Modelling an electricity market oligopoly with a competitive fringe and generation investments
MModelling an electricity market oligopoly with a competitive fringe andgeneration investments
Mel T. Devine a,b, ∗ , Sauleh Siddiqui c,d a College of Business, University College Dublin, Ireland b School of Electrical and Electronic Engineering, University College Dublin, Ireland c Department of Civil and Systems Engineering, Johns Hopkins University, Baltimore, USA d German Institute for Economic Research (DIW Berlin), Mohrenstr. 58, 10117, Berlin, Germany
Abstract
Market power behaviour often occurs in modern wholesale electricity markets. Mixed ComplementarityProblems (MCPs) have been typically used for computational modelling of market power when it is charac-terised by an oligopoly with competitive fringe. However, such models can lead to myopic and contradictorybehaviour. Previous works in the literature have suggested using conjectural variations to overcome this mod-elling issue. We first show however, that an oligopoly with competitive fringe where all firms have investmentdecisions, will also lead to myopic and contradictory behaviour when modelled using conjectural variations.Consequently, we develop an Equilibrium Problem with Equilibrium Constraints (EPEC) to model such anelectricity market structure. The EPEC models two types of players: price-making firms, who have marketpower, and price-taking firms, who do not. In addition to generation decisions, all firms have endogenousinvestment decisions for multiple new generating technologies. The results indicate that, when modelling anoligopoly with a competitive fringe and generation investment decisions, an EPEC model can represent amore realistic market structure and overcome the myopic behaviour observed in MCPs. The EPEC consideredfound multiple equilibria for investment decisions and firms profits. However, market prices and consumercosts were found to remain relatively constant across the equilibria. In addition, the model shows how it maybe optimal for price-making firms to occasionally sell some of their electricity below marginal cost in orderto de-incentivize price-taking firms from investing further into the market. Such strategic behaviour wouldnot be captured by MCP or cost-minimisation models.
Keywords:
OR in energy; Oligopoly with competitive fringe; Equilibrium Problem with EquilibriumConstraints (EPEC); Investment and generation decisions
1. Introduction
Electricity market modelling is an area of research that has attracted much attention in the OperationsResearch literature. Optimisation and equilibrium models in particular have been extensively used to betterunderstand the behaviour of electricity generators. Such tools provide insights from planning, operations ∗ Corresponding author
Email address: [email protected] (Mel T. Devine a, ) January 13, 2020 a r X i v : . [ m a t h . O C ] J a n nd regulatory perspectives. Regulators may use them to monitor market inefficiencies, profit-maximisinggenerators may use them to gain insights on possible trading strategies while policy-makers may use themto test the impact of different proposed policy mechanisms.Since the 1980s, countries have been deregulating their electricity markets with the intention of splittingownership of market activities (Pozo et al., 2017). Governments’ goals are to foster competition, increasemarket efficiencies and thus reduce consumer costs. As a result, individual market participants, also knownas market players, have been behaving selfishly by seeking to independently maximise their profits (Facchineiand Pang, 2007).Deregulation has resulted in many electricity markets showing evidence of market power (Lee, 2016).Market power is present when one (or more) seller(s) in the market can strategically maximise their profitsby influencing the selling price through the quantity they supply to the market. When such behaviouris not present in the market, the market is perfectly competitive. Accurately modelling market power inelectricity markets is a challenging area of operations research. However, there has been many electricitymarket models that have incorporated market power. For a comprehensive review of electricity marketmodels that incorporate market power, we refer the reader to Pozo et al. (2017). More recent examplesin the operations research literature include Devine and Bertsch (2018) who use a Mixed ComplementarityProblem (MCP) to study different consumer led load shedding strategies. MCPs solve multiple constrainedoptimisation problems simultaneously and in equilibrium and they allow players with market power to bemodelled as Cournot players.Fanzeres et al. (2019) proposed a Mathematical Program with Equilibrium Constraints (MPEC) forstrategic bidding in an electricity market. An MPEC model solves a bi-level optimisation problem which is amathematical program where one or more optimisation problems are embedded within another optimisationproblem. The outer optimisation is the upper-level optimisation while the inner optimisations, which arerepresented in the outer problem as constraints, are the lower-level optimisation problems. In an electricitymarket setting, MPEC problems can be used to model markets where a single player has market power. Theupper level represents the optimisation problem of the player who has market power while the lower levelproblems represents the problems of players that do not have market power.Steeger and Rebennack (2017) present a methodology that combines Lagrangian relaxation and nestedBenders decomposition to model a single hydro producer with market power. Similarly, Habibian et al. (2019)us an optimisation-driven heuristic approach to model a large electricity consumer with market power. Inboth of these works, only one market participant has a strategic advantage.The papers in the previous paragraphs do not consider strategic behaviour when the overall market wascharacterised by an by oligopoly with competitive fringe. Such a market structure occurs when more than onegenerator (the oligopolists) have market power and at least one generator does not (the competitive fringe).Many modern electricity markets are characterised by an oligopoly with competitive fringe (Bushnell et al.,2008; Walsh et al., 2016). However, when it comes to the energy market modelling literature, such markets2tructures are under-represented.Some exceptions include Huppmann (2013) and Ansari (2017), who use a Mixed Complemenatrity Prob-lem (MCP) to model an oligopoly with competitive fringe in international oil market contexts. Similarly,Devine and Bertsch (2019) develop a MCP model of an oligopoly with competitive fringe to investigate theimpact demand response has on market power in an electricity market. Huppmann (2013) highlights howmodelling an oligopoly with competitive fringe using a MCP can lead to myopic, counter-intuitive and thusunrealistic optimal decisions from the oligopolists. In a MCP framework, each oligopolist optimises its ownposition but does not take into account the optimal reaction of the competitive fringe. The oligopolistsreduce their generation levels with the intention of increasing the market price and hence increasing theiroverall profits. However, when the oligopolists do this, the competitive fringe increase their generation andfill the generation gap. This results in market prices not increasing as the oligopolists would anticipate.Huppmann (2013) proposes using conjectural variations to overcome the issue. Conjectural variations makeassumptions about how players react to other players’ quantity changes and have been widely used in theenergy market modelling literature (Egging et al., 2008; Haftendorn and Holz, 2010; Huppmann and Egging,2014; Baltensperger et al., 2016; Egging and Holz, 2016). These assumptions allow the oligopolists to some-what incorporate the reactions of the competitive into their decision making process. However, the resultingdecisions from the oligopolists may not be necessarily be optimal.Neither Huppmann (2013), Ansari (2017) nor Devine and Bertsch (2019) consider investment in newgeneration decisions. Consequently, in this work, we show that when investment decisions are incorporatedinto a MCP model of an oligopoly with competitive fringe, conjectural variations still lead to contradictorybehaviour from the oligopolists. Moreover, to overcome the modelling issue, we develop an EquilibriumProblem with Equilibrium Constraints (EPEC) model of an oligopoly with competitive fringe where botholigopolists and the competitive fringe have investment decisions. Each firm may initially hold, and investin, multiple generating technologies.An EPEC model solves multiple interconnected MPEC problems in equilibrium (Gabriel et al., 2012).In this work, each MPEC represents the optimisation problem of a different electricity generating firm whohas market power (also known as a price-making firm). The price-making firms each seek to maximise prof-its subject to capacity constraints. In addition, the equilibrium conditions representing the optimisationproblems of the competitive fringe are embedded into each price-making firm’s problem as constraints. Con-sequently, an EPEC approach can overcome the limiting assumptions associated with conjectural variations.Instead of making assumptions of how players react to other players’ quantity changes, an EPEC modelallows oligopolists to explicitly account for optimal reactions of the competitive fringe and thus to makeoptimal decisions by anticipating those reactions.There has been many examples in the literature where EPECs have been used to model electricity markets.Many of the first EPEC models for electricity markets consider electricity generators in the upper level andan Independent System Operator (ISO) in the lower level (Hu and Ralph, 2007; Ruiz et al., 2011; Pozo and3ontreras, 2011). Building on these works, Wogrin et al. (2012) and Wogrin et al. (2013) develop an EPECmodel that incorporates both capacity expansion and generation decisions amongst electricity generators. Inthe upper level, the generators decide investment decisions whilst accounting for operational decisions in thelower level. Pozo et al. (2013) and Pozo et al. (2012) developed a model similar to Wogrin et al. (2012) butadd an extra level to the model; an ISO who makes transmission expansion decisions whilst accounting forgenerators’ capacity investment and operational decisions. Jin and Ryan (2013) consider a similar EPECmodel as well but, in contrast to Pozo et al. (2013) and Pozo et al. (2012), model price-responsive demandand strategic interactions amongst the generators.Kazempour et al. (2013) and Kazempour and Zareipour (2013) use EPEC models where the upper-level problem determines the optimal investment for strategic producers while lower-level problems representdifferent market clearing scenarios. Similarly, Ye et al. (2017) use a EPEC model to investigate the impactconsumer led demand shifting has on market power and find that demand response can reduce the negativeimpacts of market power. The upper level again represents the producers’ problems while the lower-levelrepresent the market clearing process, in addition to the consumers decisions. An EPEC model is used inHuppmann and Egerer (2015) as part of a three-stage equilibrium model between a supra-national planner,zonal planners, and an ISO. Moiseeva et al. (2017) develop an EPEC model that considers generators’operational decisions in the lower level and their ramping decisions in the upper level. More recently, Guoet al. (2019) introduce another EPEC model where the upper level maximises generators’ decisions whilethe lower level represents an ISO. Interestingly, Guo et al. (2019) account for risk-averse decision making byincorporating Conditional Vale at Risk (CVaR) into their model.Despite the rich literature of EPEC models for electricity markets, none of the aforementioned EPECproblems model a market characterised by an oligopoly with competitive fringe, where some generators havemarket power and other do not. The closest work to the present paper is Zerrahn and Huppmann (2017). Theypropose a three-stage game to model transmission network expansion in an imperfectly competitive marketwhere some generators have market power while do not. They solve the model using backward induction.The third stage represents the problem of the ISO and the competitive fringe. The second stage representsthe firms who have market power and thus account for the third stage. In the first stage, social welfare ismaximised using network expansion decisions whilst accounting for the second and third stages. Significantly,we advance the work of Zerrahn and Huppmann (2017) by including generation expansion/investments forgenerating firms in our model.We apply the EPEC developed in this work to an electricity market representative of the Irish powersystem in 2025 using data from Lynch et al. (2019a) and Bertsch et al. (2018). EPEC models can bechallenging to solve computationally. We utilise the Gauss Seidel algorithm in to order use diagonalizationfor solving the EPEC, and we solve each individual MPEC using disjunctive constraints (Fortuny-Amat andMcCarl, 1981). In addition, to improve computational efficiency, we utilise the approach developed in Leyfferand Munson (2010) to provide an initial strong stationary point of the EPEC to use as a starting point for4ur diagonlization algorithm. We solve the model numerically as it is too large to be solved in closed form,which is another contribution of this work. A closed-form solution is possible using standard techniques butwe combine two techniques from the literature in order to solve our problem.Our results show that may it be optimal for generating firms with market power to occasionally operatesome of their generating units at a loss. The driving factor behind this model outcome is the fact we allowboth price-making and price-taking firms to make investment decisions. Price-taking firms’ ability to investfurther into the market motivates the price-making firms to depress prices in some timepoints. This reducesthe revenues price-taking firms could make from new investments and thus prevents them from makingsuch investments. Such strategic behaviour would not be captured by MCP or cost-minimisation models.Consequently, this result highlights the suitability of the EPEC modelling approach and the importance ofincluding investment decisions in models of oligopolies with competitive fringes.The remainder of this paper is structured as follows: firstly, in Section 2, we describe the model datainputs. Secondly, in Section 3 we demonstrate the naivety of using a MCP to model an oligopoly with acompetitive fringe where both price-making and price-taking firms have investment decisions. Thirdly, inSection 4, we introduce the EPEC model. In Sections 6 and 7, we provide some discussion and conclusions,respectively. Finally, in Appendix A, we provide additional material related to the case study.
