10 EUR/MWh, but on average the
differences are small. The higher total costs for national and zonal prices
arise from the fact that due to the omission of network constraints, an
inefficient allocation is picked. This results in the selection of generators
that would not be part of an efficient nodal dispatch. These inefficiencies
lead to higher total costs after redispatch that is required to achieve a
feasible dispatch respecting the transmission capacities. As indicated
earlier, in the past there was often an oversupply of renewable energy sources
in the north and high demand in the south of Germany, which raised
expectations that redispatch decreases significantly with two price zones.
However, our results based on data from ENTSO-E, does not confirm such
expectations.

The redispatch costs in our model do not decrease considerably with a split in
2, 3, or 4 zones. The way how these redispatch costs are computed can be
considered as a lower bound. For example, we assume that every generator can
be redispatched, but this is not the case in reality. A key goal for the
Bidding Zone Review is to find pricing zones such that the variance in prices
is small, which is seen as a proxy for network congestion. We show that the
differences in the standard deviation of prices caused by congestion in
different zones are small compared to the overall standard deviation in
prices, which explains why the differences in the redispatch costs are small
across the different zonal configurations.

We also provide an analysis of different non-uniform pricing rules such as IP
pricing (as used in U.S. power markets), Convex Hull Pricing, and the Join
pricing rule. The results illustrate the trade-offs that can be expected in
terms of quality of the congestion signals and minimum make-whole payments and
provide guidance regarding the analysis of non-uniform pricing rules in the
context of CACM 2.0 (All NEMO Committee, 2023b, ). While our analysis focuses
on Germany, the policy implications go beyond. The study highlights the
challenges in delineating price zones in a zonal market and the trade-offs
between different pricing rules based on a new and unique data set that
results from a multi-year effort by the European TSOs.

###  1.3 Limitations and Organization

Like any empirical analysis of its kind, also our study faces limitations.
First, our examination focuses on the German day-ahead market without
incorporating neighboring countries, cross-border trades, or loop flows
through other countries. Second, we do not model the intraday or forward
markets, which hold considerable significance within the European electricity
system. Third, we assume fixed demand from the ENTSO-E dataset, neglecting the
potential for demand reduction incentives under nodal prices. Moreover, the
current European market is characterized by portfolio bidding, differing from
the unit-commitment bids underlying both the study by ENTSO-E and our
analysis. Portfolio bids allow optimization of trades across assets and differ
from those feasible in nodal market designs. Incorporating such considerations
into our analysis would require making strong and potentially unwarranted
assumptions regarding market participant behavior. Finally, the data set
provided by ENTSO-E was constructed for the target year 2025, and even though
different climate years are considered, it is important to acknowledge that
analyses can vary based on differing assumptions regarding supply and demand.
However, the ENTSO-E dataset represents the best source available today for
such a study. Additional assumptions made for our analysis are described in
Section 3. We also emphasize that the discussion surrounding zonal and nodal
pricing extends beyond dispatch and prices. In the ongoing Bidding Zone
Review, a comprehensive approach is adopted which considers factors such as
liquidity, security of supply, transition costs, and investments in low-carbon
technologies, which we do not in our analysis.

The remainder of this article is structured as follows. In Section 2, we
introduce our unit commitment model and formalize nodal and zonal market
clearing. We also review the IP, CH, and Join pricing rules. Section 3
elaborates on the data sources used to implement our dispatch model and
outlines our experimental design. Section 4 presents the key results of our
analysis of different market clearing and pricing rules. Lastly, Section 5
provides a summary and conclusions.

##  2 Clearing and Pricing on Electricity Markets

###  2.1 Preliminaries

We consider an electricity market consisting of a set of buyers $B$ and a set
of sellers $S$, each located at nodes $N$ in an interconnected electricity
network. The set of power lines $L$ is encoded as pairs of nodes $(n,m)$, and
we consider multiple periods $T$. An item in this market corresponds to a unit
of electricity at a particular node $n\in N$ at a specific time $t\in T$.

A buyer $b\in B$ possesses a valuation function $v_{b}:\mathbb{R}^{N\times
T}\rightarrow\mathbb{R}$ that quantifies the buyer’s preferences and
constraints. Similarly, a seller $s\in S$ has a cost function
$c_{s}:\mathbb{R}^{N\times T}\rightarrow\mathbb{R}$.

The market operator collects these buy and sell bids to calculate a feasible
allocation. As electricity is transmitted over the network, the allocation is
subject to physical power flows, encoded as a constraint set $\Psi$. An
accurate representation of the transmission network would require $\Psi$ to be
equivalent to AC optimal power flow (ACOPF) constraints (Molzahn & Hiskens,,
2019). However, it is well known that the ACOPF is intractable for realistic
problem sizes, and electricity market operators need to employ computationally
scalable approximations of the ACOPF.

###  2.2 Market Clearing

In practice, market operators employ different bidding languages to encode
$v_{b}$ and $c_{s}$, as well as different approximations $\Psi$ of the power
flow constraints. In this work, we use the data released for the European
bidding zone review (BRZ) to compare allocation and prices under different
market clearing mechanisms.

In full generality, the market operator seeks an allocation $(x,y)$, where
$x=(x_{b})_{b\in B}$, $x_{b}\in\mathbb{R}^{N\times T}$ is the allocation
vector of buyers and $y=(y_{s})_{s\in S}$,$y_{s}\in\mathbb{R}^{N\times T}$ is
the allocation vector of sellers. The market operator seeks an allocation that
maximizes welfare, by solving the following optimization model:

$\displaystyle\max_{x,y}\quad$ $\displaystyle\sum_{b\in
B}v_{b}(x_{b})-\sum_{s\in S}c_{s}(y_{s})$ (1) subject to $\displaystyle
x,y\in\Psi.$

On European day-ahead markets, buyers and sellers submit hourly bids and block
orders (NEMO Committee,, 2019). The BZR data, however, does not contain bids
of this structure. Instead, it provides generator and load characteristics
similar to unit commitment problems in U.S. markets. Specifically, for buyers
$b$ there is only information on a fixed demand profile
$P_{b}\in\mathbb{R}^{N\times T}$, while price-elastic bids are unavailable.
Consequently, the valuation function of $b$ simplifies to strict demand
satisfaction.

$v_{b}(x_{b})=\begin{cases}-\infty,\;\;x_{b}\neq P_{b}\\\
0,\;\;\;x_{b}=P_{b}\end{cases}$ (2)

For sellers/generators $s\in S$, the BZR data release includes more detailed
information. Each generator $s$ has a minimum and maximum production quantity
$\underline{P}_{s}\in\mathbb{R}$ and $\overline{P}_{s}\in\mathbb{R}$ at their
node $n_{s}$. Moreover, once a generator has been started, there is a minimum
uptime constraint to run for at least $\underline{R}_{s}\in\mathbb{Z}_{0}^{+}$
periods. Generators incur variable costs $g_{s}$ to produce electricity and
fixed costs $h_{s}$ whenever the generator runs. We use a binary commitment
variable $u_{st}\in\\{0,1\\}$ to model a generator’s constraint set $P_{s}$.

$\displaystyle P_{s}=\\{y_{s}\in\mathbb{R}^{N\times T}:\;$ $\displaystyle
y_{s,n_{s}t}\geq\underline{P}_{st}u_{st}$ $\displaystyle\;\forall t\in T,$
$\displaystyle y_{s,n_{s}t}\leq\overline{P}_{st}u_{st}$
$\displaystyle\;\forall t\in T,$ $\displaystyle\phi_{s,n_{s}t}\geq
u_{st}-u_{s(t-1)}$ $\displaystyle\;\forall t\in T\setminus\\{T_{0}\\},$
$\displaystyle\sum_{i=t-\underline{R}_{s}+1}^{t}\phi_{s,n_{s}i}\leq u_{st}$
$\displaystyle\;\forall t\in T\setminus\\{T_{0}\\},$ $\displaystyle
y_{s,nt}=0$ $\displaystyle\;\forall n\in N\setminus\\{n_{s}\\},t\in T$
$\displaystyle u_{st}\in\\{0,1\\}$ $\displaystyle\;\forall s\in S,t\in T.\\}$

With this constraint set, we define a generator’s cost function $c_{s}$ as
follows.

$c_{s}(y_{s})=\begin{cases}\infty,\;\;y_{s}\notin P_{s}\\\ \sum_{t\in
T}g_{s}y_{s,n_{s}t}+\sum_{t\in T}h_{s}u_{st},\;\;y_{s}\in P_{s}\end{cases}$
(3)

During the 2000s and 2010s, many U.S. ISO markets moved from zonal to nodal
pricing. Recognizing that an accurate representation of the transmission
network in the form of the ACOPF is computationally infeasible, market
operators resort to linearized versions of power flow. The most common
implementation is the Direct Current OPF (DCOPF), which is based on three
simplifying assumptions (Stott et al.,, 2009; Molzahn & Hiskens,, 2019), i.e.,
(1) uniform voltage magnitudes, (2) negligibly small voltage angle
differences, and (3) neglecting line resistances and reactive power. Under
these assumptions, the flow constraints can be expressed as simple linear
constraints.

