Skip to main content

Advertisement

SpringerLink
Log in

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

  • Published:
Optimization and Engineering Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

t :

each time period, t=1,…,T.

i :

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

j :

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

m :

each thermal company, m=1,…,M.

n :

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.

y jt :

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

y :

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

x it :

production at thermal unit i for the time period t.

x :

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

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

hydroelectric production bounds for unit j.

\(y_{j}^{TOT}\) :

total hydroelectric generation for unit j.

α j :

efficiency coefficient of hydroelectric unit j.

yLOW,yUP,yTOT,α:

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

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

thermal productions bounds for unit i.

φ i ,ω i ,ψ i :

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

xLOW,xUP,φ,ω,ψ:

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

a t ,D t :

inverse demand coefficients for time t.

a,D:

corresponding vectors in ℝT.

\(\mathit{Ben}_{n}^{H}\) :

benefit obtained by the hydroelectric company n.

\(\mathit{Ben}_{m}^{T}\) :

benefit obtained by the thermal company m.

p t :

market price for period t.

f j (⋅):

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

\(c_{i}^{T}(\cdot)\) :

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.

I n :

identity matrix in ℝn×n.

1 n :

vector of ones in ℝn.

λ max (M):

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

λ min (M):

minimal eigenvalue of the SPD matrix M.

diag(v):

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

F:ℝn⇉ℝm:

operator that maps points in ℝn to subsets of ℝm.

domF :

domain of F:ℝn⇉ℝm, i.e., {x∈ℝn∣F(x)≠∅}.

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

normal cone of a convex set \(\mathcal {K}\) at x, i.e. .

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

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

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

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

References

  • Arellano MS (2004) Market power in mixed hydro-thermal electric systems. In: Econometric society Latin American meetings 211. Econometric Society

  • Baldick R (2002) Electricity market equilibrium models: the effect of parameterization. IEEE Trans Power Syst 17(4), 1170–1176

    Article  Google Scholar 

  • Bonnans JF, Gilbert JCh, Lemaréchal C, Sagastizábal C (2006) Numerical optimization: theoretical and practical aspects. Springer, Berlin, pp 177–180

    MATH  Google Scholar 

  • Burachik RS, Iusem AN, Svaiter BF (1997) Enlargement of monotone operators with applications to variational inequalities. Set-Valued Anal 5:159–180

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Chen X, Teboulle M (1994) A proximal-based decomposition method for convex minimization problems. Math Program 64:81–101

    Article  MathSciNet  MATH  Google Scholar 

  • Hobbs BF, Pang JS (2010) Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints

  • He B, Liao LZ, Han D, Yang H (2002) A new inexact alternating directions method for monotone variational inequalities. Math Program 92:103–118

    Article  MathSciNet  MATH  Google Scholar 

  • Lotito PA (2006) Issues in the implementation of the DSD algorithm for the traffic assignment problem. Eur J Oper Res 175:1577–1587

    Article  MATH  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Parente LA, Lotito PA, Solodov MV (2008) A class of inexact variable metric proximal point algorithms. SIAM J Optim 19:240–260

    Article  MathSciNet  Google Scholar 

  • Pennanen T (2002) A splitting method for composite mappings. Numer Funct Anal Optim 23:875–890

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Rockafellar RT (1970) On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 149:75–88

    Article  MathSciNet  MATH  Google Scholar 

  • Rockafellar RT, Wets J-B (1997) Variational analysis. Springer, New York

    Google Scholar 

  • Rubiales A, Mayorano F, Lotito P (2007) Optimización aplicada a la coordinación hidrotérmica del mercado eléctrico argentino. Mec Comput XXVI, 3343–3359

    Google Scholar 

  • Ruiz C, Conejo AJ, García-Bertrand R (2010) Some analytical results pertaining to Cournot models for short-term electricity markets

  • Scott TJ, Read EG (1996) Modelling hydro reservoir operation in a deregulated electricity market. Int Trans Oper Res 3:243–253

    Article  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Solodov MV, Svaiter BF (2001) A unified framework for some inexact proximal point algorithms. Numer Funct Anal Optim 22:1013–1035

    Article  MathSciNet  MATH  Google Scholar 

  • Tseng P (1997) Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J Optim 7:951–965

    Article  MathSciNet  MATH  Google Scholar 

  • Wood AJ, Wollenberg BF (1996) Power generation, operation, and control, 2nd edn. Wiley-Interscience/Wiley, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. OPTyCON, UNR, CONICET, Rosario, Argentina

    Lisandro A. Parente

  2. PLADEMA, UNICEN, CONICET, Tandil, Argentina

    Pablo A. Lotito, Fernando J. Mayorano & Aldo J. Rubiales

  3. IMPA, Rio de Janeiro, Brazil

    Mikhail V. Solodov

Authors
  1. Lisandro A. Parente
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pablo A. Lotito
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fernando J. Mayorano
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Aldo J. Rubiales
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mikhail V. Solodov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo A. Lotito.

Additional information

This work was partially supported by the STIC-AMSUD Project Optimización Energética and CONICET.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Parente, L.A., Lotito, P.A., Mayorano, F.J. et al. The hybrid proximal decomposition method applied to the computation of a Nash equilibrium for hydrothermal electricity markets. Optim Eng 12, 277–302 (2011). https://doi.org/10.1007/s11081-010-9131-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11081-010-9131-1

Keywords

Search

Navigation