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6
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7
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T
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th
e
s
ec
o
n
d
ar
y
r
eser
v
e,
an
d
m
a
n
ag
e
s
th
e
p
o
w
er
f
lo
w
b
et
w
ee
n
t
h
e
m
icr
o
g
id
an
d
th
e
g
r
id
,
an
d
w
it
h
i
n
th
e
m
icr
o
g
r
id
f
o
r
th
e
ca
s
e
o
f
is
lan
d
ed
m
o
d
e
t
h
at
lead
s
to
t
h
e
ec
o
n
o
m
ic
is
s
u
e
s
.
T
h
er
ef
o
r
e,
th
e
o
p
ti
m
al
p
o
w
er
f
lo
w
is
also
to
b
e
ca
lled
en
erg
y
ma
n
a
g
eme
n
t
th
at
i
m
p
l
e
m
en
ts
i
n
a
SC
AD
A
s
y
s
te
m
t
h
at
p
r
o
d
u
ce
s
th
e
lo
w
e
s
t c
o
s
t o
f
en
er
g
y
.
T
o
s
o
lv
e
th
e
p
r
o
b
lem
o
f
o
p
tim
al
p
o
w
er
f
lo
w
in
t
h
e
ter
tiar
y
co
n
tr
o
l
lev
el,
b
asicall
y
w
e
m
u
s
t
co
n
s
id
er
th
e
u
n
co
n
tr
o
llab
le
n
at
u
r
e
o
f
w
i
n
d
,
s
o
lar
p
o
w
er
as
w
e
ll
as
lo
ad
ch
an
g
e
r
ai
s
es
u
n
ce
r
tain
t
y
f
o
r
p
o
w
er
s
y
s
te
m
o
p
er
atio
n
o
n
th
e
o
n
e
p
ar
t,
t
h
e
ap
p
licatio
n
an
d
i
n
te
g
r
atio
n
o
f
DGs i
n
to
t
h
e
s
y
s
te
m
is
co
m
p
le
x
o
n
th
e
o
th
er
p
ar
t.
Am
o
n
g
th
e
s
e
a
s
p
ec
ts
ar
e,
th
e
i
n
v
e
s
ti
g
atio
n
o
f
i
m
p
ac
t o
f
d
is
tr
ib
u
ted
g
e
n
er
atio
n
o
n
t
h
e
p
o
w
e
r
f
l
u
ct
u
atio
n
s
f
r
o
m
p
en
etr
atio
n
o
f
w
i
n
d
,
p
h
o
to
v
o
l
taic
p
o
w
er
is
p
r
ese
n
ted
in
[
8
]
,
[
9
]
.
A
cc
o
r
d
in
g
to
[
1
0
]
,
[
1
1
]
th
e
au
th
o
r
s
h
av
e
co
n
s
id
er
ed
th
e
h
y
b
r
id
p
o
w
er
s
y
s
te
m
w
h
o
s
e
t
h
e
en
er
g
y
s
to
r
ag
e/t
h
er
m
al
u
n
it
h
a
s
a
h
i
g
h
p
o
ten
tial
f
o
r
p
r
o
v
id
in
g
r
eg
u
lat
io
n
p
o
w
er
to
m
ee
t
th
e
r
ev
er
s
e
r
eq
u
ir
e
m
e
n
ts
.
I
n
th
e
c
o
n
tex
t
o
f
u
n
ce
r
tai
n
t
y
,
th
e
w
o
r
k
in
[
1
2
]
f
o
cu
s
o
n
th
e
d
esi
g
n
o
p
ti
m
izatio
n
o
f
MG
b
y
u
s
in
g
t
h
e
s
to
ch
a
s
tic
m
et
h
o
d
to
m
o
d
el
en
er
g
y
s
y
s
te
m
s
s
u
c
h
t
h
at
th
e
p
r
o
b
a
b
ilit
y
o
f
o
cc
u
r
r
en
ce
o
f
in
cid
en
t
s
o
lar
s
o
u
r
ce
ca
lled
S
t
o
ch
a
s
tic
P
r
o
g
r
a
mmin
g
meth
o
d
.
I
n
th
e
liter
atu
r
e,
th
er
e
ar
e
m
an
y
r
esear
c
h
es t
h
at
h
av
e
b
ee
n
ai
m
ed
to
s
o
lv
e
t
h
e
p
r
o
b
lem
o
p
ti
m
alit
y
.
