Page 72 - Decision Making Applications in Modern Power Systems

Uncertainty management in decision-making Chapter | 2 43

the uncertain parameters are modeled according to the corresponding PDF.
The probabilistic programming was first introduced by Dantzig [12].
There are three different probabilistic approach techniques that are used
for uncertainty management: Monte Carlo simulation (MCS), point estima-
tion method (PEM), and scenario-based optimization method [13].
In MCS, n s samples for each uncertain parameter are generated according to
s
the corresponding PDF of each one. Assuming n s 5 n 1 ; n 2 ; ...; n s g, y 5 fðn s Þ
f
is calculated. The process is repeated for a lot of iterations, until the average
value of each uncertain parameter is obtained. Some of the valuable works that
use MCS in the probabilistic programming can be found in Refs. [13 17].
The PEM is based on the concept of statistical moments of input uncer-
tain parameters. Unlike the MCS, the PEM only generates two samples for
each parameter. Some of the valuable works that use MCS in the probabilis-
tic programming can be found in Ref. [18].
The decision-making on the basis of scenarios is another technique that is
based on probability theory. A list of scenarios is generated using the corre-
sponding PDF of each uncertain parameter. In Section 2.3 an example of
modern power system and scenario-based optimization approach is imple-
mented to uncertainty modeling. The details of this technique are presented
[19]. Some of the valuable works that use the scenario-based decision-mak-
ing method can be found in Ref. [20].

2.2.2 Information gap decision theory

In some cases the uncertain parameters do not follow a PDF or the PDF is
not known by the decision maker. In such cases the information gap decision
theory (IGDT) is used to model the uncertainty [7,21]. In IGDT the robust-
ness is defined as the inviolability of euphoria of a predefined constraint
[21]. Suppose X is the set of uncertain parameters and all the components of

this set are equal to its predicated value X 5 X , so the predicated value for
objective function yðÞ is obtained. When the value of the uncertain parameter
is not equal to the predicated value and is unknown, IGDT is implemented
to find a good solution that gives the robustness feature to the value of the
objective function against the predicting error. Some of the valuable works
that use IGDT can be found in Refs. [22 25].

2.2.3 Robust optimization

The robust optimization technique is another important uncertainty modeling
tool in the power system studies. In this method, it is assumed that the uncer-
tain parameter belongs to an uncertainty set, and it is tried to make the opti-
mal decision considering this fact. In other words the decision variables are
found in a way that the objective function remains optimal even if the uncer-
tain parameter takes its worst case value [26,27]. For example, consider a