point tracking (MPPT) applied for photovoltaic (PV) power generation systems based on boost DC-DC converter and robust control tools using Linear Matrix ...
Robust Control approach for Photovoltaic Conversion System * Menad Dahmane , Jerome Bosche and Ahmed EI-Hajjaji MIS Laboratory 33, Rue Saint Leu 80000, Amiens, France Email: (menad.dahmane.Jerome.bosche.ahmed.hajjaji)@u-picardie.fr
Abstract-This paper proposes an algorithm of maximum power point tracking (MPPT) applied for photovoltaic (PV) power
generation systems based on boost DC-DC converter and robust control tools using Linear Matrix Inequalities (LMI) technique. In this study, the nonlinearities and the parametric uncertainties of the system are taking into account. The strategy of this algorithm consists on the generation of a voltage reference which corresponds to maximum power points using a fuzzy algorithm. This is obtained from direct measurements of solar irradiation
and temperature of the PV-panel. After this; the paper proposes a robust control based on fuzzy Takagi-Sugeno (T-S) model to follow this reference and consequently force the operating point of a panel to track the maximum power point (MPP). Two objectives are covered by this control: (i) to guarantee Noo
performances and minimize the influence of disturbances on the output; (ii) to ensure stability against parametric variations of the system. These two objectives have been achieved by the resolution of LMIs system. The performances of the proposed approach
are
ensured
in
simulation,
using
real
profiles
of
irradiance and temperature measured on photovoltaic platform. Keywords-LMI; Takagi-Sugeno models; MPPT; Photovoltaic system; robustness.
I.
INTRODUCTION
The current global energy situation can be simply summarized: the demand increases so that it is more difficult to be satisfied by the offer. This increased energy demand is justified on the one hand, by a considerable technological development with the emergence of a multitude of systems that depend on energy, and secondly, to demographic changes. Yet the emergence of some countries in Asia, Latin America or in Eastern Europe, but also a growing world population, suggest energy needs increasingly important. On the other hand, these energy requirements are largely met by fossil fuels that emit greenhouse gases, and whose reserves are largely weakened by decades. Photovoltaic (PV) energy is clean energy that can be converted into electricity. It designates the electricity produced by transformation of a party of solar irradiance with a photovoltaic cell. The electrical association of several cells forms a PV panel. The problem in this characteristic is that the position of the MPP is not fixed but it varies according to the irradiance, the temperature and load [1]. Nevertheless, solar panels are not very efficient with only about 12-20% (according to the solar-cell's technology and
operating conditions) efficiency in their ability to convert sunlight to electrical power. So, in order to maximize the power derived from the PV panel, it's crucial to operate PV energy conversion systems near maximum power point to increase the output efficiency of PV. This requires a technique to search (track) the MPP called maximum power point tracking (MPPT) algorithm [11]. A number of MPPT techniques have been developed for PV systems [8, 14], and for all conventional MPPT techniques the main problem is how to obtain optimal operating points (voltage and current) automatically at maximum PV output power under variable atmospheric conditions. Several MPPT techniques can be mentioned: (i) the perturbation and observation (P&O) algorithm [1]: this is the most widely used method in commercial PV panel because it is easy to implement. This algorithm works periodically by perturbing the operating current point of the panel and observing the power variation. From the power-current characteristic of the panel, then it is to increase the value of current when the power decreases. The MPP is achieved when the power variation is almost zero; (ii) the incremental conductance (INC) algorithm [2], is implemented by periodically checking the slope of the P-V curve of a PV panel. If the slope becomes zero or equal to a pre-defined small value, the perturbation is stopped and the PV panel is forced to work at this operating point; (iii) open and short-circuit method [5]: this method assumes a constant ratio between the open-circuit voltage (Voc) and the voltage at the maximum power (VMPP). In this paper, MPPT based on fuzzy logic is considered. In fact, by using measurements of irradiation and temperature of the panel, the fuzzy algorithm generates, by using a database establishes with the manufacturer's data, the voltage which corresponds to the maximum power point. This technique is interesting because it is not based on the model of the corresponding panel. It is an direct method. It makes it easy to implement. After that, a robust control based on T-S fuzzy model is developed. The paper is structured as follows: In section II, the PV-module and the boost converter modeling are described. The development of the control tools is given in section III. The generation of the reference voltage is also given in this section. Simulation results with real irradiation and temperature profiles are proposed in section IV whereas conclusion and future works are summarized in section V.
