International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 01, January 2019, pp. 1052-1069, Article ID: IJMET_10_01_109
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=1
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
DOUBLE INTEGRAL SLIDING MODE
CONTROL APPROACH FOR A THREE-PHASE
GRID -TIED PHOTOVOLTAIC SYSTEMS
M.Vinay Kumar
Department of Electrical and Electronics Engineering, GMR Institute of Technology,
RAJAM
U.Salma
Department of Electrical and Electronics Engineering, GIT,
GITAM University, VISAKAPATNAM
ABSTRACT
This paper presents modelling and control of maximum power point tracking
(MPPT) for a three phase grid–tied photovoltaic (PV) system by using a non-linear
controller namely double integrated sliding mode controller (DISMC) to enhance
MPPT and to stabilize the output power of PV system. The non-linear I-V, P-V
characteristics of PV systems depends upon irradiation and temperature; causes
difficulty in tracking maximum power. The PV system consists of a PV panel, DC/DC
boost converter and a MPPT controller to generate pulses which are fed to converter
for tracking maximum power. In this paper, the performance of DISMC-MPPT shown
to be effective when compared to other controllers like perturb & observe (P&O)MPPT, adaptive P&O-MPPT, sliding mode controller (SMC)-MPPT and integral
SMC (ISMC)-MPPT. The presented DISMC-MPPT method is robust, provides quicker
and steady tracking maximum power with respect to the other discussed methods and
also performs well during any change in weather conditions. To validate the
effectiveness, the mathematical modelling of all the above mentioned non-linear
controller MPPT methods and their simulations are carried on Matlab/SIMULINK
Keywords: Photovoltaic (PV) System, Maximum Power Point Tracking (MPPT),
DC/DC Boost Converter, Inverter, Grid-Tied PV Systems, Matlab, SIMULINK.
Cite this Article: M.Vinay Kumar and U.Salma. Double Integral Sliding Mode
Control Approach for a Three-Phase Grid -Tied Photovoltaic Systems International
Journal of Mechanical Engineering and Technology, 10(01), 2019, pp.1052-1069.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=1
1. INTRODUCTION
The rapid decay in fossil fuels due to escalation in demand of electrical energy as a result of
vast growth in population and due to industrialization, led the researchers to search for an
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M.Vinay Kumar and U.Salma
alternative energy sources. As a result renewable energy sources (RES) came into usage of
power generation. Amongst the available RES, solar energy is gaining much attention as it is
abundant in nature, available throughout the globe and throughout the year, its free from
pollution, requires little maintenance and is noiseless. Photovoltaic (PV) system converts the
solar energy i.e, irradiation and temperature of the sun, directly into electrical energy [1]. The
I-V & P-V characteristics of a PV cell are non-linear in nature and time varying. To draw
maximum output power from PV system, tracking (MPPT) is used; generates duty cycle and
injects as switching pulses to the DC/DC boost converter. There are various controllers for
tracking maximum power point in the literature. PV system can be made to operate in standalone mode or grid-connected mode. In standalone mode, PV system is used as distributed
generator (DG); supplies the local load, or charges the battery energy storage system which
can later be used as a source. In grid-connected mode, PV systems are connected to the grid
through the inverters. In this mode power transfer can be bidirectional, depending upon the
power generated by the PV system and demand of the local load which is connected to the PV
system. The block diagram of grid-tied PV system is shown in Fig.1 below.
PCC
PV
PANELS
DC-DC
CONVERTER
GRID
INVERTER
TRANSFORMER
LCL-FILTER
SWITCH
IG
IPV
VPV
IL
GATE
PULSES
IPV
VPV
MPPT +
CONTROLLER
VSC
CONTROLLER
RLC
PARALLEL
LOAD
Figure 1 Block diagram of grid tied PV system
The rest of the paper is arranged as follows, Section II presents description and modelling
of whole system, Section III explains the MPPT controllers, Section IV discusses the
simulation results and Section V concludes the paper.
2. GRID-TIED PV SYSTEM DESCRIPTION & MODELLING
A grid-tied PV system consists of PV panel, boost converter, MPPT, inverter and a grid
2.1. PV Cell modelling
An ideal PV cell consists of a current source with a diode connected in anti-parallel to it, a
practically it consists of a series and a parallel resistance, the equivalent circuit of a solar cell
and solar array are shown in Fig.2 below.