2. Input data
Table 1: Indices and sets. f ∈ F Generating firms t ∈ T Generating technologies p ∈ P Time periods Table 2: Variables.Price-taking firms’ primal variables gen PT f,t,p Forward generation from price-taking firm f with technology t in period pinv PT f,t Investment in new generation capacity (technology t ) for price-taking firm f Price-making firms’ primal variables gen PM l,t,p Forward generation from price-making firm l with technology t in period pinv PM l,t Investment in new generation capacity (technology t ) for price-making firm l Dual variables γ p System price for time period pλ PT f,t,p Lagrange multiplier associated with price-taking firm f ’s capacity constraint for technology t and timestep pλ PM l,t,p Lagrange multiplier associated with price-making firm l ’s capacity constraint for technology t and timestep p In this section, we introduce the market we consider and describe the data inputs for models we use inSections 3 and 4. The electricity market we consider consists of two types of players: price-making firms and5 able 3: Parameters.
CAP PT f,t Initial generating capacity for price-taking firm f with technology tCAP PM f,t Initial generating capacity for price-taking firm l with technology tA p Demand curve intercept for timestep pB Demand curve slope C GEN t Marginal generation cost for technology tIC
GEN t Investment in generating technology t cost W p Weighting assigning to timestep pCV l Conjectural variation associated with firm lE t Emissions factor level for technology t price-taking firms. Price-making firms may exert market power by using generation decisions to influencethe market price. Price-taking firms do not have such ability.Each firm chooses its forward market generation decision so as to maximise its profits. Each firm mayalso hold multiple generating units with the technologies considered being baseload, mid merit and peakload.The firms are distinguished by their price-making ability and their initial generation portfolio they may hold.However, each firm may also invest in new generation capacity in any of the technologies. Table 4: Initial power generation portfolio by firm (
CAP f,t ).Technology firm 1 firm 2 firm 3 firm 4price-making price-making price-taking price-takingExisting baseload (MW) 1947 1940 - -Existing mid merit (MW) 512 - 404 -Existing peakload (MW) 270 - - 234New baseload (MW) 0 0 0 0New mid merit (MW) 0 0 0 0New peakload (MW) 0 0 0 0
We consider a electricity market that consists of four generating firms; firms l = 1 and l = 2 are price-making firms and while firms f = 3 and f = 4 are price-taking firms. We consider | T | = 6 generatingtechnologies; existing baseload, existing mid-merit, existing peaking, new baseload, new mid-merit and newpeaking. Each of the four firms hold different initial generating capacities. Firms l = 2, f = 3 and f = 4are, initially, specialised baseload, mid-merit and peaking firms, respectively. In contrast, firm l = 1 is aintegrated firm initially holding capacity across each of the existing technologies. Because of their sizes, theintegrated firm and the specialised baseload firm are modelled as the price-making firms while the specialisedmid-merit and peaking firms are the price-taking firms. Initially, each firm only holds ‘existing’ technologiesbut, through their respective optimisation problems, may invest in any of the ‘new’ technologies. Given thestylised nature of the model and following Devine et al. (2019) and Lynch et al. (2019b) , we do not explicitlymodel renewable technologies. Wind is incorporated into the model via the (net) demand intercept (seeMarket Clearing Condition (3)). We assume wind is not owned by any generation firm and its sole function6s to reduce net demand. This is because wind has a marginal cost of zero and furthermore can only bedispatched downwards, and so given an exogenously-determined level of wind capacity, wind generation itselfis unlikely to ever be strategically withheld by a generation firm (Devine et al., 2019).The initial portfolios of each firm are displayed in Table 4. These capacities follow from Lynch et al.(2019a) and Bertsch et al. (2018) and are broadly based on EirGrid (2016). Table 5: Summary of techno-economic input data of considered supply side technologies.Technology Annuity of specific invest Marginal gen. costs Spec. CO emissions( IC GEN t ) ( A GEN t ) ( E t )( e /MW y) ( e /MWh el ) (t CO /MWh el )Existing baseload - 48.87 1.17Existing mid merit - 41.10 0.36Existing peakload - 63.38 0.56New baseload 110,769 31.58 0.78New mid merit 67,268 34.00 0.30New peakload 40,363 50.50 0.45 The different characteristics associated with the technologies are displayed in Table 5. Both the marginalgeneration and investment costs again follow from Lynch et al. (2019a) and Bertsch et al. (2018). Themarginal investment costs represent annualised investment costs. We consider | P | = 5 forward time periods.Table 6 displays the demand curve intercept values which correspond to average hourly values for eachtime period. However, each time period p is assigned a weight W p = . Thus, the test case in this workrepresents one year. Following Lynch and Devine (2017), the five time periods represent summer low demand,summer high demand, winter low demand, winter high demand and winter peak demand. The demand curveslope value chosen is B = 9 . p ) 1 2 3 4 525175.993 26768.307 30429.701 34302.196 37465.783 Table 6: Demand curve intercept ( A p ) values
3. Modelling electricity market as a mixed complementarity problem
In this section, we motivate our EPEC modelling approach, by applying the above data to a MixedComplementarity Problem (MCP). This analysis demonstrates the naivety of using a MCP to model anoligopoly with a competitive fringe where both price-making and price-taking firms have investment decisions.A MCP determines an equilibrium of multiple optimisation problems by finding a point that satisfies theKKT conditions of each optimisation simultaneously as a system of non-linear equations (Gabriel et al.,7009). MCPs have used to model many energy markets (Huppmann, 2013; Egging, 2013). However, in aMCP modelling framework, a price-making firm’s optimisation problem does not contain the optimal reactionsof price-taking firms as constraints. As this subsection shows, this omission leads to myopic and contradictoryoutcomes. This is in contrast to the EPEC model described in Section 4. Appendix A describes the MCPproblem. The MCP consists of the market clearing condition (3), the price-taking firms’ KKT conditionsand the KKT conditions for all price-making firms. The parameter CV l represents the Conjectural Variation(CV) associated with firm l . CVs have been implemented in many cases MCP models (Egging et al., 2008;Haftendorn and Holz, 2010; Huppmann and Egging, 2014; Egging and Holz, 2016) as they allow price-makingfirms to somewhat account for the optimal reactions of competitors. Conjectural variations take a value inthe range [0 , CV l =1 = CV l =2 = 0, both price-making firms lose their price-makingability and thus the market outcome corresponds to perfect competition. The remaining cases correspond toan oligopoly with competitive fringe modeled through conjectural variations. Figure 1: Price-making firm l = 1’s profits using MCP setting framework. Figure 1 describes the profits of the price-making firm l = 1 for the MCP cases. It shows that firm l = 1actually make less profits in the oligopoly with competitive fringe cases compared with perfect competitioncase. Clearly, if firms have price-making ability then they should be able to to use that ability, at the very8 igure 2: Price-making firms’ investment into new mid-merit generation under MCP setting framework. least, to make the same profits as they would have in a perfect competition setting.The result can be explained by Figure 2 which shows the investment in new mid-merit generation forperfect competition case and CV l =1 = CV l =2 = 1 case (similar results are observed for the 0 < CV l < ∂γ p ∂gen PM l,t,p = − CV l × B in the oligopoly with competitive fringecase. These means that these two firms assume that if they decrease the amount of electricity they generateby one MW, then the equilibrium market price will increase by e CV l × B . In seeking to increase profits,price-making firms l = 1 and l = 2 decrease their generation in this way and hence do no invest in any newtechnology, as there is no point if they are not going to fully use that new generation.However, in the MCP with CV setting, price-making firms do not correctly account for the optimalreactions of the competitive fringe. Consequently, when price-making firms decrease their generation, theprice-taking firms increase their generation and replace the price-making firms generation. Thus, the marketprice does not increase as much the price-making firms anticipate, if it increases at all. The expandedopportunity for price-taking firms to generate enables them to invest further into new mid-merit generation,as evidenced by Figure 2.The assumption that ∂γ p ∂gen PM l,t,p = − CV l × B in the oligopoly with competitive fringe case is not valid in a9CP setting where investment decisions are also incorporated. However, this assumption is valid in a MCPsetting if all firms are price-making firms and hence behave in the same manner. In other words, when onefirms seeks to increase the market price by decreases its generation, so does the rest of the firms and no onefirm replaces the decreased generation from any other firm.This section demonstrates how the MCP modelling approach is unsuited to modelling an oligopoly withcompetitive fringe when investment decisions are also accounted for. Moreover, conjectural variations cannotovercome this modelling issue. In the following section, we show how the EPEC modelling approach overcomesthe short-sighted/myopic behaviour observed in this section.
