# Analysing the interactions between demand side and supply side investment
decisions in an oligopolistic electricity market using a stochastic mixed
complementarity problem

Mel T. Devinea,*, Valentin Bertschb<EMAIL_ADDRESS>aEnergy Institute,
College of Business, University College Dublin, Ireland
bChair of Energy Systems and Energy Economics, Ruhr-Universität Bochum,
Germany.

###### Abstract

To meet carbon emission targets, governments around the world seek electricity
consumers to invest in self-sufficiency technologies such as solar
photovoltaic and battery storage. Such behaviour is sought in markets
typically characterised by an oligopoly amongst generating firms. In this
work, we study the interactions between investment decisions on the demand
side and the supply side, and we investigate how price-making behaviour on the
supply side affects these interactions compared to a situation with perfect
competition. To do so, we introduce a novel stochastic mixed complementarity
problem to model several players in an oligopolistic electricity market. On
the supply side, we consider generating firms who make operational and
investment decisions. On the demand side, we consider both industrial and
residential consumers, each of whom may invest in self-sufficiency
technologies. The uncertainties of wind and solar generation are the sources
of the model’s stochasticity. We apply the model to a case study of a stylised
Irish electricity system in 2030. Our results demonstrate that price-making on
the supply side increases investment in self-sufficiency on the demand side,
leading to a reduction in prices and carbon emissions. We also find that both
market power and self-sufficiency alter the investment and decommissioning
decisions made by generation firms. Counter-intuitively, we also observe that
the absence of a feed-in premium increases investment in solar generation on
the demand side. Our findings highlight the importance of including both
demand and supply side investment in models of electricity markets
characterised by an oligopoly.

###### keywords:

OR in Energy; Energy Transition; Self-Sufficiency; Market Power; Stochastic
Mixed Complementarity Problem;

††journal: a journal

##  1 Introduction

To reduce carbon emissions, governments worldwide have developed
transformative climate action policies. Such policies typically involve
increasing the amount of electricity generated from renewable sources (RES-E),
primarily from wind and solar Photovoltaics (PV). Given the ambitious nature
of the political targets, significant investment in such sources will be
required on both the supply-side (generation) and the demand-side (consumer
self-sufficiency). For instance, in its “Winter Package” in 2016, the European
Commission has stated that it seeks climate-neutrality and wants to put
consumers at the centre of the energy transition [Comission, 2016].

Besides this clear political support of so-called prosumers (i.e. consumers
that are also producers) by the EU, several energy market developments have
contributed to a strong increase in consumer, as well as research interest in,
prosumption / consumer self-sufficiency [Luthander et al., 2015, Ellsworth-
Krebs and Reid, 2016, Bertsch et al., 2017, Schwarz et al., 2018b]. On the one
hand, learning rates in the growing PV market have led to strong reductions of
PV system costs over the past two decades [Candelise et al., 2013]. On the
other hand, electricity retail prices increased during the same period of
time. Consequently, solar PV has reached the so-called ‘grid parity’ in many
countries during this time period [Khalilpour and Vassallo, 2015, Hagerman et
al., 2016, Munoz et al., 2014], i.e. self-consumption of electricity generated
from rooftop PV became cheaper than using electricity from the grid at the
retail price (and typically also more attractive than feeding electricity from
rooftop PV into the grid to earn a feed-in premium). As a result, an
increasing number of consumers has started to invest in self-sufficiency in
the recent past. The price increase observed during the European energy crisis
in 2022 has further increased the profitability of self-sufficiency.

Such dynamic developments of investment and consumption decisions on the
demand side also come along with implications for the supply side. Most
importantly, these decisions on the demand side affect the load to be covered
by the supply side and hence market prices as well as optimal generation
portfolios (including the possibility of decommissioning decisions, in
particular of fossil-fuel based generation). Therefore, an in-depth analysis
of the interactions between the investment and generation/consumption
decisions on the supply and demand side is very important. Hence, in this
work, we develop and demonstrate a novel approach to study the optimal
generation and storage investment decisions from both the demand and the
supply side in addition to decommissioning decisions. Moreover, we seek to
study the impacts such decisions have on different types of consumers,
generating firms’ profits, and carbon emissions.

###  1.1 State of the art and research gap

In order to support decision making in the context of an increasingly complex
energy planning, numerous optimisation models have been developed and
published in the Operations Research and wider literature [Petropoulos et al.,
2023]. These models often adopt linear programming or mixed-integer linear
programming approaches and are used to find cost-minimal combinations of
energy supply technologies to cover a given energy demand curve under a number
of constraints. In the literature, such models are often referred to as energy
system models (ESMs). Concerning the analysis of the profitability and
implications of consumer self-sufficiency, existing applications of ESMs
usually fall into either one of the following three categories: (i) system-
scale models that assess the implications of self-sufficiency for the wider
energy system, e.g. at a regional, national or continental level [e.g. Schmidt
et al., 2012, Merkel et al., 2014, Fehrenbach et al., 2014, Senkpiel and
Hauser, 2020, Neumann, 2021], (ii) residential-scale models that mostly focus
on household self-sufficiency decisions concerning the investment in or
operation of technologies, such as solar PV, battery storage or heat pumps
[e.g. McKenna et al., 2017, Schwarz et al., 2018a, b, Hayn et al., 2018,
Aniello and Bertsch, 2023, Huckebrink et al., 2023, Aniello et al., 2024], or
(iii) approaches that soft-link different variants of models in categories (i)
and (ii) [Gissey et al., 2021, Zakeri et al., 2021, Sarfarazi et al., 2024].

The main limitations of the majority of existing models in the first category
include the following:

  * •

First, typical ESMs assume perfect competition and adopt a least-cost central-
planning approach. This is critical mainly for two reasons. On the one hand,
in today’s liberalised energy markets, private-sector investments in (e.g.
renewable) energy technologies are required to achieve political targets. To
support investors’ decisions in renewables, information on the profitability
of investment projects is therefore needed. This shortcoming is for instance
mitigated by calculating metrics such as the internal rate of return (IRR) or
the net present value (NPV) based on ESM results as part of an ex-post
analysis [e.g. Finke et al., 2023]. On the other hand, in particular in
situations where markets become tighter (e.g. as observed during the European
energy crisis starting in 2022 or because of policy-driven phase-out decisions
and a lack or delay of new investments), concerns around market power may
become more pertinent. For this purpose, game-theoretic equilibrium models
have proven useful. There have been several works in the literature that have
modelled market power in energy markets [Wogrin et al., 2013, Baltensperger et
al., 2016, Devine and Bertsch, 2018, Egging-Bratseth et al., 2020, Devine and
Siddiqui, 2023, Devine and Bertsch, 2023]. For a deeper discussion, we refer
the reader to Pozo et al. [2017]. A drawback of most existing game-theoretic
equilibrium models, however, is that they usually do not include intricate
(technical) details on either the supply or demand sides. Exceptions in the
literature include, for example, Devine and Bertsch [2023].

  * •

Second, existing models typically model one (usually hourly) market, i.e. one
revenue stream for investors. In reality, however, the profitability of
investments is often influenced by multiple revenue streams. For instance, to
encourage investment in RES-E, several countries have introduced a Feed-in
Premium (FiP), i.e. an extra payment - separate to the market price - made to
producers of renewable electricity [e.g. Bertsch et al., 2017, Farrell et al.,
2017]. In addition, an increasing number of countries has introduced a
capacity market to encourage investments in firm generation capacity. It is
therefore important to study the impact of multiple revenue streams on
investment decisions in future energy systems.

  * •

Third, the demand side is often modelled as one system-level aggregated demand
curve, i.e. no differences between consumers or consumer groups are
considered. On the one hand, this is critical because different consumers and
consumer groups have different demand profiles, which affect self-sufficiency
levels and hence the economics of self-sufficiency decisions. On the other
hand, the use of one aggregated demand curve ignores the fact that
elasticities of or costs of reducing energy demand are not uniformly
distributed across consumer groups. While game theoretic optimisation
approaches usually consider elasticity of energy demand [e.g. Lynch and
Devine, 2017b, Newbery and Greve, 2017, Lynch et al., 2019b, Devine et al.,
2019], the ignorance of differences between consumer groups is a strong
simplification [Devine and Bertsch, 2018]. Finally using just one aggregated
demand curve also means that interactions between different different types of
consumers cannot be analysed.

  * •

Fourth, if different consumers and their decisions are considered explicitly
at all, these models usually assume that all consumers participate in the
wholesale market directly (which is the only market modelled) and hence pay
wholesale prices for their energy demand ([e.g. Lynch et al., 2019a]. Again,
this is critical since retail prices in reality include several price
components in addition to wholesale prices, which has a strong impact on the
economics of self-sufficiency decisions.

  * •

Fifth, while ESMs that consider uncertainties (e.g. using stochastic
pogramming) do exist [e.g. Ambec and Crampes, 2012, Moser et al., 2016,
Wozabal et al., 2016], the majority of applications is based on deterministic
models [Pfenninger et al., 2014]. This is critical, in particular, for a high-
renewable system such as the one examined in this paper, where especially the
electricity generation of both wind and PV is highly variable and uncertain.

While existing models in the second category consider consumer-specific load
profiles and retail prices, including a number of price components on top of
wholesale prices, these models typically assume these prices as exogenous
inputs [see e.g. Schwarz et al., 2018b, Aniello and Bertsch, 2023, Nouicer et
al., 2023, Aniello et al., 2024]. This means that such models are limited in
that no interdependencies between individual consumer decisions and
generators, other consumers or the market and market prices more generally can
be analysed. In other words, none of these models considers generation
investments on both the demand-side and the supply-side and the interactions
between them.

Soft-linking approaches, such as those in the third model category, are
generally used increasingly often. Major limitations, however, include that
there is usually no general guarantee that the solution obtained will be
optimal. Moreover, it is challenging to fully capture the dynamics of the
interactions between decisions on the supply and the demand side.