Ideal Cell
IPh
D
Practical Cell
RSe IPV
(NSe/NSh )RSe
+
RSh
IPh
VPV
D
-
IPV
+
VPV
(NSe/NSh )RSh
-
Figure 2 Equivalent circuit a Solar cell, Solar array
The load current IPV of a PV cell IPV is given as
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Double Integral Sliding Mode Control Approach for a Three-Phase Grid -Tied Photovoltaic
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𝑉𝑃𝑉 +𝐼𝑃𝑉 𝑅𝑆𝐸
)−
𝑎𝑉𝑇
𝐼𝑃𝑉 = 𝐼𝑃𝐻 − 𝐼𝑂 [𝑒𝑥𝑝 (
𝑉𝑃𝑉 +𝐼𝑃𝑉 𝑅𝑆𝐸
)
𝑎𝑉𝑇
1] − (
(1)
𝐺
𝐼𝑃𝐻 = {𝐼𝑆𝐶𝑅 + 𝐾𝑖 (𝑇 − 𝑇𝑅 )} 𝐺
(2)
𝑉𝑃𝑉 +𝐼𝑃𝑉 𝑅𝑆𝐸
)}
𝑎𝑉𝑇
(3)
𝑅
𝐼𝐷 = 𝐼𝑂 {𝑒𝑥𝑝 (
𝐼𝐷 = 𝐼𝑅𝑆 (
𝑇
𝑇𝑅𝐸𝐹
𝐼𝑅𝑆 = (
𝑉𝑇 =
3
) (
𝐼𝑆𝐶𝑅
𝑒𝑥𝑝(
𝑞𝐸𝑔
1
1
−
𝑇𝑅𝐸𝐹 𝑇
)
(4)
)
(5)
𝑞𝑉𝑂𝐶
)−1
𝐴𝐾𝑇
𝐾𝑇
𝑞
(6)
The output voltage of each solar cell is around 0.5V, for getting higher voltage and
currents the cells are connected in series and parallel respectively. The load current IPV for a
PV array consisting of number of solar cells is given as
𝑁
𝑉𝑃𝑉 + 𝑆 𝐼𝑃𝑉 𝑅𝑆𝐸
𝐼𝑃𝑉 = 𝑁𝑃 𝐼𝑃𝐻 − 𝑁𝑃 𝐼𝑂 [𝑒𝑥𝑝 (
𝑁𝑃
𝑁𝑆 𝑎𝑉𝑇
𝑁𝑃
𝑉 +𝐼 𝑅
𝑁𝑆 𝑃𝑉 𝑃𝑉 𝑆𝐸
) − 1] − (
𝑎𝑉𝑇
)
(7)
IPH is the light generated by the current source, NS, NP are number of cells in series and
parallel respectively, IPH is the saturation current, VPV is the output load voltage, VT is the
thermal voltage, RSE,RSH are the series and the shunt resistance respectively [2-6].
2.2. Boost converter
A boost converter steps up the input DC voltage, it consists of a power switch Q (IGBT),
inductor L, diode D and a capacitor C. During continuous conduction mode, switch Q is in
ON state, the inductor current IL(t) linearly increases, the current flows in two loops. When
switch Q is in OFF state, the inductor stored energy gets discharged through the diode D
[7,8]. Filtering operation is done by the capacitor C, which smoothens the pulsating current
and is then fed to the inverter. The equivalent circuit of boost converter and its two modes of
operation are shown in Fig.3
Iint L
D
IL(t)
IC(t)
Vint
DC
PWM
Q1
VC(t)
RL
Vout(t)
(a)
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Iint L IL(t)
D
Iint L IL(t)
IC(t)
Vint
IC(t)
VC(t) RL Vout(t) Vint
DC
VC(t) RL Vout(t)
DC
I2
I1
I2
I1
(b)
(c)
Figure. 3. (a) Equivalent circuit of a Boost circuit, (b) Mode I - Switch is OFF, (c) Mode II - Switch is
ON
The duty ratio is ‘δ' and the control signal for the boost converter is ‘u’ , it is a series of
pulses.
When switch is in ‘OFF’ position, boost converter is modelled as
1
V̇int = C IL − C
1
Vint
1 rPV
1
1
L
(8)
1
L
iL̇ = Vint − Vout
(9)
The load voltage is Vout, dynamic resistance of PV panel is rpv and is given as
∂V
rPV = − ( ∂I int )
(10)
int
When switch is in ‘ON’ position, boost converter is modelled as
1
V̇int = C IL − C
1
Vint
1 rPV
1
(11)
1
iL̇ = L Vint
(12)
The control signal ‘u’ controls the switch by a duty ratio ‘δ’
1
V̇int = C IL − C
1
1
1 rPV
Vint
(13)
1
1
iL̇ = L Vint − δ̅ L Vout
(14)
Where δ̅ = 1 − δ,
Dynamics of boost converter in state space form considering Vint and IL as state variables
is given as
Ẋ = f(X, t) + g(X, t)u
Where X = [IL Vint
f(x) =
Vint
L
[ IL
C1
g(x) = [
(15)
]T
Vout
L
Vout
L
]
1
− C r Vout
1 PV
−
0]
(16)
T
(17)
𝑢=𝛿
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Systems
A MPPT algorithm should generate reference voltage (Vref) for tracking maximum power.
