Costos variables

The optimal response curve approach can be used to investigate the strategic reactions of market participants on the strategy of their rivals and the consequent impact on the existence of pure strategy equilibria in the presence of network constraints. This approach is straightforward for analysing models that use theα-,

β- or kF-parameterizations where only one strategic variable is involved, if the

market has only two strategic players, since the optimal response space will be a 2-dimensional graph. The optimal values for the strategic variable of one player are plotted for all the possible values of the strategic variable of the other player, and then this is repeated for the second player. The point or points at which the two reaction curves intersect indicate the pure strategy equilibria for the linear SFE game. If the reaction curves do not intersect no pure equilibrium point exists and the Nash equilibria of the SFE game are confined to mixed strategies.

Numerical algorithmic market simulations of a simple 2-bus system have been

performed in [133,120] using the kF-parameterization SFE model to show the

dependency of the existence and multiplicity of pure SFE on the transmission line constraints. The investigation in [133] considers one strategic supplier at each bus, while that in [120] considers one strategic supplier at the first bus and one strategic consumer (load) at the second bus. In both cases, the optimal response curves for the kF variables of the players are found to intersect at a

single point when no transmission line constraint is considered between the buses, indicating one pure strategy SFE for each case. When a transmission limit is induced between the two buses, in [133] there is no intersection due to a discontinuity on the response curve of the supplier at the load bus that eliminates the pure strategy SFE, and in [120] the reaction curves overlap over a range of values indicating a continuum of pure Nash equilibria. In the first case, the nonexistence of pure SFE is owed to the fact that the supplier’s profit has two local maxima and there is no preference in bidding either of the corresponding strategic variables. Therefore, any probabilistic combination of choosing either strategy is a mixed Nash SFE. In the second case, the continuum of equilibria indicate the optimum responses of the market participants in order to adjust the power flow to be exactly at the transmission limit and avoid the transmission congestion penalties from the ISO. These simple trivial cases show that the presence of network constraints may eliminate the pure strategy equilibrium or introduce SFE multiplicity.

The study in [73] investigated a 3-bus duopoly with a single load by employing

equilibrium strategies determines the output quantities and which lines will be congested. The reaction curve space is separated in two areas, corresponding to congested and uncongested operation of the transmission line and the reaction curves are compared with those for a case with no transmission constraints. For a particular topology of the system it is shown that for a relatively low constraint the equilibrium is in the uncongested area, being the same as in the absence of the transmission limit. For an intermediate constraint level a continuum of equilibria appears on the boundary that separates the two areas, while when the constraint is tightened a single equilibrium point exists in the congested region. It is then shown that for different topologies the equilibrium point for a low level constraint may be moved on the congestion boundary or eliminated, and that nonexistence of SFE may occur if the demand elasticity becomes low enough. This investigation proves that the particular topology of the network, as well as the level of the transmission line limit and demand elasticity, play a major role on the existence and nature of the SFE solution. A discussion of the same system appeared in [131] where the slope parameterization is used, and speculates that in more complicated systems there will be a continuum of equilibria if the binding network constraints induce consistent bidding behaviour from the strategic players around the constraint to eliminate congestion charges, or there will be no pure equilibrium if this behaviour is inconsistent. Furthermore, a comprehensive analysis in [116] based on the intercept parameterization method strengthens the observations of [73] by showing that nonexistence or multiplicity of SFE solutions occur in the presence of transmission constraints, depending on the properties of the particular network arrangement of the examined market.

Unlike other studies that are based on the static analysis of the market equilibrium, the analysis in [129] is concerned with the market evolution under repeated bidding and with the learning behaviour of the market participants, i.e. how the players use the available information to improve their bidding strategies from their acquired knowledge. Recalling the proof from [111] that when all players learn a unique equilibrium under slope-parameterization exists, it is shown for an unconstrained market that each supplier has incentives to learn about its opponents’ past submitted bids in order to maximise profit, since its optimum strategy depends on its rivals’ strategies.

The investigation in [113] has presented graphically the profits of the suppliers of the 2-bus system in [133] at the SFE point, as a function of the strategic variables

using the (α, β)-parameterization. It is shown that there is an enormous

(seemingly infinite) number of equilibrium points in the unconstrained case, and some of them are eliminated when the transmission constraint is introduced (those that correspond to flows higher than the line limit). The SFE points that cease to exist are among those that correspond to relatively low values of profits and therefore the binding constraint can support high equilibrium profits. The

discussion compares the SFE solutions for the (α, β)-parameterization with the

Cournot solution and shows that the profits from bids with slopes that tend to infinity and the corresponding intercepts that move towards large negative values are equal to the Cournot profits. This limiting SFE is a focal equilibrium point because it is mutually beneficial to all players and may be preferred by everyone to all the other equilibria.

Since the optimal response curve method is effective in finding SFE solutions only for duopolies, as markets with more than 3 strategies players cannot be represented graphically, more complex situations have been investigated by

applying numerical algorithmic approaches based on the (α,β)-parameterization

method. The numerical model in [89] presented results for a monopolist in an AC 3-bus network and shows that it can choose equivalent strategies from within a range of pairs of strategic variables to achieve the same level of maximum profit at the equilibrium point; this is consistent with the results in [120] where the total surplus of the market players along the SFE continuum is the same. A similar observation was made in [104] where an unconstrained market with 5 strategic firms is investigated showing that the submitted strategic variables at an SFE point are not unique. Instead, several equivalent optimum strategies exist and the corresponding profits, market price and output levels at the SFE point are unique and equal to those of the Cournot solution for the single bidding period game. In such cases the slope of the supply function for profit maximisation is required to be a very large positive value and the intercept a very large negative value, being in agreement with the discussion about the focal equilibrium in [113]. However, when a single bid was applied to multiple pricing periods in [104] the bidding

results are close to those of slope-parameterization, as discussed by Baldick in [113].

The investigation in [120] extends to analyse the behaviour of strategic firms in a 9-bus system showing that different local maxima for total profits may occur due to the presence of physical system constraints. The SFE calculated from the numerical algorithm may be attracted by a local maximum and miss the notion that a global SFE would be more desirable in terms of predicting the market behaviour. In order to deal with such complications in multi-unit networks an algorithm has been implemented in [128] to identify all possible multiple SFE points. The algorithm is searching the regions defined by all combinations of system constraints (generation and transmission limits) and identifies the different non-cooperative equilibrium points. It has successfully produced results for a 30- and a 57-bus system showing that even though the existence of multiple equilibria in realistic systems is possible they should not be expected in large numbers.