In summary, while some models have been presented in the OR and wider
literature that address several of the limitations described above, to the
best of our knowledge, there are no existing stochastic game-theoretic
equilibrium models with a sufficient level of detail on the supply and demand
side that have addressed all of them. While Devine and Bertsch [2023] mitigate
quite a few of the limitations (their model considers market power, multiple
revenue streams, energy demand profiles and elasticities for different
consumer groups, and stochasticity of renewable generation), they do not
consider retail prices for different consumer groups or investment decisions
on either the supply or demand side.

###  1.2 Research questions and contribution

In the light of the research gap outlined in the previous section, we address
the following research questions in this work:

  1. 1.

How does the presence of market power…

    1. (a)

… affect (self-sufficiency) investments, consumer tariffs and load shedding on
the demand side?

    2. (b)

… and self-sufficiency investments, in turn affect generation firms’
investment/exit decisions and profits?

  2. 2.

How does the availability/absence of a FiP as a separate revenue stream affect
the above questions?

  3. 3.

How do the interactions between market power, investment decisions, and a FiP
affect carbon emissions?

To answer these questions, we present a novel stochastic mixed complementarity
problem (MCP). MCPs have been used to model various types of energy markets
[Hobbs, 2001, Gabriel et al., 2009, Huppmann, 2013, Egging, 2013, Lynch and
Devine, 2017a, Devine et al., 2016, Devine and Bertsch, 2023]. An MCP solves
the optimisation problems of several individual players simultaneously and in
equilibrium.

The players we consider on the supply side include power generating firms with
different generation portfolios. All firms maximise their profits by
optimising the dispatch of their existing assets and making investments into
new (renewable and/or conventional) generation under consideration of the RES
uncertainty. Firms earn revenues from an energy market, a quantity-based
capacity market, and from a FiP.

On the demand side, we consider a number of different consumer groups as
players in our MCP model. We consider commercial/industrial consumers as well
as residential consumers. We also distinguish between traditional consumers
and prosumers, i.e. consumers that have their own solar PV modules. Prosumers
may make further investment decisions in PV and/or storage capacity.
Furthermore, we consider an independent battery storage operator who buys and
sells electricity from the grid in addition to considering further investments
in battery storage.

This MCP builds on the models presented in Lynch et al. [2019a] and Devine and
Bertsch [2023]. In contrast to Lynch et al. [2019a], however, we consider the
impact of market power. We also go beyond Devine and Bertsch [2023] by
including generation and storage investment decisions on both the supply side
and the demand side (i.e. including self-sufficiency investments) and by
considering retail price components for different consumer groups in addition
to wholesale prices.

Thus, to the best of our knowledge, there is no model in the energy market
literature such as the one presented in this paper. The novel model features
allow us to consider the research questions listed above.

The remainder of this paper is structured as follows: In section 2, we
introduce the mathematical model. In section 3, we describe the data of our
case study. In section 4, we present our findings. In section 5, we discuss
and interpret our findings and summarise the main conclusions.

##  2 Methodology

Table 1: Indices and sets.  $f\in F$ | Generating firms
---|---
$t\in T$ | Generating technologies
$p\in P$ | Time periods
$k\in K$ | Consumers groups
$s\in S$ | Scenarios
$h\in H$ | Time steps in storage/load shifting period
$p^{\prime}\in P^{\prime}=\\{1,|H|+1,2|H|+1,...\\}\subseteq P$ | Index representing starting points for storage period
Note: sets contain a finite amount of non-zero natural numbers while $|H|$
represents the cardinality of $H$.
Table 2: Variables. Firms’ primal variables
---
$gen_{f,t,p,s}$ |  Generation from firm $f$ with technology $t$ in period $p$ and scenario $s$
$cap^{\text{bid}}_{f,t}$ |  Capacity bid of firm $f$ with technology $t$
$inv_{f,t}$ |  Investment in new generation capacity for firm $f$ with technology $t$
$exit_{f,t}$ |  Decommissioning of old generation capacity for firm $f$ with technology $t$
Consumers’ primal variables
$g^{\text{ls}}_{k,p,s}$ |  Load shedding from consumer group $k$ in period $p$ and scenario $s$
$g^{\text{up}}_{k,p,s}$ |  Electricity stored for later time point from consumer group $k$ in period $p$ and scenario $s$
$g^{\text{down}}_{k,p,s}$ |  Electricity used from storage from consumer group $k$ in period $p$ and scenario $s$
$g^{\text{pv}}_{k,p,s}$ |  PV generation from consumer group $k$ in period $p$ and scenario $s$
$cap^{\text{STOR}}_{k}$ |  Consumer group’s $k$’s investment in electrical storage
$cap^{\text{PV}}_{k}$ |  Consumer group’s $k$’s investment in photovoltaic generation
Dual variables
$\gamma_{p,s}$ |  System price for time period $p$ and scenario $s$
$\kappa$ | Unit capacity price
$\lambda^{\\#}_{.}$ |  Lagrange multipliers associated with constraint $\\#$ of the firms’ problem
$\mu^{\\#}_{.}$ |  Lagrange multipliers associated with constraint $\\#$ of the firms’ problem consumers’ problem
Note: ’.’ is used as a place-holder as the subscripts for both Lagrange
multipliers vary depending the on constraint.
Table 3: Parameters. $PR_{s}$ |  Probability associated with scenario $s$
---|---
$MTC_{t}$ |  Maintenance cost form generating technology $t$
$CAP_{f,t}$ |  Initial generating capacity for firm $f$ with technology $t$
$D^{\text{REF}}_{k,p}$ |  Reference demand for consumer group $k$ in period $p$
$LOSS_{k}$ |  Storage loss factor for consumer group $k$
$G_{k}^{\text{LS,MAX}}$ |  Maximum load shedding for consumer group $k$ in any time period or scenario
$INT^{\text{STOR}}_{k}$ |  Electrical storage/ load shifting capacity for consumer group $k$
$INT^{\text{PV}}_{k}$ |  PV generating capacity for consumer group $k$
$FAC_{k}^{\text{STOR}}$ |  Percentage of electrical storage capacity electricity consumer group $k$ can use in each period and scenario
$X^{\text{PREM}}_{k}$ |  Retail price premium for consumer group $k$
$NORM^{\text{G}}_{f,t,p,s}$ |  Generating profile for firm $f$ with technology $t$ in period $p$ and scenario $s$
$NORM^{\text{PV}}_{p,s}$ |  PV generating profile for period $p$ and scenario $s$
$NORM^{\text{WIND}}_{t,p,s}$ |  Wind profile for wind region (technology) $t$ in period $p$ and scenario $s$
$TARGET$ |  Capacity target for overall market
$X_{t}$ |  Feed-In premium for technology $t$
$X^{\text{PREM}}_{k}$ |  Retail price premium for consumer group $k$
$DR_{t}$ |  De-rating factor for technology $t$
$A^{.}_{.}$ |  Intercept associated with marginal cost functions
$B^{.}_{.}$ |  Slope associated with marginal cost functions
$C^{\text{GEN}}_{t}$ |  Marginal cost associated with generating technology $t$
$C^{\text{PV}}_{k,p}$ |  Marginal cost of using PV generation for consumer group $k$ in period $p$
$IC^{\text{GEN}}_{t}$ |  Annualised investment in generating technology $t$ cost
$IC^{\text{STOR}}_{k}$ |  Annualised investment in electrical storage cost for consumer group $k$
$IC^{\text{PV}}_{k}$ |  Annualised investment in PV generating capacity cost for consumer group $k$
Table 4: Functions. $C^{\text{LS}}_{k,p}(.)$ |  Load shedding operational cost for consumer group $k$ in period $p$
---|---

In this section, the methodology is described. We develop a stochastic MCP to
represent an electricity market with two types of players: generation firms
and electricity consumer groups.

Firms receive revenues from energy and capacity markets as well as a FiP and
seek to maximise their profits. They may hold multiple generating units with
the technologies considered being coal, oil, Combined Cycle Gas Turbine
(CCGT), hydroelectric, wind and solar. In Section 3 we consider different wind
regions of Ireland with varying, but correlated, hourly capacity factors. Wind
for the different regions are represented in the model as different
technologies, i.e., wind from one region is considered as a different
technology from wind in another region. Firms are distinguished by the initial
generating portfolio they hold but may also invest in additional capacity in
any of these technologies. All firms may be modelled as either price-takers or
price-makers.

The capacity payment mechanism is a quantity based mechanism. Under such a
framework, a policy maker, regulator or Transmission System Operator (TSO),
exogenously selects a quantity of capacity required for a given period (eg.,
one year). An auction is held wherein generators submit capacity bids and the
total amount sold equal the MW target. Generation firms compete in the auction
to hold the options and therefore gain a fixed sum of money per the time
period considered to compensate them for each installed unit of capacity.

Consumers seek to minimise the cost of meeting their demand. They may
generally do so by utilising a range of possible demand-side flexibility
measures, such as load shedding, load shifting or electrical storage, or PV
generation. We do not model individual consumers but rather consider different
consumer groups whose decisions represent the aggregate actions of consumers
in these groups. Consumer groups are distinguished by whether they represent
industrial/commercial or residential consumers and whether or not they can
invest in additional demand-side flexibility capacity (storage and PV).

The stochasticity of the model arises from the uncertainty surrounding wind
and PV power. Thus, each scenario in our model represents different RES
generation profiles, i.e. varying levels of wind and solar power availability
at each point in time. Each of the generation firms and consumer groups
considered have separate optimisation problems that are connected through
market clearing conditions. The stochastic MCP is made up of these market
clearing conditions along with the Karush-Kuhn-Tucker (KKT) conditions for
optimality from each of the players. Thus, the MCP solves the optimisation
problem of each player simultaneously and in equilibrium. The KKT conditions
are presented in the Online
Appendix111https://figshare.com/articles/journal_contribution/Online_Appendix_for_Devine_and_Bertsch_2024/25764498.

Throughout this section the following conventions are used: lower-case Roman
letters indicate indices or primal variables, upper-case Roman letters
represent parameters (i.e., data, functions), while Greek letters indicate
prices or dual variables. The variables in parentheses alongside each
constraint in this section are the Lagrange multipliers associated with those
constraints.

