Modelling and analysis of offshore energy hubs

a]Hongyu Zhangcor1
a]Asgeir Tomasgard
b]Brage Rugstad Knudsen
b]Harald G. Svendsen
a]Steffen J. Bakker
c]Ignacio E. Grossmann
[cor1]Corresponding author's email address<EMAIL_ADDRESS>NCS, Norwegian continental shelf; OEH, Offshore energy hub; PFS, Power from shore; Base, The case with only offshore renewables, gas turbines and power from shore; S1, Scenario 1; S2, Scenario 2; MILP, Mixed-integer linear programming.

[a]Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Høgskoleringen 1, 7491, Trondheim, Norway
[b]SINTEF Energy Research, Kolbjørn Hejes vei 1B, 7491, Trondheim, Norway
[c]Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA

Clean, multi-carrier Offshore Energy Hubs (OEHs) may become pivotal for efficient offshore wind power generation and distribution. In addition, OEHs may provide decarbonised energy supply for maritime transport, oil and gas recovery, and offshore farming while also enabling conversion and temporary storage of liquefied decarbonised energy carriers for export. Here, we investigate the role of OEHs in the transition of the Norwegian continental shelf energy system towards zero-emission energy supply. We develop a mixed-integer linear programming model for investment planning and operational optimisation to achieve decarbonisation at minimum costs. We consider clean technologies, including offshore wind, offshore solar, OEHs and subsea cables. We conduct sensitivity analysis on CO$_2$ tax, CO$_2$ budget and the capacity of power from shore. The results show that (a) a hard carbon cap is necessary for stimulating a zero-emission offshore energy system; (b) offshore wind integration and power from shore can more than halve current emissions, but OEHs with storage are necessary for zero-emission production and (c) at certain CO$_2$ tax levels, the system with OEHs can potentially reduce CO$_2$ emissions by $50\%$ and energy losses by $10\%$, compared to a system with only offshore renewables, gas turbines and power from shore.

clean offshore energy hub sensitivity analysis deterministic mixed-integer linear programming model

§ INTRODUCTION

In 2020, the European Green Deal lifted the EU climate target aiming to reduce greenhouse gas emissions by at least $55\%$ by $2030$ and become the world's first carbon-neutral continent by $2050$ [European Commission, 2020]. The North Sea region may play a central role in this with a capacity target of $300$ GW for offshore wind power by $2050$ [European Commission, 2020]. Offshore Energy Hubs (OEHs) and the hub-and-spoke concept offer a transnational and cross-sector solution for better harnessing offshore wind and integration with the rest of the energy system [North Sea ind Power Hub Programme, 2021]. An energy hub is a physical energy connection point with energy storage where multiple energy carriers can be converted, conditioned [Geidl et al., 2007].

Using OEHs to effectively exploit offshore wind power to decarbonise the Norwegian Continental Shelf (NCS) energy system may contribute to meeting Norway's and Europe's climate targets. Norway was the world's third-largest exporter of natural gas in 2019 [Looney, 2020]. Offshore oil and gas extraction was responsible for $26.6\%$ ($13.3$ Mt CO$_2$ equivalent) of the total Norwegian greenhouse gases in $2020$ [Statistics Norway, 2020]. Norway steps up its climate goal to reduce emissions by $50\%$ – $55\%$ by $2030$ compared to $1990$ levels [Ministry of Climate and Environment, 2020].

This paper investigates the role of clean OEHs in reducing CO$_2$ emissions from the NCS with a focus on NCS oil and gas fields. Figure <ref> visualises a potential NCS energy system with OEHs, while Figure <ref> illustrates the functioning of an OEH. We only consider producing green hydrogen from electrolysis. A likely first use of OEHs may be to provide decarbonised energy supply for offshore oil and gas recovery, maritime cargo transport, and offshore farming [Mikkola et al., 2018]. In the longer term, the development of OEHs may also facilitate the transition to decarbonised continental energy export. After the lifetimes of oil and gas fields, OEHs may produce and export green hydrogen and connect with a North Sea offshore grid [Singlitico et al., 2021, Gorenstein Dedecca & Hakvoort, 2016, North Sea ind Power Hub Programme, 2021, Dincer et al., 2021].

Illustration of the NCS energy system with energy hubs. L1 – L5 (dotted lines) are representative HVAC cables, while L6 – L10 (solid lines) are HVDC cables. Black dots represent energy hubs and the red dots represent the onshore buses they connect to. Points with different shapes and colours represent NCS oil and gas fields.

Conceptual illustration of OEHs.

We develop a multi-carrier Mixed-Integer Linear Programming (MILP) model for investment planning optimisation with a high degree of operational details to cope with the short-term characteristics of the problem. We consider investment in clean technologies, including offshore wind, offshore solar, OEHs and Power From Shore (PFS). Offshore oil and gas platforms involve multiple energy carriers, including electricity, heat, gas, oil and water. More potential energy carriers such as hydrogen and ammonia may become critical during the offshore energy system transition [Gea Bermúdez et al., 2021]. The operational model has an hourly time resolution. It is crucial to consider hourly operational decisions in a system with higher penetration of intermittent renewable energy to systematically assess the trade-off between long-term investment decisions and short-term operational decisions [Lara et al., 2018]. This is even more critical to be an acceptable alternative to current fossil-based energy provision for installations on the NCS where the security of supply is an absolute requirement. We use a bottom-up approach and model energy demand for devices at oil and gas platforms with hourly time resolution. In this way, we can utilise the potential shifting of loads, such as for water injection, to balance volatile renewable energy better.

We conduct sensitivity analysis on CO$_2$ tax, CO$_2$ budget and the capacity of PFS. Based on the results, we address the value of OEHs in terms of system costs, energy losses and emissions.

The contributions of the paper are: (1) an integrated investment and operational model is developed with the following features: (a) OEHs are modelled for a large-scale offshore energy system and (b) the hourly device-level energy consumption of platforms is modelled, and (2) we analyse the value of OEHs in the North Sea offshore energy system transition towards zero-emission energy supply.

The outline of the paper is as follows: Section <ref> introduces the background regarding offshore energy system decarbonisation and offshore oil and gas production. Section <ref> gives the problem description followed by modelling strategies and assumption. Section <ref> presents the model for the case study. Section <ref> states the results and analysis of the case study. Section <ref> concludes the paper and suggests further research.

§ BACKGROUND

This paper tries to find the optimal system design of the NCS energy system for decarbonisation for a future point in time. We consider several decarbonisation solutions, including offshore wind, offshore solar, OEHs and PFS. As our focus is on the NCS, special attention is made to oil and gas recovery. In the following, we present background knowledge on offshore energy system decarbonisation approaches and the production process of offshore platforms.

§.§ Offshore energy system decarbonisation

CO$_2$ tax is an important instrument for stimulating offshore energy system decarbonisation. In 2021, the tax is about $55$ €/tonne in Norway with an ambition to increase it to $200$ €/tonne by 2030.

In addition, the EU Emissions Trading System is a “cap and trade” system that also includes the emissions on the NCS [Norwegian Petroleum Directorate, 2020]. Carbon tax and the emissions trading system make a total carbon price of approximately $100$ €/tonne. In this context, oil and gas companies are undertaking considerable investments in decarbonisation solutions to address climate goals, such as PFS and offshore wind. Oil and gas companies on the NCS have set climate targets. For example, Equinor and Vår Energy aim to reduce greenhouse gas emission by $40\%$ by $2030$ and near zero emission by $2050$ [Equinor, 2021, Vår-Energi, 2019].

There are four general approaches to reduce offshore CO$_2$ emissions. The first approach is to reduce CO$_2$ emissions by improving reservoir drainage and processing energy efficiency. Water injection and gas injection are common reservoir drainage strategies used on the NCS.

Pumping, compression and separation are major processes for handling produced fluids in a processing system. Injection and processing account for more than half of the power consumption at the fields on the NCS.

Secondly, it is possible to increase the energy efficiency of gas turbines. Due to security of supply requirements, gas turbines usually operate with a margin, which leads to a low efficiency of around $33\%$ [Lindegaard et al., 2014]. Adding bottoming cycles to the existing gas turbines can improve their energy efficiency.  However, unlike an onshore energy system, weight and space limitation of an offshore installation restrict extra devices like a bottoming cycle.

The third approach is to supply zero emission or low emission energy to offshore oil and gas platforms. This includes PFS, switching fuel from natural gas to ammonia or hydrogen and connecting offshore wind farms to platforms.

