11institutetext: Birla Institute of Technology and Science, Pilani, Hyderabad
Campus, Hyderabad - 500078, India 22institutetext: Saha Institute of Nuclear
Physics, HBNI, 1/AF Bidhannagar, Kolkata-700064, India

# New equation of state involving Bose-Einstein condensate of antikaon for
supernova and neutron star merger simulations

Tuhin Malik 11 Sarmistha Banik 11 Debades Bandyopadhyay 22
<EMAIL_ADDRESS>
(Received: date / Revised version: date)

###### Abstract

We compute a new equation of state table including Bose-Einstein condensate of
$K^{-}$ mesons for core collapse supernova and neutron star merger
simulations. Nuclei and interacting nucleons in non-uniform matter is
described in an extended version of the nuclear statistical equilibrium model
including excluded volume effects whereas the uniform matter at higher
densities is treated in the relativistic hadron field theory with density
dependent couplings. The equation of state table is generated for a wide range
of density ($10^{-12}$ to $\sim 1$ fm-3), positive charge fraction (0.01 to
0.60) and temperature (0.1 to 158.48 MeV). The impact of antikaon condensate
is investigated on different thermodynamic quantities for example free energy
per baryon, entropy per baryon, pressure as well as compositions of matter.
Furthermore, critical temperatures of antikaon condensation and the phase
diagram of matter are also studied in this article.

###### pacs:

97.60.Jd neutron stars - 26.60.Kp equations of state

##  1 Introduction

Equation of state (EoS) of matter is an important microphysical input for core
collapse supernova (CCSN) and binary neutron star (BNS) merger simulations.
Many EoSs were developed based on updated knowledge derived from nuclear
physics experimental data and neutron star observations hs1 ; horo ; stei ;
toga ; const ; Sophia ; Mamun ; Blaschke ; Sophia19 ; Sedra18 ; Sedra19 ;
Raduta ; Blaschke18 . One such widely used nuclear EoS known as Hempel and
Schaffner (HS) EoS describes the inhomogeneous matter within the framework of
the nuclear statistical equilibrium (NSE) model whereas the uniform dense
nuclear matter is computed in the finite temperature relativistic mean field
(RMF) model with and without density dependent (DD) couplings hs1 ; rmp . The
nuclear EoS table adopting the virial expansion for a non-ideal gas of
nucleons and nuclei was computed by Shen et al. horo . The nuclear EoS was
also constructed in a variational calculation using Argonne v18 (AV18) and
Urbana IX (UIX) bare nuclear forces toga ; const . The nuclear EoSs mentioned
above are compatible with 2 M⊙ neutron stars. The first supernova EoS table
that exploited measured masses and radii of neutron stars, was developed by
Steiner and collaborators stei .

Neutron star observations could provide important inputs in the construction
of EoS tables for CCSN and BNS merger simulations. Neutron star masses in
relativistic binaries have been estimated to very high degree of precision
owing to measurements of post-Keplerian parameters of pulsars such as orbital
decay, periastron advance, Shapiro delay, time dilation. Several heavy neutron
stars of masses $\sim 2M_{\odot}$ were discovered in the past decade. The
first among those was PSR 0348+0432 having mass 2.01$\pm 0.04$ M⊙ anto . Now
the millisecond pulsar PSR 0740+6620 is credited as the most massive neutron
star of 2.14${}^{+0.10}_{-0.09}$ M⊙ croma . Recently Neutron Star Interior
Composition Explorer (NICER) has reported the first simultaneous determination
of mass and radius of PSR J0030+0451 watts . This puts a stringent constraint
on the $\beta$-equilibrated EoS of neutron star matter.

It has been long debated that additional degrees of freedom in the form of
strange baryonsweis , quarks Blaschke18  or Bose-Einstein condensate of
antikaons kaplan ; knor  might appear in dense matter. Many model calculations
involving hyperon matter, quark matter, Bose-Einstein condensate of antikaons,
showed that the EoS of $\beta$-equilibrated matter might result in 2 M⊙ or
more massive neutron stars weis ; char14 ; dc . Several temperature dependent
EoS tables including exotic matter such as quark and hyperon matter were
constructed and employed in CCSN and BNS merger simulations ishi ; naka08 ;
irina ; sumi ; shen11 ; naka ; oertel12 ; peres ; bhb ; char15 ; rad . These
EoS tables are functions of wide range of values of three parameters - baryon
density, positive charge fraction and temperature. A few of those temperature
dependent EoS tables with exotic matter such as Banik, Hempel and
Bandyopadhyay $\Lambda$-hyperons EoS (BHB$\Lambda\phi$) where hyperon-hyperon
interaction is mediated by $\phi$ mesons, was directly compatible with the 2
M⊙ neutron star bhb .

So far, there is no EoS table involving Bose-Einstein condensate of antikaons
for CCSN and BNS merger simulations. This motivates us to construct an EoS
table with antikaon condensate. The paper is organised as follows. We describe
the EoS with an antikaon condensate within the NSE and field theoretical
models in Section 2. Results are discussed in Section 3. We conclude in
Section 4.

##  2 Equation of State

The inhomogeneous matter is described by HS EoS hs1 . In this case, the
ensemble of nuclei and nucleons are treated in an extended version of the NSE
model using the RMF model for interacting nucleons. It also considers excluded
volume effects in the thermodynamically consistent manner and excited states
of light nuclei. The medium modification of nuclei due to screening of Coulomb
energies of background electrons are taken into account hs1 . Finally the sub-
saturation density matter was matched with the dense uniform matter.

We adopt a density dependent relativistic hadron field theory for the
description of strongly interacting dense baryonic matter. Here nucleon-
nucleon interaction is mediated by exchanges of scalar $\sigma$, vector
$\omega$ and $\rho$ mesons. This is given by the Lagrangian density bhb ;
typ10 ,

$\displaystyle{\cal L}_{N}$ $\displaystyle=$
$\displaystyle\sum_{N}\bar{\psi}_{N}\left(i\gamma_{\mu}{\partial^{\mu}}-m_{N}+g_{\sigma
N}\sigma-g_{\omega N}\gamma_{\mu}\omega^{\mu}\right.$ (1)
$\displaystyle\left.-g_{\rho
N}\gamma_{\mu}{\mbox{\boldmath$\tau$}}_{N}\cdot{\mbox{\boldmath$\rho$}}^{\mu}\right)\psi_{N}$
$\displaystyle+\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2}\right)-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}{\mbox{\boldmath$\rho$}}_{\mu\nu}\cdot{\mbox{\boldmath$\rho$}}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}{\mbox{\boldmath$\rho$}}_{\mu}\cdot{\mbox{\boldmath$\rho$}}^{\mu}.$

Here $\psi_{N}$ stands for nucleon doublet, $\tau_{N}$ is the isospin operator
and density dependent meson-nucleon couplings are denoted by $g_{\alpha N}$
with $\alpha$ representing meson fields.

Figure 1: Number densities of different particle species, such as light &
heavy nuclei, neutrons, protons and antikaons, both thermal and condensate, as
a function of baryon number density for T=1, 50, 100 MeV and $Y_{q}$ = 0.1,
0.3, 0.5.

The grand-canonical thermodynamic potential per unit volume of the nuclear
phase is given by

$\displaystyle\frac{\Omega}{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}-\frac{1}{2}m_{\rho}^{2}\rho_{03}^{2}-\Sigma^{r}\sum_{i=n,p}n_{i}$
(2)
$\displaystyle-2T\sum_{i=n,p}\int\frac{d^{3}k}{(2\pi)^{3}}[\mathrm{ln}(1+e^{-\beta(E^{*}-\nu_{i})})$
$\displaystyle+\mathrm{ln}(1+e^{-\beta(E^{*}+\nu_{i})})]~{},$

where the temperature is defined as $\beta=1/T$, effective nucleon mass
$m^{*}_{N}=m_{N}-g_{\sigma N}\sigma$ and $E^{*}=\sqrt{(k^{2}+m_{N}^{*2})}$. We
calculate all thermodynamic quantities such as pressure $P=-{\Omega}/V$,
energy density and entropy density. The chemical potential is given by The
nucleon chemical potential ($\mu_{N}$) is defined as

$\mu_{N}=\nu_{N}+g_{\omega N}\omega_{0}+g_{\rho
N}\tau_{3N}\rho_{03}+\Sigma^{r}~{},$ (3)

where $\Sigma^{r}$ is the rearrangement term that takes care of many-body
effects in nuclear interaction bhb ; typ10 .

We consider a second order phase transition from the nuclear to antikaon
condensed phase. Nucleons in the antikaon condensed phase behave differently
than nucleons in the hadronic phase glen99 . Kaon-nucleon interaction is
considered in the same footing as the nucleon-nucleon interaction in Eq. (1).
The Lagrangian density for (anti)kaons in the minimal coupling scheme is
char14 ; glen99 ; banik01 ; sb08 ,

${\cal L}_{K}=D^{*}_{\mu}{\bar{K}}D^{\mu}K-m_{K}^{*2}{\bar{K}}K~{},$ (4)

where $K$ and $\bar{K}$ denote kaon and (anti)kaon doublets; the covariant
derivative is $D_{\mu}=\partial_{\mu}+ig_{\omega K}{\omega_{\mu}}+ig_{\rho
K}{\mbox{\boldmath t}}_{K}\cdot{\mbox{\boldmath$\rho$}}_{\mu}$ and the
effective mass of antikaons is $m_{K}^{*}=m_{K}-g_{\sigma K}\sigma$.