4. Equilibrium problem with equilibrium constraints
In this section we describe the Equilibrium Problem with Equilibrium Constraints (EPEC) we introducein this work. As before, it represents an electricity market with two types of players: price-making firms andprice-taking firms. Price-making firms may exert market power by using generation decisions to influencethe market price. Price-taking firms do not have such ability.Each firm chooses its forward market generation decision so as to maximise its profits. Each firm mayalso hold multiple generating units with the technologies considered being baseload, mid merit and peakload.The firms are distinguished by their price-making ability and their initial generation portfolio they may hold.However, each firm may also invest in new generation capacity in any of the technologies.The optimisation problems of each price-taking firm are embedded into the optimisation problem of eachprice-making firm. Thus, each price-making firm’s problem is a bi-level optimisation problem and can bedescribed as a Mathematical Program with Equilibrium Constraints (MPEC); the equilibrium constraints arethe optimality conditions of the price-taking firms. This problem formulation enables each price-making firmto influence the market price through their decision variables, account for the optimal reactions of price-takingfirms and, consequently, maximise profits.The overall EPEC problem is to find an equilibrium among the MPEC problems of each price-makingfirm which represent Nash Equilibria amongst them. Each MPEC problem can be represented as a MixedInteger Non-Linear Problem (MINLP) and, thus, finding Nash Equilibria is a challenging task. To do so,we employ the Gauss-Seidel algorithm . Furthermore to obtain an initial starting solution for this algorithmwe utilise the approach taken in Leyffer and Munson (2010) for solving EPEC problems (henceforth knownas the Leyffer-Munson approach). Both the Guass-Seidel algorithm and the Leyffer-Munson approach aredescribed in detail in this section.Throughout this section the following conventions are used: lower-case Roman letters indicate indicesor primal variables, upper-case Roman letters represent parameters (i.e., data), while Greek letters indicateprices or dual variables. The variables in parentheses alongside each constraint in this section are the Lagrangemultipliers associated with those constraints. Tables 1 - 3 explain the indices, variables and parameters,respectively, associated with both the price-making and price-taking firms’ optimisation problems.10 .1. Price-taking firm f ’s problem Price-taking firm f seeks to maximise its profits (revenue less costs) by choosing investments in newcapacity ( inv PT f,t ) and by choosing the amount of electricity to generate from each technology at each timeperiod ( gen PT f,t,p ). We assume each time period represents a forward time-period. Thus gen PT f,t,p representsforward market generation decisions. Firm f ’s costs include the per unit investment cost ( IC GEN t ) and themarginal cost of generation ( C GEN t ) while its revenues comes from the forward market price γ p .Price-taking firm f ’s optimisation problem is as follows:max gen PT f,t,p ,inv PT f,t (cid:88) t,p W p × gen PT f,t,p × (cid:0) γ p − C GEN t (cid:1) − (cid:88) t IC GEN t × inv PT f,t , (1)subject to: gen PT f,t,p ≤ CAP PT f,t + inv PT f,t , ∀ t, p, (2)where the parameter W p is the weight associated with timestep p . Constraint (2) ensures, for each generatingtechnology in each timestep, firm f cannot generate more than its initial capacity ( CAP PT f,t ) plus any newinvestments. The variable alongside constraint (2) ( λ PT , f,t,p ) is the Lagrange multiplier associated with thisconstraint. In addition, each of firm f ’s generation and investment decisions are constrained to be non-negative.As firm f is a price-taker, it cannot influence the market price with its generation decision. The variable γ p is exogenous to firm f ’s problem but a variable of the overall EPEC problem. When determining firm f ’sKarush-Kuhn-Tucker (KKT) conditions, it is assumed ∂γ p ∂gen PT f,t,p = 0. Moreover, firm f cannot see and henceaccount for the optimal decisions of other price-taking firms in addition to those of price-making firms. Asfirm f ’s problem is linear, solving its associated KKT conditions ensures its problem is optimised (Gabrielet al., 2012). The forward market price for each time period is determined from the following market clearing condition: γ p = A p − B × (cid:0) (cid:88) ll,tt gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1) , ∀ p, (3)where A p represents the demand curve intercept for each time period while B is the time independent demandcurve slope. Condition (3) represents a linear demand curve and allows the market price to increases as thetotal market generation decreases and vice-versa. l ’s MPEC Price-making firm l ’s optimisation problem is similar to firm f ’s problem in that it too seeks to maximiseprofits (revenues less cost) by choosing its investment ( inv PM l,t ) and forward market generation ( gen PM l,t,p )decisions. As before, firm l ’s revenues come the forward market price while its costs include marginal11eneration and investment costs. In contrast to Section 4.1 however, price-making firm l can use theirgeneration decisions to influence the market price. Price-making firm l can also account for the optimalreactions of the price-taking firms. Its objective function ismax gen PM l,t,p ,inv PM l,t gen PT f,t,p ,inv PT f,t γ p ,λ PT f,t,p (cid:88) t,p W p × gen PM l,t,p × (cid:0) γ p − C GEN t (cid:1) − (cid:88) t IC GEN t × inv PM l,t . (4)As firm l can influence the market price through its generation decisions, we re-write objective function (4)using market clearing condition (3) as follows:max gen PM l,t,p ,inv PM l,t gen PT f,t,p ,inv PT f,t γ p ,λ PT f,t,p (cid:88) t,p W p × (cid:18) A p − B × (cid:0) (cid:88) ll,tt gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1) − C GEN t (cid:19) × gen PM l,t,p − (cid:88) t IC GEN t × inv PM l,t . (5)The constraints of price-making firm l ’s problem are gen PM l,t,p ≤ CAP PM l,t + inv PM l,t , ∀ t, p, (6)where each of firm l ’s generation and investment decisions are also constrained to be non-negative. As withthe price-taking firms, constraint (6) ensures, for each technology at each time period, firm l cannot generatemore electricity than its initial initial capacity plus any new investments. In addition to constraint (6), firm l ’s constraints also include the KKT conditions of the each price-taking firm:0 ≤ gen PT f,t,p ⊥ − W p × (cid:0) γ p − C GEN t (cid:1) + λ PT f,t,p ≥ , ∀ f, t, p, (7)0 ≤ inv PT f,t ⊥ IC GEN t − (cid:88) p λ PT f,t,p ≥ , ∀ f, t, (8)0 ≤ λ PT , f,t,p ⊥ − gen PT f,t,p + CAP PT f,t + inv PT f,t ≥ , ∀ f, t, p. (9)Using market clearing condition (3) leads to condition (7) being re-written as follows:0 ≤ gen PT f,t,p ⊥ − W p × (cid:18) A p − B × (cid:0) (cid:88) ll,tt gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1) − C GEN t (cid:19) + λ PT f,t,p ≥ , ∀ f, t, p. (10)Constraints (8) - (10) represent the optimal reactions, of each price-taking firm. As firm f ’s problem (equa-tions (1) and (2)) is a linear optimisation problem, these KKT conditions are both necessary and sufficientfor optimality for the price-taking firms (Gabriel et al., 2012).Incorporating these conditions as constraints makes firm l ’s problem a bi-level optimisation problem andensures firm l correctly anticipates how each price-taking firm will react to its decisions. Thus, this allowsfirm l to adjust its decisions accordingly when seeking maximise its profits.Firm l ’s optimisation problem is affected by the generation decisions of all other price-making firms; seeobjective function (5) and constraint (10). However, when solving firm l ’s problem we assume the decisionsof all other price-making firms are fixed and exogenous to firm l ’s problem. Sections 4.4 – 4.6 describes howthe optimisation problems all price-making firms are solved such that solutions represent Nash equilibria.12s the KKT conditions (8) - (10) represent the equilibrium constraints (optimal reactions) of the price-taking firms, firm l ’s problem is a Mathematical Program with Equilibrium Constraints. We denote thisproblem as MPEC l , which is a non-linear mathematical program because of the bi-linear terms in objectivefunction (5) involving firms l ’s generation decisions and the generation decisions of all price-taking firms andbecause of the complementarity conditions incorporated into constraints (8) - (10). However, following theapproach presented in Fortuny-Amat and