On
e
o
f
i
m
p
o
r
tan
t
w
o
r
k
s
as
i
n
[
1
3
]
,
w
h
ic
h
ex
a
m
i
n
es
t
h
e
o
p
tim
a
l
p
r
o
b
lem
a
s
th
e
i
n
v
es
tig
atio
n
o
f
o
p
tim
a
l
p
o
w
er
f
lo
w
b
y
ad
o
p
t
in
g
th
e
in
ter
v
al
o
p
ti
m
izatio
n
in
w
h
ic
h
t
h
e
w
i
n
d
p
o
w
er
is
d
ef
i
n
ed
in
r
a
n
g
e
o
f
v
alu
e
s
;
th
e
in
ter
v
al
p
o
w
er
f
lo
w
p
r
o
b
le
m
f
o
r
D
C
p
o
w
er
f
lo
w
ca
n
b
e
f
o
r
m
u
lated
a
s
a
n
o
n
-
c
o
n
v
e
x
a
n
d
n
o
n
li
n
ea
r
p
r
o
g
r
am
m
i
n
g
.
I
n
co
n
s
eq
u
en
c
e,
to
en
s
u
r
e
s
y
s
te
m
r
eliab
ili
t
y
,
th
e
f
o
r
ec
asti
n
g
u
n
ce
r
tai
n
t
y
m
u
s
t
b
e
co
n
s
id
er
ed
in
to
g
en
er
at
io
n
s
ch
ed
u
li
n
g
,
a
n
d
in
ter
v
al
p
o
w
er
f
lo
w
p
r
o
v
i
d
es
p
r
o
m
is
i
n
g
ap
p
r
o
ac
h
to
ac
h
iev
e
t
h
e
b
o
u
n
d
ar
y
in
f
o
r
m
atio
n
o
f
s
y
s
te
m
s
tat
u
s
e
s
u
n
d
er
u
n
ce
r
tain
ties
.
Mo
r
e
i
m
p
o
r
ta
n
tl
y
,
th
e
co
n
tr
ib
u
t
io
n
i
n
[
1
3
]
is
a
lan
d
m
ar
k
f
o
r
class
o
f
o
p
ti
m
a
l
p
o
w
er
f
lo
w
p
r
o
b
le
m
.
Fo
r
t
h
e
lo
ad
f
lo
w
p
r
o
b
lem
w
h
ic
h
is
t
y
p
icall
y
f
o
r
m
u
lated
as
a
s
et
o
f
n
o
n
-
li
n
ea
r
eq
u
atio
n
s
a
s
f
u
n
ct
i
o
n
s
o
f
b
u
s
v
o
ltag
e
s
h
as
ta
k
e
n
s
o
m
e
ad
v
a
n
ta
g
e.
T
h
e
co
n
tr
ib
u
tio
n
s
o
f
o
p
ti
m
al
p
o
w
er
f
lo
w
ar
e
co
n
s
id
r
ed
i
n
[
1
4
]
,
th
e
co
n
tr
o
l
o
f
MG
i
n
[
1
5
]
,
an
d
q
u
ali
t
y
o
f
p
o
w
er
(
p
o
w
er
f
ac
to
r
,
to
tal
h
ar
m
o
n
ic
d
is
to
r
tio
n
)
as
i
n
[
1
6
]
.
Sto
ch
asti
c
d
e
m
a
n
d
h
as
b
ee
n
d
ev
e
lo
p
ed
an
d
p
r
esen
ted
in
[1
7]
-
[
1
9
]
,
th
o
s
e
au
th
o
r
s
u
s
ed
s
to
ch
as
tic,
f
u
zz
y
,
an
d
p
r
o
b
ab
ilit
y
p
r
o
g
r
a
m
m
i
n
g
tech
n
iq
u
es
to
m
o
d
el
th
e
u
n
ce
r
tain
t
ies.
Alth
o
u
g
h
th
e
w
o
r
k
s
in
[
1
7
]
-
[
1
9
]
h
av
e
b
ee
n
s
p
ec
i
f
ied
to
lo
ad
f
l
o
w
p
r
o
b
lem
w
it
h
s
o
m
e
alg
o
r
it
h
m
s
,
th
e
a
f
o
r
e
m
e
n
tio
n
ed
m
et
h
o
d
s
ar
e
t
y
p
icall
y
d
ep
en
d
en
t
o
n
p
r
ed
ef
i
n
ed
p
r
o
b
ab
ilit
y
d
is
tr
ib
u
tio
n
f
u
n
c
tio
n
o
r
m
e
m
b
er
s
h
ip
f
u
n
c
tio
n
o
f
u
n
ce
r
tai
n
t
y
.