###  2.1 Firm $f$’s problem

Firm $f$ maximises its expected profits (revenues less cost) by choosing the
amount of generation ($gen_{f,t,p,s}$), capacity bid
($cap^{\text{bid}}_{f,t}$), investment in new capacity ($inv_{f,t}$) and
decommissioning of existing capacity ($exit_{f,t}$). It considers revenues
received from a capacity and an energy market as well as a FiP for RES
generation. Its costs include generation costs and investment costs in
addition to costs incurred for maintaining all of its units. Firm $f$’s
generation decisions are scenario dependent while all of its other decision
variables are scenario independent. Firm $f$’s optimisation problem is:

$\begin{split}\max_{\begin{subarray}{c}gen_{f,t,p,s},cap^{\text{bid}}_{f,t},\\\
inv_{f,t},exit_{f,t}\\\ \end{subarray}}\>\>&\sum_{t,p,s}\bigg{(}PR_{s}\times
gen_{f,t,p,s}\times\big{(}\gamma_{p,s}+X_{t}-C^{\text{GEN}}_{t}\big{)}\bigg{)}\\\
-&\sum_{t}\bigg{(}IC^{\text{GEN}}_{t}\times
inv_{f,t}+\big{(}inv_{f,t}+CAP_{f,t}-exit_{f,t}\big{)}\times
MTC_{t}\bigg{)}\\\ +&\sum_{t}DR_{t}\times\kappa\times
cap^{\text{bid}}_{f,t},\\\ \end{split}$ (1a) subject to:
$gen_{f,t,p,s}\leq(CAP_{f,t}+inv_{f,t}-exit_{f,t})\times
NORM^{\text{G}}_{f,t,p,s},\>\>\forall t,p,s,\>\>(\lambda^{1}_{f,t,p,s}),$ (1b)
$cap^{\text{bid}}_{f,t}\leq CAP_{f,t}+inv_{f,t}-exit_{f,t},\>\>\forall
t,\>\>(\lambda^{2}_{f,t}),$ (1c)

where $t$ represents the different generating technologies, $p$ represents
timesteps and $s$ represents different wind and solar scenarios of the
stochastic optimisation problem. The parameters $IC_{t}$ and $MTC_{t}$ are
cost of investment and maintenance for technology $t$ respectively. The
marginal cost of generating with technology $t$ is $C^{\text{GEN}}_{t}$.

The parameters $X_{t}$, $PR_{s}$, and $DR_{t}$ represent the Feed-in Premium
associated with technology $t$, the probability associated with scenario $s$,
and the de-rating factor associated with technology $t$, respectively. In this
work a de-rating factor reflects the proportion of its overall capacity a
technology can provide to meet the capacity target.

The parameter $CAP_{f,t}$ is the initial installed capacity firm $f$ has for
technology $t$ while $NORM^{\text{G}}_{f,t,p,s}$ represents the availability
for technology $t$ in timestep $p$ and scenario $s$ for firm $f$. We assume
$NORM^{\text{G}}_{f,t,p,s}=1$ $\forall f,t,p,s$ for conventional generation.
For wind and PV generation,
$NORM^{\text{G}}_{f,t,p,s}=NORM^{\text{WIND}}_{t,p,s}$ and
$NORM^{\text{G}}_{f,t,p,s}=NORM^{\text{PV}}_{p,s}$, respectively. The values
for $NORM^{\text{WIND}}_{t,p,s}$ and $NORM^{\text{PV}}_{p,s}$ can be found in
the online appendix11footnotemark: 1. They take a value between zero and one
depending on the timestep and scenario representing the intermittency and
uncertainty of RES generation.

Constraint (1b) ensures that firm $f$ with technology $t$ cannot generate more
than its capacity times its capacity factor, while constraint (1c) ensures it
cannot make a capacity bid greater than the installed capacity. Each of firm
$f$’s primal (decision) variables are also constrained to be non-negative.

The capacity price paid for each unit of capacity accepted is $\kappa$. It is
exogenous to firm $f$’s problem but is a variable of the overall problem,
determined via the market clearing condition (6b). The energy price at each
period and scenario is $\gamma_{p,s}$. If firm $f$ is a price taker then its
decision variables cannot affect this price (i.e.,
$\frac{\partial\gamma_{p,s}}{\partial gen_{f,t,p,s}}=0$). In this case,
$\gamma_{p,s}$ is exogenous to firm $f$’s problem but is a variable of the
overall problem, determined via the market clearing condition (6a). If firm
$f$’s generating unit for technology $t$ is assumed to be a price-maker unit,
then its generation decision variable for that unit ($gen_{f,t,p,s}$) can
affect the energy price. As a result, we derive the following relationship
between the energy price and generation:

$\begin{split}gen_{f,t,p,s}=&\sum_{k}\big{(}D^{\text{REF}}_{k,p}+g^{\text{up}}_{k,p,s}-g^{\text{down}}_{k,p,s}-g^{\text{pv}}_{k,p,s}\big{)}-\sum_{k}\frac{\gamma_{p,s}-A^{\text{LS}}_{k,p}+\frac{1}{PR_{s}}(\mu^{1}_{k,p,s}+\mu^{8}_{k,p,s}-\mu^{1A}_{k,p,s})}{2B^{\text{LS}}_{k,p}}\end{split}$
(2)

where the parameters and variables not mentioned already are parameters and
variables from the consumers’ problem (section 2.2) and hence exogenous to
firm $f$’s problem. Equation (2) is determined by combining market clearing
condition (6a) with the KKT conditions11footnotemark: 1 that determine how
consumers shed their load (A.8) and the KKT conditions that determine how
firms generate (A.1) (all except firm $f$’s unit $t$). The remaining KKT
conditions cannot be used as they cannot be substituted into (6a). For price-
making firms, this relationship is substituted into firm $f$’s objective
function (1a) leading to

$\frac{\partial\gamma_{p,s}}{\partial
gen_{f,t,p,s}}=-\bigg{(}\frac{1}{\sum_{k}\frac{1}{2B^{\text{LS}}_{k,p}}}\bigg{)}.$
(3)

Equation (3) represents how firm $f$ believes it can influence $\gamma_{p,s}$
with its generation decisions. Consequently, equation (3) may over or
underestimate firm $f$’s level of market power. As outlined in Devine and
Bertsch [2023], making such assumptions around how price-making firms believe
they can influence the market price is a limitation of the modelling approach
considered in this work. However, modelling firms’ beliefs on how they can
influence the market price, also known as conjectural variations, is a
practise that has been widely used in the energy market modelling literature
[Egging et al., 2008, Haftendorn and Holz, 2010, Huppmann and Egging, 2014,
Baltensperger et al., 2016, Egging and Holz, 2016].

If firm $f$ is a price-taker, then its problem is convex. If firm $f$’s
generating unit for technology $t$ is a price-maker, then its problem is
strictly convex, assuming $B^{\text{LS}}_{k,p}>0$, $\forall k,p,t$. The firms’
KKT conditions are presented in the online appendix11footnotemark: 1. Note:
$PM_{f,t}$ is a binary parameter that is used in the KKT conditions to
indicate whether firm $f$’s generating unit for technology $t$ is a price
making unit ($PM_{f,t}=1$) or price taking unit ($PM_{f,t}=0$).

###  2.2 Consumer group $k$’s problem

Consumer group $k$ seeks to minimise the cost of meeting their expected demand
by choosing from a range of demand side flexibility measures:

  1. 1.

Load shedding: consumer group $k$ may reduce their demand at any period $p$ or
scenario $s$ The decision variable that represents the amount they do so by is
$g^{\text{LS}}_{k,p,s}$.

  2. 2.

Electrical storage: instead of obtaining electricity from the grid, consumers
may obtain electricity from electrical storage. However, they must increase
their consumption in previous timestep(s) to ensure there is availability of
electricity in storage. Storing electricity and using electricity to meet
demand in period $p$ and scenario $s$ are represented by the decision
variables $g^{\text{UP}}_{k,p,s}$ and $g^{\text{DOWN}}_{k,p,s}$, respectively.
Consumer group $k$ may also increase their electrical storage capacity which
is represented by the investment decision variable $cap^{\text{STOR}}_{k}$.

  3. 3.

Photovoltaic (PV) generation: instead of obtaining electricity from the grid,
consumers may also obtain electricity from their own PV-generation unit. The
decision variable that represents the amount they do so by is
$g^{\text{PV}}_{k,p,s}$ while $cap^{\text{PV}}_{k}$ represents their
investment in PV generation decision variable. The availability of PV power
varies across timesteps and scenarios.

Consumer group $k$’s investment decisions are scenario independent while all
their other decision variables are scenario dependent. Consumer groups $k$’s
reference demand for period $p$ is $D^{\text{REF}}_{k,p}$. This is the demand
consumer group $k$ would have in the absence of any demand side flexibility
measures. Consumer group $k$’s demand in period $p$ and scenario $s$ is its
reference demand plus any storage charging less any load shedding, PV
generation or storage discharging. The cost of meeting a unit of this demand
for consumer group $k$ is the wholesale price $\gamma_{p,s}$ plus the retail
price premium $X^{\text{PREM}}_{k}$.

