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environmental variations or externalities, which denote the evolution of the system. The
two major approaches of addressing this are either to search for the coalitional structure
that maximizes the total utility of the system (social welfare), or to find the structure
with Pareto optimal payoff distribution for the agents involved. Generally, computing
this through a centralized approach is NP-complete. The complexity of the problem
addressed is due to the fact that the utility of adding or removing actors from a coali-
tion has a dynamic valuation, as it depends on the other actors already comprising the
coalition at each moment in time. Therefore, it is desirable that the coalition formation
process takes place in a distributed manner, leveraging on the autonomy of the agents,
which spontaneously organize into topologies and functionalities to meet the desired
objectives. We believe that in order to solve these issues, the problem must be under-
stood in the context of self-organization by providing a minimum set of interaction rules
that would lead to an efficient achievement of the underlined desiderata.
Returning to our initial application, the algorithm we propose is illustrated as a case
study in the context of smart energy systems. We set to investigate the integration of
renewable energy resources to the grid in the form of virtual power plants by means
of aggregating the power generating potential of various devices in a novel way in the
context of MAS. As system designers, we choose to enable the autonomous agents with
the basic coordination primitives, and leave to the agents to self-organize and coordinate
as the situation may demand, in a fully distributed manner.
We model the problem as a dynamic coalition formation game with the following
formalization:
Let
M
=
A, G,β
i
, S,
Φ
,
v
be a multi-agent system where:
•A
=
{a
1
,a
2
, ..., a
n
}
represents the set of agents of a given portion of the distri-
bution grid. We assume that each stakeholder that is connected to the grid is repre-
sented by a software agent that manages the corresponding device (e.g., generators,
storage devices, intelligent loads).
•G
=(
A,E
)
is the underlying communication network denoted as an undirected
graph where the set of vertices is the set of agents and edges are communica-
tion links.
N
i
represents
a
i
's
set of neighbours s.t.
∀a
i
,a
j
∈A,
if
(
a
i
,a
j
)
∈
a
j
∈N
i
• β
i
is the forecast amount of electricity for the following day associated with agent
a
i
.If
E
then
a
i
∈N
j
and
β
i
>
0
, then agent
a
i
is a
provider
, whilst if
β
i
<
0
then agent
a
i
is a
consumer
(or load). Let
the set of
consumers. In this work we assume that an agent is either a provider or a load, and
therefore
P⊆A
denote the set of providers, and
L⊆A
P∪L
=
A
,
P∩L
=
∅
. We will refer onwards generically, to an agent
belonging to set
P
as
PA
, and to an agent belonging to set
L
as
LA
.
• CS
=
{S
1
,S
2
, ..., S
m
}
is the set of coalitions that partition the set
A
. We assume
that all coalitions are disjoint, and therefore:
m
S
j
=
A,S
j
∩ S
k
=
∅, ∀ j
=
k
j
=1
•
Φ=
{φ
1
,φ
2
,φ
3
}
is a set of constraints that must hold for every coalition. In this
work, we enforce that the number of members of each coalition does not exceed a