Fu
r
t
h
er
m
o
r
e,
it
is
h
ar
d
to
id
en
tify
ac
c
u
r
ate
p
r
o
b
ab
ilit
y
d
is
tr
ib
u
tio
n
f
u
n
ctio
n
s
d
u
e
t
o
d
ata
av
ailab
ilit
y
an
d
s
to
c
h
ast
ic
n
at
u
r
e
o
f
th
e
u
n
ce
r
ta
in
t
y
.
T
h
e
r
esear
ch
i
n
all
th
e
s
e
d
ir
e
ctio
n
s
[
8
]
-
[
2
2
]
f
o
r
t
h
e
ca
s
e
o
f
p
o
w
er
f
lo
w
w
a
s
b
ased
o
n
n
o
n
-
co
n
v
e
x
,
an
d
n
o
n
-
li
n
ea
r
p
r
o
g
r
am
m
i
n
g
an
d
th
e
DG
u
n
its
h
av
e
k
n
o
w
n
lo
ca
tio
n
s
an
d
r
u
n
n
i
n
g
all
t
h
e
ti
m
e.
T
o
a
d
d
t
o
th
e
co
m
p
le
x
it
y
o
f
t
h
e
p
r
o
b
lem
,
i
n
th
e
r
ea
l
li
f
e
s
y
s
te
m
s
,
t
h
e
o
p
er
atio
n
o
f
th
ese
DG
u
n
its
,
u
n
d
er
g
o
e
s
d
if
f
er
en
t
s
ce
n
ar
io
s
ac
co
r
d
in
g
to
th
e
s
tr
ateg
ies
o
f
t
h
e
elec
tr
icit
y
p
r
o
d
u
ce
r
s
an
d
th
e
n
ee
d
s
o
f
th
e
co
n
s
u
m
er
s
.
T
h
er
ef
o
r
e,
s
o
m
e
u
n
ce
r
tai
n
tie
s
ar
e
in
tr
o
d
u
ce
d
in
t
h
e
o
p
er
atio
n
o
f
s
u
c
h
u
n
i
ts
a
n
d
th
u
s
,
s
to
ch
as
tic
m
o
d
elin
g
o
f
s
y
s
te
m
s
in
v
o
l
v
i
n
g
DG
u
n
it
s
b
ec
o
m
e
s
o
f
g
r
ea
t
i
n
ter
est.
T
h
e
s
o
u
r
ce
s
o
f
t
h
e
u
n
ce
r
tain
ties
in
t
h
e
o
p
er
atio
n
o
f
t
h
e
D
G
s
y
s
te
m
s
at
a
ce
r
tai
n
h
o
u
r
o
f
t
h
e
d
a
y
i
n
cl
u
d
e.
T
h
ese
u
n
ce
r
t
ain
tie
s
af
f
ec
t
t
h
e
m
o
d
elin
g
a
n
d
ev
alu
at
io
n
o
f
th
e
s
y
s
te
m
ca
p
ac
it
y
,
p
o
w
er
lo
s
s
e
s
.
I
t’
s
n
o
t
s
i
m
ilar
to
t
h
e
r
ev
ie
w
ed
p
ap
er
s
ab
o
v
e,
th
e
ai
m
o
f
th
is
p
ap
er
d
esig
n
is
to
f
o
cu
s
o
n
in
v
e
s
ti
g
atio
n
o
f
t
h
e
p
o
w
er
f
l
o
w
o
f
h
y
b
r
id
w
i
n
d
-
PV
-
d
iesel
p
o
w
er
s
y
s
te
m
u
n
d
er
u
n
ce
r
tain
tie
s
s
u
c
h
a
s
lo
ad
ch
an
g
e,
w
i
n
d
p
o
w
er
f
lu
c
tu
at
i
o
n
,
s
u
n
ir
r
ad
iatio
n
p
o
w
er
d
is
t
u
r
b
an
ce
,
a
n
d
th
e
d
iesel
en
g
i
n
e
g
e
n
er
ato
r
f
ail
s
a
n
d
is
r
ep
air
ed
at
r
an
d
o
m
t
i
m
e
s
.
