# Contract design in electricity markets with high penetration of renewables:
A two-stage approach

Arega Getaneh Abate<EMAIL_ADDRESS>Rossana Riccardi
<EMAIL_ADDRESS>Carlos Ruiz<EMAIL_ADDRESS>Department of
Economics and Management, University of Brescia, 74/B Brescia, Italy  Scuola
Normale Superiore di Pisa, Piazza del Cavalieri, 7-56126 Pisa Italy
Department of Statistics, University Carlos III de Madrid, Avda. de la
Universidad 30, 28911-Leganés, Spain.  UC3M-BS Institute for Financial Big
Data (FiBiD), Universidad Carlos III de Madrid, 28903, Getafe, Madrid, Spain.

###### Abstract

The interplay between risk aversion and financial derivatives has received
increasing attention since the advent of electricity market liberalization.
One important challenge in this context is how to develop economically
efficient and cost-effective models to integrate renewable energy sources
(RES) in the electricity market, which constitutes a relatively new and
exciting field of research. This paper proposes a game-theoretical equilibrium
model that characterizes the interactions between oligopolistic generators in
a two-stage electricity market under the presence of high RES penetration.
Given conventional generators with generation cost uncertainty and renewable
generators with intermittent and stochastic capacity, we consider a single
futures contract market that is cleared prior to a spot market where the
energy delivery takes place. We introduce physical and financial contracts to
evaluate their performance and assess their impact on the electricity market
outcomes and examine how these depend on the level of RES penetration. Since
market participants are usually risk averse, a coherent risk measure is
introduced to deal with both risk neutral and risk averse generators. We
derive analytical relationships between contracts, study the implications of
uncertainties, test the performance of the proposed equilibrium model and its
main properties through numerical examples. Our results show that overall
electricity prices, generation costs, profits, and quantities for conventional
generators decrease, whereas quantities and profits for RES generators
increase with RES penetration. Hence, both physical and financial contracts
efficiently mitigate the impact of uncertainties and help the integration of
RES into the electricity system. However, each type of contract has different
quantitative impacts on the market outcomes. Moreover, risk aversion increases
profit and decreases generators’ risk under different levels of competition by
shielding them from the risk associated with cost, demand, and RES
uncertainties.

###### keywords:

Contract designs; Contracts for differences; Game-theoretical model; Renewable
penetration; Risk aversion

††journal: OMEGA (The International Journal of Management Science)

##  1 Introduction

The increasing concern about climate change, limited electricity generation
sources, reliability of electricity supply, and security of electricity
service while integrating renewable energy sources (RES) into the electricity
system have been extensively important topics in energy economics. Such models
have emerged after the liberalization of the electricity sector in several
countries since the early 1990s. Liberalization has not only brought the
active role of demand and supply in the electricity market but also has
increased inter-regional trades so that greater competition in electricity
generation has spurred efficient production, distribution, and transmission of
electricity [1, 2, 3]. In recent years, the transition from fossil-based to
RES and sustainable energy is favored by international agreements and national
plans with favorable economics, ubiquitous resources, scalable technology, and
significant socio-economic benefits [4, 5]. However, the transition and
integration of RES in the electricity system further complicate the
electricity system management to maintain supply-demand balance at all times.
This is because the stochasticity and intermittent natures of RES lead to
unstable supply and price volatility which, in turn, make market participants
bear increased financial risks [6, 7]. In addition, large RES penetration may
have a policy paradox in a liberalized market design due to its zero marginal
cost and non-dispatchability which may result in depressed and more volatile
electricity prices, which in turn makes renewables incentives to be more
expensive and counterproductive [8, 9]. In other words, market participants
could be exposed to more risk at all levels of the supply chain (production,
transmission, and distribution), which further complicates risk management
[10, 11]. Empirical literature in the electricity market has provided tools
and models to deal with the distinctive electricity features and risk
management through derivatives, which are common in commodity markets [12, 13,
14, 15, 16]. For instance, Philpott et al. [16] show how a risk averse social
plan is equivalent to competitive equilibrium when agents use dynamic coherent
risk measures and there are enough market instruments to hedge inflow
uncertainty.

For electricity generators, the derivatives markets work by allowing them to
commit part of their production through forward or/and bilateral contracts to
reduce the impact of major uncertainties on electricity outcomes. Therefore,
generators, retailers, and consumers can agree on prices, quantities, delivery
times, and other conditions within the framework of the contracts. There is a
vast number of different contracts, from plain forwards to interruptible
contracts which can improve competition in the spot market by applying
sophisticated risk management modeling [17, 12].

In particular, futures contracts become more liquid and relevant in
electricity trading to hedge price risks given the non-storability of
electrical energy and its inelastic demand. Futures markets allow trading
products spanning a large time horizon, such as one month or a quarter. Since
the electricity market depends on key variables such as fuel prices,
electricity demand, and weather, introducing financial contracts in
generators’ profit function with different generation technologies
(conventional and RES) under uncertainty is not an easy task. Moreover, as the
stability of power grids (frequently off equilibrium) is always an issue with
the high penetration of RES, designing electricity markets considering the
dynamic nature of renewables, which require a reserve capacity based on
controllable sources) and financial contracts liquidity (forward/futures) is a
nontrivial task [18].

From a methodological perspective, different mathematical models have been
proposed for determining optimal strategies in the electricity market with RES
penetration using financial contracts. Some of these consider deterministic
models, and others deal with uncertainty by using stochastic programming or
robust optimization approaches either considering market power and/or with
Nash competitive games [19, 20, 21, 22, 10]. After the pioneering work of
Allaz [19] that deals with oligopolistic generation traded on a forward market
for risk hedging, extensions with uncertainty and with different modeling
approaches have been introduced [20, 17, 12, 4].

However, there are still important gaps in the literature: (1) in solving the
analytical relationship between futures and spot markets contracts by
considering RES penetration and risk aversion simultaneously, (2) in
understanding the challenges entailed with large RES integration with better
modeling techniques and a coherent risk management, and (3) in considering
generators’ behavior with risk hedging via physical and financial assets. In
this paper, we contribute to fill these gaps by investigating how the risk
associated with the interactions between RES and conventional generators could
be managed, how participants risk aversion could affect electricity market
outcomes, and how financial derivatives (both physical and financial) can
hedge generators’ risk exposures under different market configuration under
uncertainty. More specifically, the main motivations are: (1) to better
understand the impact of high renewables penetration on existing liberalized
market designs where the market operates on the assumption that electricity
generation has a range of positive marginal costs. This further allows us to
examine sustainability adaptation goals (increasing RES penetration) while
addressing an increasing power demand, and (2) to explore explicitly how this
renewable penetration interacts with the relationship between spot and futures
markets under different levels of competition, generators’ risk aversion, and
different contract designs. This gives us insights on how market participants
and risk managers can efficiently approach a market under uncertainty by using
financial contracts.

To this end, we propose a two-stage game-theoretical model where each
oligopolistic generator optimizes its profit by integrating the futures
(physical and financial) and spot markets while conjecturing the impact its
production may have on other generators’ quantities and market prices.
Uncertainties are managed with contracts and a coherent risk measure is
introduced in the generators’ decision processes. The proposed equilibrium
model characterizes the interactions between generators with different levels
of market power (à la Cournot, and perfect competition) in a two-stage
electricity market. We consider a single futures market that is cleared before
a spot market by introducing both physical and financial contracts to evaluate
their impact and performance in the equilibrium market outcomes with high RES
penetration.

First, we develop a general model (GM) with a futures market that requires
physical delivery when the spot market takes place. Then, we consider a market
where the futures market makes use of contracts for differences (CFD). CFD is
a financial instrument that makes it possible for the players in the market to
hedge against the difference between the futures price and the spot market
price in a future time period, and serves as a market coordination mechanism.
For comparison we also analyze the spot market without the presence of futures
trading. To deal with both risk averse and risk neutral generators, we
introduce a coherent risk measure (Conditional Value at Risk, CVaR) which has
been extensively used in the literature after its introduction by Rockafellar
and Uryasev [23] for portfolio optimization problems111Although VaR is a well-
known risk measure in economic problems, it is a non-coherent risk measure
suffering from non-convexity, non-smoothness, subadditivity, etc., which makes
it undesirable in optimization programs. To avoid this problem, there is an
attractive alternative risk measure identified as CVaR also known as average
value at risk or mean shortfall. CVaR is a risk assessment technique often
used to reduce the probability a portfolio will incur large losses. This is
performed by assessing the likelihood (at a specific confidence level
$\alpha$) that a specific loss will exceed the value at risk. That is, CVaR is
derived by taking a weighted average between the VaR and losses exceeding the
VaR.. We derive analytical relationships between contracts and study the
implications of demand, production cost, and RES capacity uncertainties on
electricity market outcomes. Finally, the performance of the proposed
equilibrium model and its main properties are tested with extensive numerical
simulations. Our results show that overall electricity prices, generation
costs, profits, and quantities decrease for conventional generators, whereas
quantities and profits increase for RES generators with respect to the level
of RES penetration. Despite physical and financial contracts exhibit different
impacts on the resulting equilibrium market outcomes, they both efficiently
mitigate the impact of uncertainties and facilitate the integration of RES
into the electricity system. Moreover, risk aversion increases profit and
decreases generators’ risk in the general model for both the Cournot and
perfect competitions, which is a counter-intuitive result. That is because
contracts increase market participants’ willingness to participate in the
market as it shields them from the risk associated with production cost,
demand, and RES uncertainties.

The paper is organized as follows. Section 2 reviews the current state-of-the-
art in the electricity market equilibrium models with renewables and
underlines our contribution. Section 3 formulates the model for the two-stage
problem, showing how it can be reformulated from stage-two to stage-one in a
stochastic programming framework. Section 4 shows how the parameters are
generated, discusses the simulation results, and comments on the numerical
results, and their implications. Finally, Section 5 concludes with some
relevant remarks.

##  2 Literature review and contributions

###  2.1 Literature review

In the post-liberalized electricity market, the economics and regulation of
electrical energy systems that include an integral amount of stochastic
renewable generation have been studied via mathematical models using financial
derivatives under uncertainty [24, 25, 26, 27, 28, 29]. These financial
instruments are traded in over the counter (OTC) and/or by exchange markets as
energy commodities, and guarantee the delivery of an established amount of
electricity over a specific future period, with physical delivery or financial
settlement. The literature on electricity market equilibrium modeling with
financial derivatives has rapidly developed because of the growing need for
obtaining models that describe electricity market behavior accurately and the
need to integrate intermittent RES in the electricity system. Overall, three
streams of research are closely related to this study: futures and spot
markets for risk hedging in the electricity market, high renewables
penetration into the electricity systems, and stochastic models in the
electricity market. In what follows, we review studies relevant to each stream
and highlight the differences between this study and the existing literature.

