Therefore, the power
L
²
i
is more than the demand
j
Req
i
j
ð
L
²
i
9
j
Req
i
jÞ
.
L
²
i
is the solution of following quadratic
equation:
L
i
¼
P
i
0
þ j
Req
i
j ¼
R
i
0
L
2
i
U
2
0
þ
±L
i
±
Req
i
:
(4)
For a given
Req
i
, three possible solutions of eq. (4) exist,
namely none (zero), one, and two solutions. Because we
want to minimize the value of
Req
i
, if eq. (4) has two roots,
the smaller one is to be used. For the cases that eq. (4) has
no solution, we assume that the root is the same as eq. (4)
having a single root, which is
L
²
i
¼
ð
1
±
±
Þ
U
2
0
2
R
i
0
.
Because in the noncooperative case each MG can be
regarded as a coalition, the payoff of MG is equal to that of
coalition. Thus, we are able to define the noncooperative
payoff (utility) of each
MG
i
as the total power loss due to
the power transfer, as follows:
u
f
i
g
ð
Þ ¼ ±
w
2
P
i
0
;
(5)
where
w
2
is the price of a unit power in MS. Because
the objective is to minimize
u
ðf
i
gÞ
, the negative sign is
able to convert the problem into a problem of seeking
the maximum.
3.2
Cooperative Coalition Model
In the remainder of this section, the cooperative coalition
model is considered for managing the MGs acting as
‘‘buyers’’ and ‘‘sellers’’. Also, the functions of power loss
and utility in the cooperative case along with how to form
the coalitions are proposed. Toward the end of the section,
the concept of ‘‘Shapley’’ function is presented.
Besides exchanging power with the MS, the MGs can
exchange
power
with
others.
Because the
power
loss
during transmission among the neighbouring MGs are
always less than that between the MS and a MG, the MGs
Fig. 1. Construction of micro grids.
WEI ET AL.: GAME THEORETIC COALITION FORMULATION STRATEGY
2309

can form cooperative groups, referred to as coalitions
throughout this paper, to exchange power with others, so
as to alleviate the power loss in the main smart grid and
maximize their payoffs in eq. (5).
Before formally studying the cooperative behaviour of
the MGs, the framework of coalition game theory is firstly
introduced in the work in [20]. A coalition game is defined
as a pair
ðN
; v
Þ
. The game comprises three parts, namely
the set of players
N
, the strategy of players, and the
function
v
:
2
N
!
R
. In this game,
v
is a function that assigns
for every coalition
S
³ N
a real number representing the
total profits achieved by
S
. We divide any coalition
S
³ N
into two parts: the set of ‘‘sellers’’ denoted by
S
s
&
S
and
the set of ‘‘buyers’’ represented by
S
b
&
S
.
S
s
and
S
b
satisfy
that
S
s
[
S
b
¼
S
. Therefore, for a
MG
i
2
S
s
,
Req
i
9
0
and it
means that
MG
i
wants to sell power to others. On the other
hand, an arbitrary
MG
j
2
S
b
having
Req
j
G
0
indicates that
MG
j
wants to buy power from others. It is obvious that
any coalition
S
³ N
should have at least one seller and one
buyer.

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