Consumer group $k$’s optimisation problem is:

$\begin{split}\min_{\begin{subarray}{c}g^{\text{LS}}_{k,p,s},g^{\text{UP}}_{k,p,s},g^{\text{DOWN}}_{k,p,s},\\\
g^{\text{PV}}_{k,p,s}cap^{\text{STOR}}_{k},cap^{\text{PV}}_{k}\end{subarray}}&\sum_{s,p}PR_{s}\bigg{(}(\gamma_{p,s}+X^{\text{PREM}}_{k})\times\big{(}D^{\text{REF}}_{k,p}-g^{\text{LS}}_{k,p,s}+g^{\text{UP}}_{k,p,s}-(1-LOSS_{k})g^{\text{DOWN}}_{k,p,s}-g^{\text{PV}}_{k,p,s}\big{)}\\\
&+g^{\text{LS}}_{k,p,s}\times
C^{\text{LS}}_{k,p}(g^{\text{LS}}_{k,p,s})+g^{\text{PV}}_{k,p,s}\times
C^{\text{PV}}_{k,p}\bigg{)}+IC^{\text{STOR}}\times
cap^{\text{STOR}}_{k}+IC^{\text{PV}}_{k}\times
cap^{\text{PV}}_{k},\end{split}$ (4a) subject to $\displaystyle
g^{\text{LS}}_{k,p,s}$ $\displaystyle\leq$ $\displaystyle
G_{k}^{\text{LS,MAX}},\>\>\forall p,s,\>\>(\mu^{1}_{k,p,s}),$ (4b)
$\displaystyle g^{\text{LS}}_{k,p,s}$ $\displaystyle\geq$ $\displaystyle
0,\>\>\forall p,s,\>\>(\mu^{1\text{A}}_{k,p,s}),$ (4c) $\displaystyle
g^{\text{UP}}_{k,p,s}$ $\displaystyle\leq$ $\displaystyle
FAC_{k}^{\text{STOR}}\times(INT^{\text{STOR}}_{k}+cap^{\text{STOR}}_{k}),\>\>\forall
p,s,\>\>(\mu^{2}_{k,p,s}),$ (4d) $\displaystyle g^{\text{DOWN}}_{k,p,s}$
$\displaystyle\leq$ $\displaystyle
FAC_{k}^{\text{STOR}}\times(INT^{\text{STOR}}_{k}+cap^{\text{STOR}}_{k}),\>\>\forall
p,s,\>\>(\mu^{3}_{k,p,s}),$ (4e) $\displaystyle g^{\text{PV}}_{k,p,s}$
$\displaystyle\leq$ $\displaystyle
NORM^{\text{PV}}_{p,s}\times(INT^{\text{PV}}_{k}+cap^{\text{PV}}_{k}),\>\>\forall
p,s,\>\>(\mu^{5}_{k,p,s}),$ (4f)
$\displaystyle\sum^{p^{\prime}+h-1}_{e=p^{\prime}}\big{(}g^{\text{UP}}_{k,e,s}-g^{\text{DOWN}}_{k,e,s}\big{)}$
$\displaystyle\leq$ $\displaystyle
INT^{\text{STOR}}_{k}+cap^{\text{STOR}}_{k},\>\>\forall
s,p^{\prime},h,\>\>(\mu^{6}_{k,p^{\prime},h,s}),$ (4g)
$\displaystyle\sum^{p^{\prime}+h-1}_{e=p^{\prime}}\big{(}g^{\text{DOWN}}_{k,e,s}-g^{\text{UP}}_{k,e,s}\big{)}$
$\displaystyle\leq$ $\displaystyle 0,\>\>\forall
s,p^{\prime},h,\>\>(\mu^{7}_{k,p^{\prime},h,s}),$ (4h) $\displaystyle
g^{\text{LS}}_{k,p,s}+(1-LOSS_{k})g^{\text{DOWN}}_{k,p,s}+g^{\text{PV}}_{k,p,s}$
$\displaystyle\leq$ $\displaystyle
D^{\text{REF}}_{k,p}+g^{\text{UP}}_{k,p,s},\>\>\forall
p,s,\>\>(\mu^{8}_{k,p,s}),$ (4i)

where the parameter $LOSS_{k}$ is a percentage that represents the loss factor
associated with storing electricity for consumer group $k$.

The marginal cost function associated with load shedding is

$\displaystyle C^{\text{LS}}_{k,p}(x)$ $\displaystyle=$ $\displaystyle
A^{\text{LS}}_{k,p}+B^{\text{LS}}_{k,p}x,$ (5)

while $C^{\text{PV}}_{k,p}$ represents the marginal of PV generation.

The parameter $G_{k}^{\text{LS,MAX}}$ is the maximum amount of electricity
that consumer group $k$ can shed their load by in each period and scenario
while $INT^{\text{PV}}_{k}$ represent the initial PV generation capacity
consumer group $k$ holds. Furthermore, the parameter $INT^{\text{STOR}}_{k}$
represents the initial amount electricity consumer group $k$ may keep in
storage while $FAC_{k}^{\text{STOR}}$ represents the percentage that consumer
group $k$ can use of their storage in each period and scenario $s$.

Constraints (4b) - (4f) constrain the amount of electricity consumer group $k$
can generate or store in time period $p$ and scenario $s$. Constraint (4g)
ensures consumer group $k$ cannot, over a $|H|$-timestep period, store more
electricity than its storage capacity The index $h$ presents timesteps in a
$|H|$-timestep period while the index $p^{\prime}$ represents starting points
for the storage period, i.e., $p^{\prime}\in
P^{\prime}=\\{1,|H|+1,2|H|+1,...\\}\subseteq P$. Constraint (4h) ensures
consumer group $k$ cannot, over the same $|H|$-step time period, use more
electricity for meeting demand than what has already been stored. Constraints
(4g) and (4h) also ensure that any electricity stored in a $|H|$-timestep
period cannot be used in any other $|H|$-timestep period. In reality, if a
consumer stores electricity, they will be able to use it in any time period in
the future. However, because we set $|H|=48$ in Section 3, we believe this
simplification is reasonable given the daily trough and peak structure of
electricity demand. In addition, this simplification reduces the size and
complexity of the model and hence increases computational efficiency.

Constraint (4i) ensures any electricity consumer group $k$ generate on their
own, either from PV, or storage must be less than their reference demand plus
any increased consumption due to storing electricity. In other words, consumer
group $k$’s own generation cannot be used to meet other consumers’ demand. All
of consumer group $k$’s primal (decision) variables are also constrained to be
non-negative.

Consumer group $k$’s problem is convex, assuming all values for
$B^{\text{LS}}_{k,p}$ are non-negative. Its KKT conditions are presented in
the online appendix11footnotemark: 1. In Section 3 and 4 we also an
independent storage operator. Its optimisation problem is a reduced form of a
consumer group’s problem.

###  2.3 Market clearing conditions

The $|F|+|K|$ optimisation problems are connected via the following market
clearing conditions:

$\sum_{f,t}gen_{f,t,p,s}=\sum_{k}\big{(}D^{\text{REF}}_{k,p}-g^{\text{ls}}_{k,p,s}+g^{\text{up}}_{k,p,s}-(1-LOSS_{k})g^{\text{down}}_{k,p,s}-g^{\text{pv}}_{k,p,s}\big{)},\>\>\forall
p,s,\>\>(\gamma_{p,s}),$ (6a) $\sum_{f,t}DR_{t}\times
cap^{\text{bid}}_{f,t}=TARGET,\>\>(\kappa),$ (6b)

Market clearing condition (6a) ensures that the total amount of electricity
generated by the firms must equal the sum of the consumers’ demand. Consumers’
demand consists of their reference demand plus any electricity they store less
any electricity they shed or generate themselves. Market clearing condition
(6b) ensures that the capacity bids of the firms must equal the capacity
target level. The variables $\gamma_{p,s}$ and $\kappa$ are the prices and
free Lagrange multipliers associated with conditions (6a) and (6b),
respectively.

As each of the optimisation problems are convex, the KKT conditions are both
necessary and sufficient for optimality for each type of player [Gabriel et
al., 2012]. Thus, the stochastic MCP consists of conditions (A.1) - (A.21)
(see the Online Appendix11footnotemark: 1) in addition to the market clearing
conditions (6).

###  2.4 Solving the problem

For computational efficiency, the MCP is solved using Benders Decomposition.
The Benders Decomposition algorithm used in this paper is described in detail
in Online Appendix B11footnotemark: 1. It follows those used to solve MCPs in
Egging [2013], Gabriel and Fuller [2010] and Bertsch et al. [2018]; the
overall scheme is presented in Online Appendix B.1. Solving the stochastic MCP
in this paper involves two steps:

  1. 1.

The MCP is solved for a selection of 24 days in hourly resolution. These days
represent 8-day periods in winter, spring/autumn and summer, resulting in a
total of 24 $\times$ 8 $\times$ 3 = 576 hourly time steps. We choose 8-day
periods, instead of 7-day periods (weeks), to ensure an even number of days in
each period. This is because we choose $H=48$ timesteps for each storage/load
shifting period and thus require an even number of days to ensure there is
equal amount of timesteps in each storage/load shifting period. The revenues
and costs from each hour in the objective function of each player are
multiplied by weighting factors of 11.375, 22.750 or 11.625 depending on
whether that hour represents a week in winter, spring/autumn or summer,
respectively. This ensures that the 576 hourly time steps represent a full
year of 8784 hours. To determine the weighting factors we divided the whole
year into seasons and assigned 91 days (13 weeks) to winter, spring and autumn
respectively and 93 days to summer. To reduce the computational intensity of
the model, spring and autumn are represented by the same week explaining the
higher weighting factor of the time steps representing these seasons.

For this first step, a Benders Decomposition algorithm is used to solve the
MCP. Benders Decomposition is a solution algorithm that has been shown to
solve stochastic MCPs in a computational efficent manner [Egging, 2013,
Gabriel and Fuller, 2010]. Instead of solving for all variables simultaneously
it divides the MCP into multiple smaller MCPs and solves the overall problem
iteratively. At each iteration, a first-stage master problem MCP is solved for
a subset of variables. The values of the remaining variables are determined by
second-stage sub-problem MCPs.

In the model presented above, the different timesteps and scenarios are
connected through the investment, exit and capacity bid variables. If these
primal variables are fixed to specific values, the model could be solved for
each storage period and scenario separately. Hence these variables are the
first-stage variables and are also known as the complicating variables.

The master problem MCP of this work is presented in Online Appendix B.2. Its
initial constraints are those of the overall problem that involves these
variables. Constraints not involving any of these variables are not included
in the master problem MCP. At each iteration, new constraints, known as
Benders cuts, are added to the master problem MCP using the results from the
preceding sub-problems. Consequently, the solution obtained is improved at
each iteration. The algorithm stops when the first-stage variables produced by
the master problem stop changing. A metric derived in Gabriel and Fuller
[2010] is used to measure this convergence; see Online Appendix B.3.