$\Psi^{DC}=\\{(x,y):\sum_{s\in S}y_{s,nt}-\sum_{b\in B}x_{b,nt}=\sum_{m\in
N}-B_{nm}(\theta_{n}-\theta_{m})\;\forall n\in N,t\in T\\}$ (4)

In (4), $B_{nm}$ is the susceptance of the line connecting nodes $n$ and $m$.
If such a line does not exist, then $B_{nm}=0$, and $B_{nn}=0\;\forall n$. The
voltage angle decision variable for node $n$ is denoted as $\theta_{n}$. The
constraint set in (4) can be enriched by further constraints, such as thermal
line limits, box constraints on the voltage angles, and more (Li et al.,,
2022). We use such a standard DCOPF model in our analysis, similar to what was
used in the ENTSO-E study. When the representation $\Psi$ of the transmission
network considers all nodes in the transmission network, we refer to nodal
market clearing. When only a subset of the grid is considered, we speak of
zonal market clearing, as discussed in the following.

####  2.2.1 Zonal Market Clearing

The key idea of zonal pricing is to aggregate nodes into zones and only
consider transmission flows across zones. This simplifies the constraint set
$\Psi$ significantly, enhances computational scalability, and allows for
portfolio bidding of larger market participants. In the European market, two
approaches have been implemented: the available transfer capacity (ATC) model
and the flow-based market coupling (FBMC) methodology (NEMO Committee,, 2019).
Both approaches require a refined capacity calculation process before the
allocation to assess the available cross-zonal flow capacities. Furthermore,
the FBMC approach necessitates access to a power transfer distribution factor
(PTDF) matrix that indicates how zone net position changes affect cross-zonal
flows.

As these data were not included in the BZR data release, we rely on a
different approach to compute a zonal allocation. Specifically, to determine
the aggregate characteristics of a cross-zonal connection, we sum up the
thermal limits and use the mean susceptance over all relevant cross-zonal
lines. Specifically, with a set of zones $Z$ and
$B_{zz^{\prime}}=\overline{B}_{nm:n\in z,m\in z^{\prime}}$ as the mean cross-
zonal susceptance, the zonal constraint set is defined as follows.

$\Psi^{DC}_{Zonal}=\\{(x,y):\sum_{s\in S}y_{s,zt}-\sum_{b\in
B}x_{b,zt}=\sum_{z^{\prime}\in
Z}-B_{zz^{\prime}}(\phi_{z}-\phi_{z}^{\prime})\;\forall z\in Z,t\in T\\}$ (5)

In essence, (5) is a natural extension of (4) where a set of aggregate zones
is considered instead of nodes. Cross-zonal flows are feasible if they satisfy
$\Psi^{DC}_{Zonal}$.

The mapping of nodes to zones is suggested by the ENTSO-E study on locational
marginal pricing (ACER, 2022b, ). Four different zonal splits were proposed
for the German bidding zone, suggesting between two to four price zones.

####  2.2.2 Redispatch

While zonal market clearing is computationally less expensive than nodal
market clearing, it substantially simplifies the underlying physical power
flows. In fact, the economic outcome of zonal market clearing is usually
infeasible with physical power flows. As a result, transmission operators
resort to redispatch, i.e., modifying the economic allocation to provide a
physically feasible outcome.

Current redispatch mechanisms consider two paradigms: Firstly, cost-based
redispatch, as implemented in Germany, reimburses redispatched generators for
their additional costs or lost profits and makes them indifferent to the
previous market outcome. Secondly, market-based redispatch aims at a bid-based
procurement of redispatch volumes. Market-based redispatch was criticized as
prone to market power abuse and causing inc-dec gaming, i.e., strategic
overbidding in day-ahead markets to generate higher profits in subsequent
redispatch markets (Hirth & Schlecht,, 2020).

This work simulates a cost-based redispatch in line with real-world practices.
As redispatch should be cost-minimal, the fundamental idea of our approach is
to find a modified allocation that satisfies the constraint set of $\Psi^{DC}$
and, at the same time, minimizes the required compensations to sellers for
their changes in generation. Demand is excluded from redispatch, as it is
current practice in Germany. More formally, with $(x^{Zonal},y^{Zonal})$ as
the outcome of the zonal allocation, we solve the following problem:

$\displaystyle\min_{x,y}\quad$ $\displaystyle\sum_{s\in S}\lvert
c_{s}(y_{s})-c_{s}(y_{s}^{Zonal})\rvert$ (6) subject to $\displaystyle
x=x^{Zonal}$ $\displaystyle x,y\in\Psi^{DC}$

The objective of the optimization problem described in (6) is to adjust the
zonal allocation $(x^{Zonal},y^{Zonal})$ in a manner that yields a DCOPF-
feasible solution $(x,y)$ at minimal costs. This adjustment involves
compensating sellers for any changes in their dispatched quantities during
redispatch. For each seller $s$, this compensation is determined by the
difference between the costs they incur under the initial zonal allocation
$c_{s}(y_{s}^{Zonal})$ and the costs under the final DCOPF-feasible allocation
$c_{s}(y_{s})$. The objective of (6) is to minimize the total costs associated
with these adjustments. As part of our experiments we also report results for
redispatch that minimizes volumes instead of costs, i.e., the redispatch
objective function changes to $\min_{x,y}\;\sum_{s\in S}\lvert
y_{s}-y_{s}^{Zonal}\rvert$.

Problem (6) yields the final DCOPF-feasible allocation $(x,y)$ that can be
implemented as physical power flows. Note that a DCOPF-feasible solution may
still be ACOPF-infeasible and require further adjustments to the allocation,
which we disregard in this work. Market participants subject to redispatch
must be compensated for lost profits, reducing overall welfare. In our
numerical experiments, we thus consider system costs as the total cost of
generation before redispatch, redispatch costs themselves, and total costs as
the sum of system and redispatch costs.

###  2.3 Pricing on Non-Convex Markets

Once an allocation $(x,y)$ has been obtained, the market operator must provide
electricity prices $p\in\mathbb{R}^{N\times T}$ for each node and time period.
Due to the non-convexities implied by sellers’ cost functions (i.e., the
binary commitment variables $u_{st}$), this is not a trivial task. In
microeconomics, the welfare theorems provide a foundation for pricing in
markets. In their seminal paper, Arrow & Debreu, (1954) demonstrate that under
convex preferences, demand independence, and perfect competition with
divisible items, a market operator can always achieve a set of Walrasian
equilibrium prices that support the welfare-maximizing allocation. We assume
quasilinear utilities, i.e., the utility of each market participant is defined
as the difference between valuation or costs and price.

$\displaystyle u_{b}(x|p)=v_{b}(x)-\sum_{n\in N,t\in T}p_{nt}x_{b,nt}$
$\displaystyle\quad\forall\ b\in B$ (7) $\displaystyle u_{s}(y|p)=\sum_{n\in
N,t\in T}p_{nt}y_{s,nt}-c_{s}(y)$ $\displaystyle\quad\forall\ s\in S$

At given prices $p$, each market participant has some preferred bundles that
maximize utility. We call these bundles the demand set and denote the maximum
possible utility as follows.

$\displaystyle\hat{u}_{b}(p)=\max_{x}v_{b}(x)-\sum_{n\in N,t\in
T}p_{nt}x_{b,nt}$ $\displaystyle\quad\forall\ b\in B$ (8)
$\displaystyle\hat{u}_{s}(p)=\max_{y}\sum_{n\in N,t\in
T}p_{nt}y_{s,nt}-c_{s}(y)$ $\displaystyle\quad\forall\ s\in S.$

A Walrasian equilibrium consists of a market-clearing allocation and linear
prices such that no participant has an incentive to deviate.

######  Definition 1 (Walrasian Equilibrium).

A price vector $p$ and a feasible allocation $(x,y)$ form a Walrasian
equilibrium if:

  1. 1.

(Market clearing) Supply equals demand, i.e., $\sum_{s\in S,n\in
N}y_{s,nt}=\sum_{b\in B,n\in N}x_{b,nt}\;\forall t\in T$.

  2. 2.

(Envy-freeness) Every market participant maximizes their utility at the
prices, i.e., $u_{b}(x|p)=\hat{u}_{b}(p)\;\forall b\in B$ and
$u_{s}(y|p)=\hat{u}_{s}(p)\;\forall s\in S$.

  3. 3.

(Budget balance) The payments that sellers receive equal the payments that
buyers provide, i.e., $\sum_{n\in N,t\in T}p_{nt}(\sum_{s\in
S}y_{s,nt}-\sum_{b\in B}x_{b,nt})=0$.

Unfortunately, real-world electricity markets do not satisfy the assumptions
underlying the welfare theorems. As in many other markets, items are not
perfectly divisible and market participants exhibit non-convex preferences. A
large stream of literature has investigated existence conditions for Walrasian
equilibria in non-convex markets (Kelso & Crawford,, 1982; Bikhchandani &
Ostroy,, 2002; Baldwin & Klemperer,, 2019), but none of these conditions are
satisfied in electricity markets. Bikhchandani & Mamer, (1997) demonstrate
that Walrasian equilibria exist if and only if the linear relaxation of the
allocation problem has an integer solution. When this is not the case, market
participants bear (global) lost opportunity costs at any set of prices and do
not maximize their individual profits, violating envy-freeness.

Consequently, market operators have to resort to heuristic pricing rules that
aim at approximating Walrasian equilibrium prices. Many such pricing rules
have been proposed (Liberopoulos & Andrianesis,, 2016), each compromising on
the properties of Walrasian equilibria differently. In this work, we focus on
Convex Hull (CH) pricing (Hogan & Ring,, 2003; Gribik et al.,, 2007) and
Integer Programming (IP) pricing (O’Neill et al.,, 2005) as two established
pricing rules in practice, as well as on the Join pricing rule (Ahunbay et
al.,, 2022) as a novel multi-objective approach to pricing. We also consider
the Euphemia pricing rule as it is applied in European electricity markets.