####  2.1.1 Futures and spot markets in electricity

The futures market has become relevant for trading electricity as it helps to
hedge the volatility of spot market prices based on prior agreements. The
futures contracts have the spot price as an underlying reference in both
physical and financial settlements, where generators can reduce their risk
exposure for later delivery [19, 20, 26, 30, 31]. Futures contracts assure a
fixed price of electricity in the future while spot market contracts are
subject to uncertainties.

Allaz [19], with a simple two-period classical model of an oligopoly producing
a homogeneous good, shows forward contracting can be an effective tool in the
hands of noncompetitive producers. Allaz and Vila [20] develop a model with
two Cournot duopolies and forward market contracts. They show that at
equilibrium, each Cournot player will sell forward which may worsen their
position if they do not fully understand when and how to use these contracts
prior to their production. Powell [32] develops a model of contracting in the
electricity industry where the generators are price setters in the futures
market, and quantity setters in the spot market with risk aversion. He
concludes that the futures price is higher than the expected spot market
price.

Mendelson and Tunca [33] analyze forward and spot trading under uncertainties
among supply chain participants by utilizing new demand and cost information.
Bushnell [12] extends the Allaz and Vila model to multiple firms with
increasing costs to demonstrate that suppliers’ market power plays a key role
in the interaction between forward and spot trading. de Maere d’Aertrycke and
Smeers [34] analyze the interaction between financial instruments with spot
electricity markets with a generalized Nash equilibrium framework by
accounting for payoffs and an incomplete risk market endogenously.

There are other examples where financial derivatives have been used
extensively for risk management in the electricity market [35, 17, 36, 14, 37,
34, 27, 38]. These contracts have been used to achieve different objectives
from different modeling perspectives: for instance, physical and financial
contracts with different levels of competition in electricity market [17],
futures trading with retailers to achieve social welfare with market power
[14], electricity markets without price volatility [36], market equilibrium
model with asymmetric producers [35], the equilibrium in futures and spot
markets with oligopolistic generators and conjectural variations [37], effects
of futures markets on the investment decisions of a strategic electricity
producer [27], a risky design game (a stochastic Nash game) problem in a
complete risk market using financial products [38], are some relevant
contributions. However, none of these have solved the analytical relationship
between futures and spot markets contracts combining high renewables
penetration and risk aversion simultaneously. Even when they analyze contract
designs in detail, either they do not consider risk using a coherent measure
or do not address the generation mix. Thus, this paper tackles these gaps by
deriving the relationship between futures and spot markets contracts, settled
physically and financially and with different levels of competition under
uncertainty. In doing so, it accounts for the advocate of sustainable
development by considering high RES penetration into the generation mix where
financial contracts protect generators from the embedded risks of both
generation and demand.

####  2.1.2 High RES penetration

Because of the growing concern on conventional generation’s impact on the
environment and its high generation cost, the electricity generation mix has
been significantly changing with strict policies towards sustainable energy
adaptation. In particular, the increasing penetration of RES in electricity
systems is leading towards a green economy and lowering electricity prices.
Nevertheless, due to high uncertainty on RES capacity, zero marginal cost and
non-dispatchability222The assumption with renewables zero marginal cost,
mainly with wind and solar, is that once they start producing, the marginal
cost to continue operating is almost zero. This contradicts the idea of
liberalization that assumes a positive marginal production cost, and
dispatchability for the market to work properly., integrating RES into the
electricity systems requires a risk averse dispatch of resources to account
for the stochastic availability of renewable capacity and to hedge the price
volatility [39, 40].

In relation to this, governments have been creating better conditions (with
specific contracts, green certificates, feed-in-tariffs, and other market-
based approaches) towards investing in renewables to be integrated in
conventional electricity systems [41, 42, 43, 44].

The electricity market with high RES penetration has been studied with
different approaches. For instance, with a two-stage energy management
strategy under uncertainty [3], with multistage stochastic programming by
combining real options and forward contracts for risk management in renewable
investment [10], with a risk-based energy management of renewable-based
microgrids using information gap decision theory [45], are some of the
contributions. Since risk and uncertainty are the main challenges considering
RES, [46] design a CVaR model that enables the microgrid aggregator to exploit
the flexibility of the load to mitigate the negative impacts of the
uncertainties due to RES generation and electricity prices. Hence, CVaR is a
well-known risk measure that has been widely used in various energy management
problems for different entities in the electricity market such as for
retailers [13, 47], for producers [48, 49, 15, 50], for distributors [51], for
optimal power flow [52] and for coordinated energy trading problems [53, 54].
This paper departs from the literature by tackling uncertainties with
contracts, so that electricity market participants act upon the received
information, and understand the challenges entailed with large RES integration
with better modeling techniques and a coherent risk management.

####  2.1.3 Stochastic equilibrium models in the electricity market

Equilibrium models in electricity market can be applied to represent the
overall market behavior taking into consideration the competition among all
participants, such as, Nash equilibrium (Cournot and supply function
equilibria). In relation to this, Ventosa et al. [55] discuss research
developments in electricity equilibrium models based on: degree of
competition, time scope of the model, uncertainty modeling, inter-period
links, transmission constraints, generating system representation and market
modeling.

Specifically, in the optimization research community, various equilibrium
models in the electricity market and related problems under uncertainty have
drawn increasing attention, particularly on short-term operations, capacity
expansion, optimal dispatch, optimal power flow, RES integration, risk
management, and decarbonization, which are very important topics needed to
analyze electricity market designs. Novel mathematical model formulations and
numerical methods have been proposed to deal with those problems. [56, 57, 58,
59, 60, 61, 52, 62, 63, 64, 65, 44]. For example, Gürkan et al. [56] explore
the application of sample-path to the investments in gas production by
formulating a stochastic variational inequality problem, and Vehviläinen and
Pyykkönen [57] present a stochastic factor-based approach to mid-term modeling
of spot market prices in deregulated electricity markets. Galiana and Khatib
[58] propose a model on emissions trading that can be viewed as a mathematical
program subject to a Nash equilibrium problem, which in turn, is subject to
the Cournot-Nash equilibrium conditions of an hourly oligopolistic electricity
market. Aïd et al. [61] use an equilibrium model to study the relationship
between forward, spot and retail markets, and conclude both vertical
integration and forward hedging retail prices decrease under demand
uncertainty. Finally, Dvorkin [64] develop a chance-constrained stochastic
market design that enables to produce a robust competitive equilibrium and
internalize uncertainty of the RES in the price formation process. However,
modeling uncertainty in the electricity market by integrating conventional and
RES oligopolistic generators, considering generators with risk hedging via
physical and financial assets is still an important challenge. This inspired
us to develop game-theoretical stochastic models where the performance of
different contracts is analyzed in a two-stage electricity market under
uncertainty. The models are analytically and numerically solved while managing
uncertainties with a coherent risk measure and financial contracts. Moreover,
conventional and RES generators are considered, and equilibrium market
outcomes are compared for different levels of competition with respect to RES
penetration.

###  2.2 Contribution of the paper

In this paper, we propose three game-theoretical equilibrium models based on:
a futures contract with physical delivery, a contract for differences with
financial settlement, and a spot market contract with no futures. The GM and
CFD are solved based on a pre-existing futures market contract, assuming each
generator has an estimation (conjecture) of the impact that its decision may
have on the other generators’ strategies. The spot market with no futures is
derived for comparison purposes. Most importantly, the three spot market
equilibria are solved in closed form. The analytical relationship between
futures and spot contracts, and implications of uncertainties on market
outcomes, are examined from risk aversion, generation technologies, and levels
of competition perspectives. The models are tested with multiple case studies
based on calibrated data from the Spanish electricity market, and the
comparison of physical and financial contracts is made focusing on systems
with high RES penetration. Therefore, the main contributions of the paper are
summarized as follows:

  1. (1)

Modeling conventional and RES generators’ profit-maximizing problems with a
pre-existing financial contract, and risk aversion using the CVaR. Conjectural
variations and Nash-equilibrium approaches are assumed so that all generators
optimize their profit simultaneously. We consider two strands of contracts:
futures with physical delivery and contracts for differences settled
financially.

  2. (2)

Proposing game-theoretical models where each generator solves a two-stage
stochastic problem and examining generators’ exposure to uncertainties in
demand, conventional generators’ production cost, and RES generation
availability. The impact of these uncertainties on electricity market outcomes
are examined for both sets of generators with respect to high RES penetration.

  3. (3)

Deriving the spot market equilibrium in closed form to deal with the futures
market where the posterior numerical solution of the overall equilibrium is
reformulated as a nonlinear problem (NLP) by applying the method in Leyffer
and Munson [66], and the CVaR.

  4. (4)

Illustrating the analytical results using numerical examples that analyze the
quantitative and qualitative impacts of RES penetration, levels of
competition, and players’ risk aversion on market outcomes.

##  3 Problem formulation

We consider the profit optimization of two types of electricity generators
(conventional and RES, which includes solar, wind, or both) that trade their
electricity in a two-stage electricity market. In the first stage, generators
sell their electricity in a futures market. In the second stage, they
participate in a spot market where they trade the remaining electricity. We
derive three models. The first model is the general model (GM) with physical
delivery, and the second model is based on contracts for differences
(CFD)-settled financially. Finally, for comparison, we compute a third
equilibrium model without the presence of the futures market (only spot
market).

Figure 1: Modeling framework and solution approaches.

Figure 1 shows the overall modeling framework for the paper. In the spot
market, price and quantities are subject to uncertainty. The model formulation
starts from the second stage by assuming the first stage variables are already
settled. More specifically, the first stage decisions (optimal involvement in
the futures market) are taken by anticipating uncertain realizations of RES
and spot prices in the second stage and not a given RES generation level.
Thus, we explicitly model the uncertainties associated with spot prices and
RES quantities, whereby the energy generators use forward trading to hedge the
associated profit risk. In the second stage, the three models (the GM, the
CFD, and the spot market with no futures) are obtained in closed form. Then,
risk neutral and risk averse generators’ problems are solved by substituting
the spot market optimal solutions into the futures optimization problem using
the CVaR. The problem is a quadratic problem with quadratic constraints (QPQC)
so that we transform it into a NLP as proposed in Leyffer and Munson [66],
which gives an equivalent solution to the original problem.