McCarl (1981), we can remove the latter source of non-linearityusing disjunctive constraints and big M notation. Consequently, this leads to constraints (8) - (10) beingre-written as follows:0 ≤ gen PT f,t,p ≤ M × r f,t,p , ∀ f, t, p, (11)0 ≤ − W p × (cid:18) A p − B × (cid:0) (cid:88) ll,tt gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1) − C GEN t (cid:19) + λ PT f,t,p ≤ M × (1 − r f,t,p ) , ∀ f, t, p, (12)0 ≤ inv PT f,t ≤ M × r f,t , ∀ f, t, (13)0 ≤ IC GEN t − (cid:88) p λ PT f,t,p ≤ M × (1 − r f,t ) , ∀ f, t, (14)0 ≤ λ PT , f,t,p ≤ M × r f,t,p , ∀ f, t, p. (15)0 ≤ − gen PT f,t,p + CAP PT f,t + inv PT f,t ≤ M × (1 − r f,t,p ) , ∀ f, t, p. (16)where r f,t,p , r f,t and r f,t,p all represent binary 0-1 variables.When solving the overall EPEC problem using the Gauss-Seidel algorithm (see Section 4.4), MPEC l ischaracterized by objective function (5), subject to constraint (6) and constraints (11) - (16). Consequentlyprice-making firm l ’s optimisation problem is a Mixed Integer Non-Linear Problem. In Section 5, we use theDICOPT solver in GAMS to solve it. The overall EPEC can be expressed as the problem of finding Nash equilibria among the price-makers l :Find: (cid:26) inv PM l =1 ,t , ..., inv PM l = L,t ,gen PM l =1 ,t,p , ..., gen PM l = L,t,p ,inv PT f =1 ,t , ..., inv PT f = F,t ,gen PT f =1 ,t,p , ..., gen PT f = F,t,p ,λ PT f =1 ,t,p , ..., λ PT f = F,t,p ,γ p (cid:27) that solve:MPEC l for each l = 1 , ..., L. To find the such equilbiria, we implement the following Gauss-Seidel (Gabriel et al., 2012) algorithm.The algorithm iteratively solves each price-making firm’s MPEC problem by fixing every other price-making13rms’ decisions, until it converges to a point where neither leader has an optimal deviation. while (cid:80) l | x l,k − x l,k − | > T OL and k < K dofor l = 1 , .., L do Assume price maker − l ’s decision variables are fixed;Solve MPEC l ; endend Algorithm 1: Gauss-Seidel algorithmwhere
T OL and K represent a pre-defined convergence tolerance and a maximum number of allowableiterations, respectively. The vector x l,t represents the vector of all MPEC l ’s primal variables at iteration k . To improve computational efficiency, we utilise the approach to obtaining a strong stationary point forEPECs, as described in Leyffer and Munson (2010). We then use the statioanry point obtained as a startingpoint to the Gauss-Seidel algorithm. In this subsection, we describe how the Leyffer-Munson method asapplied to the EPEC presented in this work.Firstly, we re-write price-making firm l ’s problem, as defined by equations (5) - (10), using slack variables.We do this by converting firm l ’s inequality constraints into equality constraints as follows:max (cid:88) t,p W p × (cid:18) A p − B × (cid:0) (cid:88) ll,tt gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1) − C GEN t (cid:19) × gen PM l,t,p − (cid:88) t IC GEN t × inv PM l,t , (17)14ubject to: CAP PM l,t + inv PM l,t − gen PM l,t,p − s CON LR l,t,p = 0 , ∀ t, p, ( λ P Ml,t,p ) , (18) gen PM l,t,p ≥ , ∀ t, p, ( χ GEN l,t,p ) , (19) inv PM l,t ≥ , ∀ t, ( χ INV l,t ) , (20) s CON LR l,t,p ≥ , ∀ t, p, ( µ CON LR l,t,p ) , (21) − W p × (cid:18) A p − B × (cid:0) (cid:88) ll,tt gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1) − C GEN t (cid:19) + λ PT f,t,p − s KKT GEN f,t,p = 0 , ∀ f, t, p, ( α KKT GEN l,f,t,p ) , (22) gen PT f,t,p ≥ , ∀ f, t, p, ( µ KKT GEN l,f,t,p ) , (23) s KKT GEN f,t,p ≥ , ∀ f, t, p, ( µ s KKT GEN l,f,t,p ) , (24) gen PT f,t,p × s KKT GEN f,t,p = 0 , ∀ f, t, p, ( µ GEN s KKT GEN l,f,t,p ) , (25) IC GEN t − (cid:88) p λ PT f,t,p − s KKT INV f,t = 0 , ∀ f, t, ( α KKT INV l,f,t ) , (26) inv PT f,t ≥ , ∀ f, t, ( µ KKT INV l,f,t ) , (27) s KKT INV f,t ≥ , ∀ f, t, ( µ KKT s INV l,f,t ) , (28) inv PT f,t × s KKT INV f,t = 0 , ∀ f, t, ( µ INV s KKT INV l,f,t ) , (29) − gen PT f,t,p + CAP PT f,t + inv PT f,t − s CON FR f,t,p = 0 ∀ f, t, p, ( α CON f,t,p ) , (30) λ PT f,t,p ≥ , ∀ f, t, p, ( µ CON l,f,t,p ) , (31) s CON FR f,t,p ≥ , ∀ f, t, p, ( µ s CON l,f,t,p ) , (32) λ PT f,t,p × s CON FR f,t,p = 0 , ∀ f, t, p, ( µ CON s CON l,f,t,p ) . (33)The variables in brackets alongside each of these constraints are the Lagrange multipliers associated withthose constraints. Note: each of the multipliers has a subscript l associated with it showing how there areunique multipliers for each of the price-making firms problems.Secondly, we find the stationary KKT conditions of the optimisation problem (17) - (33). Let L l be theLagrangian associated with that problem. ∂ L l ∂gen PM l,t,p : − W p × (cid:18) A p − B × (cid:0) (cid:88) ll,t gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1) − B × gen PM l,t,p − C GEN t (cid:19) + λ PT f,t,p + (cid:88) ff,tt W p × B × α KKT GEN l,ff,tt,p − χ GEN l,t,p = 0 , ∀ t, p, (34) ∂ L l ∂inv PM l,t : IC GEN t − (cid:88) p λ PM l,t,p − χ INV l,t = 0 , ∀ t, (35)15 L l ∂gen PT f,t,p : (cid:88) tt W p × B × gen PM l,tt,p + (cid:88) ff,tt W p × B × α KKT GEN l,ff,tt,p − µ KKT GEN l,f,t,p + s KKT GEN f,t,p × µ GEN s KKT GEN l,f,t,p − α CON f,t,p = 0 , ∀ f, t, p, (36) ∂ L l ∂inv PT f,t : − µ KKT INV l,f,t + s KKT INV f,t × µ INV s KKT INV l,f,t − (cid:88) p α CON f,t,p = 0 , ∀ f, t, (37) ∂ L l ∂λ PT f,t,p : − α KKT GEN l,f,t,p + α KKT INV l,f,t − µ CON l,f,t,p + s CON FR f,t,p × µ CON s CON l,f,t,p = 0 , ∀ f, t, p, (38) ∂ L l ∂s CON LR l,t,p : λ P Ml,t,p − µ CON LR l,t,p = 0 , ∀ t, p, (39) ∂ L l ∂s KKT GEN f,t,p : − α KKT GEN l,f,t,p − µ s KKT GEN l,f,t,p + gen PT f,t,p × µ GEN s KKT GEN l,f,t,p = 0 , ∀ f, t, p, (40) ∂ L l ∂s KKT INV f,t : − α KKT INV l,f,t − µ KKT s INV l,f,t + inv PT f,t × µ INV s KKT INV l,f,t = 0 , ∀ f, t, (41) ∂ L l ∂s CON FR f,t,p : − α CON f,t,p − µ s CON l,f,t,p + λ PT f,t,p × µ CON s CON l,f,t,p = 0 , ∀ f, t, p. (42)In addition, each of the Lagrange multipliers associated with inequality constraints in (17) - (33) are con-strained to be non-negative.Following this, we find the complementary KKT conditions of the optimisation problem (17) - (33) asfollows: gen PM l,t,p × χ GEN l,t,p = 0 , ∀ t, p, (43) inv PM l,t × χ INV l,t = 0 , ∀ t, (44) s CON LR l,t,p × µ CON LR l,t,p = 0 , ∀ t, p, (45) gen PT f,t,p × µ KKT GEN l,f,t,p = 0 , ∀ f, t, p, (46) s KKT GEN f,t,p × µ s KKT GEN l,f,t,p = 0 , ∀ f, t, p, (47) inv PT f,t × µ KKT INV l,f,t = 0 , ∀ f, t, (48) s KKT INV f,t × µ KKT s INV l,f,t = 0 , ∀ f, t, (49) λ PT f,t,p × µ CON l,f,t,p = 0 , ∀ f, t, p, (50) s CON FR f,t,p × µ s CON l,f,t,p = 0 , ∀ f, t, p. (51)The Leyffer-Munson method, as applied to this work, is obtain a solution set that satisfies conditions(18) - (51) of each price-making firm l simultaneously. To do this, each KKT condition with bi-linear terms16equations (25), (29), (33) and (43) - (51)) are removed as constraints and are summed together to createthe following objective function:min (cid:88) f,t,p gen PT f,t,p × s KKT GEN f,t,p + (cid:88) f,t inv PT f,t × s KKT INV f,t + (cid:88) f,t,p λ PT f,t,p × s CON FR f,t,p + (cid:88) l,t,p gen PM l,t,p × χ GEN l,t,p + (cid:88) l,t inv PM l,t × χ INV l,t + (cid:88) l,t,p s CON LR l,t,p × µ CON LR l,t,p + (cid:88) f,t,p gen PT f,t,p × µ KKT GEN l,f,t,p + (cid:88) f,t,p s KKT GEN f,t,p × µ s KKT GEN l,f,t,p + (cid:88) f,t inv PT f,t × µ KKT INV l,f,t + (cid:88) f,t s KKT INV f,t × µ KKT s INV l,f,t + (cid:88) f,t,p λ PT f,t,p × µ CON l,f,t,p + (cid:88) f,t,p s CON FR f,t,p × µ s CON l,f,t,p . (52)Thus, the Leyffer-Munson optimisation problem is to minimise equation (52) subject to constraints (18) -(24), (26) - (28), (30) - (32) and (34) - (42). In addition, each of the Lagrange multipliers associated withinequality constraints in (17) - (33) are constrained to be non-negative. The Leyffer-Munson optimizationproblem is a Non-Linear Program (NLP) and, in Section 5, we use the CONOPT solver in GAMS to solveit. Algorithm 2 describes the overall algorithm for finding Nash equilibria from the EPEC problem. Foriteration i , we firstly provide a random solution set from the search space and use these as initial starting pointsolutions for the Leyffer-Munson approach. As the Leyffer-Munson approach is a non-linear optimisationproblem, the CONOPT solver does not always find a local minimum. If the Leyffer-Munson method does notconverge to a locally optimal solution, then iteration i is deemed unsuccessful and the algorithm skips aheadto iteration i + 1. If the Leyffer-Munson approach does converge however, the locally optimal solution is thenused as starting point solution for the Gauss-Seidel algorithm. If the Gauss-Seidel does (not) converge to aNash equilibrium solution, then iteration i is (not) deemed successful. This process is repeated for I iterations. for i = 1 , .., I do Provide random initial solutions;Solve Leyffer-Munson optimisation problem; if Solution from LM is locally optimal then