####  2.3.1 Convex Hull Pricing

In the absence of Walrasian equilibrium prices, there is no set of prices $p$
that maximizes the profit of every market participant. The concept of global
lost opportunity costs describes this forgone payoff.

######  Definition 2 (Global Lost Opportunity Costs (GLOCs)).

Given an allocation $(x,y)$ and prices $p$, a market participant’s global lost
opportunity costs describe the difference between their individual payoff
maximum, given $p$, and their actual payoff.

$\displaystyle\text{GLOC}_{b}=\hat{u}_{b}(p)-u_{b}(x|p)$
$\displaystyle\text{GLOC}_{s}=\hat{u}_{s}(p)-u_{s}(y|p)$

The key idea of CH pricing (Hogan & Ring,, 2003; Gribik et al.,, 2007) is to
minimize GLOCs over all market participants. Formally, CH pricing replaces
each non-convex cost function $c_{s}(y_{s})$ with its convex envelope in (1),
and derives prices from the dual of the resulting convex problem.

Computing convex envelopes and determining CH prices is generally
computationally challenging (Schiro et al.,, 2016). However, it has been shown
by Hua & Baldick, (2017) that CH pricing becomes tractable when the cost
function is described as in (3). Specifically, by relaxing the binary
constraints $u_{st}\in\\{0,1\\}$ to $u_{st}\in[0,1]$ and solving the dual of
(1), CH prices can be obtained from a single linear program. This approach,
also referred to as Extended Locational Marginal Pricing (ELMP), was also used
by the recent ENTSO-E study (ENTSO-E,, 2022).

####  2.3.2 IP Pricing

In a convex market, Walrasian equilibrium prices are equivalent to marginal
prices, i.e., the costs of the last accepted bid set the prices. Following
this notion, O’Neill et al., (2005) try to generalize marginal pricing for
non-convex markets. Their IP pricing rule assumes that generators cannot
easily deviate from their commitment status (as encoded by the binary variable
$u_{st}$) and sets prices equal to the variable costs of the marginal
committed unit (i.e., the marginal unit among all units with $u_{st}=1$).

IP pricing has gained popularity in many practical markets. It involves three
steps: (i) obtaining the optimal commitment variables $u_{st}^{*}$ from the
allocation problem in (1), (ii) fixing the commitment variables of each
generator to these optimal values, i.e., setting $u_{st}\in\\{0,1\\}$ to
$u_{st}\in[0,1]$ with $u_{st}=u_{st}^{*}$ in (3), and (iii) solving (1) with
these linearized cost functions and extracting prices from its dual.

IP prices can be considered Walrasian equilibrium prices assuming that no
generator can deviate from its commitment status. In other words, every
participant maximizes their utility locally, meaning under fixed commitment.
We define these local lost opportunity costs a subset of GLOCs.

######  Definition 3 (Local Lost Opportunity Costs (LLOCs)).

Given an allocation $(x,y)$, generator commitments $u^{*}$, and prices $p$, a
seller’s local lost opportunity costs describe the difference between their
individual payoff maximum _under fixed commitment_ , given $p$, and their
actual payoff.

$\displaystyle\text{LLOC}_{s}$
$\displaystyle=\hat{u}^{\prime}_{s}(p)-u_{s}(y|p)\text{ with
}\hat{u}^{\prime}_{s}(p)=\hat{u}_{s}(p)\text{ s.t. }u=u^{*}$

By convexity of $v_{b}(x_{b})$, a buyer’s LLOCs exactly equal their GLOCs.
Besides computational tractability, IP prices accurately reflect the marginal
value of transmission capacity (Yang et al.,, 2019). Specifically, price
differences among nodes arise solely when the network experiences congestion.
It has been shown that this property immediately corresponds minimizing LLOCs
(Ahunbay et al.,, 2022), making them a significant indicator of good
congestion signals.

####  2.3.3 Join Pricing

In most electricity markets, neither GLOCs nor LLOCs are paid out to market
participants to disincentivize deviations from the optimal allocation.
Instead, market operators merely ensure that no participant incurs losses by
participating in the market, ensuring non-negative utilities and,
equivalently, individual rationality. The payments to ensure individual
rationality are known as make-whole payments.

######  Definition 4 (Make-Whole Payments (MWPs)).

Given an allocation $(x,y)$ and prices $p$, a market participant’s make-whole
payments describe their negative payoff, if applicable.

$\displaystyle\text{MWP}_{b}$ $\displaystyle=\max\\{-u_{b}(x|p),0\\}$
$\displaystyle\text{MWP}_{s}$ $\displaystyle=\max\\{-u_{s}(y|p),0\\}$

Bids that require MWPs are also known as paradoxically accepted bids. MWPs can
be regarded as another subset of GLOCs, referring only to participants’
incentives to not participate in the market at all. However, both CH and IP
pricing can experience high levels of MWPs. Recent practical concerns have
thus motivated the development of pricing rules that ensure lower levels of
MWPs. Among them is the Join pricing rule (Ahunbay et al.,, 2022), which
proposes a dual pricing problem that combines the objective of IP pricing
(minimizing LLOCs) with minimizing MWPs. Specifically, for each participant it
considers the maximum of LLOCs and MWPs in the objective, leading to a robust
and computationally tractable pricing rule with adequate congestion signals
and low MWPs simultaneously.

###  2.4 Euphemia

CH, IP, and Join pricing all have in common that they price the welfare-
maximizing allocation. Market participants incur lost opportunity costs; some
are even paradoxically accepted and need to be compensated by MWPs.

European electricity markets follow a different approach. Paradoxically
accepted bids are not permitted, and instead, the market operator deviates
from the welfare-maximizing outcome to avoid paying any MWPs. This is the
essence of the Euphemia day-ahead clearing algorithm (NEMO Committee,, 2019).
Note that avoiding paradoxically accepted bids does not satisfy the envy-
freeness property from Definition 1. Market participants still incur GLOCs and
LLOCs, a subset of which are even rejected to participate in the market even
though it would be profitable for them to do so. Such bids are referred to as
paradoxically rejected bids.

The Euphemia algorithm combines market clearing and pricing (unlike the
pricing rules above which can be applied to any pre-computed allocation). In
European markets, it computes an allocation and zonal prices, but
theoretically, it can also be applied in nodal markets. Algorithmically, it
starts by computing the welfare-maximizing allocation, solving (1). It then
advances to examining whether prices can be established for the allocation,
i.e., they must prevent any paradoxically accepted bids, ensure the acceptance
of all in-the-money hourly bids and, optionally, certify that cross-zonal
flows occur from a lower-price to a higher-price zone. If paradoxically
accepted bids exist, Euphemia adds cuts to (1) that eliminate the current
candidate solution, constraining the allocation problem and leading to welfare
losses. These iterations between allocation and pricing continue until a
solution without paradoxically accepted bids is reached.

Subsequently, Euphemia continues by addressing some of the idiosyncrasies of
the European market (PUN search, re-insertion of certain paradoxically
rejected bids, resolving indeterminacy). If, at any of these steps, violations
are detected, the iterative procedure continues. Eventually, Euphemia outputs
allocation, zonal prices, and cross-zonal flows.

Since we aim to investigate prices and welfare losses associated with
Euphemia, we implemented a simplified version of the algorithm for our
numerical experiments. With the data provided, however, we can only replicate
the first two steps of the algorithm, i.e., the iterative allocation and
pricing procedure. When a paradoxically accepted seller $s$ in period $t$ is
detected, we add a constraint $u_{st}=0$ to the allocation problem (1). No
open-source implementation of the algorithm is available, but we followed the
public description as closely as possible.

A practical problem with Euphemia is its computational tractability. Due to
its iterative nature, multiple mixed-integer problems must be solved until
allocation and prices are obtained. With the planned 15-min market time unit
introduction in Europe, the scalability of Euphemia poses a significant
concern, and policymakers consider non-uniform pricing rules (such as the ones
introduced above) to obtain solutions faster (All NEMO Committee,, 2022).

##  3 Data and Processing

This study examines welfare, prices, GLOCs, LLOCs, and MWPs for multiple
zonal/nodal configurations, allocation rules, and pricing mechanisms. As
discussed, we leverage the data released in the context of the European
bidding zone review (ENTSO-E,, 2022; ACER, 2022b, ). We focus on Germany as
arguably one of the most important bidding zones regarding clearing volume and
its central position in the European grid. In this section, we provide an
overview of the data set and our processing of it.

###  3.1 Bidding Zone Review Data

In September 2022, ENTSO-E, on behalf of all European TSOs, released various
data sets related to their locational pricing study (ENTSO-E,, 2022, 2023).
The input data structure along two primary datasets: the grid model as a basis
for the transmission network and the input files to model generation and
demand. Data on reserves, storage, and imports/exports were not considered in
our study. Along with these input data, results files such as locational
prices and cleared generation, demand, and storage were released. The
publication refers to a total of 24 weeks over three representative climate
years (1989, 1995, and 2009). Only the climate information from 1989, 1995,
and 2009 is used, while generation and demand scenarios were generated for the
target year 2025 (ENTSO-E,, 2022).

Unfortunately, some parts of the data were aggregated before their publication
(e.g., local demand to country-wide demand), and some data were not revealed
(e.g., a mapping of generators in the input files to their location in the
grid model). We elaborate on these issues in the following. Without access to
these proprietary data, the outcome of the ENTSO-E study cannot be exactly
reproduced. However, it still provides a foundation against which we can
validate our own results.