The thermodynamic potential for antikaons is given by sb08 ; pons ,

$\frac{\Omega_{K}}{V}=T\int\frac{d^{3}p}{(2\pi)^{3}}[ln(1-e^{-\beta(\omega_{K^{-}}-\mu)})+ln(1-e^{-\beta(\omega_{K^{+}}+\mu)})]~{}.$
(5)

The in-medium energies of $K^{\pm}$ mesons are given by

$\omega_{K^{\pm}}=\sqrt{(p^{2}+m_{K}^{*2})}\pm\left(g_{\omega
K}\omega_{0}+\frac{1}{2}g_{\rho K}\rho_{03}\right)~{},$ (6)

and $\mu$ is the chemical potential of $K^{-}$ mesons and is given by
$\mu=\mu_{n}-\mu_{p}$. The threshold condition for $K^{-}$ condensation is
given by $\mu=\omega_{K^{-}}=m_{K}^{*}-g_{\omega
K}\omega_{0}-\frac{1}{2}g_{\rho K}\rho_{03}$ .

The total (anti)kaon number density ($n_{K}$) made of thermal kaons
($n_{K}^{Th}$) and the $K^{-}$ condensate ($n_{K}^{C}$), is given by
$n_{K}=n_{K}^{C}+n_{K}^{T}~{}$, where

$\displaystyle n^{C}_{K}$ $\displaystyle=$ $\displaystyle
2\left(\omega_{K^{-}}+g_{\omega K}\omega_{0}+\frac{1}{2}g_{\rho
K}\rho_{03}\right){\bar{K}}K=2m^{*}_{K}{\bar{K}}K~{},$ $\displaystyle
n_{K}^{T}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}\left({\frac{1}{e^{\beta(\omega_{K^{-}}-\mu)}-1}}-{\frac{1}{e^{\beta(\omega_{K^{+}}+\mu)}-1}}\right)~{}.$
(7)

We calculate the pressure due to thermal kaons using
$P_{K}=-{\Omega_{K}}/{V}$.

Meson field equations are solved in the mean field approximation. Meson fields
are modified in the presence of $K^{-}$ condensate glen99 ; banik01 . Finally
we obtain the pressure versus energy density known as the EoS in nuclear and
antikaon condensed phases. For density dependent couplings, the pressure of
nuclear matter includes the rearrangement term for thermodynamic consistency.
The functional forms of nucleon-meson couplings in baryon density are given by
Reftyp10 . The saturation density, mass of $\sigma$ meson, couplings at the
saturation density and unknowns of those functions are obtained by fitting
properties of finite nuclei. This parameter set is known as the DD2. Nuclear
matter properties at the saturation density are in consonance with those
obtained from nuclear physics experiments rmp . However, kaon-meson couplings
do not depend on density. In this case, kaon-vector meson coupling constants
are estimated exploiting the quark model and iso-spin counting rule i.e.
$g_{\omega K}=\frac{1}{3}g_{\omega N}$ and $g_{\rho K}=g_{\rho N}$ banik01 ;
sch96 . The scalar coupling constant is determined from the real part of
$K^{-}$ optical potential at the saturation density ($n_{0}=0.149065fm^{-3}$),

$U_{K^{-}}=-g_{\sigma K}\sigma_{0}-g_{\omega K}\omega_{0}+{\Sigma}^{r}~{}.$
(8)

Although the study of kaonic atoms indicates the an attractive $K^{-}$-nucleus
potential, there is no consensus how deep it is. The $K^{-}$ optical potential
could range from $-60$ MeV to $-200$ MeV as follows from unitary chiral model
calculations and phenomenological fit to kaonic atom data Fri94 ; Fri99 ; Tol
; Tol2 . For this calculation, we take an average antikaon optical potential
depth U${}_{K^{-}}$ = -120 MeV.

##  3 Results and Discussion

We generate the EoS table with $K^{-}$ condensate using the DD2 parameter set
and U${}_{K^{-}}$ = - 120 MeV for a wide range of values of baryon density
($n_{B}$) ($10^{-12}$ to $\sim$ 1 fm-3), temperature (T) (0.1 to 158.48 MeV)
and positive charge fraction ($Y_{q}$) defined as $Y_{q}n_{B}=n_{p}-n_{K}$,
(0.01 to 0.60). Grid spacings for baryon density is
$\Delta$log${}_{10}(n_{B})$ = 0.04; for temperature $\Delta$log${}_{10}(T)$ =
0.04 and for positive charge fraction $\Delta Y_{q}$ = 0.01. There are 301
baryon density points, 81 temperatures and 60 positive charge fractions in the
EoS table. The matching of the non-uniform matter part of nucleonic HS(DD2)
EoS 111publicly available in https://compose.obspm.fr/ fis14  with the kaon
EoS is done in the following way. We merge two EoS tables and form a kaon EoS
table including the non-uniform matter described by the extended NSE model.
This merger is guided by the minimization of free energy per baryon at fixed
$n_{B}$, T and $Y_{q}$. We impose the condition that $n_{K}^{T}$ is $>10^{-4}$
fm-3 in the non-uniform matter in addition to the above physical criterion as
the contribution of antikaons to thermodynamic observables is negligible for
$n_{K}^{T}<10^{-4}$ fm-3. Henceforth we call this merged EoS table
222https://universe.bits-pilani.ac.in/Hyderabad/sbanik/EoS with $K^{-}$ as
HS(DD2)K- .

Figure 2: Pressure is shown as a function of baryon mass density for T=1, 50,
100 MeV and $Y_{q}$ = 0.1, 0.3, 0.5. The difference in the EoS at higher
density is exhibited in the inset for $Y_{q}$ = 0.1 and T=1 MeV.

Populations of particles are shown for T=1, 50, 100 MeV and $Y_{q}$ = 0.1,
0.3, 0.5 in Fig. 1. Light ($Z\leq 5$) and heavy ($Z\geq 6$) nuclei appear in
non-uniform matter at temperatures T=1, 50 MeV. Number densities of light and
heavy nuclei are denoted by $n_{a}$ and $n_{A}$, respectively. Nuclei dissolve
into uniform matter of neutrons and protons around the saturation density. It
is evident from the left panel of T=1 MeV, no thermal (anti)kaons are
populated in the system. The Bose-Einstein condensate of $K^{-}$ mesons sets
in at higher baryon densities. It is noted that the density of $K^{-}$ mesons
($n_{K^{-}}^{C}$) in the condensate is much higher for lower values of
positive charge fractions for example at $Y_{q}$=0.1. Furthermore, the proton
number density ($n_{p}$) increases with the onset of the antikaon condensate
and even exceeds the neutron number density ($n_{n}$) for higher values of
$Y_{q}$. The middle panel of T=50 MeV exhibits similar features of T=1 MeV
panel with the population of thermal $K^{-}$ mesons. However, the density of
$K^{-}$ mesons in the condensate dominates over that of thermal $K^{-}$ mesons
($n_{K^{-}}^{Th}$) except for $Y_{q}$=0.5. The right panel shows the
significant populations of thermal $K^{-}$ mesons at T=100 MeV for all
positive charge fractions. However, in this case, the antikaon condensate
disappears except for $Y_{q}=0.1$.

We have studied various thermodynamic observables such as free energy per
baryon, entropy per baryon and pressure. The pressure is plotted in Fig. 2, as
a function of baryon mass density. Results are shown for T=1, 50, 100 MeV and
$Y_{q}=$0.1, 0.3, 0.5. In all these cases, hadronic contributions are
considered. The pressure does not include the contribution of leptons. It is
noted that results for HS(DD2) and HS(DD2)$K^{-}$ EoSs do not show any
difference for T=1, 50 MeV and three positive charge fractions, at the lower
density regions. However, at the higher density the antikaon condensates
appear, which clearly makes the HS(DD2)K- EoS softer compared to HS(DD2) EoS.
The high density portion is zoomed in the inset box for top-left panel.
However we find that there are jumps in pressure for T=100 MeV and $Y_{q}=$0.1
and 0.3 due to significant contribution of thermal $K^{-}$ mesons to the
pressure at $\sim 10^{10}$ g/cm3. A smooth transition from nuclear matter to
npK- matter is observed for temperatures below 80 MeV.

Figure 3: The phase diagram of nuclear matter with antikaon condensate is
displayed for $Y_{q}$ = 0.1, 0.3, 0.5 and U${}_{K{{}^{-}}}=-120$ MeV. The
$K^{-}$ condensed phases, represented by the shaded regions are demarcated
from the nuclear phases by the solid lines for the three $Y_{q}$ values.

It is expected that the antikaon condensate would cease to exist above a
critical temperature ($T_{c}$) sb08 . This critical temperature above which
the $K^{-}$ condensate dissolves, is a function of baryon density and positive
charge fraction. We extract critical temperatures at different values of
$n_{B}$ and $Y_{q}$ and plot the phase diagram, $T$ versus $n_{B}$, in Fig. 3
. Each line in the plot denotes a particular value of $Y_{q}$. Furthermore,
the region below each line represents the $K^{-}$ condensate phase whereas the
region above it is the nuclear phase. It is observed that the antikaon
condensate region shrinks as the value of $Y_{q}$ increases.

Figure 4: Mass versus radius of a charge neutral, $\beta$-equilibrated neutron
star is displayed for HS(DD2) and HS(DD2)K- with U${}_{K^{-}}=-120$ MeV, at
T=0.1MeV.

Finally in Fig. 4, mass of neutron star is plotted with radius for charge
neutral and $\beta$-equilibrated HS(DD2) and HS(DD2)K- EoSs at T=0.1 MeV. It
is observed that the appearance of kaon condensate make the EoS softer
resulting in lower maximum mass neutron star. Maximum mass and the
corresponding radii in two cases are 2.42 M⊙, 11.95 km and 2.19 M⊙, 12.14 km,
respectively. The tidal deformability for 1.4$M_{\odot}$ neutron star
($70\leq{\Lambda_{1.4}}\leq 580$) extracted from the BNS merger GW170817
provides radius of 1.4$M_{\odot}$ neutron star in the range
$11.9^{+1.4}_{-1.4}$ km bpa . Similarly the NICER observation of PSRJ0030+0451
gives a radius of $13.02^{+1.24}_{-1.06}$ km for the 1.44$M_{\odot}$ pulsar
mil . Our results for radius corresponding to 1.4M⊙ neutron star is slightly
higher than that of GW170817 soma  whereas it is consistent with the NICER
observation.