T
h
e
o
b
j
ec
tiv
e
f
u
n
ctio
n
is
t
o
m
i
n
i
m
ize
t
h
e
co
s
t
o
f
elec
tr
icit
y
p
r
o
d
u
ctio
n
.
T
h
er
ef
o
r
e
u
tili
ze
s
t
h
e
e
n
er
g
y
b
alan
ce
eq
u
atio
n
in
r
ea
l
ti
m
e
to
f
o
r
m
u
late
t
h
e
o
p
ti
m
al
p
o
wer
f
lo
w
p
r
o
b
le
m
a
s
o
p
tim
a
l
co
n
tr
o
l
p
r
o
b
le
m
o
f
a
lin
ea
r
s
y
s
te
m
b
y
u
s
in
g
d
y
n
a
m
ic
p
r
o
g
r
a
m
m
i
n
g
.
B
y
ad
o
p
tin
g
th
e
R
i
s
h
el
’
s
f
o
r
m
a
li
s
m
[
2
0
]
,
th
e
DE
G
f
ai
l
u
r
e
an
d
r
ep
air
ti
m
es
ar
e
ch
ar
a
cter
ized
b
y
Ma
r
k
o
v
p
r
o
ce
s
s
a
n
d
th
e
co
s
t
f
u
n
ctio
n
is
d
escr
ib
ed
at
ter
m
i
n
al
ti
m
e.
T
o
d
o
th
is
it,
w
ill
b
e
ex
p
ed
ie
n
t
to
d
er
iv
e
an
al
g
o
r
ith
m
s
i
m
i
lar
to
th
e
B
ell
m
an
p
r
in
cip
le
w
h
er
e
t
h
e
o
p
ti
m
alit
y
co
n
d
itio
n
s
s
ati
s
f
y
Ha
m
ilto
n
-
J
ac
o
b
i
-
B
ell
m
a
n
eq
u
atio
n
s
,
a
n
d
th
e
v
al
u
e
f
u
n
ctio
n
is
co
n
v
ex
.
I
n
ad
d
itio
n
,
o
n
e
o
f
t
h
e
p
r
in
cip
le
r
ea
s
o
n
s
f
o
r
in
tr
o
d
u
cin
g
f
ee
d
b
ac
k
i
n
to
an
o
p
ti
m
a
l c
o
n
tr
o
l f
o
r
p
o
w
er
f
lo
w
p
r
o
b
le
m
i
s
to
m
ak
e
th
e
r
esu
lt
in
g
s
y
s
te
m
r
elati
v
el
y
in
s
e
n
s
iti
v
e
to
f
l
u
ctu
a
tio
n
s
t
h
at
ca
n
d
ea
l
w
it
h
u
n
ce
r
tai
n
tie
s
o
f
p
o
w
er
s
y
s
te
m
co
n
s
id
er
ed
.
2.
H
YB
RID IS
L
AND
M
I
CRO
G
RI
D
Fo
r
th
e
s
i
m
p
le
s
y
s
te
m
as
s
h
o
w
n
i
n
Fi
g
u
r
e
1
,
w
e
co
n
s
id
er
t
h
e
h
y
b
r
id
p
o
w
er
s
y
s
te
m
o
n
t
h
e
is
lan
d
ed
AC
m
icr
o
g
r
id
i
n
clu
d
i
n
g
AC
lo
ad
s
,
p
h
o
to
v
o
ltaic
(
P
V)
,
w
i
n
d
tu
r
b
i
n
e
g
e
n
er
ato
r
(
W
T
G)
,
an
d
d
iesel
e
n
g
in
e
g
en
er
ato
r
(
DE
G)
.
A
l
s
o
w
e
co
n
s
id
er
a
s
to
ch
a
s
tic
co
n
tr
o
l
p
r
o
b
le
m
f
o
r
t
h
e
DE
G
s
u
b
j
e
ct
to
f
ailu
r
es
a
n
d
r
ep
air
s
in
co
n
ti
n
u
o
u
s
t
i
m
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
S
to
ch
a
s
tic
co
n
tr
o
l fo
r
o
p
tima
l p
o
w
er flo
w
in
is
la
n
d
ed
micro
g
r
id
(
Th
a
n
g
Diep
-
Th
a
n
h
)
1047
W
i
n
d
T
u
rb
i
n
e
Lo
a
d
1
P
V
a
r
r
a
y
L
o
a
d
2
Di
e
s
e
l
E
n
gi
n
e
Lo
a
d
3
T
i
e
-
l
i
n
e
Fig
u
r
e
1
.