In the past years, several offshore fields have received PFS via HVDC/HVAC cables  [Riboldi & Nord, 2017, Norwegian Petroleum Directorate, 2020].

The cost of abating CO$_2$ emissions by taking PFS can vary from less than $100$ to almost $800$ €/tonne [Norwegian Petroleum Directorate, 2020].

Many abatement projects bringing PFS, are in their planning phase highly unprofitable even considering Norway's plan to increase CO$_2$ tax to $200$ €/tonne in $2030$. Besides, due to the capacity limits of the onshore system, the available power is limited in some cases.

Offshore wind is another technology to supply clean power to platforms. Equinor's Hywind Tampen project aims to be operational by $2022$ [Equinor, 2020]. The combination of an offshore platform with a wind farm represents a potential good match for the offshore petroleum sector’s desire for renewable energy with the offshore wind power industry’s desire for an early market [Svendsen et al., 2011]. The stability and control issues for an isolated offshore energy system consisting of a wind farm and five platforms were addressed in [Svendsen et al., 2011].

Integrating large wind turbines into a stand-alone platform is theoretically possible but requires more operational and economic work to prove its feasibility [He et al., 2010]. In [Marvik et al., 2013], a simulation model that connects a petroleum installation cluster in the Norwegian part of the North Sea to the Norwegian power grid via HVDC cables was presented. They found that local wind power production matching the offshore power demand improves both voltage- and frequency-stability in offshore system.

A framework for determining optimal offshore grid structures for wind power integration and power exchange named Net-Op was presented in [Trötscher & Korpås, 2011]. The authors used a MILP model of the power grid to find optimal grid expansions for connecting offshore wind farms and increasing the transnational power exchange capacity in Europe. They pointed out that a capacity expansion model needs to include a power market model to capture the improved operation of the power system because of the transmission capacity expansion.

An extension of Net-Op that takes into account investment cost, variability of wind/demand/power prices, and the benefit of power trade between countries/price areas was presented in [Svendsen, 2013].

The fourth approach is to deploy carbon capture and storage. Storing CO$_2$ in stable underground formations, e.g., old and stable oil reservoirs, has a relatively long history. Since $1996$, nearly one million tonnes of CO$_2$ per year have been separated during the natural gas process from the Sleipner Vest field and stored in the Utsira formation [Norwegian Petroleum Directorate, 2020].

The first two approaches have a limited impact on emission reduction. Whereas the third and fourth approaches can give up to 100% reduction.

The use of OEHs may allow for better harnessing offshore wind to supply more stable energy to platforms in the short run and export clean energy to the continent in the long run. Using energy hubs for coping with wind power volatility was discussed in [Zare Oskouei et al., 2021]. Energy hubs can become sustainable pillars by improving cost-competitiveness of wind power [Thommessen et al., 2021].

In addition, the energy hub concept was also used to increase the energy flexibility in buildings [Ottesen & Tomasgard, 2015] and electricity markets [Ødegaard Ottesen et al., 2016]. Techno-economic analysis of offshore energy islands was presented in [van der Veer et al., 2019]. A separate offshore bidding zone can lead to a more efficient offshore energy system with OEHs [Kitzing & González, 2020]. An offshore artificial power-to-gas island can produce and transport hydrogen through natural gas pipelines [Gondal, 2019]. Energy-hub-based electricity system design for an offshore platform considering CO$_2$ mitigation was presented in [Zhang et al., 2017]. We refer the readers to [Mohammadi et al., 2018, Mohammadi et al., 2017] for comprehensive reviews of the research works on energy hubs.

There is a lack of research on the environmental value of OEHs. Besides research, real-world OEH projects will be carried out [Danish Energy Agency, 2021, North Sea ind Power Hub Programme, 2021, NEPTUNE ENERGY, 2019].

§.§ Offshore oil and gas fields

Offshore oil and gas field development and production involve multibillion-dollar investments and profits [Bret-Rouzaut et al., 2011]. A large number of studies have been carried out for offshore oil and gas fields production optimisation [Gunnerud et al., 2012] and planning of offshore oil and gas infrastructures [Gupta & Grossmann, 2012, Tarhan et al., 2009]. From the studies, we can see that platforms and fields vary a lot due to, amongst others, geological characteristics, reserves, and remaining lifetimes.

In the following, we present a typical composition and production process of NCS platforms.

A North Sea oil field normally consists of topside structures and subsea production systems. A topside structure typically consists of a processing plant, a utility plant, drilling facilities, and a living quarter [Voldsund, 2014], see Figure <ref>. The production plant receives and processes well streams. A visualisation of the production process is presented in  Figure <ref>. Major energy consumption takes place in the production plants. The energy demand of production plants is conventionally fulfilled by gas turbines located in the utility plant. In $2014$, gas turbines with waste heat recovery units covered approximately $90\%$ of all heat demand for operations on the NCS [Mazzetti et al., 2014].

Schematic of a topside structure of a typical North Sea oil and gas platform, adapted from [Voldsund, 2014].

Schematic of a potential decarbonised offshore field production process. A three-stage separator train separates well streams into produced water, oil, condensate and gas. Typically the first stage separator takes out most of the water and gas at arrival conditions. Fuel gas is taken from the first stage separator. The residual mix of oil, gas and water is heated before entering the second stage separator. Produced water is purified and discharged, and in some cases, reinjected into water injection wells to maintain reservoir pressure. Water lift pumps will lift seawater for reinjection if needed. Produced oil is pressurised by pumps and exported. Produced gas is used as fuel gas, compressed and exported, reinjected via dedicated wells for enhanced oil recovery or injected into the same wells for gas lift.

The grey dotted box includes the potential processes for decarbonisation. See Figure <ref> for a visualisation of the processes in an OEH.

§ PROBLEM DESCRIPTION

First, this section introduces the proposed offshore energy system planning problem with OEHs. Then we present the time and geographical structures with the aim of reducing computational time of a potentially big problem. Finally, we give the modelling assumptions.

The problem under consideration aims to make optimal investment and operational decisions for the NCS energy system with OEHs, based on the energy demand captured by the operational model. By solving such a problem, we aim to find out under what conditions OEHs may benefit the system and how OEHs operate with the rest of the system.

To model hourly energy demand, we consider: (a) separators; (b) pumps: water injection pumps, water lift pumps, oil export pumps; (c) compressors: gas injection compressors and gas export compressors. These devices have existing capacities, and no investment will be made in them. Moreover, we assume that device efficiency, flow inlet/outlet pressures and hourly mass flow are given.

For the investments in decarbonisation solutions, we consider: (a) offshore renewable energies (offshore wind and offshore solar); (b) OEHs (electrolysers, hydrogen storage facilities and fuel cells); (c) subsea cables (HVAC, HVDC and offshore and onshore converter stations); (d) electric boilers; (e) platform located batteries. The capital expenditures, fixed operational costs are assumed to be known.

The problem is to determine (a) capacities of decarbonisation technologies and (b) operational strategies that include scheduling of generators, storage and approximate power flow among regions to meet the energy demand with minimum overall investment, operational and environmental costs.

§.§ Modelling strategies and assumptions

A multilevel control hierarchy was defined in [Foss et al., 2018], and they argued that the repetitive use of static models could solve many important petroleum production optimisation problems. We develop a multi-period MILP model for an integrated investment planning and operational problem that combines short-term and long-term control hierarchies. We use aggregation, clustering and time sampling [Backe et al., 2021] to address the multi-time-scale aspects [Kaut et al., 2014] and solve a large-scale instance.

§.§.§ Time structure of the problem

The investment problem is optimised over a long-term horizon, e.g., a few decades. The operational problem is optimised on an hourly basis based on investment decisions. To combine these two control hierarchies without increasing much the computational time, we select $S$ representative slices, each containing $h$ hours, and scale them up to represent a whole operational year. A visualisation of the time structure is presented in Figure <ref>.

Illustration of combined hierarchies, (adapted from [van der Heijde et al., 2019]).

§.§.§ Geographical structure of the problem

The problem potentially consists of many regions,  and we implement a k-means cluster method based on the locations of fields to reduce the problem size. We have two considerations when deciding the number of clusters. Firstly, we assume the OEH connects the surrounding fields via HVAC cables; thus, only fields with a feasible transmission distance (up to $100$ km) are considered. Secondly, we assume that the cluster centres are the locations for OEHs. We prevent clusters with too few fields. For each cluster, we aggregate the individual fields into one bigger field with a distance to the OEH equal to the average distance of the individual fields and connect fields to OEH in hub-and-spoke form. Currently, we do not consider the interconnection among fields and clusters, resulting in reasonably simple network topology, exemplified in Figure <ref>.

