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Decision-making-based optimal generation-side Chapter | 11 281

variables, which do not grow with the number of uncertainty realizations as
in Ref. [32 37]; thus the resulting solution is less conservative compared
with [34].

11.3.1 Problem setup and reserve representation
In this work, we consider a power network consisting of N G generating units,
N w wind power plants, N L loads, N l lines, and N b buses. Taking into consid-
eration that ϒ is a set that includes the indices corresponding to outages of
all components also including the index 0 that corresponds to the base case
of no outage and denoted by ϒjj its cardinality. ϒ l , ϒ L , and ϒ G are the set of
indices representing the branch, load, and generator outages, respectively.
The following assumptions are considered for the problem formulation:
1. A DC power flow approximation is considered.
2. High-accuracy load forecasts are assumed.
3. Line outages do not lead to multiple generator/load failures.
4. The on off status of the generating units has been fixed a priori by solv-
ing a unit-commitment problem.

The first assumption is basic for these types of optimization problems
while the second and third one are meant to simplify the presentation of the
results and could still be captured by the proposed algorithm. If the last
assumption is removed by incorporating the unit-commitment problem, the
objective would give rise to a mixed-integer problem. This can be tackled
using the probabilistically robust design that can deal with a specific class of
nonconvex problems.
The results of generation-load mismatches in frequency deviations from
the nominal and reserves are used to balance the mismatches. The process is
achieved by the activation of the AGC, LFC, where its output is distributed
to certain participating generators. The set point of each generator is changed
by a certain percentage of the overall active power to be compensated. The
existing setup of the AGC loop is shown in Fig. 11.1, demonstrating the role
of the distribution vector.
This distribution vector results from the market that determines the sec-
ondary frequency control reserves, and it remains constant until the next
market auction. However, this task is performed while neglecting the net-
work constraints. Ideally, the distribution vector is the same for all possible
outages but may differ between up-spinning and down-spinning reserves.
Here different distribution vectors depending on the outage are also consid-
ered in addition to distinguishing between up-spinning and down-spinning
reserves. An optimal reserve schedule, which takes into account the network
security constraints, is determined over the distribution vectors. Using this
approach, both the minimum cost reserves per generator and also a reserve