##  4 Conclusions

We have constructed a new EoS table including thermal $K^{-}$ mesons and the
Bose-Einstein condensate of $K^{-}$ mesons known as HS(DD2)$K^{-}$ for CCSN
and BNS merger simulations. In this work, an extended version of the NSE model
with excluded volumes is adopted for the non-uniform matter whereas the
uniform matter at higher densities is described in the relativistic hadron
field theory with density dependent couplings. The role of the antikaon
condensate on various thermodynamic observables is investigated. The
compositions of matter are appreciably modified due to the onset of the
antikaon condensate at higher densities and thermal $K^{-}$ at higher
temperatures. The presence of the antikaon condensate makes the HS(DD2)K- EoS
softer than that of nucleonic HS(DD2) EoS. The phase diagram with the $K^{-}$
condensate is also investigated. Furthermore, the charge neutral and
$\beta$-equilibrated HS(DD2)K- EoS results in a lower maximum mass neutron
star compared with that of HS(DD2) EoS. We plan to include hyperons along with
the antikaon condensate in a future publication.

Acknowledgements

Authors acknowledge the DAE-BRNS grant received under the BRNS project
No.37(3)/14/12/2018-BRNS.

Author Contribution

All authors contribute equally to this work and agree to the published version
of the manuscript.

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Evaluating Offshore Electricity Market Design Considering Endogenous Infrastructure Investments: Zonal or Nodal?

Michiel Kenis,
Vladimir Dvorkin, Member, IEEE,
Tim Schittekatte,
Kenneth Bruninx, Member, IEEE,

Erik Delarue, Member, IEEE,
Audun Botterud, Member, IEEE

The work of M. Kenis is supported by a Ph.D. grant provided by the Flemish Institute for Technological Research (VITO).

M. Kenis and E. Delarue are with the Division of Applied Mechanics and Energy Conversion (TME), Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300 (box 2421), 3001, Leuven, Belgium. They are also with the EnergyVille, Thor Park 8310, 3600, Genk, Belgium. M. Kenis is also with the Flemish Institute for Technological Research (VITO), Boeretang 200, B-2400, Mol, Belgium. (e-mail: {michiel.kenis,erik.delarue}@kuleuven.be).

V. Dvorkin is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA (e-mail: dvorkin@umich.edu).

T. Schittekatte is with the MIT Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02142, USA, and is with the Florence School of Regulation, European University Institute, Firenze, FI 50133, Italy (e-mail: schtim@mit.edu).

K. Bruninx is with the Faculty of Technology, Policy and Management, TU Delft, 2628 Delft, The Netherlands. He is also with the Division of Applied Mechanics and Energy Conversion (TME), Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300 (box 2421), 3001, Leuven, Belgium (e-mail: k.bruninx@tudelft.nl).

A. Botterud is with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge MA 02142, USA (e-mail: audunb@mit.edu).

IEEE Transactions on Energy Markets, Policy and Regulation

Policy makers are formulating offshore energy infrastructure plans, including wind turbines, electrolyzers, and HVDC transmission lines. An effective market design is crucial to guide cost-efficient investments and dispatch decisions. This paper jointly studies the impact of offshore market design choices on the investment in offshore electrolyzers and HVDC transmission capacity. We present a bilevel model that incorporates investments in offshore energy infrastructure, day-ahead market dispatch, and potential redispatch actions near real-time to ensure transmission constraints are respected. Our findings demonstrate that full nodal pricing, i.e., nodal pricing both onshore and offshore, outperforms the onshore zonal combined with offshore nodal pricing or offshore zonal layouts. While combining onshore zonal with offshore nodal pricing can be considered as a second-best option, it generally diminishes the profitability of offshore wind farms. However, if investment costs of offshore electrolyzers are relatively low, they can serve as catalysts to increase the revenues of the offshore wind farms. This study contributes to the understanding of market designs for highly interconnected offshore power systems, offering insights into the impact of congestion pricing methodologies on investment decisions. Besides, it is useful towards understanding the interaction of offshore loads like electrolyzers with financial support mechanisms for offshore wind farms.

Multi-level modeling, Market design, Flow-based market coupling, Offshore wind power, Electrolysis

[A, 1]$\mathcal{G}$Generators
[A, 2]$\mathcal{E}$electrolyzers
[A, 3]$\mathcal{N}$Nodes
[A, 4]$\mathcal{L}$AC lines
[A, 5]$\mathcal{H}$DC lines
[A, 6]$\mathcal{H}'$Cross-border DC lines
[A, 7]$\mathcal{T}$Timesteps

[B, 01]$MC_g$Marginal cost of generator $g$ [/MWh]
[B, 02]$U_n$ Utility of consumption as end product at node $n$  [/MWh]
[B, 02]$U_{e,t}^{H2}$ Utility of consumption at electrolyzer $e$ [/MWh]
[B, 03]$C_e$, $C_h$ Unit investment cost of electrolyzer $e$ or line $h$ [/MW]
[B, 04]$\Bar{F}_h^{DC}$ Maximum capacity of a new DC line $h$ [MW]
[B, 05]$\Bar{D}_{n,t}$ Demand for electricity as end product at node $n$ [MW]
[B, 06]$\Bar{D}^{H2}_{e,t}$ Maximum capacity of electrolyzer $e$  [MW]
[B, 07]$LF_{g,t}$ Load factor of generator $g$ [-]
[B, 08]$nPTDF^n_l$,$zPTDF^z_l$ Nodal and zonal PTDF [-]
[B, 09]$\Bar{f}^{AC}_l$ Capacity of AC line $l$ [MW]
[B, 10]$RAM_l$ Remaining Available Margin of AC line $l$ [MW]
[B, 11]$NTC_h$ Net Transfer Capacity of DC line $h$ [MW]
[B, 12]$a$ Redispatch cost mark-up [-]
[B, 13]$I_{h,n}$, $I_{h,z}$ Connection between DC line $h$ and node $n$ or zone $z$ [-]
[B, 14]$I_{l,n}$, $I_{l,z}$ Connection between AC line $l$ and node $n$ or zone $z$ [-]

[C, 01]$TIC$ Transmission investment cost []
[C, 02]$EIC$ electrolyzers' investment cost []
[C, 03]$S_t$ Economic market surplus []
[C, 04]$R_t$ Redispatch cost []
[C, 05]$\Bar{f}^{DC}_h$ Capacity of DC line $h$ [MW]
[C, 06]$\Bar{d}^{H2}_e$ Capacity of electrolyzer $e$ [MW]
[C, 07]$y_{g,t}$ Market dispatch of generator $g$ [MW]
[C, 08]$d_{n,t}$ Market dispatch of end consumption at node $n$ [MW]
[C, 09]$d^{H2}_{e,t}$ Market dispatch of consumption at electrolyzer $e$ [MW]
[C, 10]$f^{AC}_{l,t}$,$f^{DC}_{h,t}$ Flow captured by the market on line $l$ or $h$ [MW]
[C, 11]$y^{r}_{g,t}$ Real-time production of generator $g$ [MW]
[C, 12]$d^{r}_{n,t}$ Real-time end consumption at node $n$ [MW]
[C, 13]$d^{H2,r}_{e,t}$ Real-time consumption at electrolyzer $e$ [MW]
[C, 14]$y^{U}_{g,t}$,$y^{D}_{g,t}$ Up/down redispatch of generator $g$ [MW]
[C, 15]$d^{U}_{n,t}$,$d^{D}_{n,t}$ Up/down redispatch of end consumption at node $n$ [MW]
[C, 16]$d^{H2,U}_{e,t}$,$d^{H2,D}_{e,t}$ Up/down redispatch of electrolyzer $e$ [MW]
[C, 17]$f^{AC,r}_{l,t}$,$f^{DC,r}_{h,t}$ Real-time flow on line $l$ or $h$ [MW]

§ INTRODUCTION

The total installed offshore wind power capacity worldwide amounted to 56 GW in 2021 and is expected to strongly increase in the future <cit.>. Besides, other technologies like storage and electrolyzers are expected to be deployed offshore <cit.>, partly as offshore energy hubs <cit.>. In these hubs, the wind farms are connected with an interconnector, typically using the high-voltage direct current (HVDC) technology <cit.>.

In addition to government plans, electricity prices guide investments. The market design[In this paper, we use the term `market design' to refer to methodologies used for pricing and congestion management.], which differs across regions, determines how offshore electricity wholesale prices are set. Under nodal pricing, the transmission network is incorporated in the market clearing resulting in potentially unique prices per node as used in, e.g., most U.S. electricity markets. In contrast, under zonal pricing, network constraints are simplified in the market clearing and one price per predefined bidding zone is calculated. Bidding zone borders typically align with national borders as in the EU.

Under the status quo in the European zonal day-ahead electricity marktet, offshore energy hubs would be part of the mainland zone they are (administratively) connected with. This means that the power prices in those hubs would equal mainland prices of the country the hub belongs to. However, the EU strategy for the deployment of offshore infrastructure opts for (an) off-shore market zone(s) to integrate these assets in the European electricity market <cit.>. In contrast, along the US East Coast, the natural choice is that offshore nodal pricing will be opted for to integrate future offshore infrastructure into the wholesale electricity market <cit.>. There are generally four different market design options to integrate offshore hubs as Tab. <ref> summarizes.

Overview of market design options.

