Optimization and Engineering

, Volume 12, Issue 1–2, pp 277–302 | Cite as

The hybrid proximal decomposition method applied to the computation of a Nash equilibrium for hydrothermal electricity markets

  • Lisandro A. Parente
  • Pablo A. Lotito
  • Fernando J. Mayorano
  • Aldo J. Rubiales
  • Mikhail V. Solodov


In this work, a decomposition method for computing a solution of a short-term hydrothermal scheduling problem is presented. An oligopolistic electricity market composed of two types of power generation units, thermal and hydroelectric, is considered. The hydroelectric units have also the possibility of pumping back water, paying in that case for the electricity consumed. The Nash equilibrium analytic condition is stated as a variational inclusion and it is shown that the associated operator has a structure suitable for decomposition, in particular by applying the variable metric hybrid proximal decomposition technique (VMHPDM). The application of VMHPDM is illustrated in several examples and numerical results for each example are presented.


Electric power market Nash-Cournot equilibria Variational inclusions Decomposition 



each time period, t=1,…,T.


each thermal unit, \(i=1,\ldots,\mathcal {I}\).


each hydroelectric unit, \(j=1,\ldots,\mathcal {J}\).


each thermal company, m=1,…,M.


each hydroelectric company, n=1,…,N.

\(\mathcal {C}^{Th}_{m}\)

index set for thermal units owned by company m.

\(\mathcal {C}^{H}_{n}\)

index set for hydroelectric units owned by company n.


production at hydroelectric unit j for time period t (or consumption, if this quantity is negative).


hydroelectric production vector in \(\mathbb {R}^{\mathcal {J}T}\).


production at thermal unit i for the time period t.


thermal production vector in \(\mathbb {R}^{\mathcal {I}T}\).

\(y_{j}^{LOW}, y_{j}^{UP}\)

hydroelectric production bounds for unit j.


total hydroelectric generation for unit j.


efficiency coefficient of hydroelectric unit j.


corresponding vectors in \(\mathbb {R}^{\mathcal {J}}\).

\(x_{i}^{LOW}, x_{i}^{UP}\)

thermal productions bounds for unit i.


thermal cost coefficients associated to the quadratic, linear, and independent term for unit i.


corresponding vectors in \(\mathbb {R}^{\mathcal {I}}\).


inverse demand coefficients for time t.


corresponding vectors in ℝ T .


benefit obtained by the hydroelectric company n.


benefit obtained by the thermal company m.


market price for period t.


non-smooth function representing the efficiency gap between pumping and generation for unit j.


quadratic function representing the thermal production cost for unit i.

\(\mathcal {K}^{H}_{n}\)

feasible set for the hydroelectric company n.

\(\mathcal {K}^{Th}_{m}\)

feasible set for the thermal company m.


identity matrix in ℝn×n.


vector of ones in ℝ n .


maximal eigenvalue of the symmetric positive definite (SPD) matrix M.


minimal eigenvalue of the SPD matrix M.


diagonal matrix with (diag(v))n,n=v n , for v∈ℝ n .


operator that maps points in ℝ n to subsets of ℝ m .


domain of F:ℝ n ⇉ℝ m , i.e., {x∈ℝ n F(x)≠∅}.

\(N_{\mathcal {K}}(x)\)

normal cone of a convex set \(\mathcal {K}\) at x, i.e. Open image in new window .

\(\mathrm{rint}\,\mathcal {K}\)

relative interior of the set \(\mathcal {K}\).

\(\mathrm {Proj}_{\mathcal {K}}(x)\)

(orthogonal) projection of the point x onto the set \(\mathcal {K}\).