Solve EPEC using Gauss-Seidel algorithm using solutions from LM as starting point ; if Gauss-Seidel algorithm converges then
Save solution; endendend Algorithm 2:
Overall algorithm for finding Nash Equilibria17 . Results from EPEC model
In this section, we present the results when the data presented in Section 2 is applied to the EPEC modeldescribed in Section 4. We focus on the firms’ profits, firms’ investment decisions, market prices, consumercosts and carbon emissions. To obtain these results we utilise the algorithm described in Section 4.6 for I = 2000 iterations. For the first 1000 iterations firm l = 1’s MPEC problem is solved before firm l = 2’sMPEC problem. For the subsequent 1000 iterations the opposite applies and firm l = 2’s MPEC problem issolved before firm l = 1’s MPEC problem.The algorithm did not always find a Nash Equilibrium (NE) solution. In fact, in the results to follow,only 72 of the 2000 iterations successfully found a NE solution, henceforth know as successful iterations. Ofthese, 62 iterations occurred when firm l = 1’s MPEC problem was solved before firm l = 2’s MPEC problemwhile the remaining 10 successful iterations occurred when firm l = 2’s MPEC problem was solved first. Forunsuccessful iterations the algorithm failed to find a NE solution for one of two reasons:1. For the random initial solution provided, the Leyffer Munson was found to be locally infeasible by theCONOPT solver.2. For the Gauss-Seidel algorithm, the convergence tolerance remained greater than T OL = 10 − after K = 100 iterations.For each of the 72 successful iterations, the Leyffer-Munson approach solved to a locally optimal solutionwhich implied that it is not necessarily a feasible solution to the EPEC. For 43% of these successful iterations,the objective function for the Leyffer-Munson approach (equation (52)) converged to zero. For the remaining57% of successful iterations, the objective function converged to a strictly positive objective function value.When the Leyffer-Munson approach gives a non-zero objective function value, the solutions cannot guaranteedto be a feasible point for the overall EPEC. However, the results in this section show that, despite this, suchsolutions can still provide good starting point solutions to the Gauss-Seidel algorithm.Figures 3 and 4 display price-making firms l = 1 and l = 2 profits, respectively, for each of the successfuliterations. The horizontal lines in each figure represent the profits each firm would make from the perfectcompetition case in the Section 3. Figures 3 and 4 both show that the algorithm found multiple NE solutions.Firm l = 1’s profits varies from e e l = 2’s profits ranged from e e l = 1 and l = 2 profits, respectively.Thus showing that there was no NE found where both firms’ profits were below their perfect competitionequivalent.Figure 5 display the combined profits of the two price-making firms. It shows that the combined profitsvaried across the equilibria suggesting that there was not a zero-sum game between the price-making firmson how profits were split between them. We describe below why there are multiple equilibria and why, in18 igure 3: Profits for price-making firm l = 1 for each successful iteration.Figure 4: Profits for price-making firm l = 2 for each successful iteration. igure 5: Combined profits for price-making firms for each successful iteration. some equilibria, one of the price-making firms makes a profit less than it would in a perfectly competitivemarket.Figure 6 displays the combined investments into new mid-merit generation for the two price-making firmsfor each successful iteration. In comparison to Section 3, both of these firms did not invest in any baseloador peaking generation. However, in contrast to Section 3, there was only one equilibrium point found wherethe price-taking firms invested in any new generation technology.For the majority of equilibria found (80%), the combined investments were 2941MW. But, for someequilibria, the combined investments were slightly higher with maximum combined investment reaching2967MW at one equilibrium point while the lowest combined investment was 2831MW. The first 62 successfuliterations in Figures 3 - 5 came when firm l = 1’s MPEC problem was solved before firm l = 2’s. At theseequilibria, firm l = 1 and l = 2’s investments in new mid-merit generation averaged at 2807MW and 496MW,respectively. The final 10 successful iterations occurred when firm l = 2’s MPEC problem was solved first.As a result, at these equilibria, firm l = 1 average investments in new generation decreased to 1467MW whilefirm l = 2’s increased to 2618MW.The results show that firm l = 1 makes a profit less than it would of in a perfectly competitive market,in some of the equilibria found. This is because we model two price-making firms. When price-making firm l commits to a large amount of forward generation, it can leave firm ˆ l (cid:54) = l with a reduced opportunity togenerate and hence reduced profits. We explain this in further detail in Section 5.1. Interestingly, in each ofthe successful iterations where firm l = 2 MPEC is solved first, firm l = 1’s profits are significantly below20 igure 6: Combined investment in new mid-merit for price-making firms for each successful iteration. their perfect competition result while firm l = 2 are significantly higher - this result highlights the importanceof the order in which MPEC problems of an EPEC are searching, when searching for equilibria.Despite Figures 3 - 6 presenting the multiple equilibria, the forward prices rested at one of three pricetime series. Time series one and two were observed in 81.9% and 16.7% of the equilbira found while the thirdseries was only observed at one of the equilibrium points found. Figure 7 displays these three price timeseries along with the prices from the perfect competition case of Section 3. Interestingly, for time periods p = 2 ,
3, the forward market prices in the oligopoly with competitive fringe case are less than those from theperfect competition case. This is despite half the firms having price-making ability. However, the marketprices in the oligopoly with competitive fringe case are higher at later time steps. Note: both the first andsecond equilibrium price time series were found when both firm l = 1 and firm l = 2 MPEC’s were solvedfirst while the only instance of the third equilibrium price time series occurred when firm l = 2’s MPEC wassolved first.The forward prices in Figure 7 can be explained by Figures 8 – 10. Figure 8 shows the generation mixfor the first successful iteration while Figures 9 and 10 display the revenues earned/lost by firms l = 1 and l = 2, respectively, in each time period for the same iteration. At this equilibrium point, forward pricesconverged to first time series in Figure 7 and firm l = 1 and firm l = 2 invested 2765MW and 176MW intonew mid-merit generation, respectively. In time periods p = 1 and p = 2, only firm l = 1 and l = 2’s newmid-merit units are generating leading to forward prices of γ p =1 = γ p =2 = 34, the marginal cost of newmid-merit. Consequently, neither price-making firm earns, nor loses, revenue at these time periods.21 igure 7: Equilibrium forward market prices ( γ p ) for EPEC model versus perfect competition.Figure 8: Generation mix for the first successful iteration. igure 9: Revenue earned by firm l = 1 for the first successful iteration.Figure 10: Revenue earned by firm l = 2 for the first successful iteration. p = 3 but, because the demand curve intercept is higher (see Table6), more generation is needed to meet demand. The increased demand is primarily met by firm l = 2’s newmid-merit unit. In addition however, firm l = 1’s existing mid-merit unit generates 403MW. This is despiteexisting mid-merit having a marginal cost of 41.1. Thus, as Figure 9 outlines, firm l = 1 losses revenue atthis timepoint. Firm l = 2 does not earn revenue, nor does it lose revenue, at p = 3.In time period p = 4, the forward price is 41.1 which is the marginal cost of an existing mid-merit unit.Consequently, all mid-merit units, for firm l = 1, l = 2 and f = 3 are utilised. In addition, firm l = 1 alsoutilises is existing baseload despite the marginal cost of exiting baseload being 48.87. The forward price is γ p =5 = 65 .
19 at timestep p = 5. Because this price is higher than the marginal cost of exiting baseload,both price-making firms utilise their existing baseload units and make a profit from doing so. The two price-making firms use their generation to set γ p =5 = 65 .