Market design     Onshore     Offshore

Full Nodal Pricing (FNP)     Nodal     Nodal

Offshore Nodal Pricing (ONP)     Zonal     Nodal

Offshore Zonal Pricing (OZP)     Zonal     One offshore zone

Full Zonal Pricing (FZP)     Zonal     Nodes part of onshore zone

Literature on onshore market design does not apply to an offshore context because (i) the offshore market participants mainly consist of (wind power) generators and (ii) the dominant transmission technology is HVDC as opposed to onshore AC technology. The former implies that electricity prices are set by either near-zero marginal cost wind farms or by the willingness-to-pay of a very limited set of consumers when the constraints on the exchanges between the onshore and offshore system are binding in the market clearing mechanism. The latter means that flows on HVDC lines, as well as their impact on the AC lines, can be accurately captured in the flow-based market clearing problem via Advanced Hybrid Coupling, in contrast to their AC counterparts. Note, however, that by definition only HVDC connections connecting two market zones are monitored in the market clearing problem. Intra-market zone offshore HVDC lines are not observed by the market, whereas congestion on internal AC lines can be avoided by including them in the calculation of the limitations on cross-market zone exchanges, i.e., the flow-based domains. These properties have a significant impact on the resulting prices, planned production schedule and required redispatch.

Literature on offshore electricity market design is limited and share a number of simplifying assumptions <cit.>. Firstly, while the market design determines the efficiency of the dispatch of generators, the spatial granularity of the power prices also impacts investment decisions <cit.>. The latter is ignored in existing studies on offshore pricing <cit.>. Secondly, flow-based market coupling (FBMC) is the target methodology in Europe's zonal electricity markets to calculate and allocate AC transmission capacity in wholesale electricity markets while current studies typically consider a `Net Transfer Capacities'-methodology <cit.>. Inclusion of DC transmission capacity under FBMC requires, due to operational transmission characteristics, an adjusted flow-based methodology to calculate and allocate both AC and DC transmission capacity called Advanced Hybrid Coupling <cit.>. Thirdly, novel technologies like offshore electrolyzers are not considered in current studies on the offshore pricing methodology except in Lüth et al. <cit.> who show that offshore bidding zones facilitate offshore hydrogen production. With many projects ongoing to install such technologies offshore, it is important to consider them in the analysis because they serve as the only, or at least a very important, offshore electricity demand <cit.>. Hence, electrolyzers have the potential to have a large impact on offshore prices <cit.>.

This paper addresses the gaps identified above by proposing an optimization model that considers all these three elements. Specifically, we present a bilevel optimization model that captures four temporal stages: (i) offshore HVDC transmission investment, (ii) offshore electrolyzer investment, (iii) day-ahead market clearing, and (iv) redispatch actions. We assume a fixed capacity target for offshore wind generation investment as these decisions are typically not market-driven. Our results reveal that ONP and OZP can serve as valuable alternatives to FZP. However, these schemes come at the cost of a lower profitability for existing offshore wind farms due to a lower average offshore price. The latter benefits the business case for potential offshore electricity consumers like, e.g., offshore electrolyzers, which can contribute to decrease the welfare gap between ideal FNP, on the one hand, and ONP and OZP on the other hand. As a result, policy makers should carefully re-consider support mechanisms for, e.g., offshore wind power under ONP and OZP.

The paper is structured as follows. Section <ref> discusses the existing literature. Section <ref> presents the optimization model, while Section <ref> demonstrates the model by applying it to a stylized electricity grid. Finally, Section <ref> discusses the implications of our findings and Section <ref> concludes.

§ LITERATURE

There are two main literature streams that relate to this paper: literature on (i) existing optimization models that capture both the investments in and the operation of electricity grids, and (ii) offshore electricity market design with a focus on the spatial granularity of wholesale electricity prices.

§.§ Capacity expansion models

Typical composition of optimization models in literature that capture transmission and generation investments as well as the operation of the considered electricity grid.

With regards to onshore electricity grids, the existing literature has proposed optimization models that encompass transmission and generation investments, as well as the operation of the electricity grid. Figure <ref> shows the typical composition of these models capturing four stages: transmission investment, generation investment, market clearings and redispatch actions. The structure of these optimization models varies in the literature based on underlying assumptions. The hierarchical or non-hierarchical relationship between agents, each with distinct objectives and decision variables, categorizes such models. Many papers assume that investors strategically consider the impact of their investments on the power system's operation, resulting in hierarchical models <cit.>. Non-hierarchical models, on the other hand, do not incorporate strategic anticipation <cit.>.

These approaches are, however, not suitable to analyze offshore power systems, especially when connected to zonal electricity markets (onshore) subject to FBMC as in Europe. FBMC considers the limited capacity of critical intra-zonal or inter-zonal AC lines as a constraint in the market clearing algorithm in a simplified way <cit.>. Specifically, the market representation of the flows on AC transmission network elements is capped by their Remaining Available Margins (RAMs), parameters that TSOs provide. The methodology is being rolled out in most European countries <cit.>. To consider (offshore) DC lines, an adjusted flow-based market clearing algorithm is required because the flow on DC lines depends on how the AC/DC converter is operated. Two options are being considered: Standard Hybrid Coupling (SHC – which is currently used) and Advanced Hybrid Coupling (AHC – target methodology) <cit.>. AHC, in contrast to SHC, considers the impact of DC flows on the market representation of AC flows. We refer the reader to Schonheit et al. <cit.> for an exhaustive guide on the functioning on FBMC, and to Muller et al. <cit.> for the compatibility with DC lines.

Lete et al. <cit.> includes FBMC in capacity expansion, but they do not consider DC lines. To the best of the authors' knowledge, our paper is the first to consider both AC and DC lines in a capacity expansion and operation market model by applying AHC. Moreover, an additional novelty is that we optimize investments in both HVDC transmission capacity and offshore electrolysers within the same model.

§.§ Offshore electricity market design

The literature on integrating offshore energy infrastructure into existing onshore zonal electricity markets has gained significant attention in the EU after the European Commission highlighted its “strategy to harness the potential of offshore renewable energy for a climate neutral future" <cit.>. Several studies suggest creating offshore bidding zones to enable efficient flows of renewable energy to high-cost areas, reduce redispatch costs, and enhance operational security <cit.>. Tosatto et al. <cit.> incorporate FBMC in their models and demonstrate that under offshore bidding zones, offshore electricity prices would decrease while congestion rents would increase. However, none of the aforementioned studies endogenously model how investments in generation and/or transmission assets are impacted by the different price signals under alternative market designs for congestion management.

Hardy et al. <cit.> are the first to present a capacity expansion and operational market model within the context of offshore electricity market design. They show that investors should assume that offshore bidding zones will be implemented because the resulting power system (topology as well as generating assets) also performs well under changing pricing methodologies. There are three key differences between their work and our paper. First, in contrast to <cit.>, we assume that investors in transmission capacity strategically anticipate the impact on the market clearing and redispatch actions. Second, Hardy et al. <cit.> do not consider offshore electrolysers. Lastly, their model neglects FBMC when assessing the performance of offshore bidding zones.

§ MODEL

This section presents the high-level structure of the model that is first organized as a quad-level problem, then reformulated to a bilevel problem. First, we list the assumptions that underpin our model. Next, we present the mathematical formulation of the bilevel model. Finally, we provide a solution strategy.

§.§ Assumptions

There are four stages and we assume these stages occur sequentially: HVDC transmission investment (by the TSO), electrolyzer investment (by a merchant investor), market clearing (by the market operator), and redispatch (by the TSO) (Fig. 1). Each level anticipates the impact of its decision on later stages, while taking the decision variables of all higher levels as given. We assume perfect foresight in all stages.

We assume a fixed capacity for offshore wind generation as such capacity decisions are typically not market-driven, but prescribed in tenders. Similarly, we assume an existing fleet of conventional generators connected to the onshore nodes. Investments are onshore assets is not considered.

As we seek to compare the performance of optimal system configurations under different market designs, we consider the HVDC capacity as a variable. Fixing the HVDC capacity under different market designs may lead to, e.g., over- or underestimating the benefits for FNP, depending on the grid topology and capacity of the HVDC elements. In the HVDC investment stage, we assume that there is no existing HVDC transmission capacity. We do not consider the lumpiness of network investments and only investments between predefined nodes are allowed (see Section <ref>). Electrolyzer capacity
is limited to a predefined upper bound per offshore node.

The market clearing can adopt four pricing methodologies: FNP, ONP, OZP and FZP. There is perfect competition between participants in the wholesale electricity market. Only thermal generation units face a non-zero variable cost $MC_g$. The marginal utility of demand for electricity to produce hydrogen depends on the hydrogen price and transport cost, which are assumed to be fixed. Under zonal pricing, flow-based market coupling and Advanced Hybrid Coupling are considered. We assume that all AC lines are critical and, hence, the transmission limits of all AC lines are (albeit imperfectly) considered in the market algorithm. We refer the reader to Schönheit et al. <cit.> for an exhaustive guide on flow-based market coupling.

Redispatch actions only serve to adjust the generators' dispatch in case their output as cleared in the day-ahead market would lead to infeasible power flows. This is only required if the day-ahead market does not accurately capture the transmission constraints, i.e., under ONP, OZP an FZP. We assume cost-based redispatch.

§.§.§ The quad-level problem

From a quad-level problem to a bilevel problem by assuming perfect competition between generators. The colors map to the different stages in Fig. <ref>.

Under the assumptions above, the resulting decision problem can be represented as a quad-level problem (Fig. <ref>). The first level, as shown in the left panel of Fig. <ref>, maximises the overall social welfare while deciding on the HVDC transmission capacity. The second level maximises the net utility of newly added electrolyzers while deciding on the eletrolysers' capacities. The third level maximises the surplus from the day-ahead market with the generators' dispatch (and possible load shedding) as decision variable. Finally, the fourth level minimises redispatch costs including cost mark-ups assuming cost-based redispatch.

§.§.§ From a quad-level to a bilevel problem

We transform the quad-level problem to a bilevel problem in two steps as Fig. <ref> visualises. The assumption of perfect competition between participants in the wholesale electricity market allows merging the second (electrolyzers' investments) and third (market clearing) level optimization problems into one optimization problem. The merged problem maximizes the surplus from the wholesale market less the investment cost of electrolyzers. The equivalence has been shown in literature for the case of generators' investments but is equivalent for electrolyzers' investments <cit.>.