###  3.1 The general model (GM) formulation

We consider $|I|$ conventional and $|J|$ RES generators producing and trading
their electricity in the futures and spot markets. We analyze a two-stage
single futures market where both generators can trade their futures quantities
($q_{i}^{F}/q_{j}^{F}$) at a futures price $P^{F}$, and they trade the
remaining quantities in a subsequent single spot market at a spot price
$P_{\omega}^{S}$, which depends on scenario $\omega$. Conventional generators
have a quadratic generation cost function to generate $q_{i}^{F}$ and
$q_{i\omega}^{S}$ in the futures and in the spot markets, respectively.
Electricity demand, conventional generators’ production costs, and
availability of RES are the sources of uncertainty in the model. Under this
market setting, generators have to make their first stage decision (amount of
electricity to sell in the futures market) anticipating those uncertainties.
These uncertainties are thus characterized by scenarios $\omega\in\Omega$ that
represents possible realizations of the final consumers’ demand, generation
cost, and availability of RES. Given this market setting, conventional
generators’ profit is expressed as:

$\Pi_{i\omega}=P^{F}q_{i}^{F}+P^{S}_{\omega}q_{i\omega}^{S}-\left(a_{i\omega}+b_{i\omega}(q_{i}^{F}+q_{i\omega}^{S})+\frac{1}{2}c_{i\omega}(q_{i}^{F}+q_{i\omega}^{S})^{2}\right)\quad\forall
i,\forall\omega$ (3.1)

where the first term stands for revenue from futures market selling, the
second term revenue from spot market selling, and the last term expresses the
quadratic conventional generator $i$’s cost function where $a_{i\omega}\geq
0$, $b_{i\omega}\geq 0$ and $c_{i\omega}\geq 0$ are cost parameters subject to
uncertainty. Unlike demand and RES uncertainties, generation cost uncertainty
is considered in a few previous studies [33, 67, 68]. Hence, we consider a
realistic quadratic cost function for conventional generators where quantity
is the sum of the electricity traded in the futures market and the spot
market, trading off the computational burden it adds to our model under
uncertainty. Conventional generators’ cost uncertainty is mainly caused by
fuel (gas, oil, and coal) prices (the interested reader can refer Cain and
Alvarado [69] and Gautam and Mithulananthan [70] for quadratic cost structure
and cost characteristics). Futures market decision variables $q_{i}^{F}$ and
$P^{F}$ are scenario-independent as the futures market is settled before the
uncertain parameters are observed.

Similarly, the profit for RES generator with zero generation cost
assumption333This is because once RES generators are built and operational,
their marginal generating cost can be assumed to zero. This justifies
economically to utilize the entire RES capacity available in the system to
satisfy the demand, since that capacity has already been paid for [71, 72].
is expressed as:

$\Pi_{j\omega}=P^{F}q_{j}^{F}+P^{S}_{\omega}q_{j\omega}^{S}\quad\forall
j,\forall\omega$ (3.2)

where the first term stands for revenue in the futures market and the second
term represents the revenue in the spot market. Let’s consider $Q_{j\omega}$
as a stochastic parameter representing the total amount of RES generator $j$
production with scenario $\omega$ which is traded in the futures and spot
markets. RES generators are nondispatchable so that
$Q_{j\omega}=q_{j}^{F}+q_{j\omega}^{S}$, and hence trade
$q_{j\omega}^{S}=Q_{j\omega}-q_{j}^{F}$ in the spot market.

Thus, ((3.2)) can be simplified as

$\Pi_{j\omega}=(P^{F}-P_{\omega}^{S})q_{j}^{F}+P^{S}_{\omega}Q_{j\omega}.$

The inverse demand curve in the spot market is expressed as:

$P_{\omega}^{S}=\gamma_{\omega}^{S}-\beta_{\omega}^{S}\left(\sum_{i\in{I}}(q_{i\omega}^{S}+q_{i}^{F})+\sum_{j\in{J}}(q_{j\omega}^{S}+q_{j}^{F})\right)$
(3.3)

with $\gamma_{\omega}^{S}>0$ and $\beta_{\omega}^{S}>0$ for the price-demand
function to be well-behaved. Since $Q_{j\omega}$ is a parameter, by
rearranging, we can simplify ((3.3)) as:

$P_{\omega}^{S}=\hat{\gamma}_{\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}(q_{i\omega}^{S}+q_{i}^{F})\quad\forall\omega$
(3.4)

where
$\hat{\gamma}_{\omega}^{S}=\gamma_{\omega}^{S}-\beta_{\omega}^{S}\sum_{j\in{J}}Q_{j\omega}$,
so that $\hat{\gamma}_{\omega}^{S}$ and $\beta_{\omega}^{S}$ are the component
of demand intercept and demand slope, at scenario $\omega$, respectively.
Demand uncertainty is represented by $\hat{\gamma}_{\omega}^{S}$ and
$\beta_{\omega}^{S}$, which are scenario-dependent parameters. In ((3.4)),
$P_{\omega}^{S}$ is the spot market price, defined endogenously wherein
$\sum_{i\in{I}}(q_{i\omega}^{S}+q_{i}^{F})$ are the quantities that can be
sold in the spot market. Since generators have the option to trade both in the
futures and spot markets, they can trade their unsold quantities from the
futures market in the spot market. Therefore, both futures and spot quantities
affect the spot market clearing price. Though it would be realistic to
consider the futures market over time (monthly, quarterly) and the spot market
over time (day-ahead, real-time), due to the complexity of our model and
analytical tractability, we assume a single futures market followed by a
subsequent single spot market where the electricity generation and delivery
takes place. Besides, instead of taking the usual assumption that the futures
price is the expected spot price, we endogenously define it with an inverse
demand curve as:

$P^{F}=\gamma^{F}-\beta^{F}\left(\sum_{i\in{I}}q_{i}^{F}+\sum_{j\in{J}}q_{j}^{F}\right).$
(3.5)

In ((3.5)), $P^{F}$ is the price in the futures market, and
$\sum_{i\in{I}}q_{i}^{F}+\sum_{j\in{J}}q_{j}^{F}$ are futures quantities for
conventional and RES generators.Defining the futures price by relaxing the
non-arbitrage condition is well-established in the literature as: (1) it is
possible to obtain the optimal equilibrium futures price by equating aggregate
demand and supply, and without making any assumption to express its
relationship with the spot price [73]. (2) in non-storable commodities, like
electricity, the forward price would be a biased forecast of the spot price
due to the presence of electricity forward risk premia [74, 75]. (3) since we
are dealing with a single-period spot market, it is not insightful to define
futures price as expected spot prices (in our case, we could only have
expectation over a finite set of scenarios).

To characterize the futures market decision variables (stage-one variables),
we compute the equilibrium in the spot market (stage-two), assuming the
futures market has already settled. As a result, the resulting spot market
prices and quantities, per scenario $\omega$, are parametrized with the
futures market decisions. To deal with the futures market equilibrium, we move
one step backward in time and use the spot market equilibrium decision
variables to characterize the global solution.

####  3.1.1 Spot market equilibrium (stage-two)

In this section, we analytically derive the equilibrium conditions in the spot
market starting from the equilibrium price-demand curve of generators in
Proposition 1 followed by the spot market equilibrium quantities in
Proposition 2.

###### Assumption 1

Each generator $k$ has an estimation of the impact that its generation
$q_{k\omega}^{S},q_{k}^{F}$ may have in the spot market price and its rival
($-k$) quantities $P^{S}_{\omega}=P^{S}_{\omega}(q_{k\omega}^{S},q_{k}^{F})$
and $q_{-k\omega}^{S}=q_{-k\omega}^{S}(q_{k\omega}^{S})\quad\forall
k,\forall\omega.$

This is an important assumption to derive both the equilibrium price and
quantity in the spot market. The assumption is based on the idea of
conjectural variations, an approach that is used widely to study oligopolistic
markets. It is used to model an oligopolistic market wherein firms react to
price by conjecturing (accurately or not) how changes in their decisions may
impact the production decisions of their competitors, which is a
generalization of standard Nash–Cournot models to represent imperfect
competition [76, 61, 77].

###### Proposition 1

Given Assumption 1 and the futures market quantities $q_{k}^{F},\forall k\in
I\cup J$, at each scenario $\omega$, the equilibrium spot market price is
expressed as:

$P_{\omega}^{S}=\varphi_{\omega}\left[\hat{\gamma}_{\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}(b_{i\omega}+c_{i\omega}q_{i}^{F})\right]\quad\forall\omega$
(3.6)

where $\tau_{i\omega}=\frac{1}{\beta_{\omega}^{S}(1+\delta_{i})+c_{i\omega}}$
and
$\varphi_{\omega}=\frac{1}{1+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}}.$

Proposition 1 shows the closed-form solution of the equilibrium spot price
(the spot market clearing price) which is expressed in terms of futures market
quantities. The parameter $\delta_{i}=\sum_{i\neq k}\frac{\partial
q_{k\omega}^{S}}{\partial q_{i\omega}^{S}}$ measures the level of competition
in the spot market, as analyzed in Lindh [78] where $\delta_{i}=-1$ means
generator $i$ behaves as a competitive generator, $\delta_{i}=0$ means
generator $i$ behaves as a Cournot generator, and $\delta_{i}=m-1$ means that
generator $i$ behaves as à la Cournot while being part of a cartel, where $m$
is the number of identical cartel members. The proof for this proposition is
found in (A).

###### Proposition 2

At each scenario $\omega$, the optimal spot quantity $q_{i\omega}^{S}$ of
conventional generator $i$, given the futures market quantity $q_{k}^{F},$
with knowledge of (3.3), and $P_{\omega}^{S}$ in (A) is expressed as:

$q_{i\omega}^{S}=\tau_{i\omega}\varphi_{\omega}\left[\hat{\gamma}_{\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}(b_{i\omega}+c_{i\omega}q_{i}^{F})\right]-\tau_{i\omega}\left(b_{i\omega}+c_{i\omega}q_{i}^{F}\right).$
(3.7) $\displaystyle q_{j\omega}^{S}=Q_{j\omega}-q_{j}^{F}\quad\forall
j,\forall\omega.$ (3.8)

The explicit formula for conventional generator $i$’s quantity in the spot
market is
$q_{i\omega}^{S}=\tau_{i\omega}(P_{\omega}^{S}-b_{i\omega}-c_{i\omega}q_{i}^{F})$
from (A.30). Then, substituting the equilibrium spot price, which is expressed
in ((A.31)) into (A.30) gives the optimal spot market quantity for
conventional generator $i$ ($q_{i\omega}^{S}$). Similarly, given $Q_{j\omega}$
i.e., the total RES production, the optimal spot market quantity can be
directly derived as ((3.8)). Since renewables are the first technologies to be
dispatched in the spot market based on merit order [71, 72, 79], they can only
sell at the spot market the energy left after supplying the already signed
futures contract ((3.8)). However, their strategic decision in the first stage
is taken simultaneously with the conventional generators as expressed in the
CVaR formulation in ((3.12)).