19 and hence maximise their respective profits. Thisforward price allows the two price-making firms to partially recover the investments cost associated withinvesting in new mid-merit generation. Because they both do not earn any revenue from new mid-merit intimesteps p = 1 , ,
3, the remaining investment costs are recovered in timestep p = 4 where the market priceof γ p =4 = 41 . l = 1 and l = 2 to earn enough revenue from their new mid-merit units tobreak even on their investments. If either price-making firm adjusted their generation so as to set a pricehigher than 41.1 in p = 4 or higher than 65.19 in p = 5, then the two price-taking firms would invest in newmid-merit generation also, as it would be a profitable decision. The price-making firms prevent this becauseinvestment from the price-taking fringe would erode the substantial revenues they earn in timestep p = 5.Similarly, it is profit maximising for firm l = 1 to generate using its existing mid-merit unit, at belowmarginal cost, in time period p = 3. If firm l = 1 did not do this, the remaining demand would be metby firm f = 3’s existing mid-merit unit, which would drive up the market price and thus, make investingin a new mid-merit a profitable option for both price-taking firms. Again, such a market outcome, is notprofit-maximising for firm l = 1. Instead, it is optimal for firm l = 1 to take take the small losses in timeperiod p = 3 so as to prevent the fringe from eroding its large profits in timestep p = 5. As Figure 9 shows,firm l = 1’s revenues from p = 4 and p = 5 far exceed its losses from p = 3. Additionally, in time periods p = 1 ,
2, it is optimal for firm l = 1 to ensure the market price is γ p =1 = 34. However, in these timesteps,firm l = 1 does not need its existing mid-merit unit to maintain the price at this level.These results are in contrast to the prices observed in the perfect competition case; see Figure 7. In theperfect competition setting, firms only utilise a generating unit if the market price is at or above the marginalcost of that unit. Consequently, the market price is set by the marginal cost of the most expensive unit thatis generating. Hence, there is no below marginal cost operation of units in time periods p = 3 and p = 4,which leads to higher forward prices compared with the oligopoly with a competitive fringe case. Similarly,in time period p = 5, the forward price in the perfect competition case is set by the most expensive unit thatis generating; existing baselaod. In contrast, in the oligopoly with a competitive fringe case, it is optimalfor the price-making firms to adjust their generation to ensure the forward price is higher than the perfect24ompetition case.Similar qualitative results to those in Figures 8 and 9 can be seen in the rest of the successful iterations.The exact level of revenue earned or lost in each timestep, for both price-making firms, varies in a similarmanner to Figures 3 - 6.Appendix A and equation (A.3) shows that when using a MCP model, it is never optimal for a generatorto operate one of its units at below marginal cost. In contrast, when the EPEC approach of this work isutilised, equation (34) shows that when gen PM l,t,p > γ p can be less than C GEN t . This is because of theadditional α KKT GEN l,ff,tt,p that is in equation (34) but not in equation (A.3). Moreover, this further highlightsthe benefit of the EPEC approach and the limitations of the MCP approach when modelling an oligopolywith a competitive fringe and investment decisions.The generation levels of price-taking firms f = 3 and f = 4 were similar for all equilibria that convergedto the first two equilibrium price series. Both price-taking firms utilised their existing mid-merit and peakingunits, respectively, to maximum capacity in time period p = 5 only as this was the only time period wherethe price was high enough for them to make earn profits. As the equilibrium forward prices converged to oneof only three series, the price-taking firms’ profits similarly converged to one of three levels. For equilibriathat converged on the first price time series in Figure 7, the profits were e e f = 3and f = 4, respectively, while for equilibria with the second time series, the profits were e e f = 3 also utilisedits existing at time period p = 4 in addition to p = 5. At this equilibrium point, firm f = 4 did not generateany electricity as the price was never high enough for them to so. Consequently, firm f = 4 made zero profitswhile firm f = 3 made a profit of e (cid:88) p (cid:18) W p × γ p × (cid:0) (cid:88) ll,tt gen PM ll,tt,p + (cid:88) ff,tt gen PT ff,tt,p (cid:1)(cid:19) . (53)As above, because the equilibrium prices landed at one of three price series, consumer costs also convergedto one of three levels. This is because the amount of energy consumed has a fixed relationship with marketprices; see market clearing condition (3). Figure 11 shows how the consumer costs increase by 1.7%, 2.1%,0.9% for equilibria that converged to first, second and third time series of forward prices, respectively.In previous works that use similar data, the presence of price-making behaviour was found to lead toa larger increases in consumer costs (Devine and Bertsch, 2018). However, the ability of the price-takingfringe to invest in new generation motivates the price-making firms to reduce forward market prices in sometime periods. While the market prices increase again in subsequent time periods, these consumer cost resultsshows how the presence of a competitive fringe helps mitigate against the negative effects of market power.Figures 12a and 12b display the carbon dioxide emissions level for equilibria that converged at the firstand second set of price time series, respectively. These emissions were calculated as follows: (cid:88) p,t (cid:18) W p × E t × (cid:0) (cid:88) ll gen PM ll,t,p + (cid:88) ff gen PT ff,p (cid:1)(cid:19) , (54)25 igure 11: Consumer costs as % of perfect competition case. (a) Equilibrium prices 1 (b) Equilibrium prices 2 Figure 12: C0 emission levels E t gives the emissions factor level for technology t , as displayed in Table 5. Figure 12show that, despite equilibrium prices remaining constant across subsets of the equilibria found, the emissionslevels varied across the equilibira. This is particularly evident in Figure 12b for equilibria with the secondprice time series. The reasons behind this results will now be explained. Figure 13: Generation mix for the second successful iteration.
Figures 3 - 6 show that there are multiple equilibria to the EPEC presented in this work. In this subsection,we explore the reasons behind this finding. Firstly, we look at two equilibria where the forward prices werethe same, i.e., the first equilibrium time series from Figure 7.The multiple equilibria are driven by the market’s indifference to what firm is providing electricity whenfirms are generating at the same price. For example, Figures 8 and 13 display the generation mixes forthe first two successful iterations, respectively. In the first successful iteration firms l = 1 and l = 2 invest2765MW and 176MW into new mid-merit generation, respectively. In contrast, in the second successfuliteration, they invest 2941MW and 0MW into new mid-merit generation, respectively.At time period p = 5 in the first iteration, firm l = 1 uses its new and existing mid-merit units tomaximum capacity while also generating 21MW from its baseload unit. As the same iteration, firm l = 2uses its new mid-merit unit to full capacity and also generates 2MW from its baseload unit. In the secondsuccessful iteration, firm l = 1 decreases its baseload generation at p = 5 from 21MW to 4MW but increasesits generation from new mid-merit from 2765MW to 2941MW. This allows firm l = 1 to make less profits27n Figure 13; firms break even on their new mid-merit investments but make profits from existing baseloadgeneration. Firm l = 2 increases its baselaod generation from 2MW to 19MW but decreases generation fromnew mid-merit from 176MW to 0MW. This allows firm l = 1 to make more profits in Figure 13.Because the market prices are the same across both equilibria considered, the market is indifferent towhether the electricity comes firm l = 1’s baseload or mid-merit or from firm l = 2’s baseload or mid-meritunits. Once firm l commits to forward generation decisions, firm ˆ l (cid:54) = l is not willing to adjust its generationlevels so as to either increase or decrease forward market price of γ p =5 = 65 .
19. If either price-making firmincreased any of the forward prices, then the price-taking firms would invest in new mid-merit generation,as explained in the previous subsection. It is also not profit-maximising for firm l to undercut firm ˆ l (cid:54) = l ata price lower than 65.19. To do so, would mean firm l would make a loss on its new mid-merit investment.Furthermore, if firm l adjusted its generation so as to decrease γ p =5 by e
1, then it would only be able to, atmost, increase its generation from existing baseload B = 0 . l (cid:54) = l to utilise its existing baseload at the reduced price.The small increase in generation opportunity would not make up for the decreased revenues resulting fromthe reduced price. This paragraph explains why in some of the equilbria found, firm l = 1 makes less profitsthan in the perfect competition case.Similar market in-differences are also observed in time periods p = 2-4 and in the other 57 successfuliterations that converge to the same price time series, thus explaining the multiple equilibria displayed inFigures 3 - 6. In some of other equilibria found, both price-making firms generate significant amounts fromtheir baselaod units in p = 5, thus preventing each other from generating and investing in new mid-meritgeneration. Consequently, both firms do not make as large a profit as they otherwise could. Such equilibriaare also evident in Figures 3 - 6. This particular result highlights the absence of collusion between the twoprice-making firms modelled in this work.We now examine the differences between two equilibria that converged to different forward price timeseries. Figure 14 displays the generation mix for the first successful iteration where the forward pricesconverged to the second time series in Figure 7 while Figures 15 and 16 show the revenues for firms l = 1and l = 2, respectively, for the same iteration. Firms l = 1 and l = 2 made profits of e e l = 2 generated 17MW from its existing baseload unit at time period p = 4. In contrast,in Figure 14, firm l = 2 increased its baseload generation to 93MW at time period p = 4. Following frommarket clearing condition (3), this lead to the forward price decreasing from γ p =4 = 41 . γ p =4 = 31between the two time series. This decrease in forward price meant that firm l = 1 needed to decrease itsoverall generation in p = 5 from 3456MW in Figure 8 to 3337MW in Figure 14. This resulted in a higherforward price in p = 5 for in the second time series and thus allowed both price-making firms to recover itsinvestment capital costs, despite the decreased price in p = 4.Firm l = 2 cannot make a profit from its existing baseload unit in time period p = 4 at either γ p =4 = 41 . igure 14: Generation mix for the first successful iteration that results in the second time series for forward prices.Figure 15: Revenue earned by firm l = 1 for the first successful iteration that results in the second time series for forward prices. igure 16: Revenue earned by firm l = 2 for the first successful iteration that results in the second time series for forward prices. or γ p =4 = 31. Consequently, firm l = 2 prefers a higher forward price in p = 5 as this allows it to maximiseits profits on its existing baselaod unit; see Figure 10 compared with Figure 16. In contrast, firm l = 1 prefersthe first forward price time series, i.e., a higher price in p = 4 and a slightly lower price in p = 5. In timeperiod p = 4, firm l = 1 can earn positive revenues from its new mid-merit unit and not make a loss fromits existing mid-merit unit in p = 4, if the forward price is 41.1. In contrast however, firm l = 2 does notown an exiting mid-merit unit and, in Figures 14 – 16, only invests in 1MW of new mid-merit generation.Consequently, firm l = 2 prefers a lower forward price in p = 4 and a higher price in p = 5 as this allows firm l = 2 to maximise its profits from its existing baselaod unit. For both forward price time series, the forwardprice is not high enough for existing baseload units to earn positive revenues in p = 4.In general, the equilibria resulting from the second time series represent equilibria where firm l = 2 hasinvested in a relatively small amount of new mid-merit generation, if any at all, but where firm l = 2 hasalso made forward generation decisions before firm l = 1. When firm l = 2 commits to a large amount ofgeneration in p = 4, firm l = 1 must reduce its generation in p = 5 in order to allow the forward price increaseand hence break even on its and firm l = 2’s new mid-merit investments.Interestingly, there is one equilibrium point where firm l = 1 does not invest in any new technology andconsequently, commits to a large amount of generation in p = 4. This leads to the second equilibrium pricetime series from Figure 7. This also forces firm l = 2 to reduce its generation in p = 5 and motivates it tonot make any investment decisions either. As a result, this is the only equilibrium point where the followersmake investment decisions; firm f = 3 invests 2766MW into a new mid-merit facility while firm f = 4 invest306MW into the same technology. Because of the generation commitments of the price-making firms set theequilibrium prices, both price-taking firms break exactly even on these investments. Thus, in the model,the price-making firms are indifferent to whether they do the investment at this equilibrium point or theprice-taking firms do. However, this indifference may not reflect reality. In the real-world, price-making firmsmay fear losing their price-making ability if they allow the competitive fringe to invest. The EPEC modelpresented in this work does not account for this as the price-making/price-taking characteristics of all firmsremain unchanged throughout the model.Finally, as mentioned above, there was one equilibrium point found where the prices converged to thirdequilibrium time series in Figure 7. In comparison with the second equilibrium time series, firm l = 2 commitsto investing in new mid-merit generation before l = 1. However, at this equilibrium point, firm l = 1 alsocommits to a large amount of generation in p = 5 which leads to a reduced price of γ p =5 = 47 .