The sub-problem MCPs can be represented by the overall MCP presented in
Section 2. However, each sub-problem is solved with investment, exit and
capacity bid variables fixed at the values determined by the master problem
MCP of the same iteration. The second-stage variables are all scenario
specific and, in this work, are the firms’ generation variables in addition to
each of the consumers’ load shedding, PV, charging, and discharging decision
variables.

  2. 2.

The optimal investment, exit and capacity bid variables from the first step
are fixed as parameters in the second step. The MCP is then solved 93 times
(without Benders decomposition), each time representing a different 48-hour
period in a 366-day year. As before, we choose a 366-day year to ensure an
even number of days and thus ensure there is an equal amount of timesteps in
each 48-hour storage period. As inter-temporal constraints do not exceed
beyond 48 hours, splitting the MCP into smaller problems is equivalent to
solving a single MCP with 8784 timesteps, as investment, exit, and capacity
bid decisions are fixed. However, splitting the problem up into multiple
smaller problems is more computationally efficient [Devine et al., 2016]. In
addition, we believe it is reasonable to assume that investment/exit decisions
are taken at times separate to generation, storage and load shedding
decisions.

The outputs of the model are the optimal investment, exit and capacity bid
decisions from the first step and the optimal generation, load shedding and
storage decisions and resulting prices from the second step. From this model
output, we can calculate consumer costs, generators profits and CO2 emissions.

##  3 Input data

The Irish government in its climate action plan [Government of Ireland, 2023]
has committed to significant RES-E deployment on both the demand and supply
sides. We apply the model described in section 2 to a case study based on the
future Irish power system. For this purpose, we mainly use data for 2030 from
EirGrid [2019]. The equilibrium problem is solved over $|P|=8784$ hourly time
periods. Demand side data are described in section 3.1 while in section 3.2 we
describe data for an independent storage operator. We describe input data
related to the supply side in section 3.3 and data related to renewable
generation in section 3.3.1.

###  3.1 Demand side data

We consider four different consumer groups on the demand side. These include
commercial/industrial ($k=1,2$) as well as residential consumers ($k=3,4$) and
within each of these, we distinguish between traditional consumers ($k=1,3$)
and prosumers ($k=2,4$), i.e. consumers that have their own generation from
solar PV modules and/or invest in storage capacity. Traditional consumer
groups’ investment decisions are fixed at zero.

In Section 4 we vary the percentage of consumers who are prosumers by varying
the size of the different consumer groups’ reference demand. For instance, if
the residential prosumer group’s reference demand is $\alpha\%$ of total
residential reference demand for period $p$ and scenario $s$, then the
reference demand of residential consumer group that cannot invest in demand-
side flexibility is $(1-\alpha)\%$ of total residential reference demand for
the same period. Likewise for industrial consumers.

Total reference demand profiles of the industrial and residential consumer
groups can be found in the Online Appendix11footnotemark: 1. Based on EirGrid
[2019], we consider an overall annual electricity demand of 39 TWh and a peak
demand of 6503 MW. We calculate the capacity target to be equal to the system
peak demand.

We allow consumers to shed all their load. The parameters for the marginal
load shedding cost functions (5) follow from from Devine and Bertsch [2018]
and can be found in the Online Appendix11footnotemark: 1. Devine and Bertsch
[2018] use value of lost load (VOLL) estimates by Leahy and Tol [2011] for
different types of consumers in Ireland to derive load shedding cost functions
for industrial and residential consumers.

All four consumer group are assumed to have no initial storage capacity. For
consumer groups who cannot invest in storage we fix
$cap^{\text{STOR}_{k=1,3}}=0$ while for prosumers, the storage investment
costs is $IC^{\text{STOR}}=$€32,248/MW. This assumes a lifespan of 20 years
and an interest rate of 7.1% [Pietzcker et al., 2021]. We assume that
consumers may charge (or discharge) the full storage capacity in any given
hour, i.e., $FAC^{\text{STOR}_{k=2,3}}=1$. The storage loss parameter is 10%
[Pietzcker et al., 2021], i.e., $LOSS_{k=2,4}=0.1$.

The prosuming residential and industrial consumer groups each have initial PV
capacity of 62.5MW [EirGrid, 2019]. We assume industrial prosumers invest in
large-scale PV generation while residential prosumers invest in small-scale PV
generation. That is, we assume annualised PV investment costs are
$IC^{\text{PV}}_{k=2}=$ €44,119/MW and $IC^{\text{PV}}_{k=4}=$ €61,913/MW.
These assumes a lifespan of 25 years and interest rate 0f 7.1% and 6.8%,
respectively, [Pietzcker et al., 2021, Kost et al., 2018]. Installation costs
for residential PV are higher due to the economics of scale associated with
industrial PV.

We assume that the retail price premium $X^{\text{PREM}}_{k=1,2}=$ €35/MWh for
industrial consumers and $X^{\text{PREM}}_{k=3,4}=$ €125/MWh for residential
consumers. These were obtained by comparing average Irish wholesale
prices222https://www.statista.com/statistics/1271371/ireland-monthly-
wholesale-electricity-price/ for 2022 with average industrial and residential
prices333https://www.seai.ie/data-and-insights/seai-statistics/key-
statistics/prices/ in Ireland for 2022 and 2021.

###  3.2 Independent storage operator

In this work, we consider an independent Storage Operator (SO) as a player in
the equilibrium model. The SO’s optimisation problem is a reduced form of a
consumer group’s problem (equations (4)), labelled $k=5$. The SO buys
electricity to charge its batteries and then subsequently discharges the
batteries and sells that electricity. It does so to minimise its costs
(maximise its profits). It buys and sells electricity at the wholesale price
and thus $X^{\text{PREM}}_{k=5}=0$.

The SO does not have any demand ($D^{\text{REF}}_{k=5,p,s}=0\>\forall p,s$).
Consequently, it cannot invest in, nor generate, PV generation, nor can it
shed any load. To allow the SO to sell its discharged electricity to the grid,
constraint (4i) is excluded from its optimisation problem.

In line with EirGrid [2019] values for 2030, the SO has an initial capacity of
$CAP^{\text{STOR}}_{k=5}=700$MW and may invest in further storage capacity at
the same cost of the consumers (Section 3.1). Furthermore, as with the
consumers, it may charge (discharge) its capacity fully in any given hourly
period and we assume a storage loss factor of 10%.

###  3.3 Supply side data

On the supply side, we consider $|F|=5$ generation firms who hold and may
invest in multiple generating technologies. The technologies considered are
coal, CCGT, oil, hydroelectric, wind and solar. For wind, we consider three
different investment options (wind1, wind2, and wind3), each representing
different wind regions in Ireland and hence generating profiles; see Section
3.3.1, Bertsch et al. [2018] and Devine and Bertsch [2023] for further
details.

The initial installed capacity portfolio is presented in Table 5. The maximum
capacity values for each technology are broadly based on EirGrid [2019].

Table 5: Initial power generation portfolio by firm ($CAP_{f,t}$).  Technology |  Firm 1 |  Firm 2 |  Firm 3 |  Firm 4 |  Firm 5
---|---|---|---|---|---
Coal |  570 |  0 |  476 |  0 |  0
Oil |  338 |  696 |  258 |  0 |  0
CCGT |  1992 |  464 |  648 |  1370 |  1370
Hydro |  508 |  0 |  0 |  0 |  0
Wind1 |  709 |  241 |  288 |  285 |  285
Wind2 |  882 |  300 |  358 |  355 |  355
Wind3 |  605 |  206 |  245 |  243 |  243
Solar |  49 |  17 |  20 |  20 |  20

Table 6 shows the costs and carbon emissions associated with the different
generating technologies. To calculate marginal costs for coal, oil and CCGT,
we assume power plant efficiencies 38%, 42% and 60%, respectively, and a
carbon price of €129/t CO2. Moreover, we assume the fuels associated with
these technologies cost €3/GJ, €14.3/GJ, and €7.1/GJ, respectively.

To calculated annualised investment costs, we assume investment costs of
€1800/kW, €400/kW, €900/kW, €2500/kW, €1137/kW, and €395/kW and lifespans of
45, 45, 40, 80, 25, and 25 years for coal, oil, CCGT, hydro, wind, and solar,
respectively. We also assume maintenance costs are 2%, 4%, 3%, 3%, 3%, and 2%
of fixed costs for each of these technologies, respectively. All these
parameters are based off values for 2030 from Pietzcker et al. [2021].

We assume interest rates of 10.2% for thermal generation, 7.5% for wind
generation and 7.1% for solar generation [Kost et al., 2018]. We assume a FiP
($X_{t}$) of €23/MW [Farrell et al., 2017] for wind and solar and a FiP of
zero for all other technologies (In Section 4 we also consider cases where
$X_{t}=0$ $\forall t$. ).

Table 6: Summary of techno-economic input data of considered supply side technologies.  Technology |  Annuity of specific invest |  Fixed O& M costs |  Marginal power gen. costs at intercept |  Spec. CO2 emissions
---|---|---|---|---
|  ($IC^{\text{GEN}}_{t})$ |  ($MTC_{t}$) |  ($C^{\text{GEN}}_{t}$) |  -
|  (€/GW y) |  (€/GW y) |  (€/MWhel) |  (t CO2/MWhel)
Coal |  1800 |  186 |  146.22 |  0.347
Oil |  400 |  41 |  211.64 |  0.29
CCGT |  900 |  94 |  85.60 |  0.2
Hydro |  2500 |  255 |  0.00 |  0
Wind |  1137 |  102 |  0.00 |  0
Solar |  395 |  34 |  0.00 |  0

For capacity market derating factors ($DR_{t}$), we assume a value of zero for
wind and solar, and a value of one for all other technologies. This is because
we assume hydro and fossil-fuel based technologies are always available for
generation while wind and solar generation are, at times, not available at all
for generation.