Si
m
p
li
f
ied
s
c
h
e
m
a
ti
c
o
f
an
is
la
n
d
ed
m
icr
o
g
r
id
As
in
Fi
g
u
r
e
1
,
th
e
s
y
s
te
m
c
o
n
s
is
ts
o
f
th
r
ee
s
to
ch
ast
ic
s
o
u
r
ce
s
s
u
c
h
th
a
t
W
T
G
as
s
to
ch
asti
c
w
in
d
p
o
w
er
,
P
V
as
s
to
c
h
ast
ic
s
o
lar
p
o
w
er
,
an
d
i
n
cl
u
d
es
t
h
e
DE
G
th
at
is
s
u
b
j
ec
t
to
f
ail
u
r
es
a
n
d
r
ep
air
s
.
On
th
e
o
th
er
h
an
d
,
th
e
lo
ad
d
em
a
n
d
i
s
s
to
ch
as
tic
b
ased
o
n
f
o
r
ec
asti
n
g
i
n
s
h
o
r
t
-
ter
m
.
T
h
e
DGs
ar
e
co
n
n
ec
ted
to
A
C
b
u
s
b
y
p
o
w
er
elec
tr
o
n
ic
d
ev
ic
es
u
s
ed
s
y
n
c
h
r
o
n
izatio
n
as
AC
s
o
u
r
ce
s
a
s
DE
G,
W
T
G,
an
d
P
V
w
it
h
i
n
v
er
t
D
C
v
o
ltag
e
s
i
n
to
AC
ca
lled
in
ve
r
ter
.
T
h
e
DE
G
is
u
s
ed
f
o
r
t
h
e
co
n
v
er
s
io
n
o
f
m
ec
h
an
ical
e
n
er
g
y
in
to
elec
tr
ical
en
er
g
y
as
co
n
tr
o
lla
b
le
s
o
u
r
ce
.
No
w
,
t
h
e
p
r
o
b
le
m
i
s
co
n
s
id
er
ed
in
f
i
n
ite
ti
m
e
in
o
r
d
er
to
f
in
d
t
h
e
DE
G
p
o
w
er
g
e
n
er
atio
n
.
S
u
ch
a
DE
G
is
s
u
b
j
ec
t
to
r
an
d
o
m
f
ai
l
u
r
es
a
n
d
r
ep
air
s
an
d
ca
n
i
n
j
ec
t
th
e
p
o
w
er
in
to
MG
.
T
h
e
DE
G
h
as
t
w
o
p
o
s
s
ib
le
o
p
er
atio
n
al
s
tate
s
.
T
h
e
d
y
n
a
m
ic
o
f
th
e
DE
G
(
i.e
.
,
j
u
m
p
f
r
o
m
o
n
e
s
tate
to
an
o
t
h
er
)
is
c
h
a
r
ac
ter
ized
b
y
a
t
w
o
-
s
tate
s
to
ch
a
s
tic
p
r
o
ce
s
s
(
s
)
:
s
0
d
ef
in
ed
as:
0
i
f
t
h
e
D
E
G
i
s
u
n
d
e
r
r
e
p
a
i
r
()
1
i
f
t
h
e
D
E
G
i
s
f
u
l
l
y
f
u
n
c
t
i
o
n
i
n
g
s
(
1
)
T
h
e
p
r
o
ce
s
s
(
s
)
i
s
a
co
n
tr
o
lled
p
r
o
ce
s
s
,
as
it
w
ill
d
ep
en
d
o
n
t
h
e
DE
G
ac
tio
n
w
h
e
n
t
h
e
DE
G
is
av
ailab
le
g
e
n
er
atio
n
.
T
h
u
s
,
(
s
)
:
T
s
t
is
a
Ma
r
k
o
v
p
r
o
ce
s
s
,
w
it
h
v
al
u
e
in
0
,
an
d
1
.
L
et
,,
P
F
d
en
o
te
th
e
u
n
d
er
l
y
i
n
g
p
r
o
b
ab
i
lit
y
s
p
ac
e
an
d
let
(
)
(
)
,
s
t
ts
F
s
c
t
t
=
£
£
b
e
th
e
s
ig
m
a
alg
eb
r
a.