Secondly, we note that the decision variables of the fourth level (redispatch) are independent from the second (electrolyzers' investments) or third (market clearing) level optimization problems, but not from the first (transmission investment) level. As a consequence, the redispatch problem can be considered to be happening on the same hierarchical level as the combined electrolyzer investment & market clearing problem. This is similar to the approach of Pandzic et al. <cit.> in the context of reserve capacity markets and balancing markets.

§.§ A bilevel problem

We present the mathematical equations of the bilevel problem consisting of two lower level problems. The lower level representing the combined electrolyzer investment & market clearing problem comes in two forms: either considering full nodal pricing or full zonal pricing. The cases of ONP and OZP are special cases of FZP in which each offshore node or all offshore nodes together represent (a) separate market zone(s).

§.§.§ Upper Level: transmission investment

The decision variables are the HVDC transmission capacity $\mathbf{\bar{f}^{DC}_{h}}$ of each candidate line $h \in \mathcal{H}$ and the transmission investment cost $\mathbf{TIC}$. The decision variables are written in a bold font. We assume no existing HVDC transmission capacity.

\begin{align}
& \text{max}\quad  \sum_{t \in \mathcal{T}} \Big[ S_t - R_t \Big] - EIC - \mathbf{TIC}  \label{eq:UL_obj}  \\
& \text{s.t.}\nonumber\\
& S_t = \sum_{n \in \mathcal{N}} U_n  d^{}_{n,t} + \sum_{e \in \mathcal{E}} U^{H2}_{e,t}  d^{H2}_{e,t} \nonumber\\ &- \sum_{g \in \mathcal{G}} MC_g  y^{}_{g,t} &  \forall t \label{eq:UL_con1}\\
&R_t = \sum_{n \in \mathcal{N}} U_n \cdot \big[ d_{n,t} - d^{r}_{n,t} \big] + \sum_{e \in \mathcal{E}} U^{H2}_{e,t} \cdot   \nonumber \\ & \big[ d^{H2}_{e,t} -d^{H2,r}_{e,t} \big] + \sum_{g \in \mathcal{G}} MC_g \cdot \big[ y_{g,t}^{r} - y_{g,t} \big]  &  \forall t \label{eq:UL_con2}\\
& EIC = \sum_{e \in \mathcal{E}} C_e  \bar{d}^{H2}_{e}  & \forall g, \forall t  \label{eq:UL_con3}\\
&\mathbf{TIC} = \sum_{h \in \mathcal{H}'} C_h  \mathbf{\bar{f}^{DC}_h}  &  \forall n, \forall t  \label{eq:UL_con4}\\
&0\leq \mathbf{\bar{f}^{DC}_{h}} \leq \bar{F}^{DC}_h &  \forall h \label{eq:UL_con5}
\end{align}

Objective (<ref>) maximises the overall social welfare. This is reasonable as a transmission investment is typically a strongly regulated process <cit.>. Social welfare consists of the economic surplus of the wholesale electricity market $S_t$ less redispatch costs $R_t$, electrolyzer investment cost $EIC$ and transmission investment cost $\mathbf{TIC}$. Eqs. (<ref>)-(<ref>) define each of the terms. Equation (<ref>) poses constraints to the new transmission capacity. Investments in transmission capacity only occur if the investment cost is outweighed by the operational benefits (higher welfare and/or lower redispatch costs).

The economic surplus of the wholesale electricity market $S_t$ equals the utility associated with consuming electricity less the operational generation costs. The former splits into the utility of demand for electricity as end-product $U_n \cdot d_{n,t}$ and demand for electricity as a source for electrolysis $U^{H2}_{e,t} \cdot d^{H2}_{e,t}$. The marginal utility of demand for electricity to produce hydrogen depends on the hydrogen price and transport cost. We further assume that only thermal generation units face a non-zero variable cost $MC_g$.

The redispatch cost $R_t$ is the net cost for a TSO to adjust the dispatch schedule corresponding the day-ahead market to prevent line overloading. Cost-based redispatch implies that a generator, requested by the TSO to produce more than scheduled by the market operator, is paid its marginal cost $MC_g$ by the TSO while a generator, requested by the TSO to produce less than scheduled by the market operator, pays back its avoided marginal cost to the TSO but keeps its profit margin. A similar concept holds for consumers: if the covered demand is lower than scheduled by the market operator, consumers are paid their lost utility $U_n$ or $U^{H2}_{e,t}$, and vice versa. Finally, the electrolyzers' investment cost $EIC$ and transmission investment cost $\mathbf{TIC}$ scale linearly with the installed capacity.

§.§.§ Lower Level 1: electrolyzers' investment & market clearing

Problem (<ref>) presents a first part of the first lower level problem in mathematical form. Eqs. (<ref>), in case of nodal pricing, and Eqs. (<ref>), in case of zonal pricing, present the remaining constraints. The decision variables are the new capacity of electrolyzers $\mathbf{\bar{d}^{H2}_{e}}$ as well as the dispatch of all generators $\mathbf{y_{g,t}}$. Other decision variables are the covered demand for electricity as end-product $\mathbf{d_{n,t}}$ and electricity for electrolysis $\mathbf{d^{H2}_{e,t}}$ as well as the power flows on all AC lines $\mathbf{f^{AC}_{l,t}}$ and DC-lines $\mathbf{f^{DC}_{h,l}}$.

\begin{align}
& \text{max}\quad \sum_{t \in \mathcal{T}} \mathbf{S_t} - \mathbf{EIC}  \label{eq:LL1_obj} \\
& \text{s.t.}\quad \nonumber\\
& \text{Eq. (}\ref{eq:UL_con1}\text{), Eq. (}\ref{eq:UL_con3}\text{)} \label{eq:LL1_con1}\\
&0 \leq \mathbf{\bar{d}^{H2}_{e}} \leq \bar{D}^{H2}_{e}  &  \forall e \label{eq:LL1_con4}\\
&0 \leq \mathbf{y_{g,t}} \leq LF_{g,t} \bar{y}_g  & \forall g, \forall t  \label{eq:LL1_con5}\\
&0 \leq \mathbf{d_{n,t}} \leq \bar{D}_{n,t}  &  \forall n, \forall t  \label{eq:LL1_con8}\\
&0 \leq \mathbf{d^{H2}_{e,t}} \leq \bar{d}^{H2}_{e} &  \forall e, \forall t \label{eq:LL1_con9}
\end{align}

Objective (<ref>) maximises the surplus from the wholesale market $\mathbf{S_t}$ less the electrolyzers' investment cost $\mathbf{EIC}$. Equation (<ref>) defines both terms in the objective. Equation (<ref>) puts limitations on the new capacity of offshore electrolyzers. Equations (<ref>)-(<ref>) limit the dispatch of generators and consumers to the available power capacities.

\begin{align}
& \textit{In case of nodal pricing:} \nonumber \\
& \mathbf{f^{AC}_{l,t}} = \sum_{n \in \mathcal{N}} nPTDF^{n}_{l} \Big[\sum_{g \in \mathcal{G(N)}} \mathbf{y_{g,t}} -\mathbf{d_{n,t}} \nonumber \\ & - \sum_{e \in \mathcal{E}(n)} \mathbf{d^{H2}_{e,t}} - \sum_{h \in \mathcal{H}} \mathbf{f^{DC}_{h,t}} \cdot I_{h,n} \Big] & \forall l, \forall t \label{eq:LL1_nodal_con1}\\
&|\mathbf{f^{AC}_{l,t}}|  \leq \bar{f}^{AC}_l    &  \forall l, \forall t \label{eq:LL1_nodal_con2}\\
&|\mathbf{f^{DC}_{h,t}}| \leq \bar{f}^{DC}_h &  \forall h, \forall t \label{eq:LL1_nodal_con3}\\
&\sum_{g \in \mathcal{G}(n)} \mathbf{y_{g,t}} - \mathbf{d_{n,t}} - \sum_{e \in \mathcal{E}(n)} \mathbf{d^{H2}_{e,t}} \nonumber  \\ &- \sum_{h \in \mathcal{H}} \mathbf{f^{DC}_{h,t}} \cdot I_{h,n} -\sum_{l \in \mathcal{L}} \mathbf{f^{AC}_{l,t}} \cdot I_{l,n} = 0 & \forall n, \forall t \label{eq:LL1_nodal_con4}
\end{align}

Equation (<ref>) defines the flow on each AC line using nodal PTDFs. Ignoring the DC approximation, this is an exact projection of the physical flow as opposed to the case of zonal pricing. The final term considers the injection of the DC grid into the AC grid. $I_{h,n}$ equals 1 (or -1) if the flow on DC-line $h$ goes outward (or inward) node $n$, and is otherwise 0. Eqs. (<ref>) and (<ref>) limits the flow on all AC and DC lines by their physical capacities. Finally, Eq. (<ref>) presents the nodal power balance.

\begin{align}
& \textit{In case of zonal pricing:} \nonumber \\
& \mathbf{f^{AC}_{l,t}} = \sum_{z \in \mathcal{Z}} zPTDF^{z}_{l} \Big[\sum_{g \in \mathcal{G}(z)} \mathbf{y_{g,t}} - \sum_{n \in \mathcal{N}(z)} \mathbf{d_{n,t}}  - \sum_{e \in \mathcal{E}(z)} \mathbf{d^{H2}_{e,t}} \nonumber \\ & - \sum_{h \in \mathcal{H}} \mathbf{f^{DC}_{h,t}} I_{h,z} \Big] + \sum_{h \in \mathcal{H}'} \mathbf{f^{DC}_{h,t}} \Big[ nPTDF^{n(h-)}_l \cdot I_{h,n(h-)} \nonumber \\
& + nPTDF^{n(-h)}_l \cdot I_{h,n(-h)} \Big] \hspace{29.5mm} \forall l, \forall t \label{eq:LL1_zonal_con1}\\
&|\mathbf{f^{AC}_{l,t}}|  \leq RAM_l \hspace{48mm}   \forall l, \forall t \label{eq:LL1_zonal_con2}\\
&|\mathbf{f^{DC}_{h,t}}| \leq NTC_h \hspace{39mm}   \forall h \in \mathcal{H}', \forall t \label{eq:LL1_zonal_con3}\\
&\sum_{g \in \mathcal{G}(z)} \mathbf{y_{g,t}} - \sum_{n \in \mathcal{Z}} \mathbf{d_{n,t}} - \sum_{e \in \mathcal{E}(z)} \mathbf{d^{H2}_{e,t}} \nonumber \\ & - \sum_{h \in \mathcal{H}} \mathbf{f^{DC}_{h,t}} \cdot I_{h,z} -\sum_{l \in \mathcal{L}} \mathbf{f^{AC}_{l,t}} \cdot I_{l,z} = 0  \hspace{15.5mm} \forall z, \forall t \label{eq:LL1_zonal_con4}
\end{align}

