# A MIP framework for non-convex uniform price day-ahead electricity auctions

- 151 Downloads
- 1 Citations

## Abstract

It is well known that a market equilibrium with uniform prices often does not exist in non-convex day-ahead electricity auctions. We consider the case of the non-convex, uniform-price Pan-European day-ahead electricity market “PCR” (Price Coupling of Regions), with non-convexities arising from so-called complex and block orders. Extending previous results, we propose a new primal-dual framework for these auctions, which has applications in both economic analysis and algorithm design. The contribution here is threefold. First, from the algorithmic point of view, we give a non-trivial exact (i.e., not approximate) linearization of a non-convex ‘minimum income condition’ that must hold for complex orders arising from the Spanish market, avoiding the introduction of any auxiliary variables, and allowing us to solve market clearing instances involving most of the bidding products proposed in PCR using off-the-shelf MIP solvers. Second, from the economic analysis point of view, we give the first MILP formulations of optimization problems such as the maximization of the traded volume, or the minimization of opportunity costs of paradoxically rejected block bids. We first show on a toy example that these two objectives are distinct from maximizing welfare. Third, we provide numerical experiments on realistic large-scale instances. They illustrate the efficiency of the approach, as well as the economics trade-offs that may occur in practice.

## Keywords

Day-ahead electricity market auctions Non-convexities Mixed integer programming Market coupling Equilibrium prices## Mathematics Subject Classification

90C11 90-08 90C06## Notes

### Acknowledgments

We greatly thank APX, BELPEX, EPEX SPOT, OMIE and N-Side for providing us with data used to generate realistic instances. We also thank organizers of the 11th International (IEEE) Conference on European Energy Market (Krakow, May 2014), as well as organizers of the COST Workshop on Mathematical Models and Methods for Energy Optimization (Budapest, Sept. 2014), for allowing us to present partial results developed here. This text presents research results of the P7/36 PAI project COMEX, part of the IPA Belgian Program. The work was also supported by EC-FP7 COST Action TD1207. The scientific responsibility is assumed by the authors.

## References

- Euphemia (2013) Euphemia public description v0.6. http://www.belpex.be/wp-content/uploads/Euphemia-public-description-Nov-2013.pdf
- Garcia-Bertrand R, Conejo Antonio J, Gabriel S (2006) Electricity market near-equilibrium under locational marginal pricing and minimum profit conditions. Eur J Oper Res 174(1):457–479. ISSN: 0377-2217. doi: 10.1016/j.ejor.2005.03.037. http://www.sciencedirect.com/science/article/pii/S0377221705003814
- Gribik PR, Hogan WW, Pope SL (2007) Market-clearing electricity prices and energy uplift (Technical report)Google Scholar
- Madani M, Van Vyve M (2014) Minimizing opportunity costs of paradoxically rejected block orders in european day-ahead electricity markets. In: European Energy Market (EEM), 2014 11th International Conference on the, pp 1–5. doi: 10.1109/EEM.2014.6861237
- Madani M, Van Vyve M (2015) Computationally efficient mip formulation and algorithms for european day-ahead electricity market auctions. Eur J Oper Res, 242(2):580–593. ISSN: 0377-2217. doi: 10.1016/j.ejor.2014.09.060. http://www.sciencedirect.com/science/article/pii/S0377221714007991
- Maria NS (2010) Day-ahead electricity market, proposals to adapt complex conditions in omel. Master’s thesis, Escuela Tcnica Superior De Ingeniera (ICAI), Universidad Pontificia ComillasGoogle Scholar
- Meeus L (2006) Power exchange auction trading platform design (ontwerp van een veilingsysteem voor elektrische energiebeurzen). Open Access publications from Katholieke Universiteit Leuven urn:hdl:1979/338, Katholieke Universiteit Leuven. http://ideas.repec.org/p/ner/leuven/urnhdl1979-338.html
- Meeus L, Verhaegen K, Belmans R (2009) Block order restrictions in combinatorial electric energy auctions. Eur J Oper Res 196(3):1202–1206. ISSN: 0377-2217. doi: 10.1016/j.ejor.2008.04.031. http://www.sciencedirect.com/science/article/pii/S0377221708003901
- O’Neill R, Mead D, Malvadkar P (2005a) On market clearing prices higher than the highest bid and other almost paranormal phenomena. Electr J 18(2):19–27. ISSN: 1040-6190. doi: 10.1016/j.tej.2005.02.004. http://www.sciencedirect.com/science/article/pii/S1040619005000114
- O’Neill RP, Sotkiewicz PM, Rothkopf MH (2007) Equilibrium prices in power exchanges with non-convex bids. Technical report, Working Paper (revised July)Google Scholar
- O’Neill RP, Sotkiewicz PM, Hobbs BF, Rothkopf MH, Stewart WR (2005b) Efficient market-clearing prices in markets with nonconvexities. Eur J Oper Res 164(1):269–285CrossRefGoogle Scholar
- Ruiz C, Conejo AJ (2009) Pool strategy of a producer with endogenous formation of locational marginal prices. Power Syst IEEE Trans 24(4):1855–1866. ISSN: 0885-8950. doi: 10.1109/TPWRS.2009.2030378
- Ruiz C, Conejo AJ, Gabriel SA (2012) Pricing non-convexities in an electricity pool. Power Syst IEEE Trans 27(3):1334–1342. ISSN: 0885-8950. doi: 10.1109/TPWRS.2012.2184562
- Van Vyve M (2011) Linear prices for non-convex electricity markets: models and algorithms. Technical report, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)Google Scholar
- Zak EJ, Ammari S, Cheung KW (2012) Modeling price-based decisions in advanced electricity markets. In: European Energy Market (EEM), 2012 9th International Conference on the, pp 1–6. doi: 10.1109/EEM.2012.6254813