L
et
u
s
d
ef
i
n
e
F
(
t
)
an
d
G
(
t
)
as
th
e
f
u
n
ctio
n
i
n
g
an
d
r
ep
air
ti
m
e
d
is
tr
ib
u
tio
n
s
o
f
th
e
DE
G,
r
esp
ec
ti
v
el
y
.
G
iv
en
th
a
t
th
e
DE
G
i
s
o
p
er
atio
n
al,
th
e
p
r
o
b
ab
ilit
y
o
f
a
f
ail
u
r
e
i
n
a
n
i
n
ter
v
al
o
f
le
n
g
th
t
is
t
.
T
h
e
p
r
o
b
ab
ilit
y
th
at
th
e
f
a
iled
DE
G
is
r
ep
air
ed
d
u
r
in
g
s
u
ch
a
ti
m
e
in
ter
v
a
l
is
t
.
T
h
u
s
,
t
h
e
p
ar
a
m
eter
s
a
n
d
ar
e
t
h
e
f
ail
u
r
e
an
d
r
ep
air
r
ates
f
o
r
th
e
DE
G.
No
w
,
t
h
e
d
y
n
a
m
ics
o
f
th
e
DE
G
s
tate
ar
e
g
i
v
en
b
y
:
(
)
(
)
(
)
ij
P
r
|
t
t
j
t
i
p
t
o
t
cc
éù
+
D
=
=
=
D
+
D
êú
ëû
if
j
I
(
2
)
I
n
th
i
s
ca
s
e,
p
01
=
,
p
10
=
ar
e
th
e
tr
an
s
i
tio
n
p
r
o
b
ab
ilit
ies.
B
ased
o
n
Ma
r
k
o
v
p
r
o
p
er
ties
[
2
1
]
,
th
e
tr
an
s
itio
n
p
r
o
b
ab
ilit
ies ar
e
d
ef
in
ed
b
y
(
)
(
)
(
)
(
)
ij
ij
1
i
f
P
r
|
i
f
p
t
o
t
i
j
t
t
j
t
i
p
t
o
t
i
j
cc
ì
ï
+
D
+
D
=
ï
ï
éù
+
D
=
=
=
í
êú
ëû
ï
D
+
D
¹
ï
ï
î
(
3
)
w
it
h
(
)
0
l
i
m
0
t
Ot
D®
D=
,
j
j
i
j
ij
pp
¹
=-
å
f
o
r
all
i
,
j
.
(
4
)
As
m
e
n
tio
n
ed
in
Sectio
n
1
,
t
h
e
OP
F
p
r
o
b
lem
is
co
n
s
id
er
ed
in
th
e
tert
ia
r
y
co
n
tr
o
l
leve
l
,
in
w
h
at
f
o
llo
w
in
g
,
w
e
th
e
n
d
escr
ib
e
th
e
m
o
d
elin
g
o
f
th
e
co
n
s
id
er
ed
m
icr
o
g
r
id
.
First
o
f
all,
t
h
e
to
tal
p
o
w
er
g
en
er
atio
n
o
f
DGs sat
is
f
ies t
h
e
d
e
m
a
n
d
s
u
ch
a
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
p
r
il 2
0
1
9
:
1
0
4
5
-
1057
1048
(
)
(
)
(
)
(
)
D
E
G
W
T
G
P
V
P
t
P
t
P
t
D
t
+
+
=
(
5
)
w
h
er
e
D
(
t
)
r
ef
er
s
to
t
h
e
lo
ad
d
em
a
n
d
at
ti
m
e
t
;
P
WTG
(
t
)
is
th
e
n
o
n
-
d
is
p
atc
h
ab
le
(
W
T
G)
o
u
tp
u
t
p
o
w
er
g
en
er
atio
n
at
ti
m
e
t
;
P
PV
(
t
)
is
t
h
e
P
V
o
u
tp
u
t
p
o
w
er
g
e
n
er
atio
n
at
ti
m
e
t
;
a
n
d
P
DEG
(
t
)
is
th
e
DE
G
o
u
tp
u
t
p
o
w
er
g
en
er
atio
n
at
ti
m
e
t
.
A
s
m
en
tio
n
ed
ab
o
v
e,
t
h
e
p
r
o
d
u
ce
d
p
o
w
er
b
y
P
V
an
d
W
T
G
d
ep
en
d
s
o
n
th
e
en
v
ir
o
n
m
e
n
tal
co
n
d
itio
n
s
,
a
n
d
th
e
d
e
m
an
d
d
ep
en
d
s
o
n
t
h
e
p
o
w
er
co
n
s
u
m
p
tio
n
h
ab
it
s
.
Me
an
w
h
ile,
t
h
e
DE
G
is
s
u
b
j
ec
t
to
r
an
d
o
m
ev
e
n
t
s
s
u
ch
as
f
ail
u
r
e
an
d
r
ep
air
.
A
s
r
esu
lt,
th
e
f
lu
c
tu
atio
n
s
i
n
lo
ad
s
,
P
V,
an
d
W
T
G
o
u
tp
u
t
p
o
w
er
ar
e
ad
j
u
s
ted
b
y
co
n
tr
o
l
in
t
h
e
DE
G
o
u
tp
u
t
p
o
w
er
u
n
d
er
f
ail
u
r
e
an
d
r
ep
air
ev
en
t
s
.
T
h
er
ef
o
r
e,
th
e
ex
p
r
ess
io
n
o
f
t
h
e
p
o
w
er
b
ala
n
ce
in
E
q
u
at
io
n
(
5
)
f
o
r
r
an
d
o
m
n
es
s
in
s
u
ch
DG
as
s
o
ciate
d
with
d
e
m
an
d
ca
n
b
e
d
escr
ib
ed
as f
o
llo
w
s
:
(
)
(
)
(
)
(
)
(
)
D
E
G
W
T
G
P
V
P
t
D
t
P
t
P
t
=
-
+
(
6
)
T
h
e
p
o
w
er
g
e
n
er
atio
n
o
f
DE
G
m
u
s
t
e
v
o
lv
e
th
e
f
l
u
ct
u
atio
n
s
in
lo
ad
s
an
d
DGs
in
E
q
u
a
tio
n
(
6
)
.
I
n
liter
atu
r
e
t
h
e
o
p
ti
m
al
p
o
w
er
f
lo
w
(
OP
F)
p
r
o
b
lem
h
as
b
ee
n
i
n
v
e
s
ti
g
ated
b
ased
o
n
li
n
ea
r
an
d
n
o
n
-
li
n
ea
r
p
r
o
g
r
am
m
i
n
g
s
u
ch
[
1
0
]
,
an
d
[
1
3
]
.
I
n
co
n
tr
ast
,
w
e
w
i
ll
f
o
r
m
u
late
t
h
e
co
n
s
id
er
ed
p
r
o
b
le
m
as
an
o
p
ti
m
al
co
n
tr
o
l o
n
e
in
o
r
d
er
to
f
in
d
o
u
t th
e
o
p
ti
m
al
p
o
licies
f
o
r
o
p
er
atio
n
o
f
m
icr
o
g
r
id
in
t
h
e
n
e
x
t
s
ec
tio
n
.
3.
M
AT
H
E
M
AT
I
CAL M
O
DE
L
I
n
th
is
s
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1
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f
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DE
F
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O
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1
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1
2
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F
(
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s
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f
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in
(
1
3
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T
h
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p
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e
ch
ar
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ter
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t p
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W
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(
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(
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(
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(
.
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(
,
.
)
,
i
n
f
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;
ii
Ut
V
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=
(
1
4
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T
h
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f
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e
u
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th
e
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s
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r
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ased
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m
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t
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f
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(
1
4
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T
heo
re
m
3
.
1
C
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n
tr
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l p
r
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b
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s
a
tis
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f p
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5
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Fig
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r
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2
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.
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4.
NUM
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AP
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I
n
th
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s
s
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t
io
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,
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s
r
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h
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et
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ased
o
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5
]
.
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r
ith
m
i
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[
2
8
]
,
th
e
HJ
B
eq
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at
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(
1
5
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w
h
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ir
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N
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I
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lec
&
C
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p
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
p
r
il 2
0
1
9
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0
4
5
-
1057
1052
p
ar
tial d
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iv
ativ
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r
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T
h
e
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s
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to
(
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i
v
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f
o
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s
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ep
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heo
re
m
4
.
1
Let
(
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,
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t
X
D
d
en
o
te
a
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o
lu
tio
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to
HJB
eq
u
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(
2
2
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.
A
s
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me
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a
t
th
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ro
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.
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h
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th
is
th
e
o
r
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i
s
ad
o
p
ted
f
r
o
m
th
e
o
n
e
in
[
2
9
]
f
o
r
th
e
ca
s
e
o
f
d
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m
i
n
is
tic
co
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