Equation (<ref>) defines the flow on each AC line using zonal PTDFs[Following flow-based market coupling, only the flow on critical AC lines should be monitored and capped. We assume here that all AC lines are critical.]. Zonal PTDFs are parameters, set by TSOs, that are predictive and imperfect by definition as they aggregate nodal information. As a consequence, $\mathbf{f^{AC}_l}$ does not capture the physical power flow but rather an imperfect heuristic, hereafter referred to as the commercial flow. The impact of the power flow on a DC line is captured following Advanced Hybrid Coupling <cit.>. The flow on an AC line $l$ as a result of a flow on a DC line $h$ is calculated with $nPTDF^{n(h-)}_l$ and $nPTDF^{n(-h)}_l$ with $n(h-)$ and $n(-h)$ the starting/ending end of DC-line $h$. $I_{h,n(h-)}$ and $I_{h,n(-h)}$ equal 1 (or -1) if the flow on DC-line $h$ is defined as flowing into (or away from) node $n(h-)$ or $n(-h)$, and is zero otherwise. Equation (<ref>) limits the commercial power flow on each AC line by the Remaining Available Margin (RAM), i.e., the part of the physical transmission capacity that can be traded in the wholesale market <cit.>. Equation (<ref>) limits the flow on each cross-border DC line by the Net Transfer Capacity (NTC). Finally, Equation (<ref>) presents the zonal power balance.

§.§.§ Lower Level 2: redispatch

The decision variables are the final dispatch, after redispatch, of all generators $\mathbf{y_{g,t}^{r}}$ as well as the final covered demand for electricity as end-product $\mathbf{d_{n,t}^{r}}$ and electricity for electrolysis $\mathbf{d^{H2,r}_{e,t}}$. We also consider explicit variables for the upward and downward adjustment to the wholesale market dispatch $\mathbf{y^{U}_{g,t}}$, $\mathbf{y^{D}_{g,t}}$, $\mathbf{d^{U}_{n,t}}$, $\mathbf{d^{D}_{n,t}}$, $\mathbf{d^{H2,U}_{e,t}}$ and $\mathbf{d^{H2,D}_{e,t}}$. Finally, $\mathbf{f^{AC,r}_{l,t}}$ and $\mathbf{f^{DC,r}_{h,l}}$ are the physical AC and DC power flow after redispatch.

\begin{align}
\text{min}\quad & \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} \big[ MC_g + a \big] \cdot \mathbf{y^{U}_{g,t}} + \big[ -MC_g + a \big] \cdot \mathbf{y^{D}_{g,t}} \nonumber \\
&+ \sum_{e \in \mathcal{E}} \big[ -U^{H2}_{e,t} + a \big] \cdot \mathbf{d^{H2,U}_{e,t}} + \big[ U^{H2}_{e,t} + a \big] \cdot \mathbf{d^{H2,D}_{e,t}} \nonumber \\ & + \sum_{n \in \mathcal{N}} \big[ -U_n + a \big] \cdot \mathbf{d^{U}_{n,t}} + \big[ U_n + a \big] \cdot \mathbf{d^{D}_{n,t}} \label{eq:LL2_objective}
\end{align}

\begin{align}
& \text{s.t.}\nonumber\\
&\mathbf{y_{g,t}^{r}} = y_{g,t}+\mathbf{y^{U}_{g,t}}-\mathbf{y^{D}_{g,t}} &  \forall g, \forall t \label{eq:LL2_con1}\\
&\mathbf{d_{n,t}^{r}} = d_{n,t}+\mathbf{d^{U}_{n,t}}-\mathbf{d^{D}_{n,t}} & \forall n, \forall t \label{eq:LL2_con4}\\
&\mathbf{d_{e,t}^{H2,r}} = d^{H2}_{e,t}+\mathbf{d^{H2,U}_{e,t}}-\mathbf{d^{H2,D}_{e,t}} & \forall e, \forall t \label{eq:LL2_con5}\\
&0 \leq \mathbf{y^{U}_{g,t}} \leq  LF_{g,t} \cdot \bar{y}_g - y_{g,t} & \forall g, \forall t \label{eq:LL2_con6}\\
&0 \leq \mathbf{y^{D}_{g,t}} \leq y_{g,t} & \forall g, \forall t \label{eq:LL2_con7}\\
&0 \leq \mathbf{d^{U}_{n,t}} \leq \bar{D}_n-d_{n,t} & \forall n, \forall t \label{eq:LL2_con12}\\
&0 \leq \mathbf{d^{D}_{n,t}} \leq d_{n,t} & \forall n, \forall t \label{eq:LL2_con13}\\
&0 \leq \mathbf{d^{H2,U}_{e,t}} \leq \bar{d}^{H2}_e-d^{H2}_{e,t} & \forall e, \forall t \label{eq:LL2_con14}\\
&0 \leq \mathbf{d^{H2,D}_{e,t}} \leq d^{H2}_{e,t} & \forall e, \forall t \label{eq:LL2_con15} \\
& \mathbf{f^{AC,r}_{l,t}} = \sum_{n \in \mathcal{N}} nPTDF^{n}_{l} \Big[\sum_{g \in \mathcal{G(N)}} \mathbf{y^{r}_{g,t}} \nonumber \\ & -\mathbf{d^{r}_{n,t}} - \sum_{e \in \mathcal{E}(n)} \mathbf{d^{H2,r}_{e,t}} - \sum_{h \in \mathcal{H}} \mathbf{f^{DC,r}_{h,t}} \cdot I_{h,n} \Big] &  \forall l, \forall t \label{eq:LL2_con18}\\
& |\mathbf{f^{AC,r}_{l,t}}| \leq \bar{f}^{AC}_l & \forall l, \forall t \label{eq:LL2_con19} \\
& |\mathbf{f^{DC,r}_{h,t}}| \leq \bar{f}^{DC}_h &\forall h, \forall t \label{eq:LL2_con20} \\
&\sum_{g \in \mathcal{G}(n)} \mathbf{y^{r}_{g,t}} - \mathbf{d^{r}_{n,t}} - \sum_{e \in \mathcal{E}(n)} \mathbf{d^{H2,r}_{e,t}} - \nonumber \\ & \sum_{h \in \mathcal{H}} \mathbf{f^{DC,r}_{h,t}} \cdot I_{h,n} -\sum_{l \in \mathcal{L}} \mathbf{f^{AC,r}_{l,t}} \cdot I_{l,n} = 0 & \forall n, \forall t \label{eq:LL2_con21}
\end{align}

Objective (<ref>) minimises the redispatch cost considering a cost mark-up $a$ to avoid redispatch actions when the outcome of the day-ahead market does not lead to overloading of network elements. This differs from existing papers that do not consider a mark-up <cit.>. Indeed, the aim of the optimisation problem is to determine the required adjustments to the market outcome only to avoid line overloadings, not to correct a possibly inefficient dispatch from the market. This is in line with real world redispatch actions in zonal wholesale markets <cit.>. Therefore, by imposing cost mark-ups we avoid redispatch when there is no congestion. The redispatch cost $R_t$, and thus social welfare, in the upper level problem is the true redispatch costs, i.e., without cost mark-up $a$.

Eqs. (<ref>)-(<ref>) define the final dispatch, after redispatch, of all generators $\mathbf{y_{g,t}^{r}}$ as well as the final covered demand for electricity as end-product $\mathbf{d_{n,t}^{r}}$ and electricity for electrolysis $\mathbf{d^{H2,r}_{e,t}}$. Eqs. (<ref>)-(<ref>) provide limits to the upward or downward adjustment taking the market outcome (from lower level 1) as given. Finally, similar to Eqs. (<ref>)-(<ref>), Eqs. (<ref>)-(<ref>) present the definition of the power flow on each AC line, the limitations to the AC and DC power flows, and the nodal power balance in real-time, i.e., after redispatch.

§.§ Solution Approach

We follow the approach of Kazempour and Conejo <cit.>, and apply the Benders' decomposition to a bilevel problem, considering $\Bar{f}^{DC}_h$ as the complicating variable. The approach works as follows:

* Given fixed HVDC transmission investment decisions, we solve two single-level reformulations of the bilevel problem. Specifically, after fixing the complicating variable, we first solve the mixed-integer reformulation of the simplified MPEC which now only has lower-level decision variables. Subsequently, we solve a continuous yet nonlinear version of the same problem, now formulated using the strong duality theorem, wherein the variables causing bilinear terms are fixed to the values derived from the solution of the first MPEC. This, in turn, allows the extraction of sensitivities of welfare w.r.t. HVDC investments — which is the main motivation behind solving two MPECs at every iteration, as in <cit.>.

* The sensitivities obtained in step 1 allow for formulating a so-called Benders' master problem whose solution provides updated HVDC investment decisions.

* Step 1-2 are repeated until no more economic welfare improvement can be achieved.