79, fromwhich its existing mid-merit unit profits from. Consequently, in order for firm l = 2 to break even on itsnew mid-merit investment, firm l = 2 is forced to ensure its generation in p = 4 is low enough to allow γ p =4 = 58 .
6. Discussion
The following summarises the five main findings of our research. Firstly, an Equilibrium Problem withEquilibrium Constraints (EPEC) is a prudent model choice when modelling an oligopoly with competitivefringe and investments. As outlined in Section 3, when investment decisions are included in the model, using aMixed Complementarity Problem (MCP) can lead to myopic model behaviour and thus contradictory results.Our analysis shows that an EPEC model can overcome this issue and moreover does not require the limitingassumption of conjectural variations.Secondly, our results show that may it be optimal for generating firms with market power to occasionallyoperate some of their generating units at a loss. The driving factor behind this model outcome is thefact we allow both price-making and price-taking firms to make investment decisions. The ability of price-taking firms to invest further into the market motivates the price-making firms to depress prices in sometimepoints. This reduces the revenues price-taking firms could make from new investments and thus preventsthem from making such investments. Such behaviour would not be captured by MCP or cost-minimisationunit commitment models. Consequently, this result again highlights the suitability of the EPEC modellingapproach and the importance of including investment decisions in models of oligopolies with competitivefringes.Thirdly, the analysis in Section 5 found multiple market equilibria. This led to varied investment decisions,and thus profits, for the price-making firms. These results will be of interest to generating firms, particularlythose with market power. Figures 3 - 6 highlight the benefit of making investment decisions before othercompeting price-making firms do so. In fact, our results indicate that if firms do not expand their generationportfolios, then they may face profits lower than they would if the market was perfectly competitive. The31ultiple equilbira result also indicates that that generation from existing baseload generation may be higherin some equilibria compared with others. Such market outcomes will be of interest to energy policymakerswho are concerned about carbon emission levels. Older baseload generators tend to be coal-based and thusemit higher levels of carbon. Consequently, while the market may be indifferent to where the electricitycomes from, policymakers may seek to put measures in places to encourage the equilibrium outcomes whereexisting baseload generation is reduced.Fourthly, Figure 11 showed that the presence of market power increases consumer costs by 1% – 2%.While this outcome is not surprising, the level is relatively small compared with the literature. For instance,using similar data, Devine and Bertsch (2019) estimate market power in an oligopoly with competitive fringecontext could double consumer costs compared with a perfectly competitive market. However, Devine andBertsch (2019) do not include investment decisions in their model. Thus, this result again highlights theimpact of including investment decisions in models of oligopolies with competitive fringes. It also highlightsthe importance for policymakers to encourage new entrants into electricity markets and, moreover, the benefitsof encouraging smaller generating firms to expand their portfolios, or at least threaten to.Finally, as the literature details (Pozo et al., 2017), solving EPEC problems can be computational chal-lenging. In this work we utilised the method outlined in Leyffer and Munson (2010) to obtain an initialstarting point solution to our algorithm. Using this approach our algorithm successfully found an equilib-rium from 72 of the 200 iterations attempted. When instead we used a random initial starting point solution,we found an equilibrium from only 2 of the 2000 iterations attempted.Critically reflecting on our approach, we wish to acknowledge some limitations. Firstly, because EPECproblems are challenging to solve, we choose the relatively small number of five timesteps. These representedhours in summer low demand, summer high demand, winter low demand, winter high demand and winterpeak demand. Thus, the net demand intercept values represent average values for these timesteps. In reality,particularly in systems with a large amount of renewables, these intercept values will fluctuate from hour tohour. As a result, the average values may over- or under-estimate the total profits each generating firm couldmake in each time period. This would impact investments decisions and consumer costs.Secondly, we did not account for any stochasticity in the model. Due to the intermittent and uncertainnature of wind energy, stochasticity is a feature of many electricity market models. Such stochasticity istypically introduced by making generation capacity scenario-dependent (Lynch et al., 2019a). Deterministiccapacity values may also over- or under-estimate the profits each firm may make in each timestep. However,while including further timesteps and stochastic capacity values would most likely affect the exact numberspresented in Section 5, we do not anticipate them changing the qualitative findings discussed above.Finally, we did not consider a capacity market as part of the market modelled in this work. Capacitypayments exists when firms get paid for simply owning generation units and making them available tothe grid. Capacity payments do not depend on the extent that the unit(s) are utilised. Regulators andpolicymakers include such payments so as to ensure security of supply (Lynch and Devine, 2017). The32arket we considered was an ’energy-only’ market, where the generating firms only get paid on the basis ifhow much they generate. A capacity market can affect the level of investment into new generation. Futureresearch activities will address each of these modelling limitations.
7. Conclusion
In this paper, we developed a novel mathematical model of an imperfect electricity market, one that ischaracterised by an oligopoly with a competitive fringe. We modelled two types of generating firms; price-making firms, who have market power, and price-taking firms who do not. All firms had both investmentand forward generation decisions. The model took the form of an Equilibrium Problem with EquilibriumConstraints (EPEC), which finds an equilibrium of multiple bi-level optimisation problems. The bi-levelformulation allowed the optimisation problems of the price-taking firms to be embedded into the optimisationproblems of the price-making firms. This enabled the price-making firms to correctly anticipate the optimalreactions of the price-taking firms to their decisions. We applied the model to data representative of the Irishpower system for 2025.To solve the EPEC problem, we utilised the Gauss-Seidel algorithm. Furthermore, we found the computa-tional efficiency of the algorithm was improved when the algorithm’s starting point solution was provided bythe approach detailed in Leyffer and Munson (2010). Overall, we found that an EPEC problem is a prudentmodel choice when modelling an oligopoly with competitive fringe. This is because it overcomes modellingissues previously found in the literature and requires a fewer limiting assumptions.The model found multiple equilibria. This was due to the market’s indifference to which price-makingfirm generates electricity. Although consumer costs were found to be relatively constant across the equilibriafound, this result is important to energy policymakers who who wish to avoid equilibrium outcomes thathigher carbon emission levels.We also observed that it may be strategically optimal for price-making firms to occasionally to generateat a price that is lower than their marginal cost. This is because we incorporated investment decisions intothe optimisation problems of both types of generating firms. Consequently, the price-making firms seek todepress prices occasionally so as to discourage the fringe from investing further into the market. Furthermore,we found that consumer costs only decreased by 1% – 2% when market power was removed from the model.In future research, we will study the effects of increasing the number of timesteps in the model. Moreover,we will explore the impact stochasticity, particularly from wind generation, would have. In addition, futureresearch will analyse how the introduction of a capacity market would affect equilibrium outcomes.
Acknowledgements
M. T. Devine acknowledges funding from Science Foundation Ireland (SFI) under the SFI StrategicPartnership Programme Grant number SFI/15/SPP/E3125. S. Siddiqui acknowledges funding by NSF Grant331745375 [EAGER: SSDIM: Generating Synthetic Data on Interdependent Food, Energy, and TransportationNetworks via Stochastic, Bi-level Optimization]. The authors also sincerely thank Dr. M. Lynch and Dr.S. Lyons from the Economic and Social Research Institute (ESRI) in Dublin and who provided invaluablefeedback and advice.