A switching control signal produced by a control circuit, forces the PV system to operate very
near to Vref. For a PV system, in its single input single output state space model, f is a non
linear uncertain function and its value is expected as f̂. The estimation error is given as F(x ẋ )
volts/sec
|f̂ − f| ≤ F
Control gain value is given as
0 < ĝ1,min < ĝ1 < ĝ1,max
(18)
Hence, a robust controller should be designed so as to generate a control signal ‘u’, which
leads to efficient MPPT operation in spite of variations in PV system parameter, converter and
load parameters.
2.3. Maximum Power Point Tracking
The characteristics I-V, P-V of a solar cell is non-linear; there exists a unique maximum
power point and to track it the maximum power point trackers (MPPT) are used [9-12], which
improves the efficiency of the PV system. The Fig.4 below shows the unique MPP.
Figure 4 Unique MPP on a P-V curve
A number of Maximum Power Point Tracking (MPPT) techniques have been presented in
the literature [1-2], a few of them are Perturb and observe (P & O) algorithm, Incremental
conductance (I.C) algorithm, Open Circuit Voltage (OCV) method, Short circuit current
(SCC) method, Ripple correlation control (RCC) technique, Current sweep technique, etc.,
2.4. Three Phase Inverter
The equivalent diagram of a three-phase two-level voltage source inverter (VSI) is depicted
below in Fig. 5.
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IdC
+
Iinv
IC
S1
S3
S5
S1
VdC
S3
S5
ia
ib
ic
+
C
S2
S4
Lf
Rf
Lf
Rf
Lf
Rf
ea
eb
ec
S6
-
Figure 5 Three VSI
In a VSI, each arm has two switches and requires a control signal for its operation; no two
switches in a single arm (S1 and S2) should be switched ON simultaneously, as complete
system will be short circuited in that case. The switching pulses are given by controllers. The
grid side filter is represented by Rf and Lf.
The state-space model of VSI in the abc frame as follows
𝑅
𝐿
1
𝐿
𝑖𝑎 = − 𝑖𝑎 − 𝑒𝑎 +
𝑉𝑑𝑐
(2𝑆1
3𝐿
− 𝑆2 −𝑆3 ) + ∆𝑓1
(19)
𝑅
1
𝑉𝑑𝑐
(−𝑆1
3𝐿
+ 2𝑆2 −𝑆3 ) + ∆𝑓2
(20)
𝑅
1
𝑉𝑑𝑐
(−𝑆1
3𝐿
− 𝑆2 +2𝑆3 ) + ∆𝑓3
(21)
𝑖𝑏 = − 𝐿 𝑖𝑏 − 𝐿 𝑒𝑏 +
𝑖𝑐 = − 𝐿 𝑖𝑐 − 𝐿 𝑒𝑐 +
1
𝐶
1
𝐶
𝑉𝑑𝑐 = 𝐼𝑑𝑐 − (𝑖𝑎 𝑆1 − 𝑖𝑏 𝑆2 −𝑖𝑐 𝑆3 ) + ∆𝑓4
(22)
Where
𝑆𝑖 = {
1 → 𝑆𝑖 𝐻: 𝑂𝑁, 𝑆𝑖 𝐿: 𝑂𝐹𝐹
0 → 𝑆𝑖 𝐻: 𝑂𝐹𝐹, 𝑆𝑖 𝐿: 𝑂𝑁
(23)
The transformation matrix is given as
𝑐𝑜𝑠(𝜃) 𝑐𝑜𝑠(𝜃 − 120) 𝑐𝑜𝑠(𝜃 + 120)
2
𝑎𝑏𝑐
𝑇𝑑𝑞0
= 3 [ 𝑠𝑖𝑛(𝜃) 𝑠𝑖𝑛(𝜃 − 120) 𝑠𝑖𝑛(𝜃 + 120) ]
0.5
0.5
0.5
(24)
The dynamic model (25) in dq model is obtained from [7-10] and is