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  1. Arellano MS (2004) Market power in mixed hydro-thermal electric systems. In: Econometric society Latin American meetings 211. Econometric Society Google Scholar
  2. Baldick R (2002) Electricity market equilibrium models: the effect of parameterization. IEEE Trans Power Syst 17(4), 1170–1176 CrossRefGoogle Scholar
  3. Bonnans JF, Gilbert JCh, Lemaréchal C, Sagastizábal C (2006) Numerical optimization: theoretical and practical aspects. Springer, Berlin, pp 177–180 MATHGoogle Scholar
  4. Burachik RS, Iusem AN, Svaiter BF (1997) Enlargement of monotone operators with applications to variational inequalities. Set-Valued Anal 5:159–180 MathSciNetMATHCrossRefGoogle Scholar
  5. Burachik RS, Sagastizábal CA, Svaiter BF (1999) ε-Enlargements of maximal monotone operators: Theory and applications. In: Fukushima M, Qi L (eds) Reformulation—nonsmooth, piecewise smooth, semismooth and smoothing methods. Kluwer Academic, Norwell, pp 25–44 Google Scholar
  6. Chen X, Teboulle M (1994) A proximal-based decomposition method for convex minimization problems. Math Program 64:81–101 MathSciNetMATHCrossRefGoogle Scholar
  7. Hobbs BF, Pang JS (2010) Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints Google Scholar
  8. He B, Liao LZ, Han D, Yang H (2002) A new inexact alternating directions method for monotone variational inequalities. Math Program 92:103–118 MathSciNetMATHCrossRefGoogle Scholar
  9. Lotito PA (2006) Issues in the implementation of the DSD algorithm for the traffic assignment problem. Eur J Oper Res 175:1577–1587 MATHCrossRefGoogle Scholar
  10. Lotito PA, Parente LA, Solodov MV (2009) A class of variable metric decomposition methods for monotone variational inclusions. J Convex Anal 16, 857–880. Special Issue Dedicated to Stephen Simons on the occasion of his 70th birthday MathSciNetMATHGoogle Scholar
  11. Maculan N, Santiago CP, Macambira EM, Jardim MHC (2003) An O(n) algorithm for projecting a vector on the intersection of a hyperplane and a box in ℝn. J Optim Theory Appl 117:553–574 MathSciNetMATHCrossRefGoogle Scholar
  12. Moitre D, Sauchelli V, García G (2005) Optimización Dinámica Binivel de Centrales Hidroeléctricas de bombeo en un Pool Competitivo—Parte I: Modelo y Algoritmo. Revista IEEE América Latina 3(2), 62–67 Google Scholar
  13. Parente LA, Lotito PA, Solodov MV (2008) A class of inexact variable metric proximal point algorithms. SIAM J Optim 19:240–260 MathSciNetCrossRefGoogle Scholar
  14. Pennanen T (2002) A splitting method for composite mappings. Numer Funct Anal Optim 23:875–890 MathSciNetMATHCrossRefGoogle Scholar
  15. Rivier M, Ventosa M, Ramos A (2001) A generation operation planning model in deregulated electricity markets based on the complementarity problem. In: Ferris MC, Mangasarian OL, Pang J-S (eds.) Applications and algorithms of complementarity. Kluwer Academic, Boston, pp 273–298 Google Scholar
  16. Rockafellar RT (1970) On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 149:75–88 MathSciNetMATHCrossRefGoogle Scholar
  17. Rockafellar RT, Wets J-B (1997) Variational analysis. Springer, New York Google Scholar
  18. Rubiales A, Mayorano F, Lotito P (2007) Optimización aplicada a la coordinación hidrotérmica del mercado eléctrico argentino. Mec Comput XXVI, 3343–3359 Google Scholar
  19. Ruiz C, Conejo AJ, García-Bertrand R (2010) Some analytical results pertaining to Cournot models for short-term electricity markets Google Scholar
  20. Scott TJ, Read EG (1996) Modelling hydro reservoir operation in a deregulated electricity market. Int Trans Oper Res 3:243–253 MATHCrossRefGoogle Scholar
  21. Solodov MV (2004) A class of decomposition methods for convex optimization and monotone variational inclusions via the hybrid inexact proximal point framework. Optim Methods Softw 19:557–575 MathSciNetMATHCrossRefGoogle Scholar
  22. Solodov MV, Svaiter BF (2001) A unified framework for some inexact proximal point algorithms. Numer Funct Anal Optim 22:1013–1035 MathSciNetMATHCrossRefGoogle Scholar
  23. Tseng P (1997) Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J Optim 7:951–965 MathSciNetMATHCrossRefGoogle Scholar
  24. Wood AJ, Wollenberg BF (1996) Power generation, operation, and control, 2nd edn. Wiley-Interscience/Wiley, New York Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Lisandro A. Parente
    • 1
  • Pablo A. Lotito
    • 2
  • Fernando J. Mayorano
    • 2
  • Aldo J. Rubiales
    • 2
  • Mikhail V. Solodov
    • 3
  1. 1.OPTyCONUNR, CONICETRosarioArgentina
  2. 2.PLADEMAUNICEN, CONICETTandilArgentina
  3. 3.IMPARio de JaneiroBrazil

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