K u rzfassu n g sb an d - Graz University of Technology

80 Energieinnovation 2006
3.4.4 „Hydro-Thermal Optimal Scheduling Using Decomposition
Approach“
Cherry Yuen, (ABB Schweiz), Gertrud Rossa (TU-Graz)
The study focuses on the long-term scheduling problem of hydro-thermal energy resources in big
systems. The challenge of such problems is the size of the optimization problem, mainly caused by the
stochastic nature of certain exogeneous variables, such as water inflow, energy demand, etc.
However, such problems also naturally involve multiple stages, e.g. twelve months in a year. For
planning purposes, one needs to work out the optimal usage of different resources in the system, so
that the total cost over the whole study period, e.g. one year, is minimized. The multi-stage problem,
with various scenarios to represent the possible combinations of values of the stochastic exogeneous
variables, can be formulated as one linear-programming (LP) problem. However, depending on the
available LP solver and the processing power of the computer, the so-formulated problem can take a
very long time to solve, even for a commercial solver. In some cases, due to excessive memory
requirements, the problem cannot even be solved. For example, in a big system with several hundred
reservoirs and a hundred scenarios, that can result in a LP problem with a million variables and
number of constraints in the same magnitude.
To overcome this, one can decompose the problem into multiple single-stage subproblems, each of
them is much smaller than the original problem and is therefore much faster to solve. The principle is
well-known and is called Benders Decomposition. The information in between stages is transmitted
from one stage to another by the use of Benders Cuts. These Benders Cuts form a convex piecewise
linear function which represents the approximated future cost function, which is added to the singlestage
subproblem so that the present cost can be optimized with the consideration of the future
stages. Despite of slow convergence for certain cases, this approach has proven to be performing
satisfactorily.
In the stochastic decomposition approach, a set of scenarios are used for stochastic optimization to
obtain the optimal solution. These scenarios are generated using Monte Carlo simulations. The results
so obtained are better than those obtained when an averaged scenario based on all the scenarios is
fed in a deterministic formulation. The stochastic future cost function is also a piecewise linear function
which is a weighted average representation of the Benders Cuts of all the scenarios. The results
based on several numerical examples are illustrated in the presentation.