§.§.§ Assumptions

We assume that each platform is a typical North Sea platform with production processes as presented in Figure <ref>. The energy consumption of pumps, compressors and separators can be formulated as a function of flow rate, pressure and temperature. For simplicity, we assume the pressure levels and temperatures to take values that are typical on the North Sea, leading to a linear formulation. Kirchhoff voltage law is omitted, and the model is an energy flow model. We assume no mass loss during production.

§ MATHEMATICAL MODEL

This section presents a deterministic MILP formulation for the multi-carrier energy system investment planning problem with high a degree of operational details. The complete nomenclature of the model can be found in <ref>. The model includes a long-term investment planning horizon and a short-term operational horizon. The integrated investment planning and operational model is partially based upon the LP model developed in [Mazzi et al., 2020]. To improve the representation of the fixed capacity independent investment costs, we make use of integer variables. The planner is dedicated to the NCS energy system.

§.§ Objective function

[A]\(\mathcal{P}\)set of technologies
[A]\(\mathcal{L}\)set of transmission branches
[A]\(\mathcal{Z}\)set of all locations, including platforms, OEH, and onshore buses ($\mathcal{Z}^{P}  \cup \mathcal{Z}^{H}  \cup \mathcal{Z}^{O}$)
[A]\(\mathcal{I}\)set of operational years
[A]\(\mathcal{I}_{0}\)set of investment periods
[A]\(\mathcal{I}_{i}\)set of investment periods $i$ $(i \in \mathcal{I}_{0})$ ancestor to operational year $i$ $(i \in \mathcal{I})$

[C]\(C^{PInv}_{pi}\)unitary investment cost of technology $p$ at investment period $i$ ($p \in \mathcal{P}, i \in \mathcal{I}_{0}$) [kr/MW, kr/MWh, kr/kg]
[C]\(C^{PFix}_{p}\)unitary fix operational and maintenance cost of technology $p$ ($p \in \mathcal{P}$) [kr/MW, kr/MWh, kr/kg]
[C]\(C^{PFInv}_{pi}\)fixed capacity independent investment cost of technology $p$ at investment period $i$ ($p \in \mathcal{P}, i \in \mathcal{I}_{0}$) [kr]
[C]\(C^{LInv}_{li}\)unitary investment cost of branch $l$ at investment period $i$ ($l \in \mathcal{L}, i \in \mathcal{I}_{0}$) [kr/MW/km]
[C]\(C^{LFix}_{l}\)unitary fix operational and maintenance cost of branch $l$ ($l \in \mathcal{L}$) [kr/MW/km]
[C]\(C^{LFInv}_{li}\)fixed capacity independent investment cost of branch $l$ at investment period $i$ ($l \in \mathcal{L}, i \in \mathcal{I}_{0}$) [kr]
[C]\(C^{LLInv}_{li}\)unitary investment cost of branch $l$ at investment period $i$ ($l \in \mathcal{L}, i \in \mathcal{I}_{0}$) [kr/km]
[C]\(X^{PHist}_{pz}\)historical capacity of technology $p$ in location $z$ ($p \in \mathcal{P}, z \in \mathcal{Z}$) [MW, MWh, kg]
[C]\(X^{PMax}_{pz}\)maximum capacity of technology $p$ in location $z$ ($p \in \mathcal{P}, z \in \mathcal{Z}$) [MW, MWh, kg]
[C]\(M^{P}_{p}\)maximum capacity of a unit of technology $p$ in location $z$ ($p \in \mathcal{P}, z \in \mathcal{Z}$) [MW, MWh, kg]
[C]\(\bar{N}_{pz}\)maximum number of units of technology $p$ in location $z$ ($p \in \mathcal{P}, z \in \mathcal{Z}$)
[C]\(X^{Hist}_{l}\)historical capacity of branch $l$ ($l \in \mathcal{L}$) [MW]
[C]\(X^{Lgth}_{l}\)length of branch $l$ ($l \in \mathcal{L}$) [km]
[C]\(X^{LMax}_{l}\)maximum capacity of branch $l$ ($l \in \mathcal{L}$) [MW]
[C]\(M^{L}_{l}\)maximum capacity of a single cable of branch $l$ ($l \in \mathcal{L}$) [MW]
[C]\(\bar{N}_{l}\)maximum number cables of branch $l$ ($l \in \mathcal{L}$)
[C]\(\kappa\)scaling effect depending on the number of operation years between investment periods

[E]\(x_{pzi}^{PAcc}\)accumulated capacity of device $p$ in location $z$ in operational year $i$ ($p \in \mathcal{P}, z \in \mathcal{Z}, i \in \mathcal{I}$) [MW, MWh, kg]
[E]\(x_{li}^{PAcc}\)accumulated capacity of branch $l$ in operational year $i$ ($l \in \mathcal{L}, i \in \mathcal{I}$) [MW]
[E]\(x_{pzi}^{PInst}\)newly installed capacity of device $p$ in location $z$ at investment period $i_{0}$ ($p \in \mathcal{P}, z \in \mathcal{Z}, i \in \mathcal{I}_{0}$) [MW, MWh, kg]
[E]\(x_{li}^{LInst}\)newly installed capacity of branch $l$ at investment period $i_{0}$ ($l \in \mathcal{L}, i \in \mathcal{I}_{0}$) [MW]
[E]\(\gamma_{pzi}^{PInst}\)number of newly installed device $p$ in location $z$ at investment period $i_{0}$ ($p \in \mathcal{P}, z \in \mathcal{Z}, i \in \mathcal{I}_{0}$)
[E]\(\gamma_{li}^{LInst}\)number of newly installed cables in branch $l$ at investment period $i_{0}$ ($l \in \mathcal{L}, i \in \mathcal{I}_{0}$)

[G]\(g(\cdot,\cdot)\)operational cost in operational year $i$ (kr)

\begin{equation}
\min f(\mathbf{x})+\kappa\sum_{i \in \mathcal{I}}g(x_{i}, c_{i})\label{MP_objective}
\end{equation}

The objective function Equation (<ref>) is to minimise the total investment ($f(\mathbf{x})$) and operational $(\kappa\sum_{i \in \mathcal{I}}g(x_{i}, c_{i}))$ costs over the planning horizon. The operational cost $g(x_i, c_i)$ is described in Section <ref>, where $x_{i}$ and $c_{i}$ are vectors containing capacities and costs information respectively.

§.§ Investment planning constraints

∑_i∈ℐ_0(∑_p ∈𝒫 ∑_z ∈𝒵(C^PInv_pix^PInst_pzi+C^PFInv_piγ^P_pzi).

f(x) =.+∑_l ∈ℒ(C^LInv_liX^Lgth_lx^LInst_li+(C^LLInv_liX^Lgth_l+C^LFinv_li)γ^L_li) )

f(x) =  +κ∑_i ∈ℐ(∑_p ∈𝒫∑_z ∈𝒵C^PFix_pix^PAcc_pzi+∑_l ∈ℒ C^LFix_liX^Lgth_lx^LAcc_li)

x^PAcc_pzi=X^PHist_pz+∑_i ∈ℐ_ix^PInst_pzi,    abcdefghijkl  p ∈𝒫,  z ∈𝒵,  i ∈ℐ

x^LAcc_li=X^LHist_l+∑_i ∈ℐ_ix_li^LInst,       l ∈ℒ,  i ∈ℐ

0 ≤x^PInst_pzi ≤γ^P_pziM^P_p,     p ∈𝒫,  z ∈𝒵,  i ∈ℐ_0

0 ≤x^LInst_li ≤γ^L_lziM^L_l,     l ∈ℒ, i ∈ℐ_0

0 ≤x^PAcc_pzi ≤X^PMax_pz,        p ∈𝒫,  z ∈𝒵,  i ∈ℐ

0 ≤x^LAcc_li ≤X^LMax_l,      l ∈ℒ,  i ∈ℐ

∑_i ∈ℐ_0γ^P_pzi ∈{0, 1, 2, ... , N̅_pz},   p ∈𝒫,  z ∈𝒵

∑_i ∈ℐ_0γ^L_li ∈{0, 1, 2, ... , N̅_l},   l ∈ℒ

x^PInst_pzi, x^LInst_li, x^PAcc_pzi, x^LAcc_li ∈ℝ^+_0,

γ^P_pzi, γ^L_li ∈ℤ^+_0.