####  3.1.2 Futures market analysis (stage-one)

Once the spot market equilibrium conditions are obtained in closed-form, we
can go one step backward in time to deal with the futures market equilibrium
outcomes using the optimal spot market decision variables
($P_{\omega}^{S},q_{i\omega}^{S}$ and $q_{j\omega}^{S}$). Then, we
characterize the joint maximization of all generators’ expected, or risk
averse profit (with both technologies). The backward approach is a standard
solution procedure to characterized two-stage equilibrium models. In this
regard, the initial second stage equilibrium can be viewed as the optimal
response of the market as a whole, as a function of the first stage variables.
Then, taking this response into account, generators decide their optimal
first-stage strategy [20, 77].

To that end, first we start from the conventional generator’s profit function
expressed in ((3.1)), RES generator’s profit expressed in ((3.2)), and compute
the derivative with respect to futures market quantities.

$\displaystyle\frac{\partial\Pi_{i\omega}}{\partial q_{i}^{F}}$
$\displaystyle=\frac{\partial P^{F}}{\partial
q_{i}^{F}}q_{i}^{F}+P^{F}+\frac{\partial P_{\omega}^{S}}{\partial
q_{i}^{F}}q_{i\omega}^{S}+P_{\omega}^{S}\frac{\partial
q_{i\omega}^{S}}{\partial q_{i}^{F}}-b_{i\omega}\left(1+\frac{\partial
q_{i\omega}^{S}}{\partial q_{i}^{F}}\right)+$ $\displaystyle-
c_{i\omega}\left[q_{i}^{F}+q_{i\omega}^{S}\right]\left(1+\frac{\partial
q_{i\omega}^{S}}{\partial q_{i}^{F}}\right)\forall i,\forall\omega$ (3.9a)
$\displaystyle\frac{\partial\Pi_{j\omega}}{\partial q_{j}^{F}}$
$\displaystyle=\left(\frac{\partial P^{F}}{\partial q_{j}^{F}}-\frac{\partial
P_{\omega}^{S}}{\partial
q_{j}^{F}}\right)q_{j}^{F}+P^{F}-P_{\omega}^{S}+\frac{\partial
P_{\omega}^{S}}{\partial q_{j}^{F}}Q_{j\omega}\quad\forall j,\forall\omega.$
(3.9b)

###### Assumption 2

$\forall k\in I\cup J$, each generator $k$ has an estimation of the impact its
generation $q_{k}^{F}$ may have in the futures price and competitors’
generation, where $P^{F}=P^{F}(q_{k}^{F})$ and
$q_{-k}^{F}=q_{-k}^{F}(q_{k}^{F})$.

This is considered to compute $\frac{\partial\Pi_{k\omega}}{\partial
q_{k}^{F}}$, which now depends on $\psi_{k}=\frac{\partial
q_{-k}^{F}}{\partial q_{k}^{F}}$. Therefore, the different levels of
competition in the futures market can be modeled as:
$\psi_{k}=-\frac{1}{I+J-1}$ for competitive generator, $\psi_{k}=0$ for
Cournot generator, and $\psi_{k}>0$ for generator part of a cartel. From
((3.9)), we can observe there are several partial derivatives ($\frac{\partial
P^{F}}{\partial q_{i}^{F}},\frac{\partial P_{\omega}^{S}}{\partial q_{i}^{F}}$
and $\frac{\partial q_{i\omega}^{S}}{\partial q_{i}^{F}}$ (for conventional
generators), and $\frac{\partial P^{F}}{\partial q_{j}^{F}},\frac{\partial
P_{\omega}^{S}}{\partial q_{j}^{F}}$ and $\frac{\partial
q_{j\omega}^{S}}{\partial q_{j}^{F}}$ (for RES generators), which are derived
using ((3.5)), ((A.31)), ((3.7)) and ((3.8))) that can influence the futures
market (the global solution in the CVaR formulation) as we will see in the
next section.

Given the equilibrium spot market outcomes, the partial derivatives needed to
be computed for both sets of generators are:

$\displaystyle\frac{\partial P^{F}}{\partial q_{i}^{F}}$
$\displaystyle=-\beta^{F}\left(1+\sum_{i\neq k}\frac{\partial
q_{k}^{F}}{\partial q_{i}^{F}}+\sum_{j\in{J}}\frac{\partial
q_{j}^{F}}{\partial
q_{i}^{F}}\right)=-\beta^{F}(1+(I+J-1)\psi_{i})\quad\forall i$ (3.10a)
$\displaystyle\frac{\partial P_{\omega}^{S}}{\partial q_{i}^{F}}$
$\displaystyle=\varphi_{\omega}\left[\left(-\beta_{\omega}^{S}(1+(I-1)\psi_{i})\right)\right]+\varphi_{\omega}\left[\beta_{\omega}^{S}c_{i\omega}\tau_{i\omega}+\beta_{\omega}^{S}\sum_{k\neq
i}^{I}c_{k\omega}\psi_{k}\tau_{k\omega}\right]\quad\forall i,\forall\omega$
(3.10b) $\displaystyle\frac{\partial q_{i\omega}^{S}}{\partial q_{i}^{F}}$
$\displaystyle=\tau_{i\omega}\left(\frac{\partial P_{\omega}^{S}}{\partial
q_{i}^{F}}-c_{i\omega}\right)\quad\forall i,\forall\omega$ (3.10c)
$\displaystyle\frac{\partial P^{F}}{\partial q_{j}^{F}}$
$\displaystyle=-\beta^{F}\left(1+(I+J-1)\psi_{j}\right)\quad\forall j$ (3.10d)
$\displaystyle\frac{\partial P_{\omega}^{S}}{\partial q_{j}^{F}}$
$\displaystyle=0\quad\forall j,\forall\omega$ (3.10e)
$\displaystyle\frac{\partial q_{j\omega}^{S}}{\partial q_{j}^{F}}$
$\displaystyle=-1\quad\forall j,\forall\omega.$ (3.10f)

####  3.1.3 Futures market equilibrium under the CVaR

By replacing the equilibrium spot price $P^{S}_{\omega}$ and quantities
${q^{S}_{i\omega}}$ and ${q^{S}_{j\omega}}$ in the profit functions
$\Pi_{i\omega}$ and $\Pi_{j\omega}$, we can obtain parameterized profits in
the futures market decision variables:

$\displaystyle\Pi_{i\omega}=\Pi_{i\omega}(q^{F}_{i},q^{F}_{-i},q^{F}_{j\in
J})\quad\forall i,\forall\omega$ (3.11a)
$\displaystyle\Pi_{j\omega}=\Pi_{j\omega}(q^{F}_{j},q^{F}_{-j},q^{F}_{i\in
I})\quad\forall j,\forall\omega.$ (3.11b)

The risk aversion level of generators is considered by using the CVaR measure.
CVaR is a coherent444A functional
$\mathcal{R}:\mathcal{L}^{2}\rightarrow(\infty,\infty]$ is called a coherent
measure of risk in the extended sense if (1) $\mathcal{R}(C)=C$ for all
constants $C$, (2) $\mathcal{R}((1-\lambda)X+\lambda
X^{\prime})\leq(1-\lambda)\mathcal{R}(X)+\lambda\mathcal{R}(X)$ for
$\lambda\in(0,1)$ (”convexity”),
(3) $\mathcal{R}(X)\leq\mathcal{R}(X^{\prime})$ (”monotonicity”), (4)
$\mathcal{R}(X)\leq 0$ when $||X^{k}-X||_{2}\rightarrow 0$ (”closedness”), (5)
$\mathcal{R}(\lambda X)=\lambda\mathcal{R}(X)\text{for}\lambda>0$ (”positive
homogeneity”). risk measure that calculates VaR and optimizes CVaR
simultaneously. The CVaR at $\alpha$ confidence level can be defined as the
expected value of the profit smaller than the ($1-\alpha$)-quantile of the
profit distribution. In the scenario-based stochastic optimization method,
CVaRα measures the expected profit in the $(1-\alpha)\times 100\%$ worst
scenarios. The $(1-\alpha)-$quantile of the profit distribution is known as
VaR, which is the largest value ensuring that the probability of obtaining a
profit less than that value is lower than $1-\alpha$, $\forall\alpha\in[0,1]$.
Thus, the profit maximization problem solved by the risk (neutral and averse)
generators is formulated in (3.12), where $\sigma_{k\omega}$ is the
probability assigned by generator $k$ to scenario $\omega$, and $1-\alpha$
represents the level of significance associated with the CVaR. The risk averse
problem solved by each generator is [23]:

$\displaystyle\displaystyle\max_{\xi_{k},\eta_{k\omega},q_{k}^{F}}$
$\displaystyle{(1-\phi)\sum_{\omega\in{\Omega}}\sigma_{k\omega}\Pi_{k\omega}}+\phi\left[\xi_{k}-\frac{1}{1-\alpha}\sum_{\omega\in{\Omega}}\sigma_{k\omega}\eta_{k\omega}\right]$
(3.12a) s.t. $\displaystyle\eta_{k\omega}+\Pi_{k\omega}-\xi_{k}\geq
0\quad{:\mu_{k\omega}}\quad\forall\omega$ (3.12b)
$\displaystyle\eta_{k\omega}\geq 0\quad{:\theta_{k\omega}}\quad\forall\omega$
(3.12c) $\displaystyle q_{k}^{F_{min}}\leq q_{k}^{F}\leq
q_{k}^{F_{max}}\quad{:\nu_{k}^{min},\nu_{k}^{max}}\quad\forall k$ (3.12d)

The objective function is represented by the expected profit (first term for
risk neutral generators) and the CVaR (second term for risk averse generators)
multiplied by a factor $\phi\in[0,1]$ that regulates a trade-off between the
expected profit and the CVaR for a given level $\alpha$. In this setting,
$\phi=0$ corresponds to a risk neutral generator (maximization of the expected
profits), and $\phi=1$ to the most risk averse setting in which all the weight
in the CVaR is considered, as depicted in the modeling framework in Figure 1.
Parameter $\sigma_{k\omega}$ represents the scenario occurrence probability
which is assumed to be $\frac{1}{\Omega}$. At the optimal solution, $\xi_{k}$
represents the VaR, and the auxiliary variable $\eta_{k\omega}$ equals (for
each scenario) the positive difference between the VaR and the profit
$(\Pi_{k\omega})$, which is expressed with constraint ((3.12b)).Constraints
((3.12d)) enforce the lower and upper bounds for electricity schedule of
generators (both conventional and RES generators as $k=I\cup J$). Since both
generators have capacity constraints (between zero and $q_{k}^{F_{max}}$),
they decide their production based on their benefits. The dual variables
associated with constraints ((3.12b)), ((3.12c)) and ((3.12d)) are
$\mu_{k\omega},\theta_{k\omega},\nu_{k}^{min}$, and $\nu_{k}^{max}$, which are
treated as variables in the model.

