Nicole Gröwe-Kuska, Krzysztof C. Kiwiel, Matthias P. Nowak, Werner Römisch, Isabel Wegner
Power Management in a Hydro-Thermal System under Uncertainty by Lagrangian Relaxation
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

 90C15 Stochastic programming

90C90 Applications of mathematical programming

90C11 Mixed integer programming

90C25 Convex programming

65K05 Mathematical programming, See also {90Cxx}

Abstract: We present a dynamic multi-stage stochastic programming model for the
cost-optimal generation of electric power in a hydro-thermal system
under uncertainty in load, inflow to reservoirs and prices for fuel
and delivery contracts. The stochastic load process is approximated
by a scenario tree obtained by adapting a SARIMA model to historical
data, using empirical means and variances of simulated scenarios to
construct an initial tree, and reducing it by a scenario deletion
procedure based on a suitable probability distance.
Our model involves many mixed-integer variables and individual power
unit constraints, but relatively few coupling constraints. Hence we
employ stochastic Lagrangian relaxation that assigns stochastic
multipliers to the coupling constraints. Solving the Lagrangian dual
by a proximal bundle method leads to successive decomposition into
single thermal and hydro unit subproblems that are solved by dynamic
programming and a specialized descent algorithm, respectively. The
optimal stochastic multipliers are used in Lagrangian heuristics to
construct approximately optimal first stage decisions. Numerical
results are presented for realistic data from a German power utility,
with scenario numbers ranging from 5 to 100 and a time horizon of 7
to 9 days. The corresponding optimization problems have up to
200,000 binary and 350,000 continuous variables, and more than 500,000
constraints.
Keywords: Stochastic programming, Lagrangian relaxation, unit commitment, bundle methods, scenario generation