Equation (<ref>) calculates the total capacity dependent investment costs, fixed operating and maintenance costs and fixed capacity independent investment costs. We define $\mathbf{x}$ to be a vector collecting available capacities of all technologies $(\mathcal{P})$ and branches $(\mathcal{L})$ for all operational years $(\mathcal{I})$.  Constraints (<ref>) and (<ref>) represent that the available capacity of a technology $(x^{PAcc}_{pzi})$ or a branch $(x^{LAcc}_{li})$ at an operational year equals to its historical capacity $(X^{PHist}_{pz} \text{ or } X^{LHist}_{l})$ and the sum of newly invested capacities $(x^{PInst}_{pzi} \text{ or } x^{LInst}_{li})$ in its ancestor investment periods $( \mathcal{I}_{i})$. An integer variable $\gamma^{P}_{pzi}$ gives the number of units of technology $p\in \mathcal{P}$, in location $z \in \mathcal{Z}$, in investment period $i \in \mathcal{I}_0$. We define $\gamma^{L}_{li}$ to be the number of cables of branch $l \in \mathcal{L}$. Parameters $M^{P}_{p}$ and $M^{L}_{l}$ state the maximum capacity of a technology unit and a cable, respectively. Parameters $\bar{N}_{pzi}$ and $\bar{N}_{li}$ give the maximum number of units that can be installed of technologies and cables, respectively.

§.§ Generic operational constraints

[B]\(\mathcal{S}^{T}\)set of time slices
[B]\(\mathcal{T}\)set of hours in all time slices

[B]\(\mathcal{L}\)set of subsea cables
[B]\(\mathcal{G}\)set of gas turbines
[B]\(\mathcal{R}\)set of renewable units (offshore wind, offshore solar)
[B]\(\mathcal{S}^{E}\)set of electricity storage

[B]\(\mathcal{D}^{Sep}\)set of separators
[B]\(\mathcal{B}^{E}\)set of electric boilers

[B]\(\mathcal{C}^{Exp}\)set of natural gas export compressors
[B]\(\mathcal{C}^{Inj}\)set of natural gas injection compressors

[B]\(\mathcal{P}^{O}\)set of oil export pumps

[B]\(\mathcal{P}^{WI}\)set of water injection pumps
[B]\(\mathcal{P}^{WL}\)set of seawater lift pumps

[B]\(\mathcal{E}\)set of electrolysers
[B]\(\mathcal{F}\)set of fuel cells
[B]\(\mathcal{S}^{Hy}\)set of hydrogen storage facilities

[D]\(C^{\text{CO}_{2}}\)CO$_2$ tax (kr/t)
[D]\(\mu^{E}\)yearly CO$_2$ emission limit (t)

[D]\(H_t\)number of hour(s) in one operational period $t$
[D]\(W^{S}_{s}\)weight of time slice $s$

[D]\(P_{gz}^{accG}\)accumulated capacity of gas turbine $g$ on platform $z$ ($g \in \mathcal{G}, z \in \mathcal{Z}^{P}$) [MW]
[D]\(G^{GR}_{g}\)ramping limit of gas turbine $g$ ($g \in \mathcal{G}$) [MW/MW]
[D]\(R_{rt}^{R}\)renewable generation profile of renewable unit $r$ in period $t$ ($r \in \mathcal{R}, t \in \mathcal{T}$)
[D]\(\eta_{g}^G\)efficiency of gas turbine $g$ ($g \in \mathcal{G}$)
[D]\(\eta_{l}^L\)efficiency of transmission line $l$ ($l \in \mathcal{L}$)
[D]\(\eta_{s}^{SE}\)efficiency of electricity store $s$ ($s \in \mathcal{S}^{E}$)
[D]\(H_{s}^{SE}\)power ratio of electricity store $s$ ($s \in \mathcal{S}^{E}$) [MW/MWh]
[D]\(A^{E}_{zl}\)elements of bus/line incidence matrix $A^{E}$ ($z \in \mathcal{Z}, l \in \mathcal{L}$)
[D]\(\kappa^{PWI}_{p}\)electricity demand as fraction of amount of water injected by pump $p$ ($p \in \mathcal{P}^{WI}$) [MW/kg]
[D]\(\kappa^{PWL}_{p}\)electricity demand as fraction of amount of water lifted by pump $p$ ($p \in \mathcal{P}^{WL}$) [MW/kg]
[D]\(\kappa^{PO}_{p}\)electricity demand as fraction of amount of oil pumped by pump $p$ ($p \in \mathcal{P}^{O}$) [MW/kg]
[D]\(E_{g}^{Fuel}\)emission of CO$_2$ of gas turbine $g$ burning fuel ($g \in \mathcal{G}$) [t/MWh]
[D]\(C_g^{Fuel}\)fuel cost of gas turbine $g$ burning fuel with energy content 1 MWh ($g \in \mathcal{G}$) [kr/MWh]
[D]\(C_g^G\)variable operational cost of generating 1 MW power from gas turbine $g$ ($g \in \mathcal{G}$) [kr/MW]
[D]\(C^{LShed}\)load shed penalty cost [kr/MW]
[D]\(C^{GShed}\)generation shed cost [kr/MW]
[D]\(C^{ZO}_{zt}\)electricity price in onshore bus $z$ in period $t$ ($z \in \mathcal{Z}^{O}, t \in \mathcal{T} $) [kr/MW]
[D]\(\sigma^{Res}_{z}\)spinning reserve factor on platform $z$ ($z \in \mathcal{Z}^{P}$)

[D]\(P^{accSEP}_{dz}\)capacity of separator $d^{sep}$ on platform $z$ ($d \in \mathcal{D}^{Sep}, z \in \mathcal{Z}^{P}$) [MW]
[D]\(C^{HLShed}\)heat load shed penalty cost [kr/MW]
[D]\(\eta^{Gh}_{g}\)heat recovery efficiency of gas turbine $g$ ($g \in \mathcal{G}$)
[D]\(\eta^{EB}_{b}\)efficiency of electric boiler ($b \in \mathcal{B}^{E}$)
[D]\(\rho^{SEP}_{d}\)heat demand as fraction of amount of oil heated during separation ($d \in D^{Sep}$) [MW/kg]

[D]\(V_{zt}^{NGE}\)natural gas exporting level on platform $z$ in period $t$ ($z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [kg]
[D]\(V_{zt}^{NGI}\)natural gas injection level on platform $z$ in period $t$ ($z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [kg]
[D]\(P^{accCExp}_{cz}\)capacity of compressor $c$ for exporting on platform $z$ ($c \in \mathcal{C}^{Exp}$, $z \in \mathcal{Z}^{P}$) [MW]
[D]\(P^{accCInj}_{cz}\)capacity of compressor $c$ for injection on platform $z$ ($c \in \mathcal{C}^{Inj}$, $z \in \mathcal{Z}^{P}$) [MW]
[D]\(\gamma_{c}\)compression ratio of compressor $c$ ($c \in \mathcal{C}^{Inj}  \cup \mathcal{C}^{Exp}$)

[D]\(\alpha\)polytropic exponent of compressor
[D]\(\eta^{C}\)efficiency of compressor

[D]\(P_{pz}^{accPO}\)capacity of oil pump $p$ on platform $z$ ($p \in \mathcal{P}^{O}$, $z \in \mathcal{Z}^{P}$) [MW]
[D]\(V^{OD}_{zt}\)oil exporting level on platform $z$ in period $t$ ($z \in \mathcal{Z}^{P}$, $t \in \mathcal{T}$) [kg]

[D]\(P_{pz}^{accWL}\)capacity of water lift pump $p$ on platform $z$ ($p \in \mathcal{P}^{WI}$, $z \in \mathcal{Z}^{P}$) [MW]
[D]\(P_{pz}^{accWI}\)capacity of water injection pump $p$ on platform $z$ ($p \in \mathcal{P}^{WL}$, $z \in \mathcal{Z}^{P}$) [MW]
[D]\(V^{WI}_{zt}\)water injection level on platform $z$ in period $t$  ($z \in \mathcal{Z}^{P}$, $t \in \mathcal{T}$) [kg]
[D]\(V^{WL}_{zt}\)water lift level on platform $z$ in period $t$  ($z \in \mathcal{Z}^{P}$, $t \in \mathcal{T}$) [kg]
[D]\(V^{WB}_{zt}\)produced water on platform $z$ in period $t$  ($z \in \mathcal{Z}^{P}$, $t \in \mathcal{T}$) [kg]