Our framework resembles that of Kazempour and Conejo <cit.>, with some minor differences that do not affect the solution strategy. The operational decisions in our problem are not optimized by the upper level investor, as in <cit.>, but rather by a system operator. Another distinction lies in our aim to maximize societal welfare as opposed to minimizing negative profit and the absence of scenarios due to our deterministic framework. While our model is distinct in these ways, it does not alter the solution strategy proposed in <cit.>.

§ CASE STUDY

§.§ Set-up

Figure <ref> presents the network layout. We consider three mainland zones, each consisting of three nodes, and three offshore nodes. We consider 30 time steps with varying consumption levels and load factors of offshore wind. The size of the case study is similar to related literature <cit.>. Appendix A presents the assumed values of all parameters in the algorithm. The algorithm, applying Benders' decomposition, converges after less than 20 iterations and a total run time of less than $10^4$ seconds for all cases considering a tolerance of $10^{-1}$.

Electricity grid. The symbols indicate the location of generators, consumers and transmission lines with a black color for existing elements (AC transmission lines, thermal generation units, consumers and offshore wind farms) and a yellow color for elements of which the capacity is a decision variable (DC transmission lines and electrolyzers).

We consider three unit electrolyzer investment cost levels (low, medium and high) to illustrate the possible role of offshore electrolysis under each of the four electricity market designs in Tab. <ref>.

Accuracy of monitoring the flows on the AC and DC transmission lines in the market clearing algorithm under the four different pricing methods.

Figure <ref> shows how each pricing method implies a different monitoring of the flows on the transmission lines in the market clearing. While FNP perfectly monitors all flows, the other pricing methods involve (partly) zonal pricing following FBMC. Therefore, the flows on the AC lines ($\mathcal{L}$) are imperfectly monitored using zonal PTDFs under ONP, OZP and FZP. The DC-lines follow the NTC approach and are only monitored if they are cross-border lines ($\mathcal{H}'$).

§.§ Results

We discuss market designs implications on (i) the overall welfare, (ii) the profitability of offshore wind farms and (iii) offshore electricity prices.

§.§.§ Implications on welfare

Figure <ref> presents the market surplus $\sum_t S_t$, aggregated over all time steps and normalised w.r.t. the market surplus under FNP and the lowest electrolyzer unit investment cost. The difference between the market surplus and overall welfare (Eq. (<ref>)) consists of the redispatch cost $\sum_t R_t$, the electrolyzer investment cost $GIC$, and the transmission investment cost $TIC$.

We make three observations. Firstly, ONP and OZP outperform FZP in terms of welfare despite a higher market surplus and a lower spending on electrolyzer and transmission capacities under FZP. Specifically, the market surplus under FZP amounts to 97.9%, 96.4% and 96.3% of the reference market surplus for the three investment costs of electrolyzers, while it ranges from 87.7% to 96.3% under ONP and from 88.7% to 96.6% under OZP. The market surplus is higher under FZP compared to ONP and OZP because more grid constraints are considered in the day-ahead market under the latter two cases. Besides, the combined investments in electrolyzer and transmission capacities are consistently lower under FZP (0.7%, 1.5% and 0.1% of the reference market surplus) compared to other market designs. As a consequence, the difference in overall welfare between ONP and OZP, on the one hand, and FZP, on the other hand, is entirely driven by lower redispatch costs $R_t$ due to a better grid representation in the market clearing problem (see Fig. <ref>). The redispatch cost under FZP is 20.9%, 23.3% and 25.4% of the reference market surplus for the three investment costs of electrolyzers, while it is 9.7% at most under other market designs.

Market surplus $\sum_t S_t$, normalised w.r.t. the market surplus under nodal pricing and lowest electrolyzer investment cost. The overall welfare (Eq. (<ref>)) is indicated in blue.

HVDC capacity.

Electrolyzer capacity.

A more accurate representation of the transmission constraints in the market clearing algorithm allows capturing the contribution of potential HVDC investments to increased welfare and reduced redispatch costs. Recall that intra-zonal HVDC links are not considered in the market clearing problem under OZP or FZP. Under ONP, onshore zonal pricing restricts flows on the offshore network elements, limiting the attainable wellfare gains (see also Section <ref>). In our case study a more accurate grid representation results in more HVDC capacity (Fig. <ref>) while a market clearing problem with the most crude grid representation relies on the possibility for redispatch to avoid congestion. The HVDC capacity under ONP, OZP and FZP is lower because the additional market surplus as a result of installing more HVDC capacity is limited under ONP, OZP and FZP compared to FNP. It does, however, lower the redispatch cost.

Secondly, welfare increases with a decrease in the unit investment cost of electrolyzers regardless of the market design. This is because a higher electrolyzer capacity, which is shown in Fig. <ref>, allows generating a larger market surplus, outweighing the larger electrolyzer investment cost. The market surplus grows with 2.1% (FNP), 8.6% (ONP), 8.0% (OZP) and 1.6% (FZP) of the reference market surplus while the change in combined spending in electrolyzers and HVDC lines is less than 1.4% under all market designs. In addition, a higher electrolyzer capacity allows reducing the redispatch cost, which drops with 1.3% (ONP), 2.6% (OZP) and 4.4% (FZP) of the reference market surplus.

Thirdly, the welfare gain associated with a lower unit investment cost of the electrolyzers is larger under ONP (10.1% of the reference market surplus) and OZP (11.6% of the reference market surplus) than under FNP or FZP. Lower offshore wholesale prices under ONP and OZP compared to the other market designs trigger higher investments in electrolyzers (see further) and thus an increasing impact of their investment costs on the total welfare. Hence, the electrolyzer capacity is higher under FNP, OZP and ONP than under FZP.[Note that the bilevel modeling approach inherently allows multiple optimal solutions to the lower level. The upper level problem may "exploit" this multiplicity of solutions and select the lower level solution that maximizes the upper level objective. Similarly, this characteristic of bilevel models implies that while the reported welfare and HVDC investment are optimal, this may be achieved through different electrolyzer capacity investments, day-ahead market clearing and redispatch actions.]

§.§.§ Implications on wind farm & electrolyzer profitability

Figure <ref> presents the operational profit for all offshore wind farms. Firstly, the profit is the highest under FZP closely followed by FNP, while profits are significantly lower under ONP and particularly OZP. For example, in case of a low investment cost of electrolyzers, the profit amounts to 1.72 M under FZP, followed by 1.66 M (FNP), 1.34 M (ONP) and 0.60 M (OZP). Secondly, the profit increases with a decrease in investment cost of electrolyzers regardless of the market design. Specifically, it grows with 0.10 M (FNP), 0.79 M (ONP), 0.32 M (OZP) and 0.15 M (FZP) when the investment cost decreases from high to low. Thirdly, the profit increase is the largest under ONP with 0.79 M. The drivers behind these effects are curtailment and day-ahead offshore electricity prices, which we discuss in the following.

Figure <ref> presents the wind energy curtailment as part of the market clearing and the redispatch actions, expressed as a fraction of the available offshore wind energy. Curtailment under FZP is at least 6 times higher than under other other market designs. This is a consequence of the crude representation of grid constraints in the market as the majority of the curtailment under FZP occurs as part of the redispatch actions (see also redispatch costs in Fig. <ref>). The presence of electrolyzers limits the need for curtailment as it serves as an electricity consumer at the same node at which wind power is generated. Specifically, we find a negative correlation of -0.52 between wind power curtailment and electricity consumption by electrolyzers.

Operational profit of the offshore wind farms.

Curtailment of offshore wind energy, expressed as a fraction of the available offshore wind energy. We distinguish between curtailment as part of the market clearing and the redispatch actions.

By design, the electrolyzers do not secure a net profit unless the optimal electrolyzer capacity equals the assumed upper limit. In case an electrolyzer's capacity equals this upper bound, its operating profits can outweigh its investment cost indicating that a higher electrolyzer capacity would increase social welfare. If the electrolyser capacity does not equal its upper limit, operating profits offset the investment costs and the electrolysers' net profit is zero.

§.§.§ Implications on offshore electricity prices

Figure <ref> presents the day-ahead electricity prices for offshore wind farms (node 4, 5 and 6). Offshore electricity prices are, averaged over time and offshore nodes, the highest under FZP. Specifically, the average price amounts to 71.5 /MWh under FZP compared to 65.2 /MWh under FNP at a low investment cost, to 69.3 /MWh under FZP compared to 64 /MWh under FNP at a medium investment cost and to 69.3 /MWh under FZP compared to 62.7 /MWh under FNP at a high investment cost. Average prices are the highest under FZP because the flow on HVDC lines between an offshore and an onshore node (3-4, 6-7 and 5-12) is not monitored nor constrained in the market clearing problem. Hence, price convergence with the onshore zones occurs by construction. Contrary, the flows on all HVDC lines between an offshore and an onshore node (3-4, 6-7 and 5-12) are monitored and constrained in the market clearing problem under FNP, ONP and OZP because they serve as cross-zonal HVDC lines. As a result, electricity prices in the offshore zone(s) are zero whenever the commercial transmission capacity restricts flows, lowering the average offshore prices under FNP, ONP and OZP compared to FZP.

[inner sep=0pt] (russell) at (0,0)

at (-5.2,-1.7)Low;

at (0.5,-1.7)Medium;

at (6.1,-1.7)High;
at (0.5,-2.05)Investment cost of electrolyzers [€/MW];

Prices for offshore wind farms. The boxes show the 25th and 75th quantiles and the median while the dashed line shows the average.

Despite that the flows on all HVDC lines between an offshore and an onshore node (3-4, 6-7 and 5-12) are monitored and constrained in the market clearing problem under FNP, ONP and OZP, offshore prices are different between these three pricing methods. Specifically, the average price is consistently higher under FNP (65.2 /MWh, 64 /MWh and 62.7 /MWh) than under ONP (61.8 /MWh, 43.7 /MWh and 31.6 /MWh) which in turn is higher than under OZP (46.7 /MWh, 40.8 /MWh and 25.8 /MWh). As expected, the electrolyzer capacity is inversely correlated with the average electricity price per offshore node and timestep.

