@article{KrebsScheweSchmidt2017, author = {Krebs, Vanessa and Schewe, Lars and Schmidt, Martin}, title = {Uniqueness and Multiplicity of Market Equilibria on DC Power Flow Networks}, series = {European Journal on Operations Research}, journal = {European Journal on Operations Research}, number = {271(1)}, pages = {165 -- 178}, year = {2017}, abstract = {We consider uniqueness and multiplicity of market equilibria in a short-run setup where traded quantities of electricity are transported through a capacitated network in which power flows have to satisfy the classical lossless DC approximation. The firms face fluctuating demand and decide on their production, which is constrained by given capacities. Today, uniqueness of such market outcomes are especially important in more complicated multilevel models for measuring market (in)efficiency. Thus, our findings are important prerequisites for such studies. We show that market equilibria are unique on tree networks under mild assumptions and we also present a priori conditions under which equilibria are unique on cycle networks. On general networks, uniqueness fails to hold and we present simple examples for which multiple equilibria exist. However, we prove a posteriori criteria for the uniqueness of a given solution and characterize situations in which multiple solutions exist.}, language = {en} } @article{KrebsSchmidt2017, author = {Krebs, Vanessa and Schmidt, Martin}, title = {Uniqueness of Market Equilibria on Networks with Transport Costs}, series = {Operations Research Perspectives}, journal = {Operations Research Perspectives}, number = {5}, pages = {169 -- 173}, year = {2017}, abstract = {We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a welfare maximization problem. Using the latter we prove uniqueness of the resulting equilibrium for piecewise linear and symmetric transport costs under additional mild assumptions. Moreover, we show the necessity of these assumptions by illustrating examples that possess multiple solutions if our assumptions are violated.}, language = {en} }