The equilibrium is then obtained by solving simultaneously ((3.12)) for all
generators. This is done by replacing (3.12) by its associated Karush-Kuhn-
Tucker (KKT) system of optimality conditions. Therefore, the KKT system
associated with each generator is:

$\displaystyle\frac{\partial\mathcal{L}}{\partial
q_{k}^{F}}={-(1-\phi)\sum_{\omega\in{\Omega}}\sigma_{k\omega}\frac{\partial\Pi_{k\omega}}{\partial
q_{k}^{F}}}-\sum_{\omega\in{\Omega}}\mu_{k\omega}\frac{\partial\Pi_{k\omega}}{\partial
q_{k}^{F}}-\nu_{k}^{min}+\nu_{k}^{max}=0\quad\forall k$ (3.13a)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\eta_{k\omega}}=\phi\frac{1}{1-\alpha}\sigma_{k\omega}-\mu_{k\omega}-\theta_{k\omega}=0\quad\forall
k,\forall\omega$ (3.13b)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\xi_{k}}=-\phi+\sum_{\omega}^{\Omega}\mu_{k\omega}=0\quad\forall
k$ (3.13c)
$\displaystyle(0\leq\eta_{k\omega}+\Pi_{k\omega}-\xi_{k})\perp(\mu_{k\omega}\geq
0)\quad\forall k,\forall\omega$ (3.13d)
$\displaystyle(0\leq\eta_{k\omega})\perp(\theta_{k\omega}\geq 0)\quad\forall
k,\forall\omega$ (3.13e) $\displaystyle(0\leq
q_{k}^{F}-q_{k}^{F_{min}})\perp(\nu_{k}^{min}\geq 0)\quad\forall k$ (3.13f)
$\displaystyle(0\leq q_{k}^{F_{max}}-q_{k}^{F})\perp(\nu_{k}^{max}\geq
0)\quad\forall k.$ (3.13g)

The complementarity conditions are denoted by $(0\leq x)\perp(y\geq 0)$ which
is equivalent to: $x\geq 0,y\geq 0$ and $xy=0.$ One way to solve system (3.13)
for all $k$ is by using the technique discussed in Leyffer and Munson [66]
where a nonlinear programming reformulation of KKT systems is proposed. This
technique is based on minimizing the sum of the complementarity products
subject to the remaining KKT conditions. Solutions where the objective
function is zero guarantee that all the original KKT conditions are met, as
zero is the lower bound for the objective function (sum of nonnegative terms).
Thus, we can consider the problem in the following extended form:

$\displaystyle\displaystyle\min$
$\displaystyle\sum_{i\in{I}}\sum_{\omega\in{\Omega}}\left[\mu_{i\omega}(\eta_{i\omega}+\Pi_{i\omega}-\xi_{i})+\eta_{i\omega}\theta_{i\omega}\right]+\sum_{j\in{J}}\sum_{\omega\in{\Omega}}\left[\mu_{j\omega}(\eta_{j\omega}+\Pi_{j\omega}-\xi_{j})+\eta_{j\omega}\theta_{j\omega}\right]+$
$\displaystyle+\sum_{i\in{I}}\left[\left(q_{i}^{F}-q_{i}^{F_{min}}\right)\nu_{i}^{min}+\left(q_{i}^{F_{max}}-q_{i}^{F}\right)\nu_{i}^{max}\right]$
$\displaystyle+\sum_{j\in{J}}\left[\left(q_{j}^{F}-q_{j}^{F_{min}}\right)\nu_{j}^{min}+\left(q_{j}^{F_{max}}-q_{j}^{F}\right)\nu_{j}^{max}\right]$
(3.14a) subject to
$\displaystyle\text{equalities}\quad\eqref{L_second_a}-\eqref{L_second_c}$
(3.14b) $\displaystyle\text{partial
derivatives}\quad\eqref{Par_1}-\eqref{Par_end}$ (3.14c)
$\displaystyle\text{inequalities}\quad\eqref{L_second_d}-\eqref{L_second_g}$
(3.14d)
$\displaystyle((\ref{L_1})),((\ref{L_2})),((\ref{SME_1})),((\ref{L_q_1})),((\ref{L_c_2}))$
(3.14e)

where the objective function is formulated by the sum of the complementarity
products all the remaining KKT optimality conditions entered as constraints.
((3.1)) and ((3.2)) are the profit definitions of the conventional and RES
generators, respectively. ((3.10a))-((3.10f)) are the partial derivatives with
respect to the futures market quantities. Finally, ((3.6)) is equilibrium
price in the spot market, whereas ((3.7)) and ((3.8)) are the equilibrium
quantities of conventional and RES generators, respectively. Note that the
global NLP presented in ((3.14)) represents the joint solution of several
quadratic problems with quadratic constraints (QPQC), one per player.

###  3.2 Contracts for differences (CFD) model formulation

Our second model deals with a futures market which settles financially based
on contracts for differences (CFD), that can be used by generators to protect
themselves from the price and quantity fluctuations that arises in the spot
electricity market [80, 81]. The structure of the electricity market with CFD
is similar to the market in the GM except in the CFD generators use the strike
price as a reference price and there is a financial settlement at expiry
rather than a physical delivery. For all generators $k$, the contract signed
is a CFD that fixes the electricity amount $q_{k}^{F}$, and the delivery price
$P^{F}$ (the strike price) in the first stage. When the spot market takes
place, the spot price $P_{\omega}^{S}$ can be either greater than or less than
the futures price $P^{F}$. Therefore, if the spot market price
$P_{\omega}^{S}$ is lower than $P^{F},$ generator $k$ earns the difference
between the two price times the amount of electricity agreed in the contract
$(P^{F}-P_{\omega}^{S})q_{k}^{F}$. Otherwise, generator $k$ will pay the
difference between the two prices times the amount of energy agreed in the
contract at time-one $(P_{\omega}^{S}-P^{F})q_{k}^{F}$, if the spot market
price surpasses the futures market price at the spot market. Therefore, profit
for conventional generator is expressed as:

$\Pi_{i\omega}=(P^{F}-P_{\omega}^{S})q_{i}^{F}+P_{\omega}^{S}q_{i\omega}^{S}-a_{i\omega}-b_{i\omega}q_{i\omega}^{S}-\frac{1}{2}c_{i\omega}(q_{i\omega}^{S})^{2}\quad\forall
i,\forall\omega$ (3.15)

and

$\Pi_{j\omega}=(P^{F}-P_{\omega}^{S})q_{j}^{F}+P_{\omega}^{S}q_{j\omega}^{S}\quad\forall
j,\forall\omega$ (3.16)

for RES generator. The profit formulations in the CFD are different from the
GM profit formulations for conventional generators in two respects. First,
unlike the GM, conventional generators’ revenue in the futures market with CFD
is expressed as price difference times quantity,
$(P^{F}-P_{\omega}^{S})q_{i}^{F}$ and second, the production cost function
depends only on the spot quantity ($q_{i\omega}^{S}$), as there is no
electricity generation in the futures market in the CFD. Similarly, RES
generators’ profit is different, though the first term seems similar to the GM
due to a rearrangement for convenience. So that the second term in ((3.16))
depends only on the spot quantity, rather than $P_{\omega}^{S}Q_{j\omega}^{S}$
as in the GM. The analytical derivation follows a similar procedure then in
Section 3.1.1, where we need to start from the spot market and then come back
to the futures market to compute the market equilibrium.

####  3.2.1 Spot market equilibrium (stage-two)

Since we do not have futures production ($q_{i}^{F}$ and $q_{j}^{F}$ are
financial quantities) in the CFD, the spot market price is described as:

$P_{\omega}^{S}=\gamma_{\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}Q_{j\omega}=\hat{\gamma}_{\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i\omega}^{S}\quad\forall\omega.$
(3.17)

###### Proposition 3

Taking definition of $\varphi_{\omega}$ and $\tau_{i\omega}$ from Proposition
1, the equilibrium spot price and spot quantities in the CFD are described as:

$P_{\omega}^{S}=\varphi_{\omega}\left(\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\tau_{i\omega}\right)\quad\forall\omega$
(3.18)

and

$q_{i\omega}^{S}=\tau_{i\omega}\left(\varphi_{\omega}[\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\tau_{i\omega}]-b_{i\omega}+q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\right)\quad\forall
i,\forall\omega.$ (3.19)

The proof for Proposition 3 is found in (B). As we can see, the optimal spot
market clearing price and spot market quantities for generator $i$ are
analytically derived in closed form, which are again expressed in terms of
futures market decision variables.

####  3.2.2 Futures market analysis (stage-one)

Similar to the GM, the partial derivatives with respect to futures quantities,
needed to characterize conventional generators and RES generators in CFD, are
expressed as follows:

$\frac{\partial\Pi_{i\omega}}{\partial q_{i}^{F}}=\frac{\partial
P^{F}}{\partial q_{i}^{F}}q_{i}^{F}+P^{F}-\frac{\partial
P_{\omega}^{S}}{\partial q_{i}^{F}}q_{i}^{F}-P_{\omega}^{S}+\frac{\partial
P_{\omega}^{S}}{\partial q_{i}^{F}}q_{i\omega}^{S}+\frac{\partial
q_{i\omega}^{S}}{\partial
q_{i}^{F}}P_{\omega}^{S}-b_{i\omega}-c_{i\omega}q_{i\omega}^{S}$
$=\left(\frac{\partial P^{F}}{\partial q_{i}^{F}}-\frac{\partial
P_{\omega}^{S}}{\partial
q_{i}^{F}}\right)q_{i}^{F}+q_{i\omega}^{S}\left(\frac{\partial
P_{\omega}^{S}}{\partial
q_{i}^{F}}-c_{i\omega}\right)+P_{\omega}^{S}\left(\frac{\partial
q_{i\omega}^{S}}{\partial q_{i}^{F}}-1\right)+P^{F}-b_{i\omega}\quad\forall
i,\forall\omega$ (3.20)

and

$\frac{\partial\Pi_{j\omega}}{\partial q_{j}^{F}}=\left(\frac{\partial
P^{F}}{\partial q_{j}^{F}}-\frac{\partial P_{\omega}^{S}}{\partial
q_{j}^{F}}\right)q_{j}^{F}+P^{F}-P_{\omega}^{S}+\frac{\partial
P_{\omega}^{S}}{\partial q_{j}^{F}}Q_{j\omega}\quad\forall j,\forall\omega,$
(3.21)

respectively.