The grid model for the German transmission system includes information on the
system topology, components, and operational states. Each substation comprises
multiple voltage levels, each of which can comprise several topological nodes
that are connected with grid components / transmission lines. We consider
power transformers as regular transmission lines and neglect their phase-
shifting capability. For the grid constraints of our DCOPF in (4), we derive
the per-unit susceptance $B_{nm}$ for each line from the given actual
susceptance and base admittance (Grainger & Stevenson,, 1994). We also induce
transmission limits on each power line, which we infer as the minimum of each
line’s angular stability limit (inferred from its surge impedance loading and
the St. Clair curve (St. Clair,, 1953)), voltage drop limit (inferred from
voltage and reactance (Hao & Xu,, 2008)), and thermal limit (inferred from
voltage and current on a three-phase system (Moreira et al.,, 2006)). We
assume these transmission limits to be constant over time. Missing data, e.g.,
on the length and susceptance of lines, were replaced by the data in the JAO
Static Grid Model where possible, and else inferred from the mean of all other
lines.

The topology of the grid, defined by its nodes and connections, dictates the
path electricity takes from generation to consumption, with substations
facilitating the necessary voltage transformations along the way. Substations
act as critical nodes in the network that manage voltage levels for efficient
power flow. A substation might handle multiple voltage levels. For example, a
transmission substation might receive power at a very high voltage (e.g.,
220kV or higher) and step it down to a lower high voltage (e.g., 110kV) for
further transmission or to a medium voltage (e.g., 10-35kV) for distribution
to urban or industrial areas.

The ENTSO-E dataset provides a model of Germany’s power grid, detailing 834
substations. Each of these substations encompasses one or more ”Voltage
Levels,” summing up to 1697 in the dataset. These voltage levels cover the
spectrum from transmission ($\geq$ 220kV) to distribution ($\leq$ 110kV) and
are composed of several ”Topological Nodes.” Altogether, there are 2898
topological nodes listed in the data sets. Additionally, the dataset includes
details for 100 substations located in the neighboring countries, which have
been integrated into our analysis. For our study, we manually determined the
geographical locations of all substations and computed prices for all voltage
levels on these substations. After some preprocessing, which involved removing
disconnected nodes, breakers, and disconnectors, our German grid model
consists of 1670 nodes and 3232 lines (Schmitt,, 2023). Technically, supply
and demand on different voltage levels of substations do not necessarily need
to be the same. However, it turned out that prices computed for voltage levels
on a substation were identical or very close in the ENTSO-E dataset. How to
define a node in a nodal pricing system and whether to extend such prices to
the distribution network is a separate question beyond this paper. In order to
illustrate our results, we manually derived geographical coordinates for each
substation from OpenStreetMap.

For the demand side, the input files provide hourly aggregated load profiles
for the entire country of Germany. Neither the nodal distribution of demand
nor demand valuations are provided. Demand-side response was considered in the
ENTSO-E study, but no related data was released. Consequently, we distribute
demand proportional to each consumer’s base load that is available as part of
the grid model. We thereby assume that demand fluctuations over time occur
uniformly across all consumers (Schmitt,, 2023). We further assume that demand
is exclusively price-inelastic.

The input files pertaining to the supply side are more granular and separated
by generator type, i.e., Hard Coal, Lignite, Gas, Light Oil, Solar, Hydro Run-
Of-River, Hydro Reservoir, Hydro PumpOL, Hydro PumpCL, Onshore Wind, Offshore
Wind, Other Non-RES, and Other RES. They contain operational characteristics
of thermal units, derived from the Pan-European Market Modeling Database
(PEMMDB), and hourly aggregated output of renewable energy sources, derived
from Pan European Climate Database (PECD). The operational characteristics
include – among others – minimum / maximum power and minimum runtime
requirements that we use to parameterize our generator cost functions. The
data also includes must-run obligations, start-up / shut-down times, and ramp
rates, which may be used to further extend the model. Similarly, data on
storage or demand response could leverage more detailed models.

As the maximum power of each seller is given as a fixed quantity (denoting its
nominal capacity), we need to manually incorporate the variability of
renewable energy sources. We thus distribute the aggregate dispatched energy
in each hour equally to all plants of the same type, proportional to their
nominal capacities, and set the maximum power to this value. The underlying
assumptions are that renewables are always dispatched at maximum power and
that meteorological conditions are equal across Germany. Because hydro power
plants are dispatchable, they are excluded from this logic. The data also do
not include information on variable or fixed costs for renewables, which we
derived from literature (Kost et al.,, 2021; IRENA,, 2022; ENTSO-E,, 2022).
For thermal units, variable cost information are provided, while fixed cost
information were again obtained from literature (Kost et al.,, 2021; Schröder
et al.,, 2013; Bundesnetzagentur,, 2022).

An additional challenge is the mapping of generators to corresponding network
nodes. The grid data specify the broad category of generating units at each
node (e.g., thermal, hydro, external injections) and their nominal capacities.
While renewable resources could be mapped to the seller data by matching
nominal capacities, the grid data lacks additional information regarding
thermal plant types. We developed a heuristic to map thermal generators in the
grid to their operational characteristics (such as minimum uptime constraints)
(Schmitt,, 2023). Firstly, we match identifiers with the Kraftwerksliste
(power plant list) provided by the Bundesnetzagentur to obtain each
generator’s broad thermal generation type (e.g., lignite, gas, etc.). As
nominal capacities do not match between grid and seller data, and seller data
have a higher granularity (e.g., Gas CCGT, Gas OCGT, etc.), we solve an
integer program to assign each entity in the grid model to exactly one
corresponding seller of the same plant type, with the objective to minimize
the aggregated differences in capacities and number of units per plant type.
We provide this integer problem and the outcome of the generation matching in
A While this assignment might not be exact, we argue that smaller mistakes in
the assignment of generators to nodes have little impact on the more general
questions raised in our paper that address the benefits of different zonal and
nodal configurations. We provide our code files and a data description in an
online appendix. The data are accessible through the ENTSO-E website.

###  3.2 Experimental Design

We implemented the models in Python 3.8 using Gurobi to solve optimization
problems. We conducted experiments for eight weeks in 2009, as suggested by
the ENTSO-E report (ENTSO-E,, 2022), for the German bidding zone. To maintain
computational feasibility of computing hourly allocations and prices, we
permitted a MIP gap of 5% for mixed-integer programs. Table 1 summarizes our
experimental design and the scope of our analysis.

We first employ different allocation rules in accordance with the ENTSO-E
report. These configurations range from a national single-zone model to a
fully nodal model, as well as the ACER proposals for two to four bidding zones
ACER, 2022b . Each allocation is associated with generation costs as well as
redispatch costs to obtain a physically feasible outcome. After computing an
allocation, different pricing rules, such as IP, CH, or Join pricing, can be
applied. Each pricing rule implies different price levels as well as GLOCs,
LLOCs, and MWPs for market participants. Prices were capped at EUR 100 per MWh
to limit the impact of outliers. We also consider the Euphemia allocation and
prices as the current implementation in European markets.

Calendar Weeks of 2009 | Allocation Rules | Pricing Rules | Focus Variables
---|---|---|---
04 | National | IP | Generation Costs
08 | 2 Zones (k-means) | CH | Redispatch Costs
11 | 2 Zones (spectral) | Join | Prices
15 | 3 Zones |  | GLOCs
16 | 4 Zones |  | LLOCs
21 | Nodal |  | MWPs
31 | Euphemia |
48 |  |  |
Table 1: Overview of experiments

##  4 Results

In this section, we summarize the main results of our analysis. We start with
comparing average generation and total costs, before discussing price levels,
GLOCs, LLOCs, and MWPs. We report aggregate statistics for all weeks under
consideration and further pick the days of February 18, March 11, and November
23 for an in-depth analysis of representative days with low, medium, and high
prices, respectively.

###  4.1 Generation and Redispatch Costs

Table 2 includes an overview of the average daily generation and redispatch
costs (both for cost-minimizing and volume-minimizing redispatch) for the
allocation rules under consideration.

in kEUR | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
Generation | 33,507.00 | 33,507.68 | 33,507.37 | 33,516.13 | 33,510.40 | 36,008.27
Min-Cost Redispatch | 5,124.61 | 5,120.69 | 5,127.95 | 5,117.03 | 5,145.39 | 0.00
Minimum Total Costs A | 38,631.61 | 38,628.37 | 38,635.32 | 38,633.16 | 38,655.79 | 36,008.27
Min-Volume Redispatch | 8,195.76 | 8,175.75 | 8,178.12 | 8,204.01 | 8,175.20 | 0.00
Minimum Total Costs B | 41,702.76 | 41,683.43 | 41,685.49 | 41,720.14 | 41,685.60 | 36,008.27
Table 2: Average Daily Costs

The numbers reported in Table 2 have realistic orders of magnitude as compared
to those provided by the German Bundesnetzagentur (Bundesnetzagentur,, 2024).
However, our goal is not to come up with a precise prediction of costs for
2025, but to obtain a relative comparison between different zonal
configurations and nodal pricing. Adding network constraints cannot lead to
lower objective function values in the cost minimization. As the national
allocation rule neglects all transmission constraints, their generation costs
are a lower bound for all other allocation rules. On average, the suggested
zonal configurations require only slightly more generation costs, suggesting
that adding few transmission constraints to the computation has little impact
on the cost. In contrast, the nodal allocation rule includes the entire
transmission network. Obviously, higher generation costs are required to reach
an allocation with feasible power flows. On average, generation costs increase
by 7.46% after including all network constraints for the nodal model compared
to the national configuration before redispatch. After redispatch, the
national configuration with minimum cost redispatch has 7.29% higher costs
than the nodal model, and 15.8% higher costs with minimum volume redispatch.
Note that we assume that all generators can be redispatched in our model, such
that redispatch costs in reality might be higher. In comparison, the average
daily redispatch costs in Germany between 2020 and 2022 amount to roughly kEUR
7,200 (Bundesnetzagentur,, 2022). The average daily redispatch costs in
Germany in 2022 were even at around kEUR 11,600, having increased by almost
100% compared to 2021. We assume zero redispatch for a nodal system that
satisfies $\Psi^{DC}$, although in practice there will always be some minimal
level of redispatch due to changing weather conditions and the simplified
representation of network flows in the DCOPF.