𝑅
𝑖𝑑
[ 𝑖𝑞 ] =
𝑉𝑑𝑐
−𝐿
𝜔
[−
𝑆𝑑
𝐶
𝜔
−
−
𝑅
𝐿
𝑆𝑞
𝐶
𝑆𝑑
𝐿
𝑆𝑞
1
−
𝐿
𝑖𝑑
𝑖𝑞 ] + 0
[
𝐿
𝑉𝑑𝑐
0]
[0
0
−
1
𝐿
0
0
∆𝑓𝑑
𝑖𝑑
0 [ 𝑖𝑞 ] + [ ∆𝑓𝑞 ]
1 𝑉𝑑𝑐
∆𝑓4
𝐶]
(25)
Where
𝑎𝑏𝑐
𝑎𝑏𝑐
𝑖𝑑𝑞 = 𝑇𝑑𝑞0
. 𝑖𝑎𝑏𝑐 . 𝑒𝑑𝑞 = 𝑇𝑑𝑞0
. 𝑒𝑎𝑏𝑐
(26)
𝑎𝑏𝑐
𝑎𝑏𝑐
∆𝑓𝑑𝑞 = 𝑇𝑑𝑞0
. ∆𝑓𝑎𝑏𝑐 , 𝑒𝑑𝑞 = 𝑇𝑑𝑞0
. 𝑠𝑎𝑏𝑐
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Double Integral Sliding Mode Control Approach for a Three-Phase Grid -Tied Photovoltaic
Systems
The above mentioned equations can be transformed into two phase stationary frame by
Clarke’s Transformation
2
𝑎𝑏𝑐
𝑇𝛼𝛽
= [
3
1 − 1⁄2 − 1⁄2
]
0 √3⁄2 √3⁄2
(28)
The instantaneous power S delivered to the grid is given as
S = P + jQ
Where
3
𝑃 = 2 (𝑒𝑑 𝑖𝑑 + 𝑒𝑞 𝑖𝑞 )
(29)
3
𝑄 = 2 (𝑒𝑑 𝑖𝑞 − 𝑒𝑞 𝑖𝑑 )
(30)
Where P is the active power and Q is the reactive power In synchronous dq rotating frame,
eq = 0
3
𝑃 = 2 (𝑒𝑑 𝑖𝑑 )
(31)
3
𝑄 = (𝑒𝑑 𝑖𝑞 )
2
(32)
3. PROPOSED CONTROLLERS FOR THE GRID TIED PV SYSTEM
3.1. Sliding Mode Controller
The general principle of sliding mode control (SMC) is to move the state trajectory of the
given system to a preset surface called as sliding surface. Its design has two steps, one is
defining sliding surface S and the second is developing control law. The three stages in SMC
are selecting sliding surface, finding convergence condition and the control law calculation
[13-18]. Its equivalent circuit is shown below in Fig.6
IPV
IC1
PV
PANEL
DC-DC Boost Converter
L
D
IDC
IC2
C2
Sw
Gate
IL
+
C1 VPV
+
VO R L
Signal
PWM
Vref
MPPT
-
δ
U
SMC
Figure 6 Equivalent circuit os SMC-MPPT
The control law has two parts, equivalent control (Ueq) and
non-linear law (Un)
U = Ueq + Un
(33)
Ueq : it conserves sliding surface S(x) = 0
Un : it keeps the control law constant.
Output power of a PV array
𝑃𝑃𝑉 = 𝐼𝑃𝑉 𝑉𝑃𝑉
(34)
During operation of PV array at maximum output
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𝜕𝑃𝑃𝑉
𝜕𝑉𝑃𝑉
𝜕𝑃𝑃𝑉
𝜕𝑉𝑃𝑉
𝜕𝑃𝑃𝑉
𝜕𝑉𝑃𝑉
=
𝜕(𝐼𝑃𝑉 )
𝑉
𝜕𝑉𝑃𝑉 𝑃𝑉
=
=0
𝜕(𝐼𝑃𝑉 𝑉𝑃𝑉 )
𝜕𝑉𝑃𝑉
+ 𝐼𝑃𝑉
(35)
Sliding function ‘S’ is given as
𝑆=
𝜕𝑃𝑃𝑉
𝜕𝑉𝑃𝑉
=
𝜕(𝐼𝑃𝑉 )
𝑉
𝜕𝑉𝑃𝑉 𝑃𝑉
+ 𝐼𝑃𝑉
(36)
The switch control law is given as:
0 𝑆≥0
𝑈𝑛 = {
1 𝑆<0
𝑆̇(𝑥) = −
(37)
𝜕𝑆
𝑥̇
𝜕𝑥 𝑇
=
𝜕𝑆
𝜕𝑆
𝑓(𝑥) + 𝑇 𝑔(𝑥) +
𝜕𝑥 𝑇
𝜕𝑥
𝑈𝑒𝑞 =0
𝜕𝑆
𝑇 𝑓(𝑥)
𝑈𝑒𝑞 = − 𝜕𝑥
𝜕𝑆
(38)
𝑔(𝑥)
𝜕𝑥𝑇
The equivalent control law variable:
𝑈𝑒𝑞 = −
𝐼𝑃𝑉
𝐼𝐿
(39)
3.2. Integral Sliding Mode Controller
The PV array output voltage VPV, has to follow reference voltage VREF, ISMC tunes the
MPPT for this purpose and also it works well for changing internal and external parameters.