By recalling the futures market price expressed in (3.5) and considering
Assumption 2, the partial derivatives needed to complete (3.2.2) and (3.21)
are:

$\displaystyle\frac{\partial P^{F}}{\partial q_{i}^{F}}$
$\displaystyle=-\beta^{F}(1+(I+J-1)\psi_{i})\quad\forall i$ (3.22a)
$\displaystyle\frac{\partial P^{F}}{\partial q_{j}^{F}}$
$\displaystyle=-\beta^{F}\left(1+(I+J-1)\psi_{j}\right)\quad\forall j$ (3.22b)
$\displaystyle\frac{\partial P_{\omega}^{S}}{\partial q_{i}^{F}}$
$\displaystyle=\varphi_{\omega}[-\beta_{\omega}^{S}\beta_{\omega}^{S}(1+\delta_{i})\tau_{i\omega}-\beta_{\omega}^{S}\sum_{i\neq
k}\beta_{\omega}^{S}\psi_{i}(1+\delta_{i})\tau_{i\omega}]\quad\forall
i,\forall\omega$ (3.22c) $\displaystyle\frac{\partial
q_{i\omega}^{S}}{\partial q_{i}^{F}}$
$\displaystyle=\tau_{i\omega}\frac{\partial P_{\omega}^{S}}{\partial
q_{i}^{F}}+\tau_{i\omega}\beta_{\omega}^{S}(1+\delta_{i})\quad\forall
i,\forall\omega$ (3.22d) $\displaystyle\frac{\partial P_{\omega}^{S}}{\partial
q_{j}^{F}}$ $\displaystyle=0\quad\forall j,\forall\omega$ (3.22e)
$\displaystyle\frac{\partial q_{j\omega}^{S}}{\partial q_{j}^{F}}$
$\displaystyle=-1\quad\forall j,\forall\omega.$ (3.22f)

####  3.2.3 Risk-averse futures market equilibrium under the CVaR

Once we know the equilibrium spot price $P^{S}_{\omega}$ and quantities,
${q^{S}_{i\omega}}$ and ${q^{S}_{j\omega}}$, we can obtain parameterized the
total profits in the futures decision variables:

$\displaystyle\Pi_{i\omega}=\Pi_{i\omega}(q^{F}_{i},q^{F}_{-i},q^{F}_{j\in
J})\quad\forall i,\forall\omega$ (3.23a)
$\displaystyle\Pi_{j\omega}=\Pi_{j\omega}(q^{F}_{j},q^{F}_{-j},q^{F}_{i\in
I})\quad\forall j\forall\omega.$ (3.23b)

Each player $k\in I\cup J$ maximizes its total profit (risk neutral, or risk
averse) by solving an optimization problem similar to (3.12) in the futures
market considering the CVaR. The market equilibrium can be done by following
the procedure described in Section 3.1.3 and concatenate the KKT conditions
for each generator to solve with an equivalent NLP formulation.

###  3.3 Spot market without the presence of futures contracts

Finally, we derive the equilibrium market outcomes of a market without the
presence of futures trading (a single spot market) for comparison purposes.
Thus, the profit for conventional generator in the only spot configuration is
expressed as:

$\Pi_{i\omega}=P^{S}_{\omega}q_{i\omega}^{S}-a_{i\omega}-b_{i\omega}q_{i\omega}^{S}-\frac{1}{2}c_{i\omega}(q_{i\omega}^{S})^{2}\quad\forall
i,\forall\omega$ (3.24)

and the profit for generator $j$ as:

$\Pi_{j\omega}=P_{\omega}^{S}Q_{j\omega}\quad\forall\omega.$ (3.25)

###### Proposition 4

Given Assumption 1 and the profit definitions expressed in ((3.24)) and
((3.25)), the spot market equilibrium outcomes are analytically derived as:

$P_{\omega}^{S}=\varphi_{\omega}\left[\gamma_{\omega}^{S}-\beta_{\omega}^{S}\sum\limits_{i=1}^{J}Q_{j\omega}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}\right]\quad\forall\omega.$
(3.26)
$q_{i\omega}^{S}=\tau_{i\omega}\varphi_{\omega}\left[\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}\right]-\tau_{i\omega}b_{i\omega}\quad\forall
i,\forall\omega.$ (3.27)

The proof for Proposition 4 is found in the (C). As it can be seen, both price
and quantity are expressed in terms of parameters, as there are no futures
decision variables in this case.

##  4 Numerical results

In this section, we analyze and discuss numerical examples using calibrated
data from the Spanish electricity market that combines futures contracts and
pool market structures. The reasons for us to choose this market are: first,
in the Spanish electricity market, generators can sign physical or financial
futures contracts. Then, they participate in a daily market, and the market
operator maintains the bid-offer balance of electricity power to determine the
market price as well as the generation of electricity quantities corresponding
to each generator for each hour in the schedule [30]. Second, a large
proportion of the electricity is traded by very few large generators whose
total output is the sum of several units with different technologies. This is
very important from a computational point of view since increasing the number
of generators increases the computational burden, as our problem is highly
nonlinear.

###  4.1 Data

Therefore, we consider three conventional oligopolistic generators and one RES
generator who can produce and retail their generation in the futures and spot
markets. Table 1 presents the expected values of the calibrated data for the
cost, demand and RES parameters along with the corresponding standard
deviations. From the cost parameters, $a_{i\omega}$ are fixed to arbitrary
values $a_{1\omega}=a_{2\omega}=a_{3\omega}=0$, as they do not have a direct
influence on the market results. The values of $b_{i\omega}$ and $c_{i\omega}$
are randomly generated by estimating their expected values, which are close to
the real market values (see [67, 37, 81, 77] for similar approach). Each
realization of $b_{i\omega}$ is generated from a multivariate normal
distribution with mean $\mu_{b}=[37,40,43]$ [€/MWh] and standard deviation
$\sigma_{b}=[3.5,4.55,5.59]$ [€/MWh], which is calculated using 9% coefficient
of variation (CV), as $\sigma=\mu\times CV$. Similarly, $c_{i\omega}$ is
generated randomly from a normal distribution with
$\mu_{c}=[0.013,0.003,0.019]$ [€/MWh2] and standard deviation
$\sigma_{c}=[0.000125,0.0002,0.000195]$ [€/MWh2], calculated using 5% of CV.
$150$ and $200$ equiprobable scenarios are considered for the risk neutral
case and for the CVaR, respectively. The significance level is fixed at
$\alpha=0.90.$

Table 1: Mean, CV and standard deviation used to calibrate the data applied in our simulations for conventional generators and the RES parameter.  Parameters | $i=1$ | $i=2$ | $i=3$ | $CV$ | Parameters | mean | $\sigma$
---|---|---|---|---|---|---|---
$a_{i\omega}$ | 0.00 | 0.00 | 0.00 | - | $\gamma^{F}$ | 180.00 | 18
$b_{i\omega}$ | 37.00 | 40.00 | 43.00 | 0.09 | $\beta^{F}$ | 0.005 | 0.0005
$c_{i\omega}$ | 0.013 | 0.003 | 0.019 | 0.05 | $Q_{j\omega}$ | 0-10,000 | 1000

On the other hand, the demand curve parameters are generated by approximating
the aggregated step-wise demand curve in the spot market using the mean
futures market intercept $\mu_{\gamma^{F}}=180$ and mean slope
$\mu_{\beta^{F}}=0.005.$ We assume that the parameters of the inverse demand
curve in the spot market equal, on expectation, those parameters of the
futures inverse demand, i.e, $\gamma^{F}=E[\gamma_{\omega}^{S}]$ and
$\beta^{F}=E[\beta_{\omega}^{S}]$. Note that the uncertainty associated with
consumer’s behavior is represented by both the demand intercept
($\gamma_{\omega}^{S}$) and its slope ($\beta_{\omega}^{S}$), which are
scenario dependent. Thus, the value of $\gamma_{\omega}^{S}$ for each scenario
is randomly generated with a normal distribution so that
$\gamma_{\omega}^{S}\sim N(\mu_{\gamma^{F}},\sigma_{\gamma^{F}})$ and
$\beta_{\omega}^{S}\sim N(\mu_{\beta^{F}},\sigma_{\beta^{F}})$, where
$\sigma_{\gamma^{F}}$ and $\sigma_{\beta^{F}}$ are calculated as
$\sigma=\mu\times CV$, assuming CV to be $10\%$ for the corresponding mean
values of both parameters. Total RES parameter is randomly generated with its
expected value at $5,000$MWh and a standard deviation of 1000 with CV of
$20\%$.

The simulation is done for the RES parameter that ranges from $0$ to $10,000$
MWh (we consider the amount of RES from 0-10,000MWh in a range of 1000MWhs
given the expectation of RES to be integrated into the system is increasing
realistically). The maximum amount that conventional generators are allowed to
generate in the model is arbitrarily set at 6,000, 7,000, and 5,000MWh for the
three generators, respectively. The RES generator capacity is fixed to an
increasing value within the range (0, 10,000) MWh in each iteration.

The model is implemented in JuMP version 0.21.1 [82] under the open-source
Julia language version 1.5.2 [83]. We use Artelys Knitro solver version 12.2
[84] on a CPU<EMAIL_ADDRESS>and 64.00 GB of RAM running workstation. Since
there are multiple simulation cases, we report the computational performance
of one case as a representative reference: For instance, the overall
computational time for the CVaR definition with $|\Omega|=200$ equiprobable
scenarios is 917.177904 seconds (156.46 M allocations: 3.199 GiB, 0.10% gc
time).

According to the specification of problem ((3.12)), two risk profiles
($\phi=0$ for the risk neutral and $\phi=1$ for the risk averse generators)
are considered. We analyze how the relationship between futures and spot
markets, risk aversion, and level of competition are affected with respect to
high RES penetration (level of renewables). The electricity market outcomes
(electricity price, quantity, and generators’ profit) are discussed for two
market structures (Cournot and perfect competition) and for two models (GM and
CFD) with respect to RES penetration. The x-axis in all the plots, but one
represents RES penetration that ranges from 0 to 10,000WMh, whereas the y-axis
may be price, quantity, or profit.

###  4.2 Risk averse generators’ numerical results ($\phi=1$)

For the risk averse generators, the numerical results are obtained by
simulating the model with the parameter $\phi=1$. The results are futures and
spot markets equilibrium outcomes when generators maximize their CVaR from our
spectral risk measure representation. We analyze the influence of generators’
risk aversion behavior on equilibrium market outcomes based on levels of
competitiveness (Cournot and perfect competition) under two contracts
(physical, or financial).