We identify two drivers explaining the price differences between FNP, ONP and OZP: (i) the (optimised) HVDC transmission capacity, and (ii) not considering offshore internal grid constraints under OZP. Firstly, under FNP, the flows on both AC and DC lines are perfectly monitored and limited. Contrary, under ONP and OZP, the flows on the onshore AC lines are imperfectly monitored and limited due to the characteristics of a flow-based market clearing. The zonal character of the onshore market implies that the flows on the offshore DC lines are restricted more than under FNP (see Eq. (<ref>)). This leads to less value for HVDC transmission capacity (see Fig. <ref>). Hence, there is less price convergence between the offshore nodes and the onshore zones. Secondly, the average offshore price is lower under OZP than under ONP because, under OZP, congestion between an offshore node and the connecting onshore node immediately results in a price difference between all three offshore nodes and the onshore zone. This drives down the average offshore price because the marginal cost of offshore wind farms is assumed to be zero.

A lower unit investment cost of electrolyzers consistently drives up the average offshore price under all market designs. This is the result of the combination of two drivers: (i) because thermal generation units produce more electricity and therefore can set the price more often, and (ii) because the willingness-to-pay of electrolyzers $U^{H2}_{e,t}$ (27 /MWh in this case study) sets the price more often. The increase of the average offshore price with decreasing investment costs of electrolyzers is the largest under ONP and OZP. Specifically, it increases with 96% (from 31.6 /MWh to 61.8 /MWh) under ONP and 81.4% (from 25.8 /MWh to 46.8 /MWh) under OZP compared to 4% (from 62.7 /MWh to 65.2 /MWh) under FNP and 3% (from 69.3 /MWh to 71.5 /MWh) under FZP. This is because the electrolyzers drive up the offshore electricity price such that the level of price convergence (see Fig. <ref>) increases the most under these market designs, which is reflected in the market surplus that also witnesses the largest increase under ONP and OZP with decreasing investment costs of electrolyzers. As a result, the presence of a consumer at an offshore node in the form of an electrolyzer can contribute to the profitability of offshore wind farms.

Finally, the offshore HVDC capacity impacts the level of electricity prices (<ref>) and price convergence. Table <ref> shows the percentage of time during which a specific amount of unique prices exists. One unique price implies full price convergence across all nodes/zones. The largest amount of hours in which there are one or two unique prices consistently occurs under FNP compared to ONP, OZP and FZP. Specifically, this is during 80%, 86.7% and 76.7% of time for the three levels of investment costs of electrolyzers compared to at most 20% under any other market design. This is contra-intuitive because FNP allows for up to 12 unique prices. However, FNP signals scarcity of transmission capacity better than any other market design leading to the most HVDC deployment, hence, the highest level of price convergence.

Percentage of time during which a specific amount of unique prices exists. One unique price implies full price convergence across all nodes/zones.

4 c| Low      4 c| Medium     4 c High

FNP     ONP     OZP     FZP     FNP     ONP     OZP     FZP     FNP     ONP     OZP     FZP

1 unique price     6.7%     3.3%     0%     0%     20%     0%     0%     0%     20%     0%     0%     0%

2 unique prices     73.3%     16.7%     0%     6.7%     66.7%     0%     0%     20%     56.7%     0%     0%     20%

3 unique prices      10%     60%     90%     93.3%     13.3%     43.3%     80%     80%     23.3%     13.3%     50%     80%

4 unique prices     10%     20%     10%     -     0%     46.7%     20%     -     0%     70%     50%     -

5 unique prices     0%     0%     -     -     0%     10%     -     -     0%     16.7%     -     -

§ DISCUSSION

Our results show that FNP is superior to ONP, OZP and FZP in terms of economic welfare. Yet, there exist political hurdles that prevent the implementation of FNP in many regions <cit.>. Instead, FZP is the active pricing methodology in many regions including, e.g., the EU, UK, Brazil and Australia. Many of these hurdles are rooted in concerns around the redistributional impacts for existing consumers and generators. In an offshore context, there are no or very few existing generators/loads. We would therefore expect less opposition against ONP or OZP, which our case study reveals to be valuable alternatives to FZP. Specifically, despite leading to a lower market surplus and an increased spending on HVDC transmission capacity and electrolyzer capacity, these market designs increase overall welfare by significantly decreasing offshore wind energy curtailment and redispatch costs, originating either offshore or onshore. The latter is the result of a more accurate representation of grid constraints in the market clearing algorithm under ONP and OZP compared to FZP. Offshore wind developers remain skeptical of ONP and OZP as they fear lower average electricity prices, as well as higher price and volume risks due to congestion. Our results indeed show tat the overall increase in welfare comes at the cost of a lower profitability of existing offshore wind farms. The lower profitability is explained by a lower average offshore price as congestion between onshore and offshore nodes is signaled via price differences with a certain spatial granularity. This effect is stronger under OZP compared to ONP.

Technically, the challenges with zonal and nodal offshore market design are similar, and often related to the interaction with other markets, such as the balancing market. With little demand and no "back-up" generators offshore, system operators structurally have to rely on balancing services provided by neighboring zones or nodes. Notably, the controllability of HVDC technology with regard to power flows offers Transmission System Operators (TSOs) more flexibility in managing grid constraints, potentially easing some technical challenges.

A different market design than FZP would also impact public support mechanisms for offshore wind farms like, e.g., Contract-for-Differences (CfDs). Specifically, the cost for governments would increase under CfDs assuming ceteris paribus. Under FZP, support for offshore wind developers is implicit in the high (socialized) redispatching costs, i.e., wind developers receive artificially high grid-agnostic day-ahead electricity prices but may be curtailed in real-time. Alternative to increased support for offshore wind farms, our results suggest that implementing nodal pricing onshore in addition to ONP, which is identical to FNP, also brings the profitability of wind farms close to levels under FZP.

Offshore consumers like, e.g., offshore electrolyzers can serve as a catalyst to decrease the remaining welfare gap between FNP, on the one hand, and ONP and OZP on the other hand. A lower offshore average price under ONP and OZP compared to FNP could allow offshore electrolyzers to enter the market, which in turn increases the market surplus and reduces redispatch costs. Besides, it reduces wind energy curtailment and increases the offshore average price, leading to an increased profitability of offshore wind farms. Therefore, there is an important interaction between support mechanisms for offshore wind and potential support mechanisms for electrolyzers. The latter would reduce the need for the former because of the increasing offshore prices with more electrolyzers, especially under ONP and OZP.

§ CONCLUSION

Different strategies exist, or are part of the ongoing debate, to integrate offshore energy hubs in wholesale electricity markets. We develop a bilevel optimization model to simulate the implications of four market designs: full nodal pricing, offshore nodal pricing, offshore zonal pricing, and full zonal pricing. The model captures four temporal stages: (i) offshore HVDC transmission investment, (ii) offshore electrolyzer investment, (iii) wholesale electricity market clearing, and (iv) redispatch actions. Offshore wind generation investment is exogenously determined.

We find that full nodal pricing is superior to offshore nodal, offshore zonal and full zonal pricing. Yet, there exist political hurdles that prevent the implementation of full nodal pricing in many regions. Our case study reveals that offshore nodal and offshore zonal pricing can serve as valuable alternatives to full zonal pricing. Specifically, these market designs could increase welfare as a consequence of decreased offshore wind energy curtailment and decreased redispatch costs despite a lower market surplus. However, the welfare gain comes at the cost of a lower profitability of existing offshore wind farms due to congestion between onshore and offshore nodes. The latter benefits the business case for offshore consumers like, e.g., offshore electrolyzers, which, in turn, can have a positive effect on the offshore average price. Therefore, if proven profitable in real world power systems, offshore electrolyzers can serve as a catalyst to decrease the remaining welfare gap between full nodal pricing, on the one hand, and offshore nodal and offshore zonal pricing on the other hand. Policy makers and regulators should, hence, carefully coordinate electricity market design with policies supporting the development of offshore energy infrastructure.

Future work could focus on how balancing mechanisms should be set-up in under offshore nodal and zonal or full nodal pricing schemes, in which most balancing energy would need to be provided by onshore assets in different bidding zones. Focal points in this discussion are the interaction with day-ahead electricity markets and transmission capacity allocation, the impact on offshore wind farms and the use of offshore assets such electrolyzers as a source of balancing energy.

§ APPENDICES

§.§ Appendix A: Parameters in the case study

We assume that each thermal generation unit and each offshore wind farm has an installed capacity $\bar{y}_g$ of 2000 MW and 1000 MW respectively. The operational cost $MC_g$ of each thermal generation unit is 60 /MWh at node 2, 80 /MWh at node 9 and 100 /MWh at node 10, while we assume it is zero for the offshore wind farms. Figure <ref> presents the assumed availability factor $LF_{g,t}$ at each offshore node as well as the price inelastic demand $\bar{D}_{n,t}$ at each node and time step. There are six peaks in both available wind energy and demand. However, both do not always peak at the same time steps. We assume that the utility to consume electricity as end product $U_{n}$ is 200 /MWh and to produce hydrogen $U^{H2}_{e,t}$ is 27 /MWh. The penalisation parameter in the redispatch problem $a$ is 1000.

We assume that each AC line has a thermal capacity of 700 MW and all have the same impedance. Both the RAMs and NTCs are set identical to the thermal capacity of the considered line. The zonal PTDFs are derived from the nodal PTDFs by assuming values for GSKs that are pro rata with the generation capacity in a zone <cit.>.

We assume three levels for the unit investment cost of electrolyzers $C_e$: low (50 /MW), medium (150 /MW) and high (250 /MW). The maximal electrolyzer capacity $\bar{D}^{H2}_{e,t}$ is 600 MW per offshore node. The investment cost of a DC line $C_h$ is 20 /MW. The maximal capacity of a new DC line $\bar{F}^{DC}_h$ is 2000 MW.

Availability factors of the offshore wind farms and price inelastic demand at each node and time step.