####  3.3.1 Renewable power generation data

As with Lynch et al. [2019a] and Devine and Bertsch [2023], data from the
MERRA2 reanalysis [Bosilovich et al., 2016] were used to generate input wind
and PV data. Note that wind and solar PV are not only variable but also
uncertain and their uncertainties are correlated since both depend on the
meteorological conditions. It is therefore important to take these
correlations into account when providing input data for the stochastic MCP.

To calculate, wind capacity factors ($NORM^{\text{WIND}}_{t,p,s}$), the method
described in Cradden et al. [2017] was used to convert MERRA2 wind data to
wind power. Similarly, to calculate hourly PV capacity factors
($NORM^{\text{PV}}_{p,s}$), the MERRA2 variables SWGDN and T2M were converted
into PV power using a phyiscal PV model [Ritzenhoff, 1992, Ruppert et al.,
2016, Schwarz et al., 2018b, a]444All values for $NORM^{\text{WIND}}_{t,p,s}$
and $NORM^{\text{PV}}_{p,s}$ are provided in the online
appendix11footnotemark: 1. The values for $NORM^{\text{WIND}}_{t,p,s}$ are
split into three different wind regions in Ireland - see Bertsch et al. [2018]
for details.. The hourly wind and solar capacity factor time series of 35
years were then clustered into six representative years. Thus we consider
$|S|=6$ scenarios for the MCP model described in Section 2. The renewable data
generation and the clustering procedure are described in further detail in
Bertsch et al. [2018] while the probabilities and chosen years of occurrence
are summarised in Table 7. To ensure that the spatial and temporal
correlations between wind and PV are preserved, we use these historical wind
and PV data as a basis for our analysis. We consider 250MW of solar PV
capacity, 125MW split amongst for the generating firms (Table 5) and 125MW
split evenly between amongst the two prosumer consumer groups. Furthermore,
Table 5 shows that we consider an initial 5600MW of installed wind capacity.
We base these assumptions around installed RES-E capacity estimates for 2030
from EirGrid [2019].

Table 7: Representative years chosen for RES scenarios and corresponding probabilities of occurrence; see Bertsch et al. [2018] Year | 1983 | 1998 | 2001 | 2003 | 2004 | 2015
---|---|---|---|---|---|---
Probability of occurrence ($PROB_{s}$) | 0.486 | 0.286 | 0.086 | 0.086 | 0.029 | 0.029

We assume a FiP payment of $X_{t}$ = 23 €/MWh for wind and PV generation.
These values are obtained from Farrell et al. [2017]. A FiP is not considered
for any other technology.

##  4 Results

In this section, we present the results of our study. In Section 4.1 we study
the interaction between market power and investment decisions on the demand
side and the supply side. As a sensitivity analysis, we analyse the effects of
the absence vs. presence of revenues from a Feed-in Premium for renewables in
Section 4.2. In Section 4.3 we examine carbon emissions for each of the cases
presented. In cases where market power is present in the model, all generating
firms are price-makers, modelled _á la_ Cournot, while in cases where no
market power is present, all firms are modelled as price-takers, i.e., the
market is perfectly competitive. Throughout all analyses, we consider four
different levels for the proportion of prosumers in the model, 0%, 33%, 67%,
and 100%.

###  4.1 Interaction of market power and investment decisions

Throughout Section 4.1, we assume that the generators receive a Feed-in
Premium for renewable generation in addition to the energy market revenues.

####  4.1.1 Demand side effects

(a) PV investments (MW)

(b) Storage investment (MW); market power cases only

Figure 1: Self-sufficiency investments under Market Power (MP) and Perfect
Competition (PC)

We start by examining the interaction between market power and self-
sufficiency investments. In addition, we also discuss the effects market power
has on consumer tariffs. Figures 1(a) and 1(b) display PV and storage
investments by residential and industrial consumers, respectively.

In the absence of market power, Figure 1(a) shows that residential prosumers
invest in PV with the number of investments increasing as the % of prosumers
increases, as one could expect. However, industrial prosumers do not invest in
PV in the absence of market power. This is mainly driven by the higher “retail
premium” (difference between wholesale and retail prices) of residential
prosumers as compared to industrial consumers, which basically makes PV-based
self-sufficiency more attractive for residential consumers despite the
slightly higher technology costs of residential-scale PV investments.

There are zero investments in storage when market power is absent. While more
than 2GW of conventional generation capacity are expected to be decommissioned
in both cases (with and without market power), a significant amount of
flexible and dispatchable conventional generation capacity will remain
available in the system. Therefore, in the absence of market power, there is
no need in the studied system for additional flexibility in the form of
storage. And since prices in the absence of market power are comparatively low
(see Figure 7 on page 7), there is also no sufficient incentive for prosumers
to invest in storage.

In the presence of market power, Figure 1(a) shows substantially increased
investments in PV by residential prosumers. In addition, in contrast to the
perfect competition case, industrial prosumers also now invest in PV. As the %
of prosumers goes from 33% to 67%, PV investments by industrial prosumers
increase from 2038MW to 3527MW. However once all consumers are prosumers,
industrial prosumers reduce their PV investments to 2840MW, despite extra
industrial consumers with the opportunity to invest in self-sufficiency. This
is because there are also further residential prosumers who increase the level
of PV investments in that group, which leads to reduced system prices and
consequently reduced profitability of PV-based self-sufficiency for industrial
prosumers.

Figure 1(b) displays investments in battery storage for the market power
cases. When there are no prosumers, only the storage operator invests into
storage (in addition to the initial 700MW it begins with). However, once
prosumers enter the market, residential prosumers begin to invest in battery
storage. Because all residential consumers buy electricity at the retail
price, they have more savings to gain from increased self-sufficiency by
investing in storage compared to the independent storage operator who
generates margins from arbitrage in the wholesale market – particularly in the
presence of market power. These prosumers use storage to shave peak prices and
also to increase their self-sufficiency, which helps them reduce the volume of
and costs for electricity consumed from the grid. This reduces profitability
for the independent storage operator who consequently does not invest in
further storage. While only residential prosumers invest in storage, all
consumers and prosumers still benefit from the reduced prices, particularly at
peak times.

(a) Perfect competition

(b) Market Power

Figure 2: Consumer tariffs (€/MWh ) for traditional consumers (Con) and
prosumers (Pro)

Figure 2 displays the resulting consumer tariffs. They are calculated for each
consumer group by dividing their objective function value (equation (4a)) by
their reference demand ($D^{\text{REF}}_{k,p}$). In perfect competition, there
is no difference between the traditional industrial consumers and the
industrial prosumers. This is because the industrial prosumers do not invest
in self-sufficiency, despite having the opportunity to do so. In contrast,
residential prosumers do invest in PV under perfect competition and,
consequently, their tariffs reduce as compared to the residential consumers
who do not prosume.

When market power is assumed, tariffs increase for all consumers, as expected.
For industrial consumers, the ability to prosume reduces their tariffs by 3-13
€/MWh, depending on the number of prosumers assumed. For residential
consumers, savings offered by prosuming are 24-40 €/MWh. These savings are
greater than the reductions in wholesale prices observed in Figure 7. This is
because of investment in battery storage by the prosumers, enabling them to
move their demand to lower priced periods of time. In addition, self-
sufficiency allows the prosumers to avoid paying the retail price including
the premium on top of the wholesale price.

Interestingly, as the % of prosumers increases, the tariff for traditional
consumers decreases. This shows that those who do not invest in self-
sufficiency still benefit from the investment of other consumer groups. That
is, investments by prosumers reduce the price-making ability of the firms and
hence lead to reduced prices for all.

Moreover, as the % of prosumers increases, the tariff for prosumers increases,
particularly in case of residential prosumers. This suggests that when the %
of prosumers are low, prosumers can make bigger savings but these specific
savings in €/MWh somewhat diminish as further consumers become prosumers. In
other words, while investments in self-sufficiency increase as the % of
prosumers increases, the marginal benefits of these additional investments
decrease.

Finally, we also study the impact of market power on load shedding. We find
that the increased prices under market power make load shedding an
economically attractive option and that the ability to prosume can reduce
total annual load shedding by around 8%. However, we also find that the
ability to prosume does not affect the max hourly load shedding.

####  4.1.2 Supply side effects

Figure 3: Wind investments under Market Power (MP) and Perfect Competition
(PC)

In this section we examine the impacts market power and self-sufficiency
investments have on the generation firms’ investment decisions, their profits,
and wholesale prices. Figure 3 displays investments for wind in greater detail
by firm. For each of the cases considered, we do not observe any investments
in any other generating technology (Coal, Oil, CCGT, Hydro) despite the firms
having the ability to do so.

In the absence of market power, total wind investment is similar across the
different % of prosumers cases considered. For each test case, the firms
invest the same amount as each other in perfect competition as wind investment
costs are the same for each firm.

While PV investments by prosumers increase as the % of prosumers increases,
total PV investments are also relatively constant across the % of prosumers
test cases, in the absence of market power. In the case where there are no
prosumers, only the generating firms invest in PV of course. As the % of
prosumers increases, the firms decrease their investment in PV while
residential consumers increase theirs.

In the presence of market power, Figures Figure 1(a) and 3 shows substantially
increased investments in PV but not for wind. More specifically, when there
are no prosumers, the presence of market power slightly increases total wind
investment (3203MW to 3415MW). However, the increased PV investments under
market power resulting from the increased % of prosumers, lead to decreased
investments in wind. So much so that when 100% of consumers are prosumers,
there are less wind investments (2985MW) compared to the equivalent perfect
competition case (3180MW). The decreased prices resulting from increased PV
lead to less investments in wind by the generation firms. Although, it is
important to note that, despite reduced investments by the firms, their
profits - as expected - are substantially higher when market power is present
(Figure 4).

In contrast to the perfect competition cases, there is heterogeneity in the
firms’ wind investment. In the presence of MP, firm $f=1$ does not invest in
wind. In turn firm $f=2$’s investment in wind are the highest, followed by
firm $f=3$. Firm $f=4$ and firm $f=5$’s investment in wind are similar. The
firms’ investment in wind is inversely proportional to their initial wind
generation capacity (Table 5). This is because we model the firms’ market
power _á la_ Cournot, where firms reduce their generation levels in order to
increase the system price.