Figure 2: Risk averse Cournot model in the GM Figure 3: Risk averse Cournot
CFD

Let’s examine the results starting from the Cournot model in the general
model. Looking at Figure 2, Figure 3, and Table 2, overall price decreases
(downward sloping) with respect to RES penetration. That is because when RES
penetration increases, conventional generation with generation cost decreases
and is replaced by RES generation (with no generation cost), which may
transfer part of the cost saved to consumers. If we look at the general model
results, we see that consumers must pay as much as 116€/MWh with zero level of
renewables, and as low as 101€/MWh in the futures market with higher
(10,000MWh) RES penetration. The expected spot market price without the
presence of futures contracts is always higher (ranges from 84 to 111€/MWh)
than the expected spot price in the presence of futures contracts (ranges from
75 to 100€/MWh), which overlapped in the CFD model. In the CFD, the expected
spot price without the presence of the futures market is always higher than
the futures price (coincides with normal backwardation over time) and
decreases with respect to RES penetration.

Therefore, buyers benefit from RES penetration as it decreases electricity
prices in both stages and motivates RES generators to generate and trade
higher quantities of electricity in the market. Regarding competition, there
is a counter-argument that attaining a certain level of renewable with no
marginal cost in a liberalized electricity market distorts market
competitiveness and creates contango [8]. The argument is that the further
reduction of expected spot prices beyond the speed of the futures price with
respect to RES penetration is due to market inefficiency created by the
dispatchability and zero generation cost of RES generators. The decrease in
price with RES penetration corroborates with classical economic theory, where
the higher the competition (because of RES generators competitiveness
increases) the larger is the fall in price even by conventional generators to
compete with RES generators. In perfect competition, generators are price
takers and they have no influence on the future market prices.

As far as trading quantities are concerned, conventional generators trade
quantities that range from 11,414 to 12,761MWh with RES penetration. Futures
market trading for conventional generators is not significantly affected as
much as the expected spot market quantities, which decrease across RES
penetration (see Figure 2). Following the market prices trend, the CVaR
decreases for conventional generators with respect to RES penetration, which
shows left tailed profit distributions with a low probability of low profits
with high RES penetration. This is evident as the RES generators’ trading
quantities increase both in the futures market and spot market with respect to
RES penetration (see Figure 3). As profit is the reflection of price and
quantity, the higher the RES penetration, the lower the conventional
production, which in turn lowers their profit.

Comparing the models, conventional generators’ profit is higher in the GM than
in the CFD, but it decreases in both models with respect to RES penetration.
This is due to the higher futures electricity price in the GM than the
relatively lower futures price in the CFD with the RES penetration. RES
generators’ profit, on the other hand, increases in both models as generators
trade higher electricity with respect to RES penetration.

When generators react in the market as perfect competition, the competition
squeezes their profits compared to Cournot competition. This is an intuitive
result as it shows when generators act in the market as perfect competition,
the pattern of profits follows the shape of prices, both in the GM and in the
CFD, which is consistent with existing literature [85, 81].

Total trading quantities in the CFD are much higher than the GM. This is
because the CFD model includes physical and financial quantities (Figure 3 and
Figure 5). In other words, the CFD entails larger electricity transactions in
the spot market, which is explained by the lack of a physical settlement in
the futures market that requires all quantities must be delivered in the spot
market. In addition, total trading without the presence of futures market is
higher in the GM and lower in the CFD, where the expected spot market trading
is higher. Conversely, the spot market total trading with the futures market
decreases in the GM with high RES penetration.

Figure 4: Risk averse perfect competition in the GM Figure 5: Risk averse perfect competition in the CFD Table 2: The minimum and the maximum electricity prices [€/MWh] with the two models and the two levels of competitions in the risk averse case. | General Model (GM) | Contract for differences (CFD)
---|---|---
Price |  | Cournot
---
model
| Perfect
---
Competition
| Cournot
---
model
| Perfect
---
Competition
$P^{F}$ | 101-116 | 83-100 | 78-90 | 79-90
$P_{\omega}^{S}$ | 75-100 | 76-97 | 70-96 | 76-97
$P_{\omega}^{S}$ only | 84-111 | 76-97 | 84-111 | 76-97

Figure 4 and Figure 5 show the market outcomes with perfect competition in the
GM and in the CFD, respectively. With perfect competition, the overall price
is lower in both models with respect to RES penetration (ranges from 76 to
100€/MWh in both contracts). Besides, the expected spot prices are overlapped
in both the GM and the CFD (with and without the presence of the futures
market). In the GM, total trading quantity with and without the presence of
futures contracts shows no significant difference as presented in Figure 4.
However, this is the opposite as the CFD model encompasses physical and
financial quantities. In the CFD, total trading sharply increases with no
futures market despite the trading volume in the spot market with the presence
of the futures market is not as low as it is in the GM.

###  4.3 Risk neutral generators’ numerical results ($\phi=0$)

Figure 6: Risk neutral Cournot competition in the GM with RES penetration.
Figure 7: Risk neutral Cournot competition in the CFD with RES penetration.
Figure 8: Risk neutral perfect competition in the GM with RES penetration.
Figure 9: Risk neutral perfect competition in the CFD with RES penetration.

In the risk neutral case, the numerical simulation is done by setting the risk
parameter $\phi=0$. In this case, generators maximize their expected profits,
which is expressed in ((3.13)). Figure 6 and Figure 7 show the risk neutral
results for Cournot competition, and Figure 8 and Figure 9 perfect competition
with respect to high RES penetration. Similar to the risk averse case, the
overall electricity price decreases with respect to RES penetration. With
Cournot competition, the market is in contango in the GM, where the futures
price is higher than the expected spot prices, and in normal backwardation in
the CFD model, where the futures price is lower than the expected spot market
price with respect to RES penetration.

Contrary to the risk averse generators case in the GM, conventional
generators’ futures market trading in the risk neutral case slightly increases
with RES penetration, whereas RES generators’ futures market trading is higher
in the risk averse case with Cournot competition (comparing Figure 2 and
Figure 6). In the CFD, conventional generators’ trade with Cournot competition
is always higher than the RES generators’ trading with respect to RES
penetration despite RES generator profit is as high as conventional
generators’ profit in the GM. For the perfect competition, futures prices
overlapped with expected spot market prices with respect to RES penetration,
in both the GM and the CFD model, which is inline with the standard non-
arbitrage condition that explains futures price is equal to the expected spot
market prices, particularly with risk neutrality. Total trading and profit
exhibit similar pattern with the Cournot model discussed above.

Table 3: Expected market outcomes with levels of competition in the GM and CFD
for risk neutral and risk averse cases.

| Cournot Model | Perfect competition
---|---|---
Market | Risk neutral | Risk averse | Risk neutral | Risk averse
Outcomes | GM | CFD | GM | CFD | GM | CFD | GM | CFD
$P^{F}$ | 108.28 | 108.28 | 107.68 | 107.68 | 87.26 | 87.26 | 91.46 | 91.46
$P_{i\omega}^{S}$ | 90.64 | 90.64 | 88.48 | 88.48 | 87.26 | 87.26 | 86.99 | 86.99
$\sum_{i\in{I}}q_{i}^{F}$ | 17999.99 | 10815.77 | 11791.94 | 17999.99 | 17999.99 | 14658.56 | 13924.75 | 17999.99
$\sum_{i\in{I}}q_{i}^{S}$ | 14737.43 | 2740.58 | 1656.61 | 14467.78 | 14106.44 | -550.57 | 19.05 | 13961.32
$\sum_{j\in{J}}q_{j}^{F}$ | 667.27 | 3527.72 | 2671.51 | 1065.89 | 900.89 | 3888.64 | 3782.41 | 863.93
$\sum_{j\in{J}}q_{j\omega}^{S}$ | 4321.95 | 1461.51 | 2441.14 | 4046.76 | 4251.07 | 1263.32 | 1160.34 | 4078.83
$\sum_{i\in{I}}\Pi_{i}$ | 394779.9 | 634928.70 | 636130.7 | 362298.2 | 361638.9 | 393461.36 | 446717.43 | 362277.3
$\sum_{j\in{J}}\Pi_{j}$ | 397829.4 | 483218.95 | 478013.2 | 401020.6 | 424595.6 | 424595.63 | 426842.08 | 408029.3

Our results attest to the literature that high renewables integration is
associated with exposure to electricity market risks [86, 87]. Figure 7 shows
that market outcomes are more dispersed as the RES deployment increases in the
electricity market. When intermittent generation increases, it also increases
the dispatching cost of conventional generators, as demand requires the
combination of RES with flexible powers with high variable costs.

###  4.4 Overall comparison

Table 3 shows the market outcomes, mainly the overall expected outcomes for
comparison between risk neutrality and risk aversion, levels of competition,
and the two contract settlements. Futures market and expected spot market
prices are slightly higher for risk neutral generators in the Cournot
competition than in the perfect competition. Expected spot market trading
varies in all the cases presented in Table 3, mainly for the futures market
trading which matches the expectation. RES generators trade more electricity
in the spot market compared to their trade in the futures market. The sum of
one large RES generator’s profit is almost as close as the three conventional
generators’ profit as the RES penetration displaces conventional ones from
competition. High RES penetration results in higher standard deviations as
higher levels of RES are associated with higher levels of uncertainty and
exposure to higher risks. Since RES generators are assumed with zero marginal
cost (do not have cost uncertainty) in our model setting, the increase in RES
generation quantity offsets the risks from demand and RES capacity
uncertainties. Moreover, it is evident from our simulation results that RES
generators trade much of their electricity in the spot market as contracts
manage the uncertainties raised from demand and RES capacity availability.
Another reason is due to non-dispatchability of RES, they have to trade all
the remaining quantities in the spot market so that it may contribute to the
spot market trading increments.

###  4.5 The effect of risk parameter on market outcomes

Figure 10: The effect of varying risk parameter on market outcomes in perfect
competition.

Finally, we analyze the impact of risk parameter ($\phi$) increase on market
outcomes with perfect competition, and in the GM as a representative
reference. This is done by keeping the RES parameter at its expected value
(5,000MWh) and moving the CVaR parameter (from extreme risk neutrality
($\phi=0$) to extreme risk aversion ($\phi=1$)). With this setting, Figure 10
shows that risk aversion increases electricity market prices both in the
futures market and spot market, though the effect is higher in the futures
market. The expected spot price overlapped both with and without the presence
of the futures contracts. For conventional generators, spot market quantity
slightly increases with respect to the CVaR parameter, which is reflected in
their improved profit as well. This is a counter-intuitive result as risk
neutrality is expected to render higher profit. This implies contracts give a
shield for generators from production cost, demand, and RES capacity
uncertainties risks, as the more risk averse they are the better the confident
they become to produce, and retail their electricity.

##  5 Summary and conclusions

In this paper, we propose a game-theoretical framework to model an electricity
market that considers different futures market contract designs in a two-stage
approach under uncertainty and high penetration of renewables.