[D]\(F^{FR}_{f}\)ramping limit of fuel cell $f$ ($f \in \mathcal{F}$) [MW/MW]
[D]\(\eta^{ES}\)conversion factor of electrolyser to inject hydrogen to the storage facility [MWh/kg]
[D]\(\eta^{EF}\)conversion factor of electrolyser to inject hydrogen directly to fuel cell [MWh/kg]
[D]\(\eta^{F}_{f}\)efficiency of fuel cell $f$ ($f \in \mathcal{F}$)
[D]\(\theta^{Hy}\)hydrogen energy content [MWh/kg]

[F]\(q_{sz}^{accSE}\)accumulated storage capacity of electricity store $s$ on platform $z$ ($s \in \mathcal{S}^{E}, z \in \mathcal{Z}^{P}$) [MWh]
[F]\(p_{z^{o}}^{accZO}\)accumulated capacity of shore bus $z$ ($z \in \mathcal{Z}^{O}$) [MW]
[F]\(p_{l}^{accL}\)accumulated capacity of line $l$ ($l \in \mathcal{L}$) [MW]
[F]\(p_{zt}^{D}\)total power demand on platform $z$ in period $t$ ($z \in \mathcal{Z}^{P}$) [MW]
[F]\(p_{gzt}^{G}\)power output of gas turbine $g$ on platform $z$ in period $t$ ($g \in \mathcal{G}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(p_{gzt}^{ResG}\)power reserved of gas turbine $g$ for spinning reserve requirement on platform $z^{p}$ in period $t$ ($g \in \mathcal{G}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(p_{rzt}^{R}\)power output of renewable unit $r$ near energy hub $z$ in period $t$ ($r \in \mathcal{R}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [MW]
[F]\(p_{szt}^{SE+(-)}\)charge (discharge) power of electricity store $s$ on platform $z$ in period $t$ ($s \in \mathcal{S}^{E}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(p_{szt}^{ResSE}\)power reserved in electricity store $s$ for spinning reserve requirement on platform $z$ in period $t$  ($s \in \mathcal{S}^{E}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(q_{szt}^{SE}\)energy level of electricity store $s$ on platform $z$ at the start of period $t$ ($s \in \mathcal{S}^{E}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MWh]
[F]\(p_{zt}^{GShed}\)electricity generation shed at $z$ in period $t$ ($z \in \mathcal{Z}, t \in \mathcal{T}$) [MW]
[F]\(p_{zt}^{LShed}\)electricity load shed on platform $z$ in period $t$ ($z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(p_{lt}^{L}\)power flow in line $l$ in period $t$ ($l \in \mathcal{L}, t \in \mathcal{T}$) [MW]
[F]\(p_{zt}^{ZO}\)power supply from onshore bus $z$ in period $t$ ($z \in \mathcal{Z}^{O}, t \in \mathcal{T}$) [MW]

[F]\(p^{accEB}_{bz}\)accumulated capacity of electric boiler $b$ on platform $z$ ($b \in \mathcal{B}^{E}, z \in \mathcal{Z}^{P}$) [MW]
[F]\(p_{bzt}^{EB}\)electricity input to electric boiler $b$ on platform $z$ in period $t$ ($b \in \mathcal{B}^{E}, z \in \mathcal{Z}^{P},t \in \mathcal{T}$) [MW]
[F]\(p_{dzt}^{HSEP}\)heat demand of separator $d$ on platform $z$ in period $t$ ($d \in \mathcal{D}^{Sep}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW/kg]
[F]\(p_{zt}^{HGShed}\)heat generation shed on platform $z$ in period $t$ ($z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(p_{zt}^{HLShed}\)heat load shed on platform $z$ in period $t$ ($z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]

[F]\(p_{czt}^{CExp}\)power demand of the compressor $c$ on platform $z$ in period $t$ ($c \in \mathcal{C}^{Exp}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(p_{czt}^{CInj}\)power demand of the compressor $c$ on platform $z$ in period $t$ ($c \in \mathcal{C}^{Inj}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]

[F]\(p_{pzt}^{PO}\)oil exporting pump power $p$ power consumption on platform $z$ in period $t$ ($p \in \mathcal{P}^{O}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]

[F]\(p_{pzt}^{PWI}\)water injection pump $p$ power consumption on platform $z$ in period $t$ ($p \in \mathcal{P}^{WI}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]
[F]\(p_{pzt}^{PWL}\)seawater lift pump $p$ power consumption on platform $z$ in period $t$ ($p \in \mathcal{P}^{WL}, z \in \mathcal{Z}^{P}, t \in \mathcal{T}$) [MW]

[F]\(p_{fz}^{accF}\)accumulated capacity of fuel cell $f$ on platform $z$ ($f \in \mathcal{F}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [MW]
[F]\(p_{ez}^{accE}\)accumulated capacity of electrolyser $e$ on platform $z$ ($e \in \mathcal{E}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [MW]
[F]\(v_{sz}^{accSHy}\)accumulated storage capacity of hydrogen store $s$ on platform $z$ ($s \in \mathcal{S}^{Hy}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [kg]
[F]\(p^{F}_{fzt}\)power output of fuel cell $f$ on OEH $z$ in period $t$ ($f \in \mathcal{F}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [MW]
[F]\(p^{E}_{ezt}\)power input of electrolyser $e$ on OEH $z$ in period $t$ ($e \in \mathcal{E}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [MW]

[F]\(v^{SHy+(-)}_{szt}\)injection (withdraw) of hydrogen to (from) hydrogen storage $s$ on OEH $z$ in period $t$ ($s \in \mathcal{S}^{Hy}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [kg]

[F]\(v^{SHy}_{szt}\)storage level of hydrogen storage $s$ on OEH $z$ in period $t$ ($s \in \mathcal{S}^{Hy}, z \in \mathcal{Z}^{H}, t \in \mathcal{T}$) [kg]

We now present the generic operational constraints in one operational year $i$. Note that we omit index $i$ in the operational model for ease of notation.

We model oil and gas recovery as this is the most likely use in the short to medium term. The operational constraints can be modified for other use, e.g., offshore fish farming, maritime, transport, and others. The complete description of operational constraints is presented in <ref>.

g(x,c)=∑_s ∈𝒮^T∑_t ∈𝒯∑_z ∈𝒵W_s(∑_p ∈𝒫C^P_py^P_pzt+C^Shedy^LShed_zt+ .

. ∑_g ∈𝒢C^CO_2ρ^E_gy^G_gzt )

0 ≤y^P_pzt ≤x^PAcc_pz,     p ∈𝒫,  z ∈𝒵, t ∈𝒯

-x^LAcc_l ≤y^L_lt ≤x^LAcc_l,     l ∈ℒ, t ∈𝒯

∑_g ∈𝒢y^G_gt+∑_l ∈ℒA_zly^L_lt+∑_f ∈ℱy^F_fzt+y^LShed_zt+∑_r ∈ℛy^R_rzt =

abc Y^D_zt+∑_s ∈𝒮(y^S+_szt-y^S-_szt)+∑_e ∈ℰy^E_ezt+y^GShed_zt,       z ∈𝒵,t ∈𝒯

l^S_sz(t+1)-l^S_szt=η^S_sy^S+_szt-y^S-_szt,     s ∈𝒮, z ∈𝒵, t ∈𝒯

∑_e ∈ℰy^E_ezt+∑_s ∈𝒮^Hy η^EFv^SHy-_szt =

abc∑_s ∈𝒮^Hyη^ESv^SHy+_szt+∑_f ∈ℱ η^EFy^F_fzt,     z ∈𝒵, t ∈𝒯

∑_s ∈S^T∑_z ∈𝒵∑_t ∈𝒯∑_g ∈𝒢W_sρ^E_gy^G_gzt ≤E^CO_2,

y^P_pz, y^G_gzt, y^F_fzt, y^LShed_zt, y^R_rzt, y^S+_szt, y^S-_szt, y^E_ezt ∈ℝ^+_0,

y^Gshed_zt, L^S_szt, v^SHy-_szt, v^SHy+_szt, x^PAcc_pz, x^LAcc_l ∈ℝ^+_0.