Voltage tracking speed is faster and also the voltage ripple gets reduced [19-22], ISMC holds
the system error over the integral terminal switching surface and then equates this error to
zero in a finite time.
The sliding surface ‘S’ for a grid connected PV system is given a
𝑆 = 𝑒1 + 𝛼𝑒2
(40)
Where 𝑒1 is the error between PV output voltage VPV and the reference voltage VREF
𝑒1 = 𝑉𝑃𝑉 − 𝑉𝑅𝐸𝐹
(41)
𝑒2 is integral term of 𝑒2 included with original error, 𝛼 is a positive constant,
𝑒2 = ∫(𝑉𝑃𝑉 − 𝑉𝑅𝐸𝐹 )𝑑𝑡
(42)
The sliding surface derivative 𝑆̇, also named as sliding manifold δ is given as
𝛿 = 𝑆̇ = 𝑒̇1 + 𝛼𝑒2̇ = 0
(43)
The solution is obtained by deriving it and equating it to zero
𝛿̇ = 𝑆̈ = 𝑒̈1 + 𝛼𝑒̈2 = 0
𝑉̈𝑃𝑉 − 𝑉̈𝑅𝐸𝐹 + 𝛼𝑒̇1 = 0
(44)
Substituting the values of voltages from modelling of boost converter
̇
𝐼𝑃𝑉
𝑉𝑃𝑉 𝑉𝑂𝑈𝑇
̅ ) − 𝑉̈𝑅𝐸𝐹 ] + 𝛼𝑒1
[ (
−
𝐷
𝐶𝑖𝑛 𝐶𝑖𝑛 𝐿 𝐶𝑖𝑛 𝐿 𝑒𝑞
𝐷𝐸𝑞 = 1 − 𝑉
1
𝑂𝑈𝑇
̇ . 𝐿 − 𝛼𝐿𝐶𝑖𝑛 . 𝑒1 ]
[𝑉𝑃𝑉 + 𝑉̈𝑅𝐸𝐹 . 𝐶𝑖𝑛 . 𝐿 − 𝐼𝑃𝑉
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Double Integral Sliding Mode Control Approach for a Three-Phase Grid -Tied Photovoltaic
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Due to external disturbance, the switching control is
𝐷𝑑𝑖𝑠 = 𝐺. 𝑠𝑖𝑔𝑛(𝑠)
(46)
Hence, the ultimate control equation is given as
𝐷 = 𝐷𝐸𝑞 + 𝐷𝑑𝑖𝑠
(47)
3.3. Double Integral Sliding Mode Controller
The DISMC based MPPT consists of DISMC, a boost converter and an MPPT algorithm [2328]. The operational circuit is shown in Fig.8 below.
IPV
PV
PANEL
IL
+
IC1
C1 VPV
DC-DC Boost Converter
L
D
IDC
IC2
C2
Sw
Gate
Signal
-
G
T
IL
VO
IPV
VPV
MPPT
Sliding Surface
Calculation
Vref
1/α
V
U
α
e
-
Calculation
of Ueq
Sat
X
U
+
+
Calculation
of K
Calculation
of e
VO R L
PWM
V0 ref
Vpv
+
Ueq
Figure 7 Operational circuit of DISMC-MPPT
The signal flow in the circuit is shown by dashed line. The PV current (IPV), PV voltage
(IPV) are tuned by DISMC in such a way that maximum power can be drawn from PV
system. The steps for calculation of VREF are shown in Table.I below
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Table 1
SL.
NO.