Firm $f=1$ has the largest generating portfolio and thus strategically reduces
its generation the most. Consequently, if firm $f=1$ was to invest in wind, it
would not use that generation all that often. Hence, in the presence of market
power, it is not profitable for firm $f=1$ to invest in wind. Similarly, the
next largest firms ($f=4$ and $f=5$) do not invest as much in wind as firms
$f=2$ and $f=3$ as they do would not use it as much, due to similar strategic
behaviour. Another effect of assuming all generating firms behave _á la_
Cournot competition is that we observe an equalizing effect. That is, the
difference in the size of the firms’ portfolios post-investment is smaller
than pre-investment.

(a) Market Power

(b) Perfect competition

Figure 4: Firm profits per MW installed (€hundreds of thousands)

Figure 4 displays the firms’ profits per MW installed. Firm $f=1$ has the
largest profit per MW installed. This is because firm $f=1$ has the largest
portfolio of renewable generators (wind, PV, and hydro; see Table 5), which
make the largest profits due to their low variable power generation costs.

In the absence of market power, each firms’ profits per MW installed increase
as the % of prosumers grows. This is because firms decrease investment in PV
as more prosumers enter the market. While this has little effect on the firms’
absolute profits, it increases their profits per MW installed.

As expected, once market power is assumed in the model, profits for all firms
increase substantially. Again, once prosumers enter the market, profits of
firms 2 - 5 increase. However, this increase is not monotonic as these firms’
capacity after investment and decommissioning changes very little, as the % of
prosumers grows from 33% to 100%.

The exception to this is firm $f=1$’s profits under market power. Firm $f=1$
does not invest in PV when market power is present, regardless of the number
of prosumers. Hence, as the % of prosumers increases, firm $f=1$’s profits
decrease due to declining prices.

The absolute profits of all generating firms increase when the assumption of
perfect competition changes to market power, regardless of what (fixed)
percentage of prosumers is assumed. When the level of prosumers increases, the
profit of all firms (absolute) decreases, regardless of whether market power
is assumed or not.

Figure 5: Storage operator’s profits (€Millions)

Figure 5 displays the independent storage operator’s profits. In perfect
competition, they do not invest into further storage. The profits they make
are from the initial 700MW of battery storage we assume they have (Section
3.2). Moreover, in perfect competition, the % of prosumers has little effect
on the storage operator’s profits. This is because the storage operator
generates margins from arbitrage in the wholesale market and, under perfect
competition, the presence of prosumers does not have a massive impact on
wholesale prices.

When market power is present, the storage operator’s profits increase by €14
to €29 million, depending on the number of prosumers. However, the arrival of
prosumers reduces the storage operator’s profits. On the one hand, the self-
sufficiency by prosumers leads to reduced price peaks, which explains reduced
storage profits. On the other hand, as shown in Figure 1(b), investment in
storage switches from the storage operator to residential prosumers when
market power is present.

With regard to exit decisions, we observe that all coal generation is
decommissioned in all test cases considered. The presence of market power
increases the decommissioning of CCGT generation, particularly for firm $f=1$,
which consequently leads to decreased decommissioning of oil-fired generation.
This is because when firm $f=1$ behaves strategically, it utilises its CCGT
generation less often and because oil-fired generation has cheaper maintenance
costs than CCGT.

###  4.2 Sensitivity analysis: removal of FiP

In recent years, the capital costs associated with renewables have been
decreasing, thus questioning the need for Feed-in Premia [Finke et al., 2023].
In this section, we examine the effect that the FiP has on investments,
prices, consumer tariffs, and load shedding.

(a) Market power

(b) Perfect competition

Figure 6: Impact of FiP on system-wide investments in storage (stor), wind,
and PV (GW)

Figure 6 shows, as one might suspect, that the presence of a FiP increases
wind investment. And this holds regardless of the number of prosumers. In
contrast, the presence of a FiP has a mixed effect on PV investments. When
there are no prosumers, the FiP increases PV investments from the generating
firms. The FiP increases the firms’ revenues and hence the profitability
associated with such investments.

However, the arrival of prosumers means PV investment starts to switch from
the generating firms to the prosumers. When there is a majority of prosumers
(67% and 100%), the absence of the FiP increases PV investment compared to a
case where a FiP is available. This result appears counter-intuitive. However,
the absence of a FiP decreases wind investments and hence increases wholesale
prices (Figure 7). Consequently, the increased prices lead to increased
profitability of PV-based self-sufficiency with the level of PV investments
increasing as the % of prosumers grows. As Figure 6 shows, this result holds
regardless of whether the market is perfectly competitive or not.

The effect that the FiP has on storage investments is correlated with the
effect on PV. That is, as PV investment increases, so does storage. When
prosumers are present in the market, both investments are mainly observed on
the demand side, where storage investment can only be observed when market
prices are elevated due to market power. Particularly under market power, such
investments enable prosumers to cover a relatively large share of their
electricity demand by own generation, which helps protect them against high
market prices.

Figure 7: Average wholesale prices (€/MWh) under Market Power (MP) and Perfect
Competition (PC)

Figure 7 displays average wholesale prices with and without the FiP. The FiP
decreases wholesale prices by €9 - €15 /MWh in perfect competition and by €8 -
€14 /MWh when market power is present. This is due to increased wind
investments. The effect is largest when there are no prosumers - as discussed
above. In these cases, the FiP also increases PV investments. As the % of
prosumers grows, the size of the effect decreases. This is because of the
decreases in PV investments from prosumers when the FiP is present (see Figure
6), which can be explained by the slightly reduced attractiveness of self-
sufficiency investments resulting from higher wind investments which in turn
lead to reduced wholesale prices.

(a) 0% Prosumers

(b) 100% Prosumers

Figure 8: Impact of FiP on firms’ profits (€billions); market power cases only

Figure 8 examines the effect that the FiP has on the firms’ profits. While the
absolute profits are high, the relative changes resulting from the FiP are
generally rather small. In both the absence and presence of market power, the
FiP increases profits for firms $f=2$ \- $f=5$ as the FiP provides extra
revenues to these firms.

The FiP has a mixed effect on the largest firm, firm $f=1$. When there are no
prosumers, the FiP decreases firm $f=1$’s profits. This is because the FiP
encourages the other firms to invest in wind generation and hence this reduces
firm $f=1$’s market share. Firm $f=1$ does not invest in wind when the firms
are assumed to have market power.

However, as the % of prosumers increases, the impact of the FiP on firm
$f=1$’s profits flips. PV investments by prosumers increase as the % of
prosumers grows. The presence of a FiP switches investments back from
prosumers to firms $f=2$ \- $f=5$; increased wind from the firms and decreased
PV investment from the prosumers (Figure 6). In contrast to the prosumers, the
firms behave _á la_ Cournot with this generation. This means that the relative
decrease in wholesale prices due the FiP is not as big when there is a
majority of prosumers. This benefits firm $f=1$.

When there are no prosumers, the absence of a FiP increases the generating
firms’ profit per MW installed. This is because the firms invest in less
capacity when there is no FiP. Increasing the % of prosumers leads to the
absence/presence of a FiP having little effect on the firms’ profit per MW
installed.

Finally, we analyse the effect that the FiP has on consumer tariffs. As
expected, we find that the FiP reduces consumer tariffs across all levels of
prosumers and regardless of whether market power is present or not. Note,
however, that this finding does not account for who pays for the FiP. That is,
is it electricity consumers via their bills or is it paid via general taxation
through the exchequer. In addition, it is important to note that the impact of
the FiP on consumer tariffs is much smaller than that of market power.

###  4.3 Impact of market power, self-sufficiency and FiP on Carbon Emissions

Figure 9: Carbon emissions (Million tonne CO2/annum) under Market Power (MP)
and Perfect Competition (PC) in presence (FiP) and absence (No FiP) of a FiP

In this section, we examine the effects that both market power and the Feed-in
Premium have on carbon emissions. Figure 9 shows that market power decreases
carbon emissions by 6% - 22%, depending on the test case considered. This
effect is present when the FiP is included and excluded from the model. Market
power increases investment in PV when the FiP is present (Figure 1(a)) and
increases investment in both PV and wind when the FiP is not present (Figure
6(a)).

In the presence of market power, as the % of prosumers and hence PV and
storage investments increase, emissions decrease. PV and storage enable
prosumers to cover a relatively large share of their demand by own, climate-
friendly electricity generation, which in the end displaces fossil-fuel
generation on the supply side.

Similarly, without market power and without the FiP, carbon emissions decrease
as the % of prosumers increases. However, without market power but with FiP,
the % of prosumers does not affect carbon emissions. With the FiP, there are
high levels of investment in wind (3200MW) which lead to relatively low prices
(Figure 7). Without the FiP, investment in wind decreases by ($\tilde{}$
1800MW). This leads to higher prices and consequently increased levels of PV
investments for self-sufficiency, which increases as the % of prosumers
increases, resulting in lower emissions (Figure 6(b)).

Figure 10: Consumer costs for carbon mitigation by introducing a FiP in €/t
CO2 under market power (MP) as well as perfect competition (PC)

Figure 10 puts the total additional consumer costs from introducing a FiP into
perspective with the emissions saved (increase in consumer costs divided by
decrease in emissions). As opposed to our analysis of consumer tariffs in
Section 4.2, the total consumer costs here include the capacity costs and FiP
costs borne by the consumers in addition to the consumer objective. The
results in Figure 10 reveal that, in the absence of any prosumers, the
introduction of a FiP enables carbon emission reduction at rather low costs to
consumers (27€/t CO2 in the presence and 12€/t CO2 in the absence of market
power).555For comparison: Actual CO2 market prices in the EU’s ETS varied
roughly within a range of 50-100 €/t CO2 in 2023 and the beginning of 2024. As
soon as consumers are able to invest in PV and storage, however, the
additional benefits in terms of emission reduction that can be achieved by
introducing a FiP become increasingly expensive. This effect is even more
pronounced when market power is present. This is because the demand side
investments in PV and storage lead to carbon emission reductions and, as
discussed above, in the absence of a FiP and in the presence of market power,
market prices are highest, which makes these self-sufficiency investments
economically very attractive for consumers. As described above, the
introduction of a FiP mainly leads to higher investments in wind power, which
displaces PV investments on the demand side. That is, when demand side
investments are possible, the introduction of a FiP leads to a displacement of
one low-carbon technology by another one.