We analyze the profit-maximizing problems of risk neutral (using expected
value) and risk aversion (using the CVaR) generators with different contracts
(physical and financial) under different generation technologies (RES and
conventional). This is done by taking conjectural variations assumption where
each generator has an estimation of the impact of its production may have on
market prices and rival quantities. In this regard, generators have the option
to trade their generation first, in the futures market (stage-one), and
subsequently in the spot market (stage-two) given the inherent uncertainties.
Since demand, conventional generators’ production cost, and RES generation
availability are uncertain, a coherent risk measure with the CVaR is
introduced to model risk aversion. We also introduce different types of
contracts in the futures market to evaluate their performance and impact in
the equilibrium market outcomes, which in turn depend on the levels of RES
generation in the system.

The analytical derivation of the market equilibrium starts at the second stage
(spot market) where we parameterize the spot market equilibrium as a function
of the futures market decision variables. In the first stage, the global
equilibrium of the market is computed from the joint solution of all the
generators’ profit maximization problems using the CVaR. The global
optimization is solved by concatenating and minimizing the product of the
complementarity constraints subject to the remaining equality and inequality
KKT optimality conditions from the CVaR formulation, and the closed-form
solutions obtained from the spot market equilibrium.

The analytical results obtained are tested with numerical results by analyzing
different levels of competition (Cournot and perfect competition), different
contracts (physical and financial) with risk neutral and risk averse
strategies for players.

We can conclude that the level of risk aversion and competition have a strong
impact on electricity market outcomes with respect to RES penetration.
Therefore, RES penetration increases overall quantity traded and decreases
overall electricity prices, which is better for social welfare. RES
generators’ profit increases as their trading quantities increase and
contracts provide a way for generators to manage the risk associated with the
inherent volatility of the spot market. The simulation results show that
generators can increase their profit when they act as risk averse in the
market, which is a counter-intuitive result of the model.

This may be because of the positive impact financial contracts have in risk
hedging which, in turn, increases RES generators trading quantity mainly from
RES capacity. Combining financial contracts with a coherent risk measure can
help managers to deal with electricity markets with high uncertainty levels,
while achieving the energy transition targets. The proposed approach can help
decision makers to identify an appropriate modeling strategy, and to choose
the best type of contracts to hedge risk in such two-stage oligopolistic
markets. To this end, as a contribution to the existing literature, we provide
a risk-management strategy via financial derivatives for improving the
viability of energy transition, as well as policy insights regarding
application to liquid futures market.

Comparing contracts, the CFD (financial trading) model renders better consumer
surplus as it has lower electricity prices while the GM (physical delivery) is
better for electricity generators as it renders higher electricity prices and
higher profits.

There are several possible extensions to the model and methods proposed in
this paper. First, the conventional generators’ profit function could be
modified to study emission trading scheme in the electricity market in a bid
to plummet greenhouse gas emission. Second, by relaxing conventional
generators’ production cost function to be linear, the model could be extended
to study the interaction between market power, and risk aversion considering
different contracts over time. Finally, the model can be applied for problems
that incorporate financial contracts and risk aversion to study equilibrium
solutions.

## Acknowledgements

The authors sincerely thank the editor and the reviewers for their
constructive comments and suggestions that improve this study. Carlos Ruiz
gratefully acknowledges the financial support from the Spanish government
through projects PID2020-116694GB-I00 and from the Madrid Government
(Comunidad de Madrid) under the Multiannual Agreement with UC3M in the line of
“Fostering Young Doctors Research” (ZEROGASPAIN-CM-UC3M), and in the context
of the V PRICIT (Regional Programme of Research and Technological Innovation.

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##  Appendix A Proof for Proposition 1

###### Proof 1

The profit for conventional generator $i$, in scenario $\omega$, in the spot
market is:

$\Pi_{i\omega}^{S}=P^{S}_{\omega}q_{i\omega}^{S}-a_{i\omega}-b_{i\omega}(q_{i}^{F}+q_{i\omega}^{S})-\frac{1}{2}c_{i\omega}(q_{i}^{F}+q_{i\omega}^{S})^{2}.$
(A.28)

From (A.28), we can derive the spot market equilibrium by computing the first-
order optimality conditions for all generators simultaneously as:

$\frac{\partial\Pi_{i\omega}}{\partial q_{i\omega}^{S}}=0=\frac{\partial
P_{\omega}^{S}}{\partial
q_{i\omega}^{S}}q_{i\omega}^{S}+P_{\omega}^{S}-b_{i\omega}-c_{i\omega}(q_{i}^{F}+q_{i\omega}^{S})=0\quad\forall
i,\forall\omega$ (A.29)

where $\frac{\partial P_{\omega}^{S}}{\partial
q_{i\omega}^{S}}=-\beta_{\omega}^{S}\left(1+\sum_{i\neq k}\frac{\partial
q_{k\omega}^{S}}{\partial
q_{i\omega}^{S}}\right)=-\beta_{\omega}^{S}(1+\delta_{i}).$ By substituting
$\frac{\partial P_{\omega}^{S}}{\partial
q_{i\omega}^{S}}=-\beta_{\omega}^{S}(1+\delta_{i})$ into (A.29), we can
simplify and rearrange it to solve for $q_{i\omega}^{S}$:

$q_{i\omega}^{S}=\frac{1}{\beta_{\omega}^{S}(1+\delta_{i})+c_{i\omega}}(P_{\omega}^{S}-b_{i\omega}-c_{i\omega}q_{i}^{F})=\tau_{i\omega}(P_{\omega}^{S}-b_{i\omega}-c_{i\omega}q_{i}^{F})\quad\forall
i,\forall\omega.$ (A.30)

At this level, we can substitute $P_{\omega}^{S}$ from (3.4) into ((A.30)) to
obtain the clearing price in the spot market in terms of futures market
decision variables which can be rearranged as:
$P_{\omega}^{S}=\hat{\gamma}_{\omega}^{S}-\beta^{S}_{\omega}\sum_{i\in{I}}q_{i\omega}^{S}-\beta^{S}_{\omega}\sum_{i\in{I}}q_{i}^{F}$,
where we can collect price as:
$P_{\omega}^{S}+\beta^{S}_{\omega}\sum_{i\in{I}}\tau_{i\omega}P_{\omega}^{S}=\hat{\gamma}_{\omega}^{S}-\beta^{S}_{\omega}\sum_{i\in{I}}q_{i}^{F}+\beta^{S}_{\omega}\sum_{i\in{I}}\tau_{i\omega}\left(b_{i\omega}+c_{i\omega}q_{i}^{F}\right).$
Finally, the equilibrium spot market price is:

$\displaystyle
P_{\omega}^{S}=\varphi_{\omega}\left[\hat{\gamma}_{\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}\left(b_{i\omega}+c_{i\omega}q_{i}^{F}\right)\right]\quad\forall\omega.$
(A.31)

##  Appendix B Proof for Proposition 3

###### Proof 2

From the profit function expressed in (3.15), we can compute the spot
equilibrium by deriving the first-order optimality conditions for all
generators:

$\frac{\partial\Pi_{i\omega}}{\partial
q_{i\omega}^{S}}=-q_{i}^{F}\frac{\partial P_{\omega}^{S}}{\partial
q_{i\omega}^{S}}+\frac{\partial P_{\omega}^{S}}{\partial
q_{i\omega}^{S}}q_{i\omega}^{S}+P_{\omega}^{S}-b_{i\omega}-c_{i\omega}q_{i\omega}^{S}=0$
(B.32)

Rearranging ((B.32)) and solving for $q_{i\omega}^{S}$ as:

$q_{i\omega}^{S}=\tau_{i\omega}\left(P_{\omega}^{S}-b_{i\omega}+q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\right).$
(B.33)

Then, substituting (B.33) into (3.17) and simplifying it gives:
$P_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}P_{\omega}^{S}=\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\tau_{i\omega}$,
from which the optimal price at the spot market is expressed as:

$P_{\omega}^{S}=\varphi_{\omega}\left[\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\tau_{i\omega}\right]\quad\forall\omega.$
(B.34)

Finally, by substituting (B.34) into (B.33), the optimal spot quantity for
generator $i$, at scenario $\omega$, is expressed as:

$q_{i\omega}^{S}=\tau_{i\omega}\left[\varphi_{\omega}[\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\tau_{i\omega}]-b_{i\omega}+q_{i}^{F}\beta_{\omega}^{S}(1+\delta_{i})\right]\quad\forall
i,\forall\omega.$

##  Appendix C Proof for Proposition 4

###### Proof 3

The price-demand curve is:

$\displaystyle
P_{\omega}^{S}=\gamma^{S}_{\omega}-\beta^{S}_{\omega}\left(\sum_{i\in{I}}q_{i\omega}^{S}+\sum_{j\in{J}}Q_{j\omega}\right)\quad\forall\omega$
(C.35)

which can be further simplified as:
$P_{\omega}^{S}=\hat{\gamma}_{\omega}^{S}-\beta_{\omega}^{S}\sum_{i\in{I}}q_{i\omega}^{S}.$

The equilibrium in the spot market is reached when all conventional generators
maximize their profits simultaneously, which is expressed by the first-order
optimality conditions as:

$\frac{\partial\Pi_{i\omega}}{\partial q_{i\omega}^{S}}=\frac{\partial
P_{\omega}^{S}}{\partial
q_{i\omega}^{S}}q_{i\omega}^{S}+P_{\omega}^{S}-b_{i\omega}-c_{i\omega}q_{i\omega}^{S}=0.$
(C.36)

Simplifying (C.36), gives
$q_{i\omega}^{S}=\tau_{i\omega}(P_{\omega}^{S}-b_{i\omega})$ hence we can
substitute $q_{i\omega}^{S}$ into ((C.35)), and solving for the spot price as
$P_{\omega}^{S}\left(1+\beta^{S}_{\omega}\sum_{i\in{I}}\tau_{i\omega}\right)=\hat{\gamma}_{\omega}^{S}+\beta^{S}_{\omega}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}$
gives the equilibrium price:

$P_{\omega}^{S}=\varphi_{\omega}\left[\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}\right]$,
which can be expressed by substituting $\hat{\gamma}_{\omega}^{S}$ as:

$P_{\omega}^{S}=\varphi_{\omega}\left[\gamma_{\omega}^{S}-\beta_{\omega}^{S}\sum\limits_{i=1}^{J}Q_{j\omega}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}\right]\quad\forall\omega.$
(C.37)

Finally, substituting (C.37) into (C.36) gives the equilibrium quantity for
conventional generators as:

$q_{i\omega}^{S}=\tau_{i\omega}\varphi_{\omega}\left[\hat{\gamma}_{\omega}^{S}+\beta_{\omega}^{S}\sum_{i\in{I}}\tau_{i\omega}b_{i\omega}\right]-\tau_{i\omega}b_{i\omega}\quad\forall
i,\forall\omega.$ (C.38)