STEP
ACTION
1
1
Using Temperature sensors, measure solar
cell working temperature (T) and
environmental temperature (Tenv)
Calculate irradiation (G);
2
𝐺=
2
(𝑇−𝑇𝑒𝑛𝑣 )∗800 𝑊/𝑚2
(𝑇𝑁𝑂𝐶𝑇 −200)
;
where TNOCT = 400C
Calculate thermal voltage VT ;
3
3
𝑉𝑇 =
𝐾𝑇
𝑞
Calculate load current IPV, no load saturation
current IO ;
𝐼𝑃𝑉 = 𝐼𝑃𝐻 − 𝐼𝑂 [𝑒𝑥𝑝 (
4
4
(
𝐼𝑂 = 𝐼𝑂,𝑅𝐸𝐹 (
𝑇
298
𝑉𝑃𝑉 +𝐼𝑃𝑉𝑅𝑆𝑒
𝑎𝑉𝑇
𝑉𝑃𝑉 +𝐼𝑃𝑉 𝑅𝑆𝑒
𝑅𝑆ℎ
3
) 𝑒𝑥𝑝 (
) − 1] −
);
𝐸𝑔
(
1
𝑁𝑆 𝑉𝑇 298
1
− ))
𝑇
Calculate open circuit voltage VOC;
5
5
6
6
𝐼𝑃𝑉 + 𝐼𝑂
𝑉𝑂𝐶 = 𝑁𝑆 𝑉𝑇 𝑙𝑛 (
)
𝐼𝑂
Calculate reference voltage VREF ;
𝑉𝑅𝐸𝐹 = 𝐾𝑂𝐶 𝑉𝑂𝐶
The operation of DISMC-MPPT works as follows
Step 1: The percentage duty ratio of the boost converter in terms of load voltage VO and
PV panel voltage VPV is given as
%δ =
(VO −VPV )
(48)
VO
Reference duty ratio is given as
%δREF =
(VO −VREF )
(49)
VO
Step 2: The switching surface [15,16] can be obtained from
d
S(X) = [dt + β]
n−1
e(x)
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(50)
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e(x) is the tracking error, n is the order of sliding surface,
for n=1; S(x) = e(x)
The tracking error e is
e(x) = e(x1 ) + e(x2 ) + e(x3 )
(51)
Where
e(x1 ) = (VREF − VPV )
e(x2 ) = ∫(VREF − VPV ) dt
e(x3 ) = ∫ {∫(VREF − VPV ) dt} dt
Step 3: applying invariance control, equivalent control (ueq) is obtained
𝑆(𝑥) = 0 ; 𝑆̇(𝑥) = 0 ⇒ 𝑢 ≅ 𝑢𝑒𝑞
(52)
Sliding motion is possible over switching surface and there exists equivalent control (ueq)
when s(x)=0
By solving the following equation, equivalent signal (ueq) can be calculated as 𝑆̇(𝑥) = 0
Hence equivalent signal (ueq) can be obtained as
𝑢𝑒𝑞 = − 𝑔̂−1 [𝑓̂ + 𝑒(𝑥1 ) + 𝑒(𝑥2 )]
(53)
Step 4: Lyapunov’s stability criterion is applied to nonlinear switching control or the input
signal (un)
𝑆(𝑥)𝑆̇(𝑥) < 0
(54)
A nonlinear switching control or the input signal (un) takes care of external disturbance
and is given as
𝑢𝑛 = − 𝑔̂−1 𝐾|𝑆(𝑥)|𝛼 )𝑠𝑎𝑡
𝑆(𝑥)
𝛼
(55)
0<𝛼<1
𝑆(𝑥)
𝑠𝑎𝑡 𝛼
𝑆(𝑥)
=𝑠𝑎𝑡 𝛼
=
𝑆(𝑥)
|≤1
𝛼
{
𝑆(𝑥)
𝑠𝑖𝑔𝑛 ( ) > 1
𝛼
1; |
α
The exponential term | S(x)| permits input signal (un) to increase the reaching speed when
state is at a far distant from sliding surface, and the reaching speed decreases when state is
near the sliding surface.
By incorporating input signal (un) in a boundary of thickness φ, chattering magnitude can
be decreased
To satisfy the reaching condition
𝑆(𝑥)𝑆̇(𝑥) < 0,
(56)
the gain K should be large and is calculated by
1 𝑑
𝑆(𝑥)2
2 𝑑𝑡
≤ −𝜂|𝑆(𝑥)|
Integrating within limits 0 and treach, the teaching time can be found as
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𝑆(𝑥)𝑡=0
𝜂
𝑡𝑟𝑒𝑎𝑐ℎ ≤
(57)
The state trajectory reaches to sliding surface within a definite time lesser than
𝑆(𝑥)𝑡=0
𝜂
Step 5: The switching control signal is calculated as
𝐾 ≥ 𝑔̂𝑔−1 (𝑓̂ − 𝐹) + 𝑓̂ + 𝑔̂𝑔−1 (𝑒(𝑥1 ) + 𝑒(𝑥2 ))
(58)
Step 6: From load voltage VDC and PV panel voltage VPV, % δ is calculated.