The operational cost function $g(x, c)$, which is included in the objective function Equation (<ref>) for each operational year $i$, is described by Equation (<ref>) that includes total operating costs of all devices, energy load shedding costs and CO$_2$ emissions costs. Vectors $x$ and $c$ contain capacities and costs information, respectively. Constraints (<ref>) and (<ref>) ensure devices $(y^{P}_{pzt})$ and transmission lines $(y^{L}_{lt})$ are within their capacities $(x^{PAcc}_{pz}, x^{LAcc}_{l})$. Constraint (<ref>) gives the energy balance at each region, where $y^{G}_{gt}, y^{F}_{fzt} \text{ and } y^{R}_{rzt}$ are power generation of generators, fuel cells and renewables respectively. Moreover, we define $y^{E}_{ezt}$ to be the power that goes into electrolysers and $l^{S}_{szt}, y^{S+}_{szt}$ and $y^{S-}_{szt}$ represent the storage level, input and output energy of storage facilities. The energy demand $Y^{D}_{zt}$ can be modelled corresponding to the specific sector, such as offshore platforms. The modelling of offshore platforms is described in details in <ref>. Constraint (<ref>) states the storage balance of storage facilities. Constraint (<ref>) gives the hydrogen nodal balance of OEHs, where $v^{SHy-}_{szt}$ and $ v^{SHy+}_{szt}$ are the hydrogen output and input of hydrogen storage facilities. Constraint (<ref>) restricts the total emissions. We only consider emissions from the generators, but the model can easily be extend to include other emissions. The complete MILP problem consists of Equations (<ref>)-(<ref>).

§ RESULTS

We demonstrate the results of a static integrated investment planning and operational problem given by Equations (<ref>)-(<ref>) for a future point in time. The problem consists of $461,108$ continuous variables, $100$ integer variables and $980,013$ constraints. We implemented the model in Julia $1.6.1$ using JuMP [Dunning et al., 2017] and solved with Gurobi $9.1.2$ [Gurobi Optimization, LLC, 2021]. We run the code on a MacBook Pro with 2.4 GHz 8-core Intel Core i9 processor, with 64 GB of RAM, running on macOS 11.6 Big Sur. The Julia code and data for the case study have been made publicly available [Zhang, 2021]. We solve the integrated investment and operational model given by Equations (<ref>)-(<ref>) to conduct sensitivity analysis on CO$_2$ tax, CO$_2$ budget and the capacity of PFS.

§.§ Case study

The case study is carried out on the North Sea part of the NCS, considering $66$ fields. The problem consists of $77$ regions, divided into $66$ fields, $5$ OHEs and $5$ onshore buses. By using the clustering approach described in Section <ref> we find that the system can be represented using $5$ clusters and henceforth go from $77$ regions to $15$ regions. We assume that the power demand of platforms initially is entirely supplied by gas turbines, as only a limited number of platforms receives PFS. We construct four representative months with hourly resolution and scale them up to represent a whole year.

The operational data of oil and gas industry is sensitive and usually not disclosed to the public. Aggregated data such as monthly or yearly production of petroleum on the NCS can be obtained from [Norwegian Petroleum Directorate, 2020]. One can also find monthly production and injection data for each field from [Norwegian Oil and Gas Association, 2021] and [Norwegian Petroleum Directorate, 2021]. Neither of these can be directly used as inputs for this study due to the time resolution difference.

Therefore, reasonable data generation is necessary. Raw data is collected from (a) Norne ($1998-2006$) and Volve ($2008-2016$) fields with hourly production and injection data from [The LowEmission Research Centre, SINTEF, 2020] and (b) monthly production and injection data of each field from [Norwegian Oil and Gas Association, 2021]. We come up with a data generation method that considers the lifetimes of offshore fields
[Zhang, 2021].

We define a base case (Base) with offshore renewables, electric boiler, battery and PFS as investment options. We then use this case as a benchmark to check against the case with OEHs. The full model given by Equations (<ref>)-(<ref>) takes approximately 2 hours.

§.§ Energy system analysis

Power consumption and supply (Only two lines are observable since power supply and demand match exactly. OCGT power equals power demand at all times).

Heat consumption and supply.

In this section, we present results on energy consumption and CO$_2$ emission of the initial system. By post-processing, we verify that the energy consumption of platforms is of the same order of magnitude as reported numbers. The resulted CO$_2$ emission is $5.51$ Mt/yr. In comparison, the reported total emission of the relevant fields was $6.89$ Mt in $2019$ [Norwegian Petroleum Directorate, 2021]. We expect emissions from the model to be lower than $6.89$ Mt since not all emission sources are considered. Based on [Nguyen et al., 2014], one could assume that the major processes considered in this study make up about $80\%$ of the total load. Therefore, $5.51$ Mt yearly emission is within the correct range, implying that the energy load modelling is relatively accurate.

Power demand in a year.

cluster1     cluster2     cluster3     cluster4     cluster5
Emission distribution     7.2%        5.5%        44.7%      11.2%       31.3%

Emission distribution by cluster.

From Figure <ref>, we can see that the power output of the Open Cycle Gas Turbine (OCGT) matches power demand at every operational period. Heat recovery of OCGTs is assumed the only heat source. Figure <ref> shows that heat recovery of OCGTs provides more than enough heat due to high electricity generation.

We can also see that energy consumption can vary significantly. A breakdown of electricity load is shown in Figure <ref>, gas export compressors dominate the power consumption in clusters $3-5$. Water injection is the largest power consumer in cluster $2$ since there are some mature fields (e.g., Ekofisk) whose reservoir pressures are mainly maintained by water injection. OCGT is the only energy and emission source in the initial setup. Therefore, emission breakdown includes the emissions from the total energy consumption of each region. Cluster $1$ has the second smallest share of the total energy consumption, with a considerable amount of power consumed by gas injection. The fields in cluster $1$, such as Grane, have the third-highest gas injection level among the $66$ fields. From Table <ref>, we find that emission mainly comes from the northern part of the North Sea.

§.§ Sensitivity analysis of  tax

This section presents the results of sensitivity analysis of CO$_2$ tax. We introduce CO$_2$ tax and still keep the carbon budget inactive. We increase the carbon tax from $55$ to $500$ €/tonne with a step size of $5$ €/tonne. PFS capacity limits are estimated from [Norwegian Petroleum Directorate, 2020, Statistics Norway, 2020].  Note that the cost of PFS may be underestimated since we only consider the costs of subsea cables, onshore and offshore converter stations and electricity bills. In reality, PFS projects may also face investment in onshore transmission lines or onshore power system capacity expansion.

We analyse the results from two metrics, CO$_2$ emission and energy loss. Energy losses are from conversions, transmission, and generation shed. The calculation is presented in <ref>.

Emission and cost comparison (CO$_2$ tax sensitivity analysis).

Energy loss (CO$_2$ tax sensitivity analysis).

Capacities of technologies in each cluster (CO$_2$ tax sensitivity analysis),  hydrogen storage is measured in tonne.

From Figure <ref>, we can see that CO$_2$ tax as a single instrument may not be enough to motivate a zero emission system. We also find that near zero emission can be achieved with a very high CO$_2$ tax. Therefore, a hard carbon cap may be necessary for stimulating a zero emission system. When CO$_2$ tax is $55$ €/tonne, the system reduces about 64% of the initial emissions. Approximately $84\%$ of the emissions can be cut if CO$_2$ tax is increased to $200$ €/tonne as planned. As OCGTs are replaced by renewable energy, energy loss is reduced as well. OEHs can potentially reduce up to around $50\%$ more CO$_2$ emission and $5\%$ total cost than the case with only offshore wind and PFS (Base) at certain CO$_2$ tax levels. From Figure <ref>, we find that energy loss during production counts for $11\%$ of the energy loss. OCGTs lose $18$ GWh of energy during an operational year.
As production from wind turbines replaces gas turbines, energy loss from OCGT is reduced. However, due to the lack of energy storage, electricity generation shedding increases because wind power is shed. We find that OEHs can effectively reduce electricity generation shedding although, it loses energy during conversion. Overall, energy loss is up to $10\%$ lower in the case of OEHs compared with Base at certain tax levels.

From Figure <ref>,
we find that different clusters show different levels of sensitivity to CO$_2$ tax. Offshore wind is the first renewable energy solution that is added to the system. Electric boilers are soon needed as offshore wind replaces gas turbines partially. OEHs are installed when CO$_2$ tax is above $290$ €/tonne. Offshore solar is only added in clusters $3$ and $5$ under very high CO$_2$ tax levels. OCGTs still operate even CO$_2$ tax increases to $500$ €/tonne.

