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  <doc>
    <id>389</id>
    <completedYear>2021</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>29</pageNumber>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2021-05-31</completedDate>
    <publishedDate>2021-05-31</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium</title>
    <abstract language="eng">Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems.</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Julia Grübel</author>
    <author>Olivier Huber</author>
    <author>Lukas Hümbs</author>
    <author>Max Klimm</author>
    <author>Martin Schmidt</author>
    <author>Alexandra Schwartz</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Energy markets</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Nonconvex games</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Existence</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Equilibrium computation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Perfect competition</value>
    </subject>
    <collection role="institutes" number="">Friedrich-Alexander-Universität Erlangen-Nürnberg</collection>
    <collection role="institutes" number="">Technische Universität Berlin</collection>
    <collection role="institutes" number="">Weierstraß-Institut für Angewandte Analysis und Stochastik</collection>
    <collection role="subprojects" number="">A05</collection>
    <collection role="subprojects" number="">A07</collection>
    <collection role="subprojects" number="">B02</collection>
    <collection role="subprojects" number="">B07</collection>
    <collection role="subprojects" number="">B08</collection>
    <collection role="institutes" number="">Universität Trier</collection>
    <collection role="subprojects" number="">B09</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/389/nonconvex-gas-markets_preprint_v2.pdf</file>
  </doc>
  <doc>
    <id>492</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>doctoralthesis</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2022-05-27</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>2021-10-20</thesisDateAccepted>
    <title language="eng">Existence, Uniqueness, and Algorithms for Equilibria in Competitive Energy Markets</title>
    <abstract language="eng">Due to the transition towards climate neutrality, energy markets are rapidly evolving. New technologies are developed that allow electricity from renewable energy sources to be stored or to be converted into other energy commodities. As a consequence, new players enter the markets and existing players gain more importance. Market equilibrium problems are capable of capturing these changes and therefore enable us to answer contemporary research questions with regard to energy market design and climate policy. &#13;
&#13;
This cumulative dissertation is devoted to the study of different market equilibrium problems that address such emerging aspects in liberalized energy markets. In the first part, we review a well-studied competitive equilibrium model for energy commodity markets and extend this model by sector coupling, by temporal coupling, and by a more detailed representation of physical laws and technical requirements. Moreover, we summarize our main contributions of the last years with respect to analyzing the market equilibria of the resulting equilibrium problems. &#13;
&#13;
For the extension regarding sector coupling, we derive sufficient conditions for ensuring uniqueness of the short-run equilibrium a priori and for verifying uniqueness of the long-run equilibrium a posteriori. Furthermore, we present illustrative examples that each of the derived conditions is indeed necessary to guarantee uniqueness in general. &#13;
&#13;
For the extension regarding temporal coupling, we provide sufficient conditions for ensuring uniqueness of demand and production a priori. These conditions also imply uniqueness of the short-run equilibrium in case of a single storage operator. However, in case of multiple storage operators, examples illustrate that charging and discharging decisions are not unique in general. We conclude the equilibrium analysis with an a posteriori criterion for verifying uniqueness of a given short-run equilibrium. Since the computation of equilibria is much more challenging due to the temporal coupling, we shortly review why a tailored parallel and distributed alternating direction method of multipliers enables to efficiently compute market equilibria. &#13;
&#13;
For the extension regarding physical laws and technical requirements, we show that, in nonconvex settings, existence of an equilibrium is not guaranteed and that the fundamental welfare theorems therefore fail to hold. In addition, we argue that the welfare theorems can be re-established in a market design in which the system operator is committed to a welfare objective. For the case of a profit-maximizing system operator, we propose an algorithm that indicates existence of an equilibrium and that computes an equilibrium in the case of existence. Based on well-known instances from the literature on the gas and electricity sector, we demonstrate the broad applicability of our algorithm. Our computational results suggest that an equilibrium often exists for an application involving nonconvex but continuous stationary gas physics. In turn, integralities introduced due to the switchability of DC lines in DC electricity networks lead to many instances without an equilibrium. Finally, we state sufficient conditions under which the gas application has a unique equilibrium and the line switching application has finitely many. &#13;
&#13;
In the second part, all preprints belonging to this cumulative dissertation are provided. These preprints, as well as two journal articles to which the author of this thesis contributed, are referenced within the extended summary in the first part and contain more details.</abstract>
    <identifier type="doi">10.25353/ubtr-xxxx-4c49-7f53</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Julia Grübel</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Energy markets</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Equilibrium computation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Existence</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Perfect competition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Uniqueness</value>
    </subject>
    <collection role="subprojects" number="">B08</collection>
    <thesisGrantor>Universität Trier</thesisGrantor>
    <file>https://opus4.kobv.de/opus4-trr154/files/492/Dissertation_Gruebel.pdf</file>
  </doc>
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