Step 7: Lyapunov function V (x) for verifying the sliding mode of DISMC is given as
1
2
𝑉(𝑥) = 𝑆 𝑇 (𝑥)𝑆(𝑥)
(59)
To confirm this existence condition, 𝑉̇ (x) must be negative definite such as
𝑉̇ (𝑥) = 𝑆̇(𝑥)𝑆(𝑥){𝑓(𝑥)} < 0
(60)
DISMC-MPPT for a PV system is given as
𝑆̇(𝑥) =
≅ −[
𝑖𝐿
𝐶1
−
𝑑
[𝑒(𝑥1 ) + 𝑒(𝑥2 ) + 𝑒(𝑥3 )]
𝑑𝑡
1
𝑉 ] [𝑉𝑅𝐸𝐹
𝐶1 𝑟𝑃𝑉 𝑃𝑉
− 𝑉𝑃𝑉 + ∫(𝑉𝑅𝐸𝐹 − 𝑉𝑃𝑉 )𝑑𝑡]
(61)
Chattering point h is given as the difference of higher and lower value
ℎ = ℎ1 − ℎ2
(62)
SSE is given as
𝑆𝑆𝐸 = |𝑉𝑅𝐸𝐹 − [ℎ2 +
(ℎ1 −ℎ2 )
2
]|
(63)
4. SIMULATION RESULTS
The Voc and Vref are calculated at different solar irradiances..
The performance of different MPPT controllers at irradiation of 800 W/m2 and operating
temperature of 50 °C are presented below.
i. Scenario I
By using P&O-MPPT, the PV voltage is shown below in Fig.8.
Figure 8 PV Voltage Vs. Time
The inverter voltage and current is shown in Fig.9.
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(a)
(b)
Figure 9 Inverter Output Voltage and Current
The active and reactive power is shown In Fig.10
Figure 10 Inverter Output (a) Voltage and (b) Current
ii.
Scenario II
By using SMC-MPPT, the PV voltage is shown below in Fig.11.
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Figure 11 PV Voltage Vs. Time
The inverter voltage and current is shown in Fig.12
(a)
(b)
Figure 12 Inverter Output (a) Voltage and (b) Current
The active and reactive power is shown In Fig.13
Figure. 13. Inverter Active and Reactive Power
iii. Scenario III
By using ISMC-MPPT, the PV voltage is shown below in Fig.14.
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Double Integral Sliding Mode Control Approach for a Three-Phase Grid -Tied Photovoltaic
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Figure 14 PV Voltage Vs. Time
The inverter voltage and current is shown in Fig.15
(a)
(b)
Figure 15 Inverter Output (a) Voltage and (b) Current
The active and reactive power is shown In Fig.16
Figure 16 Inverter Active and Reactive Power
iv.
Scenario IV
By using DISMC-MPPT, the PV voltage is shown below in Fig.17
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Figure. 17. PV Voltage Vs. Time
The inverter voltage and current is shown in Fig.18
(a)
(b)
Figure 18 Inverter Output (a) Voltage and (b) Current
The active and reactive power is shown In Fig.18
Figure 19 Inverter Active and Reactive Power
5. CONCLUSION
In this paper, a three-phase grid tied to photovoltaic system is described using MATLAB
simulation. All the components of the presented system were modelled. The boost converter
operation in two modes was presented; it extracts the maximum power from the PV system.
Four MPPT controllers PO-MPPT, SMC-MPPT, ISMC-MPPT and DISMC-MPPT for a grid
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tied PV system were presented in the paper. Among them DISMC-MPPT displayed better
tracking performance when compared with other MPPTs.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
Andújar, J. M., F. Segura, and T. Domínguez. 2016. “Study of a Renewable Energy
Sources-based Smart Grid. Requirements, Targets and Solutions.” 2016 3rd Conference
on Power Engineering and Renewable Energy (ICPERE), Yogyakarta, 45–50.
Jan Leuchter ; Vladimir Rerucha ; Ahmed F. Zobaa, “Mathematical modeling of
photovoltaic systems”, Proceedings of 14th International Power Electronics and Motion
Control Conference EPE-PEMC 2010, sept, 2010
Villalva M.G, Gazoli J.R and Filho E.R, “Comprehensive approach to modeling and
simulation of photovoltaic Array’’, IEEE Trans on Power Electronics, Vol. 24, n°5, pp.
1198- 1208, May 2009.
T. Esram and P. L. Chapman, “Comparison of photovoltaic array maximum power point
tracking techniques,” IEEE Trans. Energy Conv., vol. 22, no. 2, pp. 439–449, Jun. 2007.
