Optimization and Engineering

, Volume 12, Issue 1–2, pp 277–302

# 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
Article

## 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.

### Keywords

Electric power market Nash-Cournot equilibria Variational inclusions Decomposition

### 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.

yjt

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

y

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

xit

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.

αj

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.

φi,ωi,ψi

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

xLOW,xUP,φ,ω,ψ

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

at,Dt

inverse demand coefficients for time t.

a,D

corresponding vectors in ℝ T .

$$\mathit{Ben}_{n}^{H}$$

benefit obtained by the hydroelectric company n.

$$\mathit{Ben}_{m}^{T}$$

benefit obtained by the thermal company m.

pt

market price for period t.

fj(⋅)

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.

In

identity matrix in ℝn×n.

1n

vector of ones in ℝ n .

λmax(M)

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

λmin(M)

minimal eigenvalue of the SPD matrix M.

diag(v)

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

F:ℝn⇉ℝm

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

domF

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. .

$$\mathrm{rint}\,\mathcal {K}$$

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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## 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