####  4.1.1 Welfare losses with Euphemia - Impact of Non-Uniform Pricing

Euphemia maximizes welfare subject to linear and anonymous prices, which is
different from unconstrained welfare maximization. In our analysis, the
average daily welfare loss associated with Euphemia compared to the national
configuration (without any network constraints) is 0.13% – or EUR 40,587.32 in
absolute terms. This is comparable with previous estimates of 0.05% as a
relative loss (Meeus et al.,, 2009), and with an absolute loss of EUR 130,000
for the entire SDAC region reported by All NEMO Committee, 2023a .
Disregarding market coupling, the cuts introduced by the Euphemia algorithm to
get to linear prices do not deteriorate the outcome by a lot. After
redispatch, Euphemia suffers an average daily loss of 0.18% in total costs (or
EUR 60,942.65) compared to the national allocation rule and a loss of 1.49%
(or EUR 525,587.01) compared to the nodal allocation rule. We argue that the
national, zonal, or Euphemia models differ only marginally in terms of
allocation, and only a transition to a nodal allocation rule would
substantially affect the average generation and redispatch costs.

###  4.2 Price Levels

Table 3 provides average prices over all hours of the weeks under
consideration, while Table 4 illustrates median prices and Table 5 price
standard deviations for all computed prices under the respective allocation
and pricing rule.

in EUR/MWh | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
IP | 42.68 | 42.54 | 42.04 | 42.20 | 43.30 | 44.34
CH | 37.19 | 43.41 | 37.23 | 37.23 | 37.25 | 43.43
Join | 40.85 | 42.55 | 42.12 | 42.29 | 41.44 | 43.48
Euphemia | 43.41 |  |  |  |  |
Table 3: Average Prices in EUR/MWh | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
IP | 28.56 | 28.55 | 28.55 | 28.53 | 28.59 | 45.82
CH | 30.81 | 30.81 | 30.81 | 30.81 | 30.81 | 46.52
Join | 28.41 | 28.55 | 28.55 | 28.52 | 28.45 | 44.27
Euphemia | 28.58 |  |  |  |  |
Table 4: Median Prices in EUR/MWh | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
IP | 19.85 | 19.71 | 19.40 | 19.62 | 20.05 | 19.44
CH | 14.18 | 14.13 | 14.13 | 14.11 | 14.10 | 16.92
Join | 21.53 | 20.21 | 19.96 | 20.17 | 21.78 | 21.08
Euphemia | 20.63 |  |  |  |  |
Table 5: Price Standard Deviation

Note that the mean BZR price for the weeks under consideration was EUR/MWh
40.17 and the standard deviation EUR/MWh 18.51, computed bi-hourly based on a
linear relaxation.

Average CH prices are slightly lower compared to the other pricing rules under
consideration, with lower standard deviation and irrespective of the
allocation rule. In contrast, median prices are at comparable levels across
all pricing rules. Since CH prices are unaffected by pre-determined generator
commitments, they tend to be lower and more stable. In particular, prices can
be set by a cheaper – yet uncommitted – generator (Schiro et al.,, 2016). In
contrast, average Euphemia prices tend to be slightly higher than zonal
prices, while median prices are almost equal. This finding implies that the
price-setting generator is usually identical for Euphemia and zonal prices,
yet sometimes the Euphemia algorithm introduces a cut that disallows a
generator that would otherwise be dispatched. As a result, a higher-price
generator may set the price.

Price distributions are generally similar across the different zonal
configurations for a specific pricing rule. Due to the simplified
representation of the transmission network, cross-zonal flow constraints are
rarely tight and thus zonal prices are identical to national (single-zone)
prices in many hours. This observation suggests that splits into only a few
zones, on average, have little impact on prices and thus provide few
locational incentives. Table 6 illustrates that, on average, no major price
discrepancies between zones can be expected. The zones are depicted in Figure
1.

in EUR/MWh | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones
---|---|---|---|---|---
Zone 1 | 42.68 (19.85) | 41.91 (19.47) | 42.19 (19.64) | 41.41 (19.05) | 42.80 (20.31)
Zone 2 |  | 43.17 (19.93) | 41.88 (19.16) | 42.75 (19.81) | 44.45 (20.66)
Zone 3 |  |  |  | 42.46 (19.95) | 42.97 (19.60)
Zone 4 |  |  |  |  | 42.99 (19.56)
Table 6: IP Zonal Prices Average and Standard Deviation per Zone

(a) 2 Zones (k)

(b) 2 Zones(s)

(c) 3 Zones

(d) 4 Zones

Figure 1: Zone Configurations

A more in-depth examination of zonal pricing reveals the impact of cross-zonal
flows in equalizing prices across different zones. Specifically, while
southern zones face a relative shortage of generation capacity compared to
demand, the capacity of cross-zonal transmission lines is typically high
enough to transmit electricity from northern zones. Our sensitivity analysis
revealed that if cross-zonal flows were limited or prohibited, zonal prices
can display differences of more than 30 EUR/MWh between northern and southern
regions on some days. The small differences in zonal prices observed in Table
6 are due to the cross-zonal transmission capacity (calculated as the sum of
thermal limits of all cross-zonal lines) that equalizes prices across zones.

Under nodal pricing (ca. 40,000 prices per day for 1670 nodes – compared to,
e.g., 24 national prices per day) more price outliers were observed than under
zonal pricing. For instance, 0.08% of IP prices had to be capped at EUR 100
per MWh under the zonal configurations, compared to 2.26% under nodal pricing.
Without capping, the maximum observed IP and Join price would be as high as
EUR 70,000 per MWh for an hour of extreme scarcity. Note that SDAC also
imposes a price cap of EUR 5,000 per MWh for such periods. CH prices, in
contrast, are less prone to outliers, peaking at EUR 416 per MWh. Thus tight
flow constraints can increase price levels in the short run. The zonal
allocation rules oversimplify the transmission network to an extent that nodal
prices can hardly be lower, even without congestion. In contrast, when
congestion is present in parts of the grid, nodal prices will be higher,
resulting in higher average and median prices. Upon further analysis, it was
found that nodal prices have high sensitivity to transmission line
susceptances, a phenomenon previously described in Bichler & Knörr, (2023).

Figure 2 illustrates nodal IP, CH, and Join prices, averaged over all hours
and sorted in ascending order. The dashed horizontal and vertical lines mark
the 5th and 95th percentiles for prices and nodes, respectively. Additionally,
the figure shows the average national prices and the national average
including a uniform EUR/MWh price adder for redispatch costs. This price adder
is calculated as the total redispatch costs under min-volume redispatch (ca.
EUR 459.0 million) divided by the total demand (ca. 83.3 TWh), and equals 5.51
EUR/MWh. Notably, with redispatch taken into account, higher average prices
are expected at some of the nodes. For the majority of nodes, however, the
change in average prices is minor or even negative under nodal pricing.

Figure 3 maps these average prices to their geographical locations. As much of
the electricity supply, particularly wind energy, is located in Northern
Germany, we observe that average nodal prices tend be higher in Southern
Germany. Generally, this price gap is moderate and provides locational
incentives for market participants. The choice of pricing rule does not seem
to have a major effect on average price levels.

We observe the highest prices in Northern Central Germany in a region with a
high share of expensive biomass energy and the lowest prices in the Alps
region with inexpensive hydro energy. The associated nodes are relatively
isolated with frequent congestion. This behavior might follow from imprecise
estimations of generation costs and line limits described in Section 3, yet
price outliers can also be observed in the ENTSO-E report (ENTSO-E,, 2022).

(a) IP Prices

(b) CH Prices

(c) Join Prices

Figure 2: Sorted Nodal Prices [EUR/MWh]

(a) IP Prices

(b) CH Prices

(c) Join Prices

Figure 3: Average Nodal Prices [EUR/MWh]

(a) IP Prices

(b) CH Prices

(c) Join Prices

Figure 4: Standard Deviation Nodal Prices [EUR/MWh]

Nodal prices typically vary across locations for each hour, and thereby set
locational incentives for flexible units or storage. As seen in Figure 4,
standard deviations of nodal prices resemble each other across nodes and
pricing rules. Figure 5 illustrates the distribution of hourly standard
deviations of prices across nodes, with mean and median standard deviations as
solid and dashed lines, respectively. The nodal price variances can be
substantial. For example, in the extreme first hour of 2009/11/24, IP prices
vary across the country between EUR/MWh 3.45 and EUR/MWh 100. This sets
incentives for demand response and short-term adjustments in particular in
energy-intensive industrial production. Persistent high nodal prices signal
transmission bottlenecks and direct investments in grid expansion.