Vinay Kumar, U.Salma, “Mathematical Modelling of a Solar Cell and its Performance
Analysis under Uniform and Non-Uniform Insolation”, International Journal of
Engineering Research & Technology (IJERT), ISSN: 2278-0181, Vol. 6 Issue 12,
December – 2017
M.G.Villalva, J.R.Gazoli and E. Ruppert F “Modeling and circuit based simulation of
photovoltaic arrays,” Power Electronics Conference, 2009. COBEP'09. Brazilian. IEEE,
2009.
Saharia, Barnam Jyoti, and Kamala Kanta Saharia. "Simulated Study on Nonisolated DCDC Converters for MPP Tracking for Photovoltaic Power Systems." Journal of Energy
Engineering: 04015001(2015)
Hart, Daniel W. Power electronics. Tata McGraw-Hill Education, 2011
Ma, L., Ran,W., Zheng, T.Q. ‘Modeling and control of three-phase grid-connected
photovoltaic inverter’. In: IEEE ICCA 2010. (IEEE, 2010. pp. 2240–2245
Levron Y., Shmilovitz D.: ‘Maximum power point tracking employing sliding-mode
control’, IEEE Trans. Circuits Syst. (I), 2013, 60,(3), pp. 724–731
M. Ciobotaru, T. Kerekes, R. Teodorescu, Senior A. Bouscayrol , “ PV inverter simulation
using MATLAB/Simulink graphical environment and PLECS block set,” IEEE Industrial
Electronics 32nd Annual Conference on, pp. 5313-5318, 2006.
M. Vinay Kumar, U. Salma, “A novel fault ride-through technique for grid-connected
Photo Voltaic energy systems”, International Journal of Ambient Energy, Feb,2018.
Slotine, J.J., Li, W.: ‘Applied Nonlinear Control’. (Pearson, 1991).
Hanifi G. Sliding mode control of DC-DC boost converter. Journal of Applied sciences 5
(3), pp.588-592, 2005.
EI Fadil, H.; Giri, F.; Guerrero and losep M. "Grid-connected of photovoltaic module
using nonlinear control", 3rd IEEE International Symposium on Power Electronics for
Distributed Generation Systems (PEDG) 2012.
Chu C., Chen C.: ‘Robust maximum power point tracking method for photovoltaic cells A
sliding mode control approach’, Sol. Energy, 2009, 8, (1), pp. 1370–1378
Tse, C.: ‘Sliding Mode Control of Switching Power Converters’. (CRC Press, 2011)
M.Vinay Kumar, U.Salma, S.Hemanth, “A Review of Sliding ModeController
Application For Maximum Power Point Tracking In Photo VoltaicSystems”, International
Journal of Academic Engineering Research, Vol.2, Issue8, pp. 219-224, August 2018.
http://www.iaeme.com/IJMET/index.asp
1068
editor@iaeme.com
M.Vinay Kumar and U.Salma
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
S. Yan, Shuo, S. C. Tan and S. R. Hui. “Sliding mode control for improving the
performance of PV inverter with MPPT—A comparison between SM and PI control,”
Power Electronics and Applications (EPE'15 ECCE-Europe), 2015 17th European
Conference on. IEEE, 2015.
Chan C.Y.: ‘A nonlinear control for DC-DC converters’, IEEE Trans. Power Electron.,
2007, 22, (1), pp. 216–222
J.-J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ, USA: PrenticeHall, 1991.
M. Vinay Kumar, G.V. Nagesh Kumar, “Field Programmable Gate Array Chip Design for
Hybridized PSO-BFO Based Maximum Power Point Tracker”, I J C T A, 9(10), pp. 1-11,
Sept.2016.
B. Subudhi and S. S. Ge, “Sliding-mode-observer-based adaptive slip ratio control for
electric and hybrid vehicles,” IEEE Trans. Intell. Transp. Syst., vol. 13, no. 4, pp. 1617–
1626, Dec. 2012.
Raseswari Pradhan and Bidhyadhar Subudhi, “Double integral sliding mode MPPT
control of photovoltaic system,” IEEE Trans. Control. Syst. Technol., vol. 24,NO.
1,pp.285-292, Dec 2016.
Chinchilla-Guarin, J., J. Rosero. 2016. “Impact of Including Dynamic Line Rating Model
on Colombian Power System.” 2016 IEEE Smart Energy Grid Engineering (SEGE),
Oshawa, ON, 36–40.
Pradhan R., Subudhi B.: ‘A new digital double integral sliding mode maximum power
point tracker for photovoltaic power generation application’. Ninth IEEE ICSET, Nepal,
2012.
http://www.iaeme.com/IJMET/index.asp
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