caa a22 4500 445878150 CHVBK 20180317145533.0 cr unu---uuuuu 170323e20110301xx s 000 0 eng 10.1007/s11081-010-9131-1 doi (NATIONALLICENCE)springer-10.1007/s11081-010-9131-1 The hybrid proximal decomposition method applied tothe computation of a Nash equilibrium forhydrothermal electricity markets [Elektronische Daten] [Lisandro Parente, Pablo Lotito, Fernando Mayorano, Aldo Rubiales, Mikhail 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. Springer Science+Business Media, LLC, 2011 Electric power market nationallicence Nash-Cournot equilibria nationallicence Variational inclusions nationallicence Decomposition nationallicence t : each time period, t=1, ,T nationallicence i : each thermal unit, $i=1,\ldots,\mathcal {I}$ nationallicence j : each hydroelectric unit, $j=1,\ldots,\mathcal {J}$ nationallicence m : each thermal company, m=1, ,M nationallicence n : each hydroelectric company, n=1, ,N nationallicence $\mathcal {C}^{Th}_{m}$ : index set for thermal units owned by company m nationallicence $\mathcal {C}^{H}_{n}$ : index set for hydroelectric units owned by company n nationallicence y jt : production at hydroelectric unit j for time period t (or consumption, if this quantity is negative) nationallicence y : hydroelectric production vector in $\mathbb {R}^{\mathcal {J}T}$ nationallicence x it : production at thermal unit i for the time period t nationallicence x : thermal production vector in $\mathbb {R}^{\mathcal {I}T}$ nationallicence $y_{j}^{LOW}, y_{j}^{UP}$ : hydroelectric production bounds for unit j nationallicence $y_{j}^{TOT}$ : total hydroelectric generation for unit j nationallicence α j : efficiency coefficient of hydroelectric unit j nationallicence y LOW , y UP , y TOT , α : corresponding vectors in $\mathbb {R}^{\mathcal {J}}$ nationallicence $x_{i}^{LOW}, x_{i}^{UP}$ : thermal productions bounds for unit i nationallicence φ i , ω i , ψ i : thermal cost coefficients associated to the quadratic, linear, and independent term for unit i nationallicence x LOW , x UP , φ , ω , ψ : corresponding vectors in $\mathbb {R}^{\mathcal {I}}$ nationallicence a t , D t : inverse demand coefficients for time t nationallicence a , D : corresponding vectors in ℝT nationallicence $\mathit{Ben}_{n}^{H}$ : benefit obtained by the hydroelectric company n nationallicence $\mathit{Ben}_{m}^{T}$ : benefit obtained by the thermal company m nationallicence p t : market price for period t nationallicence f j (⋅) : non-smooth function representing the efficiency gap between pumping and generation for unit j nationallicence $c_{i}^{T}(\cdot)$ : quadratic function representing the thermal production cost for unit i nationallicence $\mathcal {K}^{H}_{n}$ : feasible set for the hydroelectric company n nationallicence $\mathcal {K}^{Th}_{m}$ : feasible set for the thermal company m nationallicence I n : identity matrix in ℝn×n nationallicence 1 n : vector of ones in ℝn nationallicence λ max ( M ) : maximal eigenvalue of the symmetric positive definite (SPD) matrix M nationallicence λ min ( M ) : minimal eigenvalue of the SPD matrix M nationallicence diag( v ) : diagonal matrix with (diag(v))n,n=v n, for v∈ℝn nationallicence F :ℝ n ⇉ℝ m : operator that maps points in ℝn to subsets of ℝm nationallicence dom F : domain of F:ℝn⇉ℝm, i.e., {x∈ℝn∣F(x)≠∅} nationallicence $N_{\mathcal {K}}(x)$ : normal cone of a convex set $\mathcal {K}$ at x, i.e nationallicence $\mathrm{rint}\,\mathcal {K}$ : relative interior of the set $\mathcal {K}$ nationallicence $\mathrm {Proj}_{\mathcal {K}}(x)$ : (orthogonal) projection of the point x onto the set $\mathcal {K}$ nationallicence Parente Lisandro OPTyCON, UNR, CONICET, Rosario, Argentina aut Lotito Pablo PLADEMA, UNICEN, CONICET, Tandil, Argentina aut Mayorano Fernando PLADEMA, UNICEN, CONICET, Tandil, Argentina aut Rubiales Aldo PLADEMA, UNICEN, CONICET, Tandil, Argentina aut Solodov Mikhail IMPA, Rio de Janeiro, Brazil aut Optimization and Engineering Springer US; http://www.springer-ny.com 12/1-2(2011-03-01), 277-302 1389-4420 12:1-2<277 2011 12 11081 https://doi.org/10.1007/s11081-010-9131-1 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11081-010-9131-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Parente Lisandro OPTyCON, UNR, CONICET, Rosario, Argentina aut NATIONALLICENCE 700 1- Lotito Pablo PLADEMA, UNICEN, CONICET, Tandil, Argentina aut NATIONALLICENCE 700 1- Mayorano Fernando PLADEMA, UNICEN, CONICET, Tandil, Argentina aut NATIONALLICENCE 700 1- Rubiales Aldo PLADEMA, UNICEN, CONICET, Tandil, Argentina aut NATIONALLICENCE 700 1- Solodov Mikhail IMPA, Rio de Janeiro, Brazil aut NATIONALLICENCE 773 0- Optimization and Engineering Springer US; http://www.springer-ny.com 12/1-2(2011-03-01), 277-302 1389-4420 12:1-2<277 2011 12 11081 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer