System and method for controlling a hybrid microgrid system
View Patent ↗A system and a method for controlling a hybrid microgrid system (HMS) is disclosed. The HMS includes a WTG, an RSC, a GSC, a DC-link connecting the RSC and the GSC, a PV system that outputs a DC current to the DC-link, a rechargeable battery, a bidirectional BBC connected between the DC-link and the rechargeable battery, and a controller. The method for controlling the HMS includes: preparing a definition set including a characteristic element c i and equations defining desired value c i *, a fractional order sliding mode surface ζ i , and a control law element u i cnt ; monitoring c i (t) and the HMS status; calculating the equations based on monitored information; and controlling the HMS based on the u i cnt (t) calculated and in accordance with a global sliding mode control with fractional order terms. The ζ i comprises a fractional time integral and fractional time derivative of e i (t), where e i (t)=c i (t)−c i *(t). The u i cnt (t) satisfies ζ i ( t ) d ζ i ( t ) dt < 0 , when ζ i (t)≠0.
1. A method for controlling a hybrid microgrid system (HMS), the HMS comprising: a renewable energy source; a grid side converter (GSC) configured to output a power to a point of common coupling (PCC); a DC-link configured to receive a power from the renewable energy source and to supply a power to the GSC; a rechargeable battery configured to exchange a power via the DC-link; a load configured to receive a power via the PCC; a utility grid configured to exchange power via the PCC; and a controller configured to control the HMS by executing a program and in accordance with a global sliding mode control with fractional order terms (GSMCFO) method, wherein the program comprises a definition set customized for the HMS and to be referred in applying the GSMCFO method to the HMS, wherein the method comprises:
preparing the definition set,
wherein the definition set comprises a characteristic element c i to be measured; and equations defining a desired value c i * of the characteristic element c i , a fractional order sliding mode (FOSM) surface ζ i for the characteristic element c i , and a control law element u i cnt of the characteristic element c i ;
monitoring the characteristic element c i (t) and a related status of the HMS;
calculating at least one of the equations based on the characteristic element c i (t) monitored and the related status of the HMS monitored; and
controlling the HMS based on the control law element u i cnt (t) calculated and in accordance with the GSMCFO method,
wherein, the equation defining the FOSM surface ζ i (t) of the characteristic element c i (t) comprises a fractional time integral of a tracking error e i (t) and a fractional time derivative of the tracking error e i (t) defined as,
e i ( t )= c i ( t )− c i *( t ), (1)
and wherein the equation defining the control law element u i cnt (t) is configured to satisfy a condition,
ζ
i
(
t
)
d
ζ
i
(
t
)
d
t
<
0
,
(
2
)
so far as ζ i (t) is not zero.
2. The method of claim 1 , wherein the definition set further comprises a minimum value SOC min and a maximum value SOC max of the state of charge (SOC) of the BESS; and equations defining a power imbalance ΔP and a power balance condition of the HMS, given respectively as
Δ
P
=
P
re
+
P
u
g
-
P
d
e
m
-
P
b
,
(
3
)
Δ
P
=
0
,
(
4
)
wherein P re represents a total power generated by the renewable energy source, P ug , a grid power exchanged between the utility grid and the PCC, P dem , a load demand, P b , a battery power exchanged between the BESS and the DC-link,
and wherein the method further comprises:
monitoring elements required to calculate a power imbalance ΔP and a SOC;
calculating a power imbalance AP with the equation given in the definition set; and
controlling the battery power P b and the grid power P ug to satisfy and maintain the power balance condition, under a restriction that the SOC of the BESS satisfies a condition,
SOC min ≤SOC≤SOC max . (5)
3. The method of claim 2 , wherein the HMS further comprises:
a wind turbine (WT) generator having a wind turbine (WT) and an electric generator having a rotor and a stator;
a rotor side converter (RSC) configured to receive an AC power from the electric generator and output an RSC output DC current to the DC-link;
a solar photovoltaic (PV) system configured to output a PV output DC current to the DC-link; and
a bidirectional buck-boost converter (BBBC) connected between the DC-link and the rechargeable battery and configured to control a power exchanged between the BESS and the DC-link,
the definition set further comprises an equation defining an equivalent control law element u i eqv for the characteristic element c i , wherein the equivalent control law element u i eqv comprises a maximum disturbance term δ i , representing a possible maximum value of a lumped external disturbances and parametric perturbations to the tracking error e i (t), wherein, represents a Riemann-Liouville fractional integration, and δ i , a positive function, and
wherein the equation defining the control law element u i cnt (t) comprises a function SG (ζ i (t)) given by a signum function sgn(ζ i (t)) or one of its smooth approximations including
tanh
(
ζ
i
(
t
)
θ
)
,
wherein, θ(>0).
4. The method of claim 3 , wherein the characteristic elements c i (t)(i=1, 2) are an angular frequency ω r of the WT for i=1, and a d-axis stator current I ds of a stator of the electric generator for i=2, respectively,
wherein the desired value ω r * of the angular frequency ω r of the WT is given as,
c
1
*
(
t
)
=
ω
r
*
=
λ
*
V
w
R
,
(
6
)
wherein, λ* denotes a desired tip speed ratio giving a maximum power coefficient for the WT with a blade radius R and at a wind speed V w ,
the desired value I ds * for the d-axis stator current I ds of the stator is given as,
c 2 *( t )=I ds *=0, (7)
and wherein the equations defining the FOSM surfaces ζ i (t) for the angular frequency ω r (i=1) and the d-axis stator current I ds (i=2) are given as
ζ
1
(
t
)
=
k
1
R
𝒥
t
μ
e
1
(
t
)
+
σ
1
R
𝒟
t
1
-
μ
e
1
(
t
)
+
R
𝒟
t
2
-
μ
e
1
(
t
)
,
(
8
)
ζ
2
(
t
)
=
k
2
R
𝒥
t
μ
e
2
(
t
)
+
R
𝒟
t
1
-
μ
e
2
(
t
)
,
(
9
)
wherein, μ∈(0, 1), k 1 , k 2 , and σ 1 are positive constants, denotes a Riemann-Liouville fractional integration, denote Riemann-Liouville fractional derivations, and
wherein the equations defining the equivalent control law element u 1 eqv (t) and the control law element u 1 cnt (t) of the angular frequency ω 4 of the WT are given by q-axis stator voltages V qs eqv (t) and V qs cnt (t), and defined respectively as,
u
1
e
q
v
(
t
)
=
V
q
s
e
q
v
(
t
)
=
-
1
a
(
k
1
e
1
(
t
)
+
σ
1
e
˙
1
(
t
)
+
R
𝒥
t
1
-
μ
δ
1
-
ω
¨
r
*
+
a
[
L
q
ω
r
I
d
s
+
R
s
I
q
s
+
λ
r
ω
r
]
)
,
(
10
)
u
1
c
n
t
(
t
)
=
V
q
s
c
n
t
(
t
)
=
V
q
s
e
q
v
(
t
)
-
1
a
[
ϱ
1
R
𝒥
t
1
-
μ
SG
(
ζ
1
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
1
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
1
R
𝒥
t
1
-
μ
ζ
1
(
t
)
]
,
(
11
)
wherein the equations defining the equivalent control law element u 2 eqv (t) and the control law element u 2 cnt (t) of the d-axis stator current I ds of the rotor are given by d-axis stator voltages V ds eqv (t) and V ds cnt (t), and defined respectively as,
u
2
e
q
v
(
t
)
=
V
d
s
e
q
v
(
t
)
=
-
L
d
(
k
2
e
2
(
t
)
+
[
L
q
ω
r
I
q
s
-
R
s
I
d
s
]
L
d
+
R
𝒥
t
1
-
μ
δ
2
-
I
.
d
s
*
)
,
(
12
)
u
2
c
n
t
(
t
)
=
V
d
s
c
n
t
(
t
)
=
V
d
s
e
q
v
(
t
)
-
L
d
[
ϱ
2
𝒥
t
1
-
μ
S
G
(
ζ
2
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
2
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
2
R
𝒥
t
1
-
μ
ζ
2
(
t
)
]
,
(
13
)
wherein α is given by,
a
=
3
P
2
Λ
r
2
J
L
q
,
(
14
)
wherein P denotes a number of pole pairs of the rotor, Λ r , a rotor flux, J, an inertia of mechanical shaft of the WT generator, L q , a q-axis self-inductance of the stator, Rs, a stator resistance, I qs , a q-axis stator current, Λ r , a rotor flux, L d , a d-axis self-inductance of the stator, δ 1 and δ 2 represents the maximum disturbance terms, respectively, α∈(0, 1), and γ i (i=1, 2) are positive constants.
5. The method of claim 4 , wherein the DC-link further comprises a DC-bus and a DC-link capacitor, and wherein the characteristic element c 3 is a DC-link voltage V dc ,
the desired value V dc * defined for the DC-link voltage V dc is given as
c
3
*
(
t
)
=
V
d
c
*
=
{
V
p
v
MPPT
;
when
V
dc
min
≤
V
p
v
MPPT
≤
V
dc
max
,
V
dc
n
o
m
;
when
V
p
v
MPPT
<
V
dc
min
,
or
V
p
v
MPPT
>
V
dc
max
,
(
15
)
wherein, V pv MPPT represents an output voltage of the PV system under a maximum power point tracking (MPPT) operation, V dc min , V dc max and V dc nom represent a minimum allowable value, a maximum allowable value, and a nominal value of the DC-link voltages, each predetermined respectively,
the equation defining the FOSM surfaces ζ 3 (t) of the DC-link voltage is given as,
ζ 3 ( t )= k 3 e 3 ( t )+ e 3 ( t ), (16)
wherein, μ∈(0, 1) and k 3 are positive constants, denotes a Riemann-Liouville fractional integration, denotes a Riemann-Liouville fractional derivation, and
wherein the equations defining the equivalent control law element u 3 eqv (t) and the control law element u 3 cnt (t) of the DC-link voltage V dc are given by d-axis AC output currents I d eqv (t) and I d cnt (t) from the GSC, and defined respectively as,
u
3
e
q
v
(
t
)
=
I
d
e
q
v
(
t
)
=
2
3
C
dc
V
dc
V
d
(
k
3
e
3
(
t
)
+
P
w
C
dc
V
dc
+
I
pv
C
dc
+
(
1
-
D
)
C
dc
I
b
+
R
𝒥
t
1
-
μ
δ
3
-
V
˙
dc
*
(
t
)
)
,
(
17
)
u
3
c
n
t
(
t
)
=
I
d
c
n
t
(
t
)
=
I
d
e
q
v
(
t
)
+
2
3
C
d
c
V
d
c
V
d
[
ϱ
3
R
𝒥
t
1
-
μ
S
G
(
ζ
3
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
3
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
3
R
𝒥
t
1
-
μ
ζ
3
(
t
)
]
,
(
18
)
wherein, C dc denotes a capacitance of the DC-link capacitor, V d , a voltage of the AC output from the GSC, P w , an output power from the RSC, I pv , an output current from the PV system, D, a duty cycle ratio of the BBBC, I b , a battery output current, δ 3 is the maximum disturbance term, α∈(0, 1), and γ 3 are positive constants.
6. he method of claim 5 , wherein the GSC is further configured to output an AC output to a point of common coupling (PCC) via a grid side filter, and
wherein the characteristic elements c i (t)(i=4, 5) are a d-axis AC current I d (i=4) of the AC output from the GSC, and a q-axis AC current I q (i=5) of the AC output from the GSC for, respectively,
the desired values defined for the d-axis AC current and the q-axis AC current are given respectively as,
c
4
*
(
t
)
=
I
d
*
=
I
d
=
2
3
P
d
e
m
-
P
u
g
V
d
,
(
19
)
c
5
*
(
t
)
=
I
q
*
=
0
,
(
20
)
wherein P dem and P ug represent a load demand at the load and a power exchanged between the PCC and the utility grid, respectively, wherein P ug >0, when provided from the utility grid to the PCC, P ug <0, when provided from the GSC to the utility grid, V d , a d-axis AC voltage of the AC output from the GSC,
the equations defining the FOSM surfaces ζ i (t) for the d-axis AC current I d (i=4) and the q-axis AC current I q (i=5) are given respectively as,
ζ
4
(
t
)
=
k
4
R
𝒥
t
μ
e
4
(
t
)
+
R
𝒟
t
1
-
μ
e
4
(
t
)
,
(
21
)
ζ
5
(
t
)
=
k
5
R
𝒥
t
μ
e
5
(
t
)
+
R
𝒟
t
1
-
μ
e
5
(
t
)
,
(
22
)
wherein, μ∈(0, 1), k 4 , and k 5 are positive constants, and each denotes a Riemann-Liouville fractional integration and a Riemann-Liouville fractional derivation, respectively,
wherein, the equations defining the equivalent control law element u 4 eqv (t) and the control law element u 4 cnt (t) of the d-axis AC current I d are given by d-axis AC voltages V d eqv (t) and V d cnt (t) of the AC output from the GSC, and defined respectively as,
u
4
e
q
v
(
t
)
=
V
d
e
q
v
(
t
)
=
-
L
f
(
k
4
e
4
(
t
)
-
I
.
d
*
-
1
L
f
[
R
f
I
d
+
U
d
-
L
f
ω
g
I
q
-
R
𝒥
t
1
-
μ
δ
4
]
)
,
(
23
)
u
4
c
n
t
(
t
)
=
V
d
c
n
t
(
t
)
=
V
d
e
q
v
(
t
)
-
L
f
[
ϱ
4
R
𝒥
t
1
-
μ
S
W
(
ζ
4
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
4
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
4
R
𝒥
t
1
-
μ
ζ
4
(
t
)
]
,
(
24
)
wherein the equations defining the equivalent control law element u 5 eqv (t) and the control law element u 5 cnt (t) of the q-axis AC current I q are given by q-axis AC voltages V q eqv (t) and V q cnt (t) of the AC output from the GSC, and defined respectively as,
u
5
e
q
v
(
t
)
=
V
q
e
q
v
(
t
)
=
-
L
f
(
k
5
e
5
(
t
)
-
I
.
q
*
-
1
L
f
[
R
f
I
q
+
U
q
+
L
f
ω
g
I
d
-
R
𝒥
t
1
-
μ
δ
5
]
)
,
(
25
)
u
5
c
n
t
(
t
)
=
V
q
c
n
t
(
t
)
=
V
q
e
q
v
(
t
)
-
L
f
[
ϱ
5
R
𝒥
t
1
-
μ
S
G
(
ζ
5
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
5
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
5
R
𝒥
t
1
-
μ
ζ
5
(
t
)
]
,
(
26
)
wherein L f and R f denote a grid side filter inductance and a grid side filter resistance, respectively, U d and U q , a d-axis and a q-axis voltages at a point of common coupling (PCC), respectively, ω g , an electrical angular frequency of the AC output from the GSC, δ i , (i=4, 5) are the maximum disturbance terms, α∈(0, 1), and γ i (i=4, 5) are positive constants.
7. The method of claim 6 , wherein the BBBC is configured to facilitate charging of the rechargeable battery while operating as a buck converter, and to facilitate discharging to the DC- link while operating as a boost converter,
wherein the characteristic element c i (i=6) is a battery current I b ,
the desired value I b * defined for the battery current I b is given as,
c
6
*
(
t
)
=
I
b
*
=
P
b
V
b
=
P
r
e
+
P
u
g
-
P
d
e
m
V
b
,
(
27
)
wherein P re represents a sum of powers generated by the renewable energy sources, P ug , a power supplied by the utility grid, P dem , a load demand, and V b , a battery voltage,
the equation defining the FOSM surfaces ζ 6 (t) of the battery current is given as,
ζ 6 ( t )= k 6 e 6 + , (28)
wherein, μ∈(0, 1) and k 6 are positive constants, denotes a Riemann-Liouville fractional integration, denotes a Riemann-Liouville fractional derivation, and
wherein the equations defining the equivalent control law element u 6 eqv (t) and the control law element u 6 cnt (t) of the battery current I b are given by duty cycles D eqv (t) and D cnt (t) of the BBBC, and defined respectively as,
u
6
e
q
v
(
t
)
=
D
e
q
v
(
t
)
=
L
b
V
dc
(
k
6
e
6
(
t
)
+
V
b
L
b
-
I
b
R
b
L
b
+
R
𝒥
t
1
-
μ
δ
6
-
I
b
*
)
(
29
)
u
6
c
n
t
(
t
)
=
D
c
n
t
(
t
)
=
D
e
q
v
(
t
)
-
L
b
[
ϱ
6
R
𝒥
t
1
-
μ
S
G
(
ζ
6
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
6
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
6
R
𝒥
t
1
-
μ
ζ
6
(
t
)
]
,
(
30
)
wherein, L b denotes a battery inductance, R b , a battery resistance, V dc , a DC-link voltage, δ 6 represents the maximum disturbance term, δ 6 , α∈(0, 1), and γ 6 are positive constants.
8. A hybrid microgrid system (HMS) comprising:
a renewable energy source;
a grid side converter (GSC) configured to output a power to a point of common coupling (PCC);
a DC-link configured to receive a power from the renewable energy source and to supply a power to the GSC;
a rechargeable battery configured to exchange a power between the DC-link;
a load configured to receive a power via the PCC;
a utility grid configured to exchange power via the PCC; and
a controller comprising: a processor; a memory; a bus-line; and I/O port, wherein, the controller is configured to control the HMS by executing a program installed in the memory and in accordance with a global sliding mode control with fractional order terms (GSMCFO) method, and wherein, the program comprises a definition set customized for the HMS and to be referred in applying the GSMCFO method to the HMS,
wherein the definition set comprises: a characteristic element c i to be measured; and equations defining a desired value c i * of the characteristic element c i , a fractional order sliding mode (FOSM) surface ζ i of the characteristic element c i , and a control law element u i cnt of the characteristic element c i , and
wherein the controller is further configured to
monitor the characteristic element c i (t) and a related status of the HMS,
calculate at least one of the equations defined in the definition set based on the characteristic element c i (t) monitored and the status of the HMS monitored, and
control the HMS based on the control law element u i cnt (t) calculated,
wherein the equation defining the FOSM surface ζ i (t) for the characteristic element c i (t) comprises a fractional time integral of a tracking error e i (t) and a fractional time derivative of the tracking error e i (t), wherein the tracking error e i (t) for the characteristic element c i (t) is defined as,
e i ( t )= c i ( t )− c i *( t ), (31)
wherein the equation defining the control law element u i cnt (t) is configured to satisfy a condition
ζ
i
(
t
)
d
ζ
i
(
t
)
d
t
<
0
,
(
32
)
so far as ζ i (t) is not zero.
9. The hybrid microgrid system of claim 8 , wherein the definition set further comprises: a minimum value SOC min and a maximum value SOC max of the state of charge (SOC) of the rechargeable battery; and equations defining a power imbalance ΔP and a power balance condition of the HMS, given respectively as,
Δ
P
=
P
re
+
P
u
g
-
P
d
e
m
-
P
b
,
(
33
)
Δ
P
=
0
,
(
34
)
wherein P re represents a total power generated by the renewable energy source, P ug , a grid power exchanged between the utility grid and the PCC, P dem , a load demand, P b , a battery power exchanged between the rechargeable battery and the DC-link,
wherein the controller is further configured to
monitor elements required to calculate a power imbalance ΔP and a SOC,
calculate a power imbalance ΔP with the equation given in the definition set, and
control the battery power P b and the grid power P ug to satisfy and maintain the power balance condition, under a restriction that the SOC of the rechargeable battery satisfies a condition,
SOC min ≤SOC≤SOC max . (35)
10. The hybrid microgrid system of claim 9 , wherein the HMS further comprises:
a wind turbine (WT) generator further comprising a wind turbine (WT) and an electric generator further comprising a rotor and a stator;
a rotor side converter (RSC) configured to receive an AC power from the electric generator and output an RSC output DC current to the DC-link;
a solar photovoltaic (PV) system configured to output a PV output DC current to the DC-link; and
a bidirectional buck-boost converter (BBBC) connected between the DC-link and the rechargeable battery and configured to control a power exchanged between the rechargeable battery and the DC-link,
and wherein, the definition set further comprises an equation defining an equivalent control law element u i eqv for the characteristic element c i , wherein the equivalent control law element u i eqv comprises a maximum disturbance term δ i , representing a possible maximum value of a lumped external disturbances and parametric perturbations to the tracking error e i (t), wherein, represents a Riemann-Liouville fractional integration, and δ i , a positive function,
wherein, the equation defining the control law element u i cnt (t) comprises a function SG(ζ i (t)) given by a signum function sgn(ζ i (t)) or one of its smooth approximations including
tanh
(
ζ
i
(
t
)
θ
)
,
wherein, θ(>0).
11. The hybrid microgrid system of claim 10 , wherein the characteristic element c 1 is an angular frequency ω r of the WT, and the characteristic element c 2 is a d-axis stator current I ds , wherein the desired value ω r * for the angular frequency ω r of the WT is defined as,
c
1
*
(
t
)
=
ω
r
*
=
λ
*
V
w
R
,
(
36
)
wherein λ* denotes a desired tip speed ratio, giving a maximum power coefficient for the turbine with a blade radius R, at a wind speed V w ,
wherein the desired value I ds * for the d-axis stator current I ds of the rotor is given as,
c 2 *( t )=I ds *=0 (37)
wherein the equations defining the FOSM surfaces ζ i (t) for the characteristic elements c i (i=1, 2) are given as,
ζ
1
(
t
)
=
k
1
R
𝒥
t
μ
e
1
(
t
)
+
σ
1
R
𝒟
t
1
-
μ
e
1
(
t
)
+
R
𝒟
t
2
-
μ
e
1
(
t
)
,
(
38
)
ζ
2
(
t
)
=
k
2
R
𝒥
t
μ
e
2
(
t
)
+
R
𝒟
t
1
-
μ
e
2
(
t
)
,
(
39
)
wherein, 0<μ<1, k 1 , k 2 , and of are positive constants, denotes a Riemann-Liouville fractional integration, denote Riemann-Liouville fractional derivations,
wherein the equivalent control law element u 1 eqv (t) and the control law element u 1 cnt (t) of the angular frequency ω r are given by q-axis stator voltages V qs eqv (t) and V qs cnt (t), and defined respectively as,
u
1
e
q
v
(
t
)
=
V
q
s
e
q
v
(
t
)
=
-
1
a
(
k
1
e
1
(
t
)
+
σ
1
e
˙
1
(
t
)
+
R
𝒥
t
μ
δ
1
-
ω
¨
r
*
+
a
[
L
q
ω
r
I
d
s
+
R
s
I
q
s
+
λ
r
ω
r
]
)
,
(
40
)
u
1
c
n
t
(
t
)
=
V
q
s
c
n
t
(
t
)
=
V
q
s
e
q
v
(
t
)
-
1
a
[
ϱ
1
R
𝒥
t
1
-
μ
S
G
(
ζ
1
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
1
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
1
R
𝒥
t
1
-
μ
ζ
1
(
t
)
]
,
(
41
)
wherein the equivalent control law element u 2 eqv (t) and the control law element u 2 cnt (t) of the d-axis stator current I ds of the rotor are given by d-axis stator voltages V ds eqv (t) and V ds cnt (t), and defined respectively as
u
2
e
q
v
(
t
)
=
V
d
s
e
q
v
(
t
)
=
-
L
d
(
k
2
e
2
(
t
)
+
[
L
q
ω
r
I
q
s
-
R
s
I
d
s
]
L
d
+
R
𝒥
t
1
-
μ
δ
2
-
I
.
ds
*
)
,
(
42
)
u
2
c
n
t
(
t
)
=
V
ds
c
n
t
(
t
)
=
V
ds
e
q
v
(
t
)
-
L
d
[
ϱ
1
R
𝒥
t
1
-
μ
S
G
(
ζ
2
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
2
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
2
R
𝒥
t
1
-
μ
ζ
2
(
t
)
]
,
(
43
)
wherein α is given by
a
=
3
P
2
Λ
r
2
J
L
q
,
(
44
)
wherein P denotes a number of pole pairs of the rotor, Λ r , a rotor flux, J, an inertia of mechanical shaft of the wind turbine generator, L q , a q-axis self-inductance of the stator, Rs, a stator resistance, I qs , a q-axis stator current, Λ r , a rotor flux, L d , a d-axis self-inductance of the stator, δ 1 and δ 2 represent the maximum disturbance terms, α∈(0, 1), and γ i (i =1, 2) are positive constants.
12. The hybrid microgrid system of claim 10 , wherein the DC-link further comprises a DC-bus and a DC-link capacitor, and wherein the characteristic element c 3 is a DC-link voltage V dc ,
the desired value V dc * defined for the DC-link voltage V dc is given as
c
3
*
(
t
)
=
V
d
c
*
=
{
V
p
v
MPPT
;
when
V
d
c
min
≤
V
p
v
MPPT
≤
V
dc
max
,
V
dc
n
o
m
;
when
V
p
v
MPPT
<
V
d
c
min
,
or
V
p
v
MPPT
>
V
dc
max
,
(
45
)
wherein V pv MPPT represents an output voltage of the PV system under a maximum power point tracking (MPPT) operation, V dc min , V dc max and V dc nom represent a minimum allowable value, a maximum allowable value, and a nominal value of the DC-link voltages, each predetermined respectively,
the equation defining the FOSM surfaces ζ 3 (t) of the DC-link voltage is given as,
ζ 3 ( t )= k 3 e 3 ( t )+ e 3 ( t ), (46)
wherein, μ∈(0, 1) and k 3 are positive constants, denotes a Riemann-Liouville fractional integration, denotes a Riemann-Liouville fractional derivation,
wherein the equations defining the equivalent control law element u 3 eqv (t) and the control law element u 3 cnt (t) of the DC-link voltage V dc are given by d-axis AC output currents I d eqv (t) and I d cnt (t) from the GSC, and defined respectively as,
u
3
e
q
v
(
t
)
=
I
d
e
q
v
(
t
)
=
2
3
C
d
c
V
d
c
V
d
(
k
3
e
3
(
t
)
+
P
w
C
d
c
V
d
c
+
I
pv
C
d
c
+
(
1
-
D
)
C
d
c
I
b
+
R
𝒥
t
1
-
μ
δ
3
-
V
˙
d
c
*
(
t
)
)
,
(
47
)
u
3
c
n
t
(
t
)
=
I
d
c
n
t
(
t
)
=
I
d
e
q
v
(
t
)
+
2
3
C
d
c
V
d
c
V
d
[
ϱ
3
R
𝒥
t
1
-
μ
S
G
(
ζ
3
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
3
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
3
R
𝒥
t
1
-
μ
ζ
3
(
t
)
]
,
(
48
)
wherein C dc denotes a capacitance of the DC-link capacitor, V d , a voltage of the AC output from the GSC, P w , an output power from the RSC, I pv , an output current from the PV system, D, a duty cycle ratio of the BBBC, I b , a battery output current, δ 3 is the maximum disturbance term, α∈(0, 1), and γ 3 are positive constants.
13. The hybrid microgrid system of claim 10 , wherein the GSC is further configured to output an AC output to a point of common coupling (PCC) via a grid side filter,
wherein the characteristic elements c i (t)(i=4, 5) are a d-axis AC current I d =4) of the AC output from the GSC, and a q-axis AC current I q (i=5) of the AC output from the GSC for, respectively,
the desired values defined for the d-axis AC current and the q-axis AC current are given respectively as
c
4
*
(
t
)
=
I
d
*
=
I
d
=
2
3
P
d
e
m
-
P
u
g
V
d
,
(
49
)
c
5
*
(
t
)
=
I
q
*
=
0
,
(
50
)
wherein P dem and P ug represent a power demand at the load and a power exchanged between the PCC and the utility grid, respectively, wherein P ug >0, when provided from the utility grid to the PCC, P ug <0, when provided from the GSC to the utility grid, V d , a d-axis AC voltage of the AC output from the GSC,
the equations defining the FOSM surfaces ζ i (t) for the d-axis AC current I d (i=4) and the q-axis AC current I q (i=5) are given respectively as,
ζ
4
(
t
)
=
k
4
R
𝒥
t
μ
e
4
(
t
)
+
R
𝒟
t
1
-
μ
e
4
(
t
)
,
(
51
)
ζ
5
(
t
)
=
k
5
R
𝒥
t
μ
e
5
(
t
)
+
R
𝒟
t
1
-
μ
e
5
(
t
)
,
(
52
)
wherein μ∈(0, 1), k 4 , and k 5 are positive constants, and each denotes a Riemann- Liouville fractional integration and a Riemann-Liouville fractional derivation, respectively,
wherein the equations defining the equivalent control law element u 4 eqv (t) and the control law element u 4 cnt (t) of the d-axis AC current I d are given by d-axis AC voltages V d eqv (t) and V d cnt (t) of the AC output from the GSC, and defined respectively as,
u
4
e
q
v
(
t
)
=
V
d
e
q
v
(
t
)
=
-
L
f
(
k
4
e
4
(
t
)
-
I
.
d
*
-
R
f
I
d
+
U
d
-
L
f
ω
g
I
q
R
𝒥
t
1
-
μ
δ
4
]
)
,
(
53
)
u
4
c
n
t
(
t
)
=
V
d
c
n
t
(
t
)
=
V
d
e
q
v
(
t
)
-
L
f
[
ϱ
4
R
𝒥
t
1
-
μ
SW
(
ζ
4
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
4
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
4
R
𝒥
t
1
-
μ
ζ
4
(
t
)
]
,
(
54
)
wherein the equations defining the equivalent control law element u 5 eqv (t) and the control law element u 5 cnt (t) of the q-axis AC current I q are given by q-axis AC voltages V q eqv (t) and V q cnt (t) of the AC output from the GSC, and defined respectively as,
u
5
e
q
v
(
t
)
=
V
q
e
q
v
(
t
)
=
-
L
f
(
k
5
e
5
(
t
)
-
I
.
q
*
-
1
L
f
[
R
f
I
q
+
U
q
+
L
f
ω
g
I
d
-
R
𝒥
t
1
-
μ
δ
5
]
)
,
(
55
)
u
5
c
n
t
(
t
)
=
V
q
c
n
t
(
t
)
=
V
q
e
q
v
(
t
)
-
L
f
[
ϱ
5
R
𝒥
t
1
-
μ
S
G
(
ζ
5
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
5
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
5
R
𝒥
t
1
-
μ
ζ
5
(
t
)
]
,
(
56
)
wherein L f and R f denote a grid side filter inductance and a grid side filter resistance, respectively, U d and U q , a d-axis and a q-axis voltages at a point of common coupling (PCC), respectively, ω g , an electrical angular frequency of the AC output from the GSC, δ i , (i=4, 5) are the maximum disturbance terms, α∈(0, 1), and γ i (i=4, 5) are positive constants.
14. The HMS of claim 10 , wherein the BBBC is configured to facilitate charging of the rechargeable battery while operating as a buck converter, and to facilitate discharging to the DC- link while operating as a boost converter,
and wherein the characteristic element c i (i=6) is a battery current I b ,
the desired value I b * defined for the battery current I b is given as,
c
6
*
(
t
)
=
I
b
*
=
P
b
V
b
=
P
r
e
+
P
u
g
-
P
dem
V
b
,
(
57
)
wherein P re represents a sum of powers generated by the renewable energy sources, P ug , a power supplied by the utility grid, P dem , a load demand, and V b , a battery voltage,
the equation defining the FOSM surfaces ζ 6 (t) of the battery current is given as,
ζ 6 ( t )= k 6 e 6 + e 6 , (58)
wherein, μ∈(0, 1) and k 6 are positive constants, denotes a Riemann-Liouville fractional integration, denotes a Riemann-Liouville fractional derivation,
wherein the equations defining the equivalent control law element u 6 eqv (t) and the control law element u 6 cnt (t) of the battery current I b are given by duty cycles D eqv (t) and D cnt (t) of the BBBC, and defined respectively as,
u
6
e
q
v
(
t
)
=
D
e
q
v
(
t
)
=
L
b
V
d
c
(
k
6
e
6
(
t
)
+
V
b
L
b
-
I
b
R
b
L
b
+
R
𝒥
t
1
-
μ
δ
6
-
I
b
*
)
,
(
59
)
u
6
c
n
t
(
t
)
=
D
c
n
t
(
t
)
=
D
e
q
v
(
t
)
-
L
b
[
ϱ
6
R
𝒥
t
1
-
μ
SG
(
ζ
6
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
6
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
6
R
𝒥
t
1
-
μ
ζ
6
(
t
)
]
,
(
60
)
wherein L b denotes a battery inductance, V b , a battery voltage, R b , a battery resistance, V dc , a DC-link voltage, δ 6 represents the maximum disturbance term, α∈(0, 1), and γ 6 are positive constants.
15. A controller of a hybrid microgrid system (HMS), the HMS comprising: a renewable energy source; a grid side converter (GSC) configured to output a power to a point of common coupling (PCC); a DC-link configured to receive a power from the renewable energy source and to supply a power to the GSC; a rechargeable battery configured to exchange a power via the DC-link; a load configured to receive a power via the PCC; a utility grid configured to exchange power via the PCC; and a controller comprising: a processor; a memory; a bus-line; and I/O port,
wherein the controller is configured to control the HMS by executing a program installed in the memory and in accordance with a global sliding mode control with fractional order terms (GSMCFO) method,
wherein the program comprises a definition set customized for the HMS and to be referred in applying the GSMCFO method to the HMS,
wherein the definition set comprises: a characteristic element c i to be measured; and equations defining a desired value c i * of the characteristic element c i , a fractional order sliding mode (FOSM) surface ζ i of the characteristic element c i , and a control law element u i cnt of the characteristic element c i ,
wherein the controller is further configured to
monitor the characteristic element c i (t) and a related status of the HMS,
calculate at least one of the equations defined in the definition set based on the characteristic element c i (t) monitored and the related status of the HMS monitored, and
control the HMS based on the control law element u i cnt (t) calculated, and in accordance with the GSMCFO,
wherein the equation defining the FOSM surface ζ i (t) for the characteristic element c i (t) comprises a fractional time integral of a tracking error e i (t) and a fractional time derivative of the tracking error e i (t), wherein the tracking error e i (t) for the characteristic element c i (t) is defined as,
e i ( t )= c i ( t )− c i *( t ), (61)
wherein the equation defining the control law element u i cnt (t) is configured to satisfy a condition
ζ
i
(
t
)
d
ζ
i
(
t
)
d
t
<
0
,
(
62
)
so far as ζ i (t) is not zero.
16. The controller of claim 15 , wherein the definition set further comprises: a minimum value SOC min and a maximum value SOC max of the state of charge (SOC) of the rechargeable battery; and equations defining a power imbalance ΔP and a power balance condition of the HMS, given respectively as
Δ
P
=
P
re
+
P
u
g
-
P
d
e
m
-
P
b
,
(
63
)
Δ
P
=
0
,
(
64
)
wherein P re represents a total power generated by the renewable energy source, P ug , a grid power exchanged between the utility grid and the PCC, P dem , a load demand, P b , a battery power exchanged between the rechargeable battery and the DC-link,
and wherein the controller is further configured to
monitor elements required to calculate a power imbalance AP and a SOC,
calculate the power imbalance AP with the equation given in the definition set, and
controlling the battery power P b and the grid power P ug to satisfy and maintain the power balance condition, under a restriction that the SOC of the rechargeable battery satisfies a condition,
SOC min ≤SOC≤SOC max . (65)
17. The controller of claim 16 , wherein the HMS further comprises:
a wind turbine (WT) generator further comprises a wind turbine (WT) and an electric generator further, wherein the electric generator further comprises a rotor and a stator;
a rotor side converter (RSC) configured to receive an AC power from the electric generator and output an RSC output DC current to the DC-link;
a solar photovoltaic (PV) system configured to output a PV output DC current to the DC-link; and
a bidirectional buck-boost converter (BBBC) connected between the DC-link and the rechargeable battery and configured to control a power exchanged between the rechargeable battery and the DC-link, and
wherein the definition set further comprises an equation defining an equivalent control law element u i eqv for the characteristic element c i , wherein the equivalent control law element u i eqv comprises a maximum disturbance term δ i , representing a possible maximum value of a lumped external disturbances and parametric perturbations to the tracking error e i (t) , wherein, represents a Riemann-Liouville fractional integration, and δ i , a positive function,
and wherein the equation defining the control law element u i cnt (t) comprises a function SG(ζ i (t)) given by a signum function sgn(ζ i (t)) or one of its smooth approximations including
tanh
(
ζ
i
(
t
)
θ
)
,
wherein, θ(>0),
and wherein the characteristic element c 1 is an angular frequency ω r of the WT, and the characteristic element c 2 is a d-axis stator current I ds ,
wherein the desired value ω r * for the angular frequency ω r of the WT is defined as,
c
1
*
(
t
)
=
ω
r
*
=
λ
*
V
w
R
,
(
66
)
wherein λ* denotes a desired tip speed ratio, giving a maximum power coefficient for the turbine with a blade radius R, at a wind speed V w , and wherein,
the desired value I ds * for the d-axis stator current I ds of the rotor is given as,
c 2 *( t )= I ds *=0 (67)
and wherein the equations defining the FOSM surfaces ζ i (t) for the characteristic elements c i (i=1, 2) are given as,
ζ
1
(
t
)
=
k
1
R
𝒥
t
μ
e
1
(
t
)
+
σ
1
R
𝒟
r
1
-
μ
e
1
(
t
)
+
R
𝒟
t
2
-
μ
e
1
(
t
)
,
(
68
)
ζ
2
(
t
)
=
k
2
R
𝒥
t
μ
e
2
(
t
)
+
R
𝒟
r
1
-
μ
e
2
(
t
)
,
(
69
)
wherein, 0<μ<1, k 1 , k 2 , and σ i are positive constants, denotes a Riemann-Liouville fractional integration, denote Riemann-Liouville fractional derivations,
and wherein the equivalent control law element u 1 eqv (t) and the control law element u 1 cnt (t) of the angular frequency ω r are given by q-axis stator voltages V qs eqv (t) and V qs cnt (t), and defined respectively as,
u
1
e
q
v
(
t
)
=
V
q
s
e
q
v
(
t
)
=
-
1
a
(
k
1
e
1
(
t
)
σ
1
e
.
1
(
t
)
+
R
𝒥
t
1
-
μ
δ
1
-
ω
¨
r
*
+
a
[
L
q
ω
r
I
d
s
+
R
s
I
q
s
+
λ
r
ω
r
]
)
,
(
70
)
u
1
c
n
t
(
t
)
=
V
q
s
c
n
t
(
t
)
=
V
q
s
e
q
v
(
t
)
-
1
a
[
ϱ
1
R
𝒥
t
1
-
μ
S
G
(
ζ
1
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
1
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
1
R
𝒥
t
1
-
μ
ζ
1
(
t
)
]
,
(
71
)
and wherein the equivalent control law element u 2 eqv (t) and the control law element u 2 cnt (t) of the d-axis stator current I ds of the rotor are given by d-axis stator voltages V ds eqv (t) and V ds cnt (t), and defined respectively as,
u
2
e
q
v
(
t
)
=
V
d
s
e
q
v
(
t
)
=
-
L
d
(
k
2
e
2
(
t
)
+
[
L
q
ω
r
I
q
s
-
R
s
I
d
s
]
L
d
+
R
𝒥
t
1
-
μ
δ
2
-
I
.
d
s
*
)
,
(
72
)
u
2
c
n
t
(
t
)
=
V
d
s
c
n
t
(
t
)
=
V
d
s
e
q
v
(
t
)
-
L
d
[
ϱ
2
R
𝒥
t
1
-
μ
S
G
(
ζ
2
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
2
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
2
R
𝒥
t
1
-
μ
ζ
2
(
t
)
]
,
(
73
)
wherein, α is given by,
a
=
3
P
2
Λ
r
2
J
L
q
,
(
74
)
wherein P denotes a number of pole pairs of the rotor, Λ r , a rotor flux, J, an inertia of mechanical shaft of the wind turbine generator, L q , a q-axis self-inductance of the stator, Rs, a stator resistance, I qs , a q-axis stator current, Λ r , a rotor flux, L d , a d-axis self-inductance of the stator, δ 1 and δ 2 represent the maximum disturbance terms, α∈(0, 1), and γ i (i =1, 2) are positive constants.
18. The controller of claim 17 , wherein the DC-link further comprises a DC-bus and a DC-link capacitor, and wherein the characteristic element c 3 is a DC-link voltage V dc ,
the desired value V dc * defined for the DC-link voltage V dc is given as
c
3
*
(
t
)
=
V
d
c
*
=
{
V
p
v
M
P
P
T
;
when
V
dc
min
≤
V
p
v
MPPT
≤
V
dc
max
,
V
d
c
n
o
m
;
when
V
p
v
MPPT
<
V
dc
min
,
or
V
p
v
MPPT
>
V
dc
max
,
(
75
)
wherein, V pv MPPT represents an output voltage of the PV system under a maximum power point tracking (MPPT) operation, V dc min , V dc max and V dc nom represent a minimum allowable value, a 10 maximum allowable value, and a nominal value of the DC-link voltages, each predetermined respectively,
the equation defining the FOSM surfaces ζ 3 (t) of the DC-link voltage is given as,
ζ 3 ( t )= k 3 e 3 ( t )+ e 3 , (76)
wherein, μ∈(0, 1) and k 3 are positive constants, denotes a Riemann-Liouville fractional integration, denotes a Riemann-Liouville fractional derivation,
and wherein the equations defining the equivalent control law element u 3 eqv (t) and the control law element u 3 cnt (t) of the DC-link voltage V dc are given by d-axis AC output currents I d eqv (t) and I d cnt (t) from the GSC, and defined respectively as,
u
3
e
q
v
(
t
)
=
I
d
e
q
v
(
t
)
=
2
3
C
dc
V
dc
V
d
(
k
3
e
3
(
t
)
+
P
w
C
dc
V
dc
+
I
p
v
C
dc
+
(
1
-
D
)
c
dc
I
b
+
R
𝒥
t
1
-
μ
δ
3
-
V
˙
dc
*
(
t
)
)
,
(
77
)
u
3
c
n
t
(
t
)
=
I
d
c
n
t
(
t
)
=
I
d
e
q
v
(
t
)
+
2
3
C
dc
V
dc
V
d
[
ϱ
3
R
𝒥
t
1
-
μ
S
G
(
ζ
3
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
3
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
3
R
𝒥
t
1
-
μ
ζ
3
(
t
)
]
,
(
78
)
wherein, C dc denotes a capacitance of the DC-link capacitor, V d , a voltage of the AC output from the GSC, P w , an output power from the RSC, I pv , an output current from the PV system, D, a duty cycle ratio of the BBBC, I b , a battery output current, δ 3 is the maximum disturbance term, α∈(0, 1), and γ 3 are positive constants.
19. The controller of claim 17 , wherein the GSC is further configured to output an AC output to a point of common coupling (PCC) via a grid side filter,
and wherein the characteristic elements c i (t)(i=4, 5) are a d-axis AC current I d (i=4) of the AC output from the GSC, and a q-axis AC current I q (I=5) of the AC output from the GSC for, respectively,
the desired values defined for the d-axis AC current and the q-axis AC current are given respectively as,
c
4
*
(
t
)
=
I
d
*
=
I
d
=
2
3
P
d
e
m
-
P
u
g
V
d
,
(
79
)
c
5
*
(
t
)
=
I
q
*
=
0
,
(
80
)
wherein P dem and P ug represent a power demand at the load and a power exchanged between the PCC and the utility grid, respectively, wherein P ug >0, when provided from the utility grid to the PCC, P ug <0, when provided from the GSC to the utility grid, V d , a d-axis AC voltage of the AC output from the GSC,
the equations defining the FOSM surfaces ζ i (t) for the d-axis AC current I d (i=4) and the q-axis AC current I q (i=5) are given respectively as,
ζ
4
(
t
)
=
k
4
R
𝒥
t
μ
e
4
(
t
)
+
R
𝒟
r
1
-
μ
e
4
(
t
)
,
(
81
)
ζ
5
(
t
)
=
k
5
R
𝒥
t
μ
e
5
(
t
)
+
R
𝒟
r
1
-
μ
e
5
(
t
)
,
(
82
)
wherein, μ∈(0, 1), k 4 , and k 5 are positive constants, and each denotes a Riemann- Liouville fractional integration and a Riemann-Liouville fractional derivation, respectively,
and wherein the equations defining the equivalent control law element u 4 eqv (t) and the control law element u 4 cnt (t) of the d-axis AC current I d are given by d-axis AC voltages V d eqv (t) and V d cnt (t) of the AC output from the GSC, and defined respectively as,
u
4
e
q
v
(
t
)
=
V
d
e
q
v
(
t
)
=
-
L
f
(
k
4
e
4
(
t
)
-
I
.
d
*
-
1
L
f
[
R
f
I
d
+
U
d
-
L
f
ω
g
I
q
-
R
𝒥
t
1
-
μ
δ
4
]
)
,
(
83
)
u
4
c
n
t
(
t
)
=
V
d
c
n
t
(
t
)
=
V
d
e
q
v
(
t
)
-
L
f
[
ϱ
4
R
𝒥
t
1
-
μ
S
W
(
ζ
4
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
4
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
4
R
𝒥
t
1
-
μ
ζ
4
(
t
)
]
,
(
84
)
and wherein the equations defining the equivalent control law element u 5 eqv (t) and the control law element u 5 cnt (t) of the q-axis AC current I q are given by q-axis AC voltages V q eqv (t) and V q cnt (t) of the AC output from the GSC, and defined respectively as,
u
5
e
q
v
(
t
)
=
V
d
e
q
v
(
t
)
=
-
L
f
(
k
5
e
5
(
t
)
-
I
.
q
*
-
1
L
f
[
R
f
I
q
+
U
q
+
L
f
ω
g
I
d
-
R
𝒥
t
1
-
μ
δ
5
]
)
,
(
85
)
u
5
c
n
t
(
t
)
=
V
q
c
n
t
(
t
)
=
V
q
e
q
v
(
t
)
-
L
f
[
ϱ
5
R
𝒥
t
1
-
μ
SG
(
ζ
5
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
5
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
5
R
𝒥
t
1
-
μ
ζ
5
(
t
)
]
,
(
86
)
wherein L f and R f denote a grid side filter inductance and a grid side filter resistance, respectively, U d and U q , a d-axis and a q-axis voltages at a point of common coupling (PCC), respectively, ω g , an electrical angular frequency of the AC output from the GSC, δ i , (i=4, 5) are the maximum disturbance terms, α∈(0, 1), and γ i (i=4, 5) are positive constants.
20. The controller of claim 17 , wherein the BBBC is configured to facilitate charging of the rechargeable battery while operating as a buck converter, and to facilitate discharging to the DC-link while operating as a boost converter, and
wherein:
the characteristic element c i (i=6) is a battery current I b ,
the desired value I b * defined for the battery current I b is given as,
c
6
*
(
t
)
=
I
b
*
=
P
b
V
b
=
P
r
e
+
P
u
g
-
P
d
e
m
V
b
,
(
87
)
wherein P re represents a sum of powers generated by the renewable energy sources, P ug , a power supplied by the utility grid, P dem , a load demand, and V b , a battery voltage,
the equation defining the FOSM surfaces ζ 6 (t) of the battery current is given as,
ζ 6 ( t )= k 6 e 6 + e 6 , (88)
wherein, μ∈(0, 1) and k 6 are positive constants, denotes a Riemann-Liouville fractional integration, denotes a Riemann-Liouville fractional derivation,
and wherein the equations defining the equivalent control law element u 6 eqv (t) and the control law element u 6 cnt (t) of the battery current I b are given by duty cycles D eqv (t) and D cnt (t) of the BBBC, and defined respectively as,
u
6
e
q
v
(
t
)
=
D
e
q
v
(
t
)
=
L
b
V
d
c
(
k
6
e
6
(
t
)
+
V
b
L
b
-
I
b
R
b
L
b
+
R
𝒥
t
1
-
μ
δ
6
-
I
b
*
)
,
(
89
)
u
6
c
n
t
(
t
)
=
D
c
n
t
(
t
)
=
D
e
q
v
(
t
)
-
L
b
[
ϱ
6
R
𝒥
t
1
-
μ
S
G
(
ζ
6
(
t
)
)
❘
"\[LeftBracketingBar]"
ζ
6
(
t
)
❘
"\[RightBracketingBar]"
α
+
γ
6
R
𝒥
t
1
-
μ
ζ
6
(
t
)
]
,
(
90
)
wherein, L b denotes a battery inductance, R b , a battery resistance, V dc , a DC-link voltage, δ 6 represents the maximum disturbance term, δ 6 , α∈(0, 1), and γ 6 are positive constants.