We can see that in a static planning problem, if CO$_2$ tax is the only instrument and increases to $200$ €/tonne as the government's plan in 2030, OEHs may not be necessary. However, CO$_2$ tax combined with the EU emissions trading system may likely increase the total CO$_2$ price to around $250-300$ €/tonne, which is about the breakeven price of OEHs. In addition, the potential benefits of the OEHs may realise once they provide services to more sectors, such as exporting hydrogen for industries or transportation.

§.§ Sensitivity analysis of  budget

Emission and cost comparison (CO$_2$ budget sensitivity analysis).

Energy loss (CO$_2$ budget sensitivity analysis)

Capacities of technologies in each cluster (CO$_2$ budget sensitivity analysis),  hydrogen storage is measured in tonne.

For the CO$_2$ budget, we use initial emission as the starting point and reduce it by $5\%$ until it hits $0$. From Figure <ref>, we find that carbon cap is binding most of the time and we rarely see that emissions reduce more than the carbon cap. Thus, there is no difference in actual emission in Base and the system with OEHs. However, the cost is 25% lower in a zero emission system with OEHs compared with Base.

We find that in a zero emission system without OEHs, energy loss is around 477 TWh due to 81 GW of wind power capacity without storage. This may not be likely from happening since some forms of storage would be added to compensate for offshore wind in reality. From Figure <ref>, we can see a large amount of energy loss when reaching near zero emission system in Base. The energy loss in Base is 10,832 GWh in a near zero emission system that is about twice higher than that with OEHs. A large amount of wind power is installed to meet power demand at any time. Therefore, the same capacity of wind that can cope with peak demand hours will also generate surplus power during normal hours. This leads to increased energy losses as more wind replaces OCGT without proper energy storage. In the case of OEHs, wind power can be stored when excess power is generated. It is also worth noticing that in the energy system without an OEH, energy storage is the battery on the platforms, which can be infeasible due to space and weight limitations. We observe that investments in batteries are only needed when approaching zero emission in Base. No battery is needed in a system with OEHs. In addition, the energy loss of OEHs is 27% of the total loss, and the loss during production is about 50% of the total.

From Figure <ref>, we find that cluster 3 receives PFS after a 10% reduction of the carbon cap. Cluster 3 has the highest emission level but the shortest distance from shore. Therefore, taking PFS and partially electrify the fields in cluster 3 can help the system reduce 10% emission in a cost effective way. The system does not cut emissions proportionally in each cluster but cut emissions from clusters with the highest emission, such as cluster 3 and cluster 5. Therefore, it may be necessary to consider the whole NCS when conducting system planning rather than consider each cluster separately and reach sub-optimality. Cluster 2 is the most remote, more than 300 km from the shore; PFS is less economical than offshore wind. Therefore, offshore wind is added to cluster 2 when the carbon cap drops to 2.75 Mt/yr.

When the CO$_2$ budget reduces to below $0.83$ Mt/yr, CO$_2$ emissions are nearly zero in clusters $1$ and $2$. However, the carbon cap needs to reduce to zero to shut down OCGTs completely in all clusters. Nearly $4,200$ tonnes of hydrogen storage capacity is needed in a zero emission NCS energy system, and nearly half is installed in cluster 3.

§.§ Sensitivity analysis of the capacity of PFS

We now present the results of sensitivity analysis of the capacity of PFS. The capacity of PFS affects the investments in offshore technologies. An onshore system has a limited capacity to transmit power offshore. Furthermore, onshore system expansion can vary the limitation. However, we do not consider onshore system expansion in this paper. Therefore, we conduct sensitivity analysis to reveal the relationship between onshore power system capacity and offshore decarbonisation technologies.

§.§.§ Scenario 1 (S1)

The first scenario is to fix the CO$_2$ tax to $300$ €/tonne and increase the PFS capacity of each onshore node from 0 MW to 1000 MW with a 10 MW step. We find that the investment decisions remain the same if the PFS capacity is higher than 710 MW. Therefore, we only present the results from 0 MW to 710 MW.

Emission and cost (PFS capacity sensitivity analysis, S1).

Capacities of technologies in each cluster (PFS capacity sensitivity analysis, S1), hydrogen storage is measured in tonne.

Energy loss (PFS capacity sensitivity analysis, S1).

From Figure <ref>, we can see that by having 710 MW capacity in each onshore node, the system can achieve 0.05 Mt/yr emission and reduce nearly 50% of the total cost. However, increasing the capacity further does not cut emissions or costs further. Figure <ref> shows that energy loss during transmission makes up 16% of the total energy loss as we increase the onshore capacity. Electricity generation shed decreases as onshore capacity increases because PFS gradually replaces offshore wind, and less energy is lost from wind turbines. From Figure <ref>, we find that for onshore nodes that connect to cluster 1 and cluster 2, the needed onshore capacities are about 130 MW and 110 MW, respectively. There are also upper limits on the installed capacity of PFS in the other clusters. We also notice that OEHs are still needed in clusters 1, 3, 5 as we increase the onshore capacity. However, eventually, OEHs are not needed since PFS can provide more stable power and OEH with storage becomes less important.

§.§.§ Scenario 2 (S2)

Emission and cost (PFS capacity sensitivity analysis, S2).

Capacities of technologies in each cluster (PFS capacity sensitivity analysis, S2), hydrogen storage is measured in tonne.

Energy loss (PFS capacity sensitivity analysis, S2).

In the second scenario, we fix the CO$_2$ tax to $400$ €/tonne. We increase the onshore capacity from 0 MW to 1000 MW and present the results until 740 MW.
From <ref>, we can see that without PFS, the system can achieve 0.63 Mt/yr emissions under S2 condition. Increasing the onshore capacity brings down 43% of the cost and also cut emission further to near zero. Figure <ref> shows that about 20% of the energy loss is from OEHs initially. OEHs are not needed when the onshore capacity increase to around 400 MW for each node. By adding the installed PFS capacity shown in Figure <ref>, we find that a total onshore capacity of 1.75 GW may help the offshore energy system achieve near zero emission. We notice that the onshore system needs to provide an averagely of 1.4 GW By checking the average power transmission of PFS, which might not be feasible without onshore system expansion.

§ CONCLUSIONS AND FUTURE WORK

This paper presents a multi-carrier offshore energy system investment planning optimisation model with a high degree of operational detail to find cost-optimal solutions for decarbonising NCS energy supply. The major novelties and contributions are (1) we formulate OEHs in a MILP model for large-scale offshore energy system planning; (2) we model the device-level energy consumption of the offshore platforms with hourly time resolution on a large scale; and (3) We use an integrated MILP investment and operational model to conduct a large-scale analysis of the value of OEHs in the North Sea offshore energy system transition towards decarbonised energy supply. Results from our case study indicate that (1) OEHs can reduce up to $10\%$ of the energy loss and $50\%$ of the emissions with CO$_2$ tax above 290 €/tonne; (2) OEHs can reduce energy loss by $65\%$ in a near zero emission system; (3) a carbon budget may be necessary to enable a zero emission energy system in addition to CO$_2$ tax; (4) the system cuts about 63% of the initial emissions when CO$_2$ tax is 55 €/tonne, and approximately 84% of the CO$_2$ emissions can be cut if CO$_2$ tax is increased to Norway's target of 200 €/tonne; and (5) it may be necessary to consider the whole NCS when doing system planning towards zero emission rather than consider each region separately and reach sub-optimality.

Although the deterministic MILP model in this paper has led to many insights, there are possibilities for extensions. A deterministic optimisation model is not capable of representing load and supply uncertainties. Therefore, we aim to develop a stochastic optimisation model [Birge & Louveaux, 2011] and incorporate long-term and short-term uncertainties in future work. In addition, multiple investment stages are needed to represent the investment planning problem more realistically. Besides, we only consider using OEHs for fields decarbonisation, which makes OEHs seem less attractive than other technologies due to their high costs. However, OEHs can have various utilities such as energy provision to offshore fish farming, maritime and transport, and using the infrastructure past the lifetime of the oil and gas fields for purposes such as exporting hydrogen. These may motivate the investments in OEHs, and we aim to include some of the aspects in future. Finally, more work can be done on offshore network topology and the representation of the onshore power system.