(a) IP Prices

(b) CH Prices

(c) Join Prices

Figure 5: Histograms of Hourly Standard Deviations Across Nodes [EUR/MWh]

(a) IP Prices

(b) CH Prices

(c) Join Prices

Figure 6: Histograms of Nodal Standard Deviations Across Hours [EUR/MWh]

A primary goal in the design of new bidding zones is the reduction of price
variance within a zone (ACER, 2022b, ), which signals congestion. Tables 5 and
6 indicate that the standard deviation of zonal prices does not decrease with
the number of zones. However, varying prices can have two main reasons:
congestion and temporal changes in supply and demand. Figure 5 considers
hourly standard deviations of prices across all nodes, meaning it illustrates
the price variance based on grid congestions in each hour, which we refer to
as congestion-based standard deviation. In contrast, Figure 6 includes
histograms of the standard deviation of nodal prices across all hours, meaning
it illustrates the price variance based on temporal effects of supply and
demand at each node, which we refer to as time-based standard deviation.
Together, Figures 5 and 6 represent a composition of the standard deviations
reported for nodal prices in Table 5. Note that prices at a single node can
vary substantially over time due to changes in supply and demand. This time-
based standard deviation exceeds the congestion-based standard deviation. For
example, IP prices exhibit a mean congestion-based standard deviation of
EUR/MWh 12.09, compared to a mean time-based standard deviation of EUR/MWh
16.95. These numbers provide evidence that temporal fluctuations of supply and
demand have greater impact on price variance compared to congestion.

Tables 7 and 8 provide a more detailed variance decomposition for IP prices
under all zonal configurations. In particular, Table 7 indicates the
congestion-based standard deviation of nodal IP prices, averaged over all
nodes assigned to a zone. Equivalently, Table 8 summarizes the time-based
standard deviations grouped by zone. Specifically, the time-based standard
deviation always exceeds the congestion-based standard deviation, implying
that temporal effects drive price variance more than congestion effects. At
the same time, the time-based standard deviation varies only little across
allocation rules, suggesting that nation-wide fluctuations in supply and
demand, on average, affect all price zones equally. In contrast, the
congestion-based standard deviation has a slightly decreasing trend with more
zones, indicating that zonal splits can reduce price variance at least for
some of the zones (e.g., Zone 3). This effect, however, is not consistent
across all zones, implying that the cross-zonal lines fail to encompass all
congested lines. Consequently, the computed dispatch does not account for the
remaining intra-zonal congestion, necessitating costly redispatch under all
zonal models, as shown in Table 2. In a complementary paper, we demonstrate
that the proposed zone configurations lack stability and fail to effectively
segregate nodes along congested lines As a result, the congestion-based
standard deviation exhibits minimal reduction as the number of zones
increases, and redispatch costs remain high.

in EUR/MWh | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones
---|---|---|---|---|---
Zone 1 | 12.09 | 12.29 | 12.31 | 11.51 | 12.52
Zone 2 |  | 10.45 | 10.53 | 12.47 | 10.97
Zone 3 |  |  |  | 6.42 | 6.44
Zone 4 |  |  |  |  | 12.85
Table 7: Average Congestion-Based Standard Deviation of Nodal IP Prices in EUR/MWh | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones
---|---|---|---|---|---
Zone 1 | 16.95 | 16.92 | 17.04 | 16.46 | 17.47
Zone 2 |  | 16.99 | 16.83 | 17.40 | 17.22
Zone 3 |  |  |  | 16.86 | 16.70
Zone 4 |  |  |  |  | 16.11
Table 8: Average Time-Based Standard Deviation of Nodal IP Prices

###  4.3 GLOCs, LLOCs, MWPs

The average daily GLOCs, LLOCs, and MWPs are summarized in Tables 9–11,
respectively.

in EUR | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
IP | 2,743,625.97 | 2,715,067.90 | 2,581,180.46 | 2,665,434.36 | 3,073,576.90 | 6,667,208.89
CH | 141,338.05 | 141,785.33 | 142,372.81 | 153,430.35 | 145,411.76 | 361,031.42
Join | 2,784,666.32 | 2,773,482.10 | 2,680,900.33 | 2,719,728.69 | 3,115,596.04 | 4,665,204.10
Euphemia | 2,772,814.13 |  |  |  |  |
Table 9: Average Daily GLOCs in EUR | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
IP | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
CH | 33,683.28 | 32,956.96 | 33,554.87 | 40,266.34 | 35,536.04 | 179,887.45
Join | 989.05 | 812.56 | 1,547.75 | 1,795.06 | 1,057.30 | 41,476.85
Euphemia | 0.00 |  |  |  |  |
Table 10: Average Daily LLOCs in EUR | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
IP | 19,812.81 | 25,662.91 | 21,626.78 | 28,348.30 | 21,558.54 | 338,926.64
CH | 37,345.65 | 37,658.44 | 37,975.21 | 45,054.94 | 39,156.73 | 102,818.68
Join | 766.2 | 1,308.67 | 1,446.57 | 1,094.99 | 859.76 | 23,840.19
Euphemia | 0.00 |  |  |  |  |
Table 11: Average Daily MWPs

By definition, CH prices minimize GLOCs, and on average, they are
substantially lower compared to both IP and Join prices, irrespective of the
chosen allocation rule. Similar orders of magnitude for GLOCs are observed
across all zonal allocation rules, including Euphemia. In contrast, nodal
pricing results in the highest GLOCs, aligning with the higher average prices
presented in Table 3. Higher prices imply greater forgone profits for
generators and increase GLOCs. Generators in nodal markets face stronger
incentives to deviate from the optimal outcome, necessitating penalties – a
common practice in many U.S. markets (Bichler et al.,, 2022).

As a natural subset of GLOCs, LLOCs follow a similar pattern, with the highest
values occurring under nodal pricing, once again correlated with higher
average prices. IP pricing ensures zero LLOCs, as discussed in Section 2.3.2,
ensuring reliable congestion signals where price differences between nodes
only arise during network congestion. In this case, the price difference
precisely reflects the marginal value of transmission capacity. In contrast,
CH pricing consistently results in the highest LLOCs, supporting observations
that congestion signals may be flawed (Schiro et al.,, 2016). This is
especially relevant under a nodal allocation rule considering all transmission
lines. The Join pricing rule yields substantially fewer LLOCs than CH pricing,
indicating a superior quality of congestion signals.

In terms of average MWPs to compensate generator losses, only the Euphemia
algorithm generates zero MWPs, albeit at the expense of welfare losses. An
interesting observation is that the welfare losses of Euphemia – as discussed
in Section 4.1 – exceed, on average, the MWPs required under a national
welfare-maximizing allocation, irrespective of the pricing rule. This implies
that the additional generated welfare could potentially cover the MWPs under a
national allocation, resulting in a net welfare gain. Under nodal pricing,
MWPs tend to increase compared to zonal pricing, particularly for IP pricing.
This can be attributed to the fact that under a national allocation rule,
there exists a single hourly IP price set at the variable cost of the marginal
generator. With positive fixed costs and constant variable costs, such a
generator will inevitably incur a loss and necessitate MWPs under IP pricing.
In contrast, nodal pricing involves several hourly IP prices and price-setting
generators across the network, leading to higher overall MWPs. Conversely, the
Join pricing rule generates lower MWPs than IP and CH pricing, constituting a
negligible share of the total costs outlined in Table 2.

In summary, GLOCs, LLOCs, and MWPs follow similar patterns across different
zonal configurations, but increase with nodal pricing. Concerning pricing
rules, IP, CH, and Euphemia each minimize a specific class of lost opportunity
costs. However, the Join pricing rule stands out by striking a remarkable
balance between low MWPs and LLOCs. At the expense of slightly increased
GLOCs, Join prices facilitate marginal side-payments and maintain a high
quality of congestion signals.

###  4.4 Analysis of Representative Days

To corroborate our findings, we examine three selected days in greater detail.
Specifically, we analyze November 23 for its low price levels, March 11 for
medium price levels, and February 18 for high price levels.

The generation costs for each of these days are presented in Table 12. On
2009/11/23, characterized by warm and windy conditions with low demand,
generation costs were minimal, and no congestion occurred in the zonal models,
resulting in identical allocations. In contrast, February 18, as a cold and
cloudy day, saw substantially higher generation costs, but cross-zonal flow
constraints had little impact on the allocation. Across all days, the shift
from zonal to nodal pricing led to a similar increase in generation costs.

Table 13 indicates that minimum cost redispatch costs are not correlated with
the generation costs presented in Table 12. This observation underscores that
zonal models lack a representation of transmission bottlenecks, necessitating
redispatch regardless of the zonal configuration or day.

in kEUR | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
2009/11/23 | 26,672.26 | 26,672.26 | 26,672.26 | 26,672.26 | 26,672.26 | 29,593.76
2009/03/11 | 30,494.25 | 30,507.06 | 30,502.60 | 30,507.54 | 30,496.97 | 33,068.16
2009/02/18 | 45,481.38 | 45,483.18 | 45,483.04 | 45,666.68 | 45,486.15 | 48,718.75
Table 12: Generation Costs: Nov 25, Mar 11, Feb 18 in kEUR | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
2009/11/23 | 3,457.50 | 3,457.50 | 3,457.50 | 3,457.50 | 3,457.50 | 0.00
2009/03/11 | 2,969.30 | 2,989.12 | 2,955.14 | 2,965.60 | 2,990.37 | 0.00
2009/02/18 | 3,649.39 | 3,664.49 | 3,639.52 | 3,561.45 | 3,596.63 | 0.00
Table 13: Min-Cost Redispatch Costs: Nov 25, Mar 11, Feb 18

In summary, as outlined in Table 14, the nodal allocation rule consistently
yields lower total costs than any zonal configurations. Despite modest cost
savings, nodal allocation enables locational price signals and incentivizes a
long-run equilibrium.