ReferencesReferences
Ansari, D., 2017. Opec, saudi arabia, and the shale revolution: Insights from equilibrium modelling and oilpolitics. Energy Policy 111, 166–178.Baltensperger, T., F¨uchslin, R. M., Kr¨utli, P., Lygeros, J., 2016. Multiplicity of equilibria in conjecturalvariations models of natural gas markets. European Journal of Operational Research 252 (2), 646–656.Bertsch, V., Devine, M. T., Sweeney, C., Parnell, A. C., 2018. Analysing long-term interactions betweendemand response and different electricity markets using a stochastic market equilibrium model. Tech. rep.,ESRI Working Paper.Bushnell, J. B., Mansur, E. T., Saravia, C., 2008. Vertical arrangements, market structure, and competition:An analysis of restructured us electricity markets. American Economic Review 98 (1), 237–66.Devine, M. T., Bertsch, V., 2018. Examining the benefits of load shedding strategies using a rolling-horizonstochastic mixed complementarity equilibrium model. European Journal of Operational Research 267 (2),643–658.Devine, M. T., Bertsch, V., 2019. The role of demand response in mitigating market power - a quantitativeanalysis using a stochastic market equilibrium model. Tech. rep., ESRI Working Paper.Devine, M. T., Nolan, S., Lynch, M. ´A., OMalley, M., 2019. The effect of demand response and windgeneration on electricity investment and operation. Sustainable Energy, Grids and Networks 17, 100190.Di Cosmo, V., Hyland, M., 2013. Carbon tax scenarios and their effects on the irish energy sector. EnergyPolicy 59, 404–414.Egging, R., 2013. Benders decomposition for multi-stage stochastic mixed complementarity problems–appliedto a global natural gas market model. European Journal of Operational Research 226 (2), 341–353.Egging, R., Gabriel, S. A., Holz, F., Zhuang, J., 2008. A complementarity model for the european naturalgas market. Energy policy 36 (7), 2385–2414.Egging, R., Holz, F., 2016. Risks in global natural gas markets: investment, hedging and trade. Energy Policy94, 468–479. 34irGrid, 2016. All-island generation capacity statement 2016-2025.Facchinei, F., Pang, J.-S., 2007. Finite-dimensional variational inequalities and complementarity problems.Springer Science & Business Media.Fanzeres, B., Ahmed, S., Street, A., 2019. Robust strategic bidding in auction-based markets. EuropeanJournal of Operational Research 272 (3), 1158–1172.Fortuny-Amat, J., McCarl, B., 1981. A representation and economic interpretation of a two-level program-ming problem. Journal of the operational Research Society 32 (9), 783–792.Gabriel, S. A., Conejo, A. J., Fuller, J. D., Hobbs, B. F., Ruiz, C., 2012. Complementarity modeling in energymarkets. Vol. 180. Springer Science & Business Media.Gabriel, S. A., Zhuang, J., Egging, R., 2009. Solving stochastic complementarity problems in energy marketmodeling using scenario reduction. European Journal of Operational Research 197 (3), 1028–1040.Guo, H., Chen, Q., Xia, Q., Kang, C., 2019. Electricity wholesale market equilibrium analysis integratingindividual risk-averse features of generation companies. Applied Energy 252, 113443.Habibian, M., Downward, A., Zakeri, G., 2019. Multistage stochastic demand-side management for price-making major consumers of electricity in a co-optimized energy and reserve market. European Journal ofOperational Research.Haftendorn, C., Holz, F., 2010. Modeling and analysis of the international steam coal trade. The EnergyJournal, 205–229.Hu, X., Ralph, D., 2007. Using epecs to model bilevel games in restructured electricity markets with locationalprices. Operations research 55 (5), 809–827.Huppmann, D., 2013. Endogenous shifts in opec market power: a stackelberg oligopoly with fringe.Huppmann, D., Egerer, J., 2015. National-strategic investment in european power transmission capacity.European Journal of Operational Research 247 (1), 191–203.Huppmann, D., Egging, R., 2014. Market power, fuel substitution and infrastructure–a large-scale equilibriummodel of global energy markets. Energy 75, 483–500.Jin, S., Ryan, S. M., 2013. A tri-level model of centralized transmission and decentralized generation expansionplanning for an electricity marketpart i. IEEE Transactions on Power Systems 29 (1), 132–141.Kazempour, S. J., Conejo, A. J., Ruiz, C., 2013. Generation investment equilibria with strategic producersparti: Formulation. IEEE Transactions on Power Systems 28 (3), 2613–2622.35azempour, S. J., Zareipour, H., 2013. Equilibria in an oligopolistic market with wind power production.IEEE Transactions on Power Systems 29 (2), 686–697.Lee, C.-Y., 2016. Nash-profit efficiency: A measure of changes in market structures. European Journal ofOperational Research 255 (2), 659–663.Leyffer, S., Munson, T., 2010. Solving multi-leader–common-follower games. Optimisation Methods & Soft-ware 25 (4), 601–623.Lynch, M., Devine, M. T., Bertsch, V., 2019a. The role of power-to-gas in the future energy system: Marketand portfolio effects. Energy 185, 1197–1209.Lynch, M. A., Devine, M. T., 2017. Investment vs. refurbishment: examining capacity payment mechanismsusing stochastic mixed complementarity problems. The Energy Journal 38 (2).Lynch, M. ´A., Nolan, S., Devine, M. T., OMalley, M., 2019b. The impacts of demand response participationin capacity markets. Applied Energy 250, 444–451.Moiseeva, E., Wogrin, S., Hesamzadeh, M. R., 2017. Generation flexibility in ramp rates: Strategic behaviorand lessons for electricity market design. European Journal of Operational Research 261 (2), 755–771.Pozo, D., Contreras, J., 2011. Finding multiple nash equilibria in pool-based markets: A stochastic epecapproach. IEEE Transactions on Power Systems 26 (3), 1744–1752.Pozo, D., Contreras, J., Sauma, E., 2013. If you build it, he will come: Anticipative power transmissionplanning. Energy Economics 36, 135–146.Pozo, D., Sauma, E., Contreras, J., 2017. Basic theoretical foundations and insights on bilevel models andtheir applications to power systems. Annals of Operations Research 254 (1-2), 303–334.Pozo, D., Sauma, E. E., Contreras, J., 2012. A three-level static milp model for generation and transmissionexpansion planning. IEEE Transactions on Power systems 28 (1), 202–210.Ruiz, C., Conejo, A. J., Smeers, Y., 2011. Equilibria in an oligopolistic electricity pool with stepwise offercurves. IEEE Transactions on Power Systems 27 (2), 752–761.Steeger, G., Rebennack, S., 2017. Dynamic convexification within nested benders decomposition using la-grangian relaxation: An application to the strategic bidding problem. European Journal of OperationalResearch 257 (2), 669–686.Walsh, D., Malaguzzi Valeri, L., Di Cosmo, V., 2016. Strategic bidding, wind ownership and regulation in adecentralised electricity market.Wogrin, S., Barqu´ın, J., Centeno, E., 2012. Capacity expansion equilibria in liberalized electricity markets:an epec approach. IEEE Transactions on Power Systems 28 (2), 1531–1539.36ogrin, S., Centeno, E., Barquin, J., 2013. Generation capacity expansion analysis: Open loop approximationof closed loop equilibria. IEEE Transactions on Power Systems 28 (3), 3362–3371.Ye, Y., Papadaskalopoulos, D., Strbac, G., 2017. Investigating the ability of demand shifting to mitigateelectricity producers market power. IEEE Transactions on Power Systems 33 (4), 3800–3811.Zerrahn, A., Huppmann, D., 2017. Network expansion to mitigate market power. Networks and SpatialEconomics 17 (2), 611–644.
Appendix A. Alternative Price-making firm l ’s problem When the problem is solved as a Mixed Complementarity Problem (MCP), price-making firm l ’s op-timisation problem takes the following form, where all variables an parameters are as defined previously:max gen PM l,t,p ,inv PM l,t gen PT f,t,p ,inv PT f,t γ p ,λ PT f,t,p (cid:88) t,p W p × gen PM l,t,p × (cid:0) γ p − C GEN t (cid:1) − (cid:88) t IC GEN t × inv PM l,t . (A.1)subject to: gen PM l,t,p ≤ CAP PM l,t + inv PM l,t , ∀ t, p. (A.2)The KKT conditions associated with this optimisation problem are0 ≤ gen PM l,t,p ⊥ − W p × (cid:0) γ p + ∂γ p ∂gen PM l,t,p × gen PM l,t,p − C GEN t (cid:1) + λ PM l,t,p ≥ , ∀ t, p, (A.3)0 ≤ inv PM l,t ⊥ IC GEN t − (cid:88) p λ PM l,t,p ≥ , ∀ t, (A.4)0 ≤ λ PM , l,t,p ⊥ − gen PM l,t,p + CAP PM l,t + inv PM l,t ≥ , ∀ t, p, (A.5)where ∂γ p ∂gen PM l,t,p = − CV l × B, ∀ l, t, p, (A.6)is determined via market clearing condition (3). Furthermore, the parameter CV l ∈ [0 ,
1] represents theConjectural Variation associated with firm l . When firm l ’s problem is described by equations (A.1) and(A.2), it is a convex optimisation problem and hence the KKT conditions (A.3) - (A.5) are both necessaryand sufficient for optimality (Gabriel et al., 2012).When the overall market problem is solved as a MCP, the problem consists of the market clearing condition(3), the price-taking firms’ KKT conditions (equations (7) - (9)) and the KKT conditions for all price-makingfirms (equations (A.3) - (A.5)).It is important to note that when gen PM l,t,p >
0, then condition (A.3) is only satisfied if γ p ≥ C GEN tt