§ CREDIT AUTHOR STATEMENT

Hongyu Zhang: Conceptualisation, Methodology, Software, Validation, Formal analysis, Investigation, Visualisation, Data curation, Writing - original draft, Writing - review & editing. Asgeir Tomasgard: Conceptualisation, Supervision, Writing - review & editing, Funding acquisition. Brage Rugstad Knudsen: Conceptualisation, Supervision, Writing - review & editing. Harald G. Svendsen: Conceptualisation, Writing - review & editing. Steffen J. Bakker: Conceptualisation, Writing - review & editing. Ignacio E. Grossmann: Conceptualisation, Supervision, Writing - review & editing.

§ DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

§ ACKNOWLEDGEMENTS

This work was supported by the Research Council of Norway through PETROSENTER LowEmission (project code 296207).

§ COMPLETE OPERATIONAL CONSTRAINTS

g(x,c)=∑_s ∈𝒮^TW^S_s(∑_t ∈𝒯H_t(∑_z∈𝒵^P(∑_g ∈𝒢(C_g^G+C_g^Fuel+C^CO_2E_g^Fuel/η_g^G)p_gzt^G+...

g(x_i.. .C_g^Fuel/η_g^GC^LShedp_zt^LShed+C^HLShedp^HLShed_zt)
+∑_z ∈𝒵^OC^ZO_ztp^ZO_zt))

0 ≤p_gzt^G+p_gzt^ResG≤P_gz^accG,     g ∈𝒢, z ∈𝒵^P, t ∈𝒯

-P_gz^accGG^GR_g ≤p_gzt^G+p_gzt^ResG

abc -p_gz(t-1)^G-p_gz(t-1)^ResG ≤P_gz^accGG^GR_g,    ab g ∈𝒢, z ∈𝒵^P, t ∈𝒯

p_rzt^R=R^R_rztp_rz^accR,      r ∈ℛ, z ∈𝒵^H, t ∈𝒯

0 ≤p_zt^ZO≤p_z^accZO,      z ∈𝒵^O,t ∈𝒯

q_sz(t+1)^SE=q_szt^SE+H_t(η_s^SEp_szt^SE+-p_szt^SE-),     s ∈𝒮^E, z ∈𝒵^P, t ∈𝒯

H_t(p_szt^ResSE + p_szt^SE-)≤q_szt^SE,     s ∈𝒮^E, z ∈𝒵^P, t ∈𝒯

0 ≤p_szt^SE+ ≤H^SE_sq_sz^accSE,     s ∈𝒮^E, z ∈𝒵^P, t ∈𝒯

0 ≤p_szt^SE- + p_szt^ResSE ≤H^SE_sq_sz^accSE,     s ∈𝒮^E, z ∈𝒵^P, t ∈𝒯

0 ≤q_szt^SE ≤q_sz^accSE,     s ∈𝒮^E, z ∈𝒵^P, t ∈𝒯

∑_g ∈𝒢p_gzt^G+
∑_l ∈ℒA^E_zlη^L_lp_lt^L

abc+∑_s ∈𝒮^Ep_szt^SE-+p_zt^LShed=

abcμ^Dp_zt^D+∑_s ∈𝒮^ℰp_szt^SE++
p_zt^GShed,      z ∈𝒵^P, t ∈𝒯

∑_r ∈ℛp_rzt^R+
∑_l ∈ℒA^E_zlη^L_lp_lt^L+∑_f ∈ℱp^F_fzt=

abcp_zt^GShed+∑_e ∈ℰp^E_ezt,    z ∈𝒵^H, t ∈𝒯

p_zt^ZO+∑_l ∈ℒA^E_zlη^L_lp_lt^L= p_zt^GShed,
z ∈𝒵^O, t ∈𝒯

-p_l^accL ≤p_lt^L ≤p_l^accL,     l ∈ℒ, t ∈𝒯

σ^Resp^D_zt ≤∑_g ∈𝒢 p^ResG_gzt+ ∑_s ∈𝒮^Ep^ResSE_szt,     z ∈𝒵^P, t ∈𝒯

p^D_zt=∑_b ∈ℬ^Ep^EB_bzt+∑_c ∈𝒞^Expp^CExp_czt

abc+∑_c ∈𝒞^Injp^CInj_czt+∑_p ∈𝒫^Op^PO_pzt

abc+∑_p ∈𝒫^WIp^PWI_pzt+∑_p ∈𝒫^WLp^PWL_pzt,     z ∈𝒵^P, t ∈𝒯

∑_s ∈𝒮^TW^S_s∑_z ∈𝒵^P ∑_t ∈𝒯 ∑_g ∈𝒢E_g^Fuelp_gzt^GH_t/η_g^G ≤μ^E,

sss.        0 ≤p^EB_bzt ≤p_bz^accEB,     b ∈ℬ^E, z ∈𝒵^P,     t ∈𝒯

0 ≤p^HSEP_dzt ≤P^accSEP_dz,     d ∈𝒟^Sep, z ∈𝒵^P, t ∈𝒯

∑_g ∈𝒢η^Gh_gp^G_gzt+∑_b∈ℬ^Eη^Eb_bp^EB_bzt+p^HLShed_zt

abc=∑_d ∈𝒟^Sepp^HSEP_dzt+p^HGShed_zt,    z ∈𝒵^P, t ∈𝒯

p^HSEP_dzt=ρ^SEP_dV^OD_zt,     d ∈𝒟^Sep,  z ∈𝒵^P, t ∈𝒯

0 ≤p_czt^CExp ≤P_cz^accCExp,     c ∈𝒞^Exp, z ∈𝒵^P, t ∈𝒯

0 ≤p_czt^CInj ≤P_cz^accCInj,     c ∈𝒞^Inj, z ∈𝒵^P, t ∈𝒯

p_czt^CInj=V_zt^NGI/η^CHρ[(γ_c)^α-1/α-1],       c ∈𝒞^Inj, z ∈𝒵^P, t ∈𝒯

p_czt^CExp=V_zt^NGE/η^CHρ[(γ_c)^α-1/α-1],       c ∈𝒞^Exp, z ∈𝒵^P, t ∈𝒯

0 ≤p_pzt^PO ≤P_pz^accPO,      p ∈𝒫^O, z ∈𝒵^P, t ∈𝒯

p_pzt^PO = κ^PO_pV^OD_zt,      p ∈𝒫^O, z ∈𝒵^P, t ∈𝒯

0 ≤p_pzt^PWI ≤P_pz^accPWI,     p ∈𝒫^WI, z ∈𝒵^P, t ∈𝒯

0 ≤p_pzt^PWL ≤P_pz^accPWL,    p ∈𝒫^WL, z ∈𝒵^P, t ∈𝒯

p_pzt^PWI = κ^PWI_pV^WI_zt,     p ∈𝒫^WI, z ∈𝒵^P, t ∈𝒯

p_pzt^PWL = κ^PWL_pV^WL_zt,     p ∈𝒫^WL, z ∈𝒵^P, t ∈𝒯

min        0≤p_ezt^E ≤p_ez^accE,        e ∈ℰ, z ∈𝒵^H, t ∈𝒯

0 ≤p_fzt^F ≤p_fz^accF,    f ∈ℱ, z ∈𝒵^H,  t ∈𝒯

0 ≤v_szt^SHy ≤v_sz^accSHy,      s ∈𝒮^Hy, z ∈𝒵^H,  t ∈𝒯

|p_fzt^F- p_fz(t-1)^F|≤F^FR_fp_fz^accF,       f ∈ℱ, z ∈𝒵^H,  t ∈𝒯

v^SHy_sz(t+1)=v^SHy_szt+v^SHy+_szt-v^SHy-_szt,     s ∈𝒮^Hy, z ∈𝒵^H, t ∈𝒯

η^EF(∑_f ∈ℱp^F_fztH_t/η^F_fθ^Hy-∑_s ∈𝒮^Hyv^SHy-_szt)=

abc∑_e ∈ℰp^E_eztH_t-η^ES∑_s ∈𝒮^Hyv^SHy+_szt,       z ∈𝒵^H,  t ∈𝒯.

(<ref>)–(<ref>) give the electricity system constraints. (<ref>)–(<ref>) give the platform production constraints. (<ref>)–(<ref>) represent hydrogen system constraints of OEHs. All variables except for power transmission is non-negative. Power transmission is a free variable. All variables are also indexed by operational year $i$, and we omit them for ease of notation.

§ CALCULATION OF ENERGY LOSS

The indices, summation and multiplication of one hour are omitted.

\begin{equation}
\begin{split}
\end{split}\notag
\end{equation}

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