On 2009/11/23 and 2009/03/11, the allocation obtained from the Euphemia
algorithm is identical to the national configuration, and no welfare loss
occurs. On 2009/02/18, costs under the Euphemia algorithm are slightly higher
by 0.36%. This validates our assertion that differences between zonal models –
including Euphemia – are minor and nuanced, while nodal pricing constitutes a
more substantial change in generation and redispatch costs.

in kEUR | National | 2 Zones (k) | 2 Zones (s) | 3 Zones | 4 Zones | Nodal
---|---|---|---|---|---|---
2009/11/23 | 30,129.75 | 30,129.75 | 30,129.75 | 30,129.75 | 30,129.75 | 29,593.76
2009/03/11 | 33,463.55 | 33,496.18 | 33,457.73 | 33,473.14 | 33,487.34 | 33,068.16
2009/02/18 | 49,130.77 | 49,147.67 | 49,122.56 | 49,228.13 | 49,082.78 | 48,718.75
Table 14: Total Costs: Nov 25, Mar 11, Feb 18

Figure 7 illustrates the locational IP, CH, and Join prices for the three
selected days, averaged over 24 hours. As previously discussed, the prices on
these days exhibit a positive correlation with generation costs, with the
exception of Northern Central and Southern Germany with consistent price
outliers due to local supply structures. Notably, even though at lower price
levels, a more distinct north-south price gradient can be observed on
2009/11/23 compared to 2009/03/11. Specifically, on 2009/11/23, demand is low
nationwide, yet inexpensive electricity from the north cannot be transmitted
to the south, leading to uneven price levels. In contrast, on 2009/03/11, with
higher demand but a balanced grid, prices are uniform across the country. This
phenomenon holds for all three pricing rules. The simplified zonal models fail
to detect any transmission bottlenecks on 2009/11/23, i.e., cross-zonal
constraints are not tight and there are no price variations between zones.
Consequently, if the zone configurations are suboptimal, zonal prices do not
signal scarcity appropriately. B includes an analysis of hourly prices
observed on the three days.

(a) 2009/11/23 - Avg IP Prices

(b) 2009/11/23 - Avg CH Prices

(c) 2009/11/23 - Avg Join Prices

(d) 2009/03/11 - Avg IP Prices

(e) 2009/03/11 - Avg CH Prices

(f) 2009/03/11 - Avg Join Prices

(g) 2009/02/18 - Avg IP Prices

(h) 2009/02/18 - Avg CH Prices

(i) 2009/02/18 - Avg Join Prices

Figure 7: Average Nodal Prices

Table 8 summarizes the GLOCs, LLOCs, and MWPs for each selected day. On the
low-demand day of 2009/11/23, a Walrasian equilibrium as in Definiton 1 is
attainable for the zonal models, resulting in zero GLOCs, LLOC, and MWPs.
Crucially, all three pricing rules successfully obtain these equilibrium
prices. The Walrasian equilibrium, however, can only persist before redispatch
is considered and ceases to exist when nodal transmission constraints are
taken into account.

GLOCs, LLOCs, and MWPs do not necessarily increase with average price levels.
For nodal pricing, they remain relatively consistent across all days. There is
a substantial increase for zonal pricing on the medium-price day of
2009/03/11, compared to the high-price day of 2009/02/18. This indicates that
incentives to deviate hinge more on the quality of the price signal than the
actual price level.

The simplification inherent in zonal models, which leads to redispatch, does
not necessarily imply better economic properties of prices. For example, on
2009/03/11 and 2009/02/18, zonal IP and Join prices result in higher GLOCs
than nodal pricing. Specifically, a few zonal prices are insufficient to
incentivize many market participants to adhere to the allocation.

As discussed above, CH prices always minimize GLOCs, and IP prices always
yield zero LLOCs. Although GLOCs are reduced by CH pricing compared to both IP
and Join pricing, their LLOCs, and often MWPs, experience a slight increase.
In practical terms, this implies that congestion signals are more distorted
and higher side-payments are required. In contrast, the Join pricing rule
achieves a favorable trade-off between LLOCs and MWPs, requiring only slightly
higher penalties (corresponding to GLOCs) than IP pricing.

in kEUR |  | IP | CH | Join
---|---|---|---|---
National | GLOC | 0.00 | 0.00 | 0.00
LLOC | 0.00 | 0.00 | 0.00
MWP | 0.00 | 0.00 | 0.00
2 Zones (k) | GLOC | 0.00 | 0.00 | 0.00
LLOC | 0.00 | 0.00 | 0.00
MWP | 0.00 | 0.00 | 0.00
2 Zones (s) | GLOC | 0.00 | 0.00 | 0.00
LLOC | 0.00 | 0.00 | 0.00
MWP | 0.00 | 0.00 | 0.00
3 Zones | GLOC | 0.00 | 0.00 | 0.00
LLOC | 0.00 | 0.00 | 0.00
MWP | 0.00 | 0.00 | 0.00
4 Zones | GLOC | 0.00 | 0.00 | 0.00
LLOC | 0.00 | 0.00 | 0.00
MWP | 0.00 | 0.00 | 0.00
Nodal | GLOC | 1,277.13 | 272.47 | 1,492.66
LLOC | 0.00 | 147.44 | 26.79
MWP | 167.32 | 94.76 | 34.70

(a) GLOC, LLOC, MWP for 2009/11/23

in kEUR |  | IP | CH | Join
---|---|---|---|---
National | GLOC | 4,323.93 | 43.73 | 4,323.93
LLOC | 0.00 | 10.68 | 0.00
MWP | 0.00 | 9.04 | 0.00
2 Zones (k) | GLOC | 8,009.06 | 56.54 | 8,009.06
LLOC | 0.00 | 23.49 | 0.00
MWP | 0.00 | 23.03 | 0.00
2 Zones (s) | GLOC | 5,419.69 | 52.08 | 5,419.69
LLOC | 0.00 | 20.17 | 0.00
MWP | 0.00 | 17.05 | 0.00
3 Zones | GLOC | 7,097.32 | 57.03 | 7,097.32
LLOC | 0.00 | 23.98 | 0.00
MWP | 0.00 | 23.53 | 0.00
4 Zones | GLOC | 5,152.01 | 46.45 | 5,152.01
LLOC | 0.00 | 13.35 | 0.00
MWP | 0.00 | 11.10 | 0.00
Nodal | GLOC | 1,146.96 | 178.49 | 1,159.88
LLOC | 0.00 | 96.05 | 37.93
MWP | 94.90 | 29.73 | 22.33

(b) GLOC, LLOC, MWP for 2009/03/11

in kEUR |  | IP | CH | Join
---|---|---|---|---
National | GLOC | 1,157.36 | 290.42 | 1,177.70
LLOC | 0.00 | 30.30 | 9.38
MWP | 59.64 | 22.53 | 9.29
2 Zones (k) | GLOC | 1,798.28 | 288.22 | 1,772.59
LLOC | 0.00 | 28.16 | 9.58
MWP | 132.61 | 18.85 | 9.50
2 Zones (s) | GLOC | 1,430.45 | 285.19 | 1,455.50
LLOC | 0.00 | 25.13 | 9.36
MWP | 58.69 | 13.73 | 9.27
3 Zones | GLOC | 2,387.52 | 471.72 | 2,493.28
LLOC | 0.00 | 170.36 | 9.20
MWP | 27.32 | 164.81 | 9.02
4 Zones | GLOC | 1,778.31 | 291.19 | 1,860.59
LLOC | 0.00 | 30.96 | 9.41
MWP | 61.81 | 20.96 | 9.33
Nodal | GLOC | 1,024.32 | 274.54 | 1,495.84
LLOC | 0.00 | 86.48 | 14.44
MWP | 72.94 | 34.68 | 9.78

(c) GLOC, LLOC, MWP for 2009/02/18

Figure 8: Daily GLOCs, LLOCs, MWPs

###  4.5 Discussion

Our numerical experiments provide insights into different market clearing
models, pricing rules, and their economic implications on day-ahead markets.
Note that the final dispatch is affected by additional factors such as
intraday trading or long-term capacity allocations. Nodal pricing tends to
result in slightly higher average prices compared to zonal pricing,
irrespective of the chosen zonal configuration. However, Transmission
constraints can be violated and necessitate costly redispatch measures. As a
result, a zonal pricing will always pick an inefficient dispatch, and even if
redispatch is conducted at minimal costs, the total system costs will be
substantially higher than under nodal pricing. Given the steep increase of
redispatch costs in recent years, this effect is likely to magnify in the
future.

Interestingly, the choice of zone configuration did not substantially impact
prices. All zonal configurations suggested in the Bidding Zone Review simplify
the network to an extent that opting for two, three, or four zones yields
similar results regarding the average prices and the price standard deviation.
In contrast to nodal pricing, zonal pricing leads to substantial redispatch
and higher total costs. While nodal prices may vary between nodes, these
differences incentivize short-term demand response on nodes where this is most
helpful for overall system stability. If such price differences persist over
more extended time periods, they also set investment incentives for generators
and grid expansion.