###  4.4 Quantifying further effects of self-sufficiency investments: a back
of the envelope analysis

Besides the interactions between market power, FiP, self-sufficiency
investments and emissions presented in the previous subsections, there are
further implications involved. Naturally, self-sufficiency leads to reduced
grid demand by prosumers. In most jurisdictions, all consumers and prosumers
do not just pay the wholesale price for their electricity demand from the
grid, but typically pay different (usually volumetric) charges (e.g., grid
charges, as well as other levies, taxes, or procurement margins of retailers).
In this paper, we have considered these charges in a simplified way as a
volumetric lumpsum of 35 €/MWh for industrial and 125 €/MWh for residential
consumers, which we called ‘retail premium’. It is important to note that
these charges constitute a crucial element for recovering the costs for
maintaining the electricity system as a whole. Without changing the way in
which these charges are collected (i.e. when they remain volumetric charges),
this means that reduced grid demand resulting from self-sufficiency also leads
to a cost recovery gap for the system infrastructure [cf. e.g. Bertsch et al.,
2017, Schwarz et al., 2018b, Mehigan et al., 2018, Aniello and Bertsch, 2023].

For cases, where 100% of consumers are able to invest in PV and/or storage,
Figure 11 shows the relative grid demand reduction that can be expected.
Relative grid demand reduction is defined as follows:

$\frac{\sum_{k,p}\big{(}D^{\text{REF}}_{k,p}-D^{\text{GRID}}_{k,p}\big{)}}{\sum_{k,p}D^{\text{REF}}_{k,p}},$
(7a) where consumer group $k$’s grid demand is their reference demand less any
self-consumption:
$D^{\text{GRID}}_{k,p}=D^{\text{REF}}_{k,p}+g^{\text{UP}}_{k,p,s}-(1-LOSS_{k})g^{\text{DOWN}}_{k,p,s}-g^{\text{PV}}_{k,p,s}.$
(7b)

In line with the findings presented so far, self-sufficiency and hence grid
demand reduction are higher when market power is present than under perfect
competition and they are higher for residential compared to industrial
consumers. Overall the grid demand reduction amounts to roughly 15-30% for
residential consumers and to roughly 10-15% for industrial consumers. When
multiplying the absolute grid demand reduction by the retail premia for
residential and industrial consumers respectively, this results in a total
cost recovery gap (industrial and residential consumers together) of roughly
290-620 Million €/a depending on whether market power and a FiP are present or
not. As a reference point, in the cases, where no prosumers are present in the
system at all, i.e. the retail premia would need to be paid for the complete
reference demand, the entire retail premia collected would amount to roughly
2.5 Billion €/a.

Figure 11: Relative reduction of grid demand following from self-sufficiency
under cases with market power (MP) and perfect competition (PC), each in
presence and absence of a Feed-in Premium (FiP)

##  5 Discussion, conclusions and outlook

In this work, we introduce a stochastic Mixed Complementarity Problem to model
an electricity market characterised by an oligopoly. On the supply side, we
consider a number of generating firms, each of whom hold and may invest in
multiple generating technologies. On the demand side, we consider both
industrial and residential consumers. Furthermore, we distinguish between
traditional consumers and consumers who may prosume, that is prosumers you can
invest in self-sufficiency such as PV and battery storage. In addition, we
consider an independent storage operator.

We go beyond the existing literature by studying the interactions between
market power, self-sufficiency investments on the demand side, and supply-side
investments. That is, we analyse how market power affects self-sufficiency
investments and, in turn, if self-sufficiency can help mitigate against market
power. Moreover, we carry out our analyses in a multi-revenue stream setting
where generators receive remuneration in both an energy and capacity market in
addition to receiving a FiP for renewables.

We study how both market power and self-sufficiency impact investment
decisions made by generation firms. Furthermore, we study how self-sufficiency
impacts each of the considered consumer groups differently. In addition, we
use the model to analyse the impacts of the presence/absence of a FiP. We
examine consumer tariffs, firms’ profits, wholesale prices, load shedding, and
carbon emissions and we apply our model to a stylised version of the Irish
power system for 2030.

In relation to the research questions outlined in Section 1, we can discuss
and interpret our findings as follows. First, prosumers investing in self-
sufficiency can help make markets more efficient. That is, the more consumers
that prosume and invest in PV and battery storage, the more we observe the
negative effects of market power being mitigated against. While this finding
is in line with the literature [Devine and Bertsch, 2023], the level of
mitigation observed in this work is much greater due to our consideration of
investment decisions. However, it should also be noted that, even when all
consumers prosume, self-sufficiency is still not enough to fully mitigate
against price increases resulting from market power.

Second, we observe that battery storage investments only occur when prices are
elevated due to market power. When the market is efficient (perfectly
competitive), there is no need for additional storage, particularly not on the
demand side, from a systems perspective and for the levels of renewables
considered. When market power is assumed, it is only the residential prosumers
who invest in storage. They do so in order to protect themselves against the
negative effects of market power. Despite this, they would still be better off
in a system without market power and thus without the need to invest in
storage.

Third, the presence of both market power and self-sufficiency affects the
investment decisions of generation firms. The presence of market power
increases investment for smaller generation firms but decreases investment for
the largest firm we considered. In the absence of prosumers, the assumption of
market power increases overall investment in wind and PV by the firms.
However, when prosumers are present, market power decreases overall investment
by the generation firms. These findings highlight the importance of
considering investment decisions on both the demand and supply side in
addition to market power. Moreover, the presence of market power alters firms
exit decisions, with less decommissioning of conventional peak generation and
increased decommissioning of conventional mid-merit generation. This is
because market power leads to increased investment in PV and storage on the
demand side so that the demand that needs to be covered by the supply-side
generators is reduced.

Fourth, self-sufficiency, while constituting and attractive option for
consumers/prosumers to reduce their electricity bills, can lead to challenges
concerning the cost recovery of the electricity system infrastructure. When
continuing to collect these charges (considered as retail premia in this
paper) on a volumetric basis and assuming that the gap presented was to be
filled with the reduced grid demand, retail premia would need to increase by
10-20% for industry and 20-40% for residential consumers. In turn, one would
expect that such increased retail premia would further increase the economic
attractiveness of self-sufficiency investments, which could easily lead to
spiral effect [cf. Mehigan et al., 2018]. Overall, this can result in enormous
cost recovery challenges for grid operators and retailers, but also for
consumers who cannot invest in self-sufficiency - for whatever reason [Bertsch
et al., 2017]. This calls for investigating alternative consumer tariff
structures as part of future research.

Fifth, we observe that the absence of a FiP, as expected, decreases investment
in renewables (particularly wind) by the generation firms. It is interesting
to note though that this in turn leads to increased wholesale prices and
consequently, increased investment in renewables (particularly PV) by
prosumers. Thus, the absence of a FiP leads to investment in renewables by the
prosumers. This finding is counter-intuitive as FiPs are typically introduced
to systems in order to incentivize investments in renewables. This also
explains why the absence of a FiP only has an attentuated effect on emissions.

Sixth, the presence of market power decreases carbon emissions as market power
encourages prosumers to invest in renewables. However, the effects are not as
large when all consumers are prosumers. This finding is in contrast to Devine
and Bertsch [2023] who, using similar data but do not consider investment
decisions, suggest that the presence of market power increases carbon
emissions. This finding highlights the importance of considering investment
decisions when analysing energy market models. In addition, we find that a FiP
reduces carbon emissions, but in the presence of market power and prosumers,
the effect of the FiP on emissions becomes relatively small.

As both liberalisation and self-sufficiency are features of modern electricity
markets, all of the above findings highlight the importance of including
market power in energy market models in addition investment decisions on both
the demand side and supply side. Excluding such modelling features may give
misleading results. This will be of interest to policymakers, system
operators, regulatory, i.e., those that operate on behalf of consumers, as
well as generating firms.

Critically reflecting on our approach, we wish to acknowledge some
limitations. First, modelling all generating firms as Cournot players using a
MCP may be unrealistic, particularly given that firm $f=1$ has a generating
portfolio larger than all the other firms. As suggested in Section 4,
modelling the largest firm as a Stackelberg leader may be more appropriate.
However, the MCP methodology cannot model investment decisions in an oligopoly
[Devine and Siddiqui, 2023]. Furthermore, MCPs cannot model integer decision
variables and thus several aspects of power systems such as online and start-
up costs.

Alternative methodologies in the energy market literature include Equilibrium
Problem with Equilibrium Constraints (EPECs) and Mathematical Programs with
Equilibrium Constraints (MPECs). However, such models can be computationally
difficult to solve [Pozo et al., 2017, Fanzeres et al., 2019] and can only be
solved for relatively small problems. Hence, EPECs and MPECs would not be
appropriate for the current modelling work where we consider a capacity
market, 8760 hourly time steps, inter-temporal demand-response constraints, 6
RES-E scenarios, generation firms with multiple technologies and investment
decisions, multiple types of consumers who prosume and who may also make
investment decisions. The MCP approach outlined in this paper allows us to
consider all of these market features together, which is a significant
progress beyond the state of the art.

Second, each of the players we model is considered to be risk-neutral. In
reality, it is more likely that they would make decisions in a risk-averse
paradigm [Russo et al., 2022], both in terms of operational and investment
decisions. However, modelling risk aversion in equilibrium models is only an
emerging area of Operations Research [Egging-Bratseth and Siddiqui, 2023].
Consequently, and given the complexity of the current work, we leave modelling
risk-aversion for future work.

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