been verified in [
22
] and [
23
]. Each level is formulated as a
stochastic two-stage problem with the first stage to optimize
the base generation and power exchanges of all entities based
on the forecasted outputs of RES-based DGs and the second
stage to adjust generations according to the variations of real-
ized RES-based DG outputs. The uncertain power outputs of
wind turbines and PVs are described by scenarios generated
from Monte Carlo simulations (MCs). The simultaneous back-
ward scenario reduction method [
24
] is applied to increase
the calculation speed while maintaining the accuracy of the
solution.
The major contributions of this paper are summarized as
follows.
1) Optimal coordinated control of networked MGs with
distinct economic and operational objectives in a dis-
tribution system is a new topic with limited existing
works.
2) Uncertainty and variability of RES-based DG outputs
are fully considered.
3) Stochastic bi-level formulation of the control framework
with each level modeled as a two-stage problem.
The
remainder
of
this
paper
is
organized
as
follows.
Section II presents the local optimization problems of the
DNO and MGs. Section III introduces the coordinated con-
trol scheme of multiple MGs and transforms the coordinated
control problem into a stochastic MPCC formulation and pro-
poses the solution methodology. In Section IV, the numerical
results are provided. Section V concludes the paper with the
major findings.
II. M
ATHEMATICAL
M
ODELING OF
I
NDIVIDUAL
S
YSTEMS
This section introduces a widely used electrical network
model and provides the local optimization formulation for
individual systems, DNO and MGs.
A. Distribution System Model
Consider an electrical network as shown in Fig.
1
, there
are
n
buses indexed by
i
=
0
,
1
, . . . ,
n
. DistFlow [
25
]
equations can be used to describe the complex power flows at
each node
i
P
i
+
1
=
P
i
−
r
i
(
P
2
i
+
Q
2
i
)
V
2
i
−
p
i
+
1
(1)
Q
i
+
1
=
Q
i
−
x
i
(
P
2
i
+
Q
2
i
)
V
2
i
−
q
i
+
1
(2)
V
2
i
+
1
=
V
2
i
−
2
(
r
i
P
i
+
x
i
Q
i
)
+
(
r
2
i
+
x
2
i
)(
P
2
i
+
Q
2
i
)
V
2
i
(3)
p
i
=
p
D
i
−
p
g
i
,
q
i
=
q
D
i
−
q
g
i
.
(4)
In the above equations, we assume
p
g
i
is generated by both
RES-based DG units which are subject to uncertainties and
controllable DG units,
q
g
i
is generated by controllable DG
units [
26
]. The DistFlow equations can be simplified using
linearization. The linearized power flow equations have been
extensively used and justified in both traditional distribution
systems and MGs [
7
], [
27
], [
28
]
P
i
+
1
=
P
i
−
p
i
+
1
(5)
Q
i
+
1
=
Q
i
−
q
i
+
1
(6)
V
i
+
1
=
V
i
−
(
r
i
P
i
+
x
i
Q
i
)
V
2
1
(7)
p
i
=
p
D
i
−
p
g
i
,
q
i
=
q
D
i
−
q
g
i
.
(8)
B. Optimization Problem for DNO
It is assumed that the DNO also owns both dispatchable
DGs (MTs in this paper) and RES-based DGs (WTs in this
paper) [
29
]. The optimization problem of a DNO can be
formulated as follows (denote the formulation as
M
):
min
i
∈
D
c
G
p
G
i
+
c
B
,
DNO
η
1
+
m
c
S
,
MG
θ
m
1
−
c
S
,
DNO
θ
1
−
m
c
B
,
MG
η
m
1
+
s
γ
s
i
∈
D
c
G
p
G
i
,
s
+
C
rd
i
,
s
+
s
γ
s
c
B
,
DNO
η
1
,
s
+
m
c
S
,
MG
θ
m
1
,
s
−
c
S
,
DNO
θ
1
,
s
−
m
c
B
,
MG
η
m
1
,
s
(9)
s
.
t
.
P
i
+
1
=
P
i
−
p
D
i
+
1
+
g

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