Given the absence of Walrasian equilibria, any pricing rule – regardless of
zonal or nodal pricing – must balance economic properties such as welfare
gains and the participants’ lost opportunity costs. The currently used
Euphemia algorithm leads to welfare losses and is computationally expensive
but avoids paradoxically accepted bids. Our experiments suggest that non-
uniform pricing rules achieve low MWPs for paradoxically accepted bids while
maintaining the optimal allocation and scaling in polynomial time. Our
findings thus provide support for non-uniform pricing as they are currently
being discussed in the context of the Capacity Allocation and Congestion
Management (CACM) Regulations (All NEMO Committee, 2023b, ).

CH pricing minimizes GLOCs, but our results confirm previous findings that
congestion signals are distorted (evidenced by high LLOCs) and high MWPs are
necessary. IP pricing, the prevailing rule in U.S. markets, ensures effective
congestion signals but requires very high MWPs. The Join pricing rule strikes
a balance between LLOCs and MWPs, ensuring low side-payments and good
congestion signals. GLOCs are less of a concern, because already today, in
U.S. markets with IP pricing, such incentives to deviate from the computed
dispatch are mitigated by penalties for deviations.

##  5 Conclusion

The discussion surrounding zonal and nodal pricing has a long history. The
European electricity market has employed zonal pricing for many years, often
with large, nationwide price zones. However, the ongoing energy transition
requires a reevaluation of price zones. Based on recent data released by
ENTSO-E in the context of the EU Bidding Zone Review process, we compare
various allocation and pricing rules for the German power market. Our findings
indicate that, in terms of system costs and price signals, a nodal allocation
rule with non-uniform pricing leads to the lowest total costs. Costly
redispatch is mitigated, and the average prices increase only slightly.
Additionally, pricing rules such as Join pricing require very low side-
payments, while efficiently signaling transmission bottlenecks. In contrast,
the proposed zonal configurations only differ marginally regarding system
costs and the differences in average prices across zones are low.
Interestingly, the effect of zonal splits on price standard deviations is low,
and we do not find evidence for significant reductions in redispatch costs
resulting from such splits, all other factors held constant.

## Acknowledgements

We would like to thank Karsten Neuhoff and Maximilian Rinck for many helfpul
comments. This project was funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Grant No. BI 1057/10.

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##  Electronic Companion A Matching of Generators to Grid Locations

The BZR grid data specify the nominal capacity of thermal generating units at
each node, but information on the specific plant type and its operating
restrictions are not provided. A different data set provides the aggregate
capacity of each specific plant type (e.g., Hard Coal old, Hard Coal new, Gas
CCGT, etc.) as well as their operating characteristics (e.g., minimum uptime
constraints). Unfortunately, no straightforward mapping exists between the two
datasets.

As discussed in Section 3, we first identify each thermal unit’s broad plant
type (e.g., hard coal, gas, etc.) by matching IDs with the Kraftwerksliste
(power plant list) provided by the Bundesnetzagentur. Next, we need to
distribute these plants to the sub-categories provided in the seller data. For
example, all hard coal units must be distributed between Hard Coal old1, Hard
Coal old2, and Hard Coal new, with each sub-category having different
operational characteristics.

To perform this mapping, we solve an integer program (Schmitt,, 2023). Let $K$
be the set of broad plant types (e.g., hard coal, gas) and $A_{k}$ be the set
of sub-categories (e.g., Hard Coal old1, Hard Coal old2, Hard Coal new) for
each type $k$. The total capacity and number of units of each specific plant
type $a\in A_{k}$ is given as $P_{a}$ and $n_{a}$ from the seller data. From
the grid data, we obtain a set of generators $I_{k}$ for each broad plant type
$K$, each having a nominal capacity of $p_{i},i\in I_{k}$. We denote as
$x_{ia}\in\\{0,1\\}$ as the desired mapping of each generator to a specific
plant type.

With this notation, the mapping problem looks as follows:

$\displaystyle\min_{x}\quad$ $\displaystyle\sum_{k\in K,a\in
A_{k}}-n_{a}(P_{a}-\sum_{i\in I_{k}}p_{i}x_{ia})+\sum_{k\in K,a\in
A_{k}}-P_{a}(n_{a}-\sum_{i\in I_{k}}x_{ia})$ (9) subject to
$\displaystyle\sum_{a\in A_{k}}x_{ia}=1\quad\forall i\in I_{k},k\in K$
$\displaystyle 0\leq P_{a}-\sum_{i\in I_{k}}p_{i}x_{ia}\geq 600\quad\forall
a\in A_{k},k\in K$ $\displaystyle 0\leq N_{a}-\sum_{i\in I_{k}}x_{ia}\geq
2\quad\forall a\in A_{k},k\in K$

The objective function minimizes the aggregate differences in both capacities
and number of units per plant type, with equal weight. The first constraint
ensures that each unit of broad type $k$ (e.g., hard coal) is assigned to
exactly one specific plant type $a\in A_{k}$ (e.g., Hard Coal old1). The last
two constraints bound the maximum allowed deviations in capacities and the
number of units for each specific plant type $a$. Note that a broad type $k$
could not be identified for all units, and such unidentified units could be
matched freely to any specific type $a$. After solving the optimization
problem, we obtain a specific type $a$ for each unit in the grid model and can
use its respective operating characteristics as input to the clearing problem
in (1).

The result of this mapping is presented in Table 15. Especially hard coal,
lignite, and oil plants could be mapped almost perfectly. Concerning gas
units, the grid and seller datasets exhibited more significant differences,
which is reflected in the mapping.

| Seller Data | Grid Data
---|---|---
Fuel / PlantType | Capacity [MW] | Units | Capacity [MW] | Units
Hard Coal / old1 | 1,905.1 | 8 | 1,905 | 8
Hard Coal / old2 | 5,172.4 | 27 | 5,165.4 | 27
Hard Coal / new | 5,178 | 8 | 5,178 | 7
Lignite / old1 | 4,989.3 | 13 | 4,989 | 13
Lignite / old2 | 3,111.1 | 11 | 3,111 | 11
Lignite / new | 6,437.8 | 11 | 6,437 | 9
Gas Conventional / old1 | 776.01 | 26 | 775.97 | 26
Gas Conventional / old2 | 1215.4 | 23 | 1,215.33 | 23
Gas CCGT / old1 | 3,601.049 | 78 | 3,001.054 | 78
Gas CCGT / old2 | 5,356.22 | 40 | 5,055.723 | 38
Gas CCGT / new | 7,389.1 | 45 | 6,789.321 | 43
Gas OCGT / old | 760.1 | 15 | 760.025 | 15
Gas OCGT / new | 2,613.168 | 66 | 2,013.224 | 66
Gas CCGT / present1 | 64 | 2 | 64 | 2
Gas CCGT / present2 | 282 | 2 | 282 | 2
Light oil | 998.33 | 21 | 997.43 | 21
Table 15: Generator Mapping Results

Finally, Figure 9 illustrates the geographical distribution of different
generation technologies as the basis for our clearing model.

Figure 9: Generation Capacity Distribution (Schmitt,, 2023)

##  Electronic Companion B Hourly Prices for 2009/11/23, 2009/03/11, and
2009/02/18

Figure 10 illustrates the behavior of IP, CH, and Join prices across different
days and allocations. The depicted lines represent the hourly prices of
different allocations, averaged over all zones or nodes. On 2009/11/23, when
demand is low and no cross-zonal congestion is observed, IP, CH, and Join
prices are identical for all zonal models, including Euphemia. Nodal prices
are slightly higher and show marginal differences among the three pricing
rules.

On 2009/03/11 and 2009/02/18, price levels are generally higher, corresponding
to increased generation costs. Moreover, average zonal prices exhibit
fluctuations between hours, both below and above average nodal prices.
Specifically, average nodal prices are less affected by intertemporal effects
and exhibit a much smoother price curve. IP and Join prices show similar
patterns and price levels, while CH prices form a smoother curve with less
susceptibility to temporal fluctuations. This distinction arises from the fact
that CH pricing, unlike IP and Join pricing, is independent of the allocation
and especially intertemporal constraints. Specifically, under IP pricing a
committed generator with high minimum uptime can be price-setting for multiple
hours, even though cheaper generators would be committed in the absence of
minimum uptimes. We refer to Bjørndal & Jörnsten, (2008) for a critical
assessment of the volatility of IP prices due to fixed integer variables.
Particularly, as discussed, non-convex optimization problems lack
straightforward equilibrium pricing solutions. Bjørndal & Jörnsten, (2008)
show that finding alternative valid inequalities to determine equilibrium
prices requires reformulating the problem such that coefficients from the
Benders sub-problem serve as coefficients in valid inequalities. When integer
variables are fixed under IP pricing, the resulting coefficients may behave
unpredictably. In some instances, these coefficients may contribute to forming
valid inequalities that support stable prices. However, in other cases, the
supporting valid inequality may not eliminate any feasible solutions. This
inconsistency leads to fluctuating IP prices.

(a) 2009/11/23 - IP Prices

(b) 2009/11/23 - CH Prices

(c) 2009/11/23 - Join Prices

(d) 2009/03/11 - IP Prices

(e) 2009/03/11 - CH Prices

(f) 2009/03/11 - Join Prices

(g) 2009/02/18 - IP Prices

(h) 2009/02/18 - CH Prices

(i) 2009/02/18 - Join Prices

Figure 10: Average Hourly Prices

Because Euphemia does not alter the national allocation for 2009/11/23 and
2009/03/11, the obtained prices also coincide with national IP prices. When
Euphemia requires cuts to avoid paradoxically accepted bids, as on 2009/02/18,
prices vary compared to national IP prices. On average, Euphemia prices are
2.78% higher, and the mean absolute price difference to national IP prices is
EUR/MWh 5.81.