Research supported by European Regional Development Fund and "Ie conseil regional de Picardie" within the framework of the project "GEO· ECOHOME"
978-1-4673-6374-7/13/$31.00 ©2013
IEEE
II.
PV-MODULE AND BOOST MODELING
A. PV Module Model The electrical equivalent circuit of PV-panel is usually represented by the single or the double diode. In this study, the single diode model is considered as shown fig. I.
1
'ph
1
Rsh
v..,
Voltage V[V]
'd
Fig.3. I-V characteristic ofPV Module influenced by Temperature
---.
B. Boost Con verter Model
Fig.l. Electrical Equivalent schema ofPV Module
"\;- ,
L
I
• - -
The expression of the current in term of voltage, current and temperature of cells is given by equations:
I
I
I�
: rpvi�
I
I [I. - J ( ) o
=
sc
V oc Rp
exp - � AVT
kT Vr=q
L-
(1)
_�--�---_--_--� _ ---..-�-r� _
PV
Boost
The current-voltage characteristic of a photovoltaic generator is highly nonlinear, in addition to this; it changes considerably depending on the variation of atmospheric conditions. Consequently, the maximum power points (MPPs) of photovoltaic modules change with the solar radiation as illustrated in fig. I. Besides insolation, another important factor that influences the characteristics of a photovoltaic module is cell temperature, as shown in fig.2.
.
•
According to Fig. 4, the state model of the system is deduced from the well-known of laws of Kirchhoff. It leads to the following stat equations:
1�iL =-iiLL ---=-
m
d -v v p dt
Where
R,
Consider
xT
-
L -1
+
C]
dv r v = ----Pp dl v p =
(
I -U
L
I
C].r v p
) ( vFw+ vba, )+ �L vpv (2)
v pv
is the dynamic resistance of PV-Panel.
[iL, vpv] E JP?2 as the state vector,
A
E
IR2+2
the
dynamic matrix, BE IR2+' the input matrix and WE JP?2+j an exogenous input matrix. The stat representation is such as: x =Ax+Bu+W R[
I
[::l= l [�lf:'� }r�i'�)l f. C]
U E VoltageVIV]
)
•
Battery
Fig.4: photovoltaic system.
Where 1ph is the photo-generated current; 10 the dark saturation current, R" cell series resistance, Rp, the cell shunt resistance; T the diode quality factor; q,the electron charge (I.6xlO-19C); k, the Boltzmann's constant (l.38xlO-23 11K) and T, the ambient temperature in kelvin. The values of Isc, Voc, Tre! and Go are given by the PV module manufacturers in the datasheet [19].
•
- Vc1
I
(3)
C1.rpv
[0, l]is the control input corresponding to the duty cycle. III.
THE STRATEGY OF CONTROL
Fig.2. I-V characteristic ofPV Module influenced by irradiance
The control strategy consists of two steps and is illustrated in Fig. 5: • From the weather conditions in terms of temperature of the panel surface's T and irradiance G, the "ANFIS." block allows to generate a voltage reference Vret
corresponding to the voltage •
Vmp
at the maximum power.
The "ROBUST CONTROL" block allows tracking the voltage reference Vrefby generating a control signal u for the boost converter.
G� � T
resistance
Fig.5: Diagram of control strategy
A.
through it, in the present study; this element is simply the photovoltaic cell. However, this ratio varies depending on the voltage or current. The ratio of the change in voltage to the change in current is known as the dynamic resistance, which represents the slope of the I-V curve of PV-panel [16]. For fixed cells temperature and irradiation, the I-V characteristics of the PV (T=25°C, G=1000W) system is represented in fig 8. To obtain the fuzzy model, we have approximated this curve by a fuzzy model with 4 Takagi Sugeno (TS) rules using the Levenberg-Marquardt optimization algorithm. In fact the I-V curve of the photovoltaic output is divided into four regions. For each region, the slope of the corresponding curve is calculated. The value of the slope corresponds to the
The closed-loop state model
In order to ensure a smooth tracking of the reference current, a new state variable is introduced. It corresponds to the integral of the tracking error E as shown Fig. 6.
rpv in each region.
The equation of the approximated curve for each region can be written as a first degree polynomial: (6) ipV_i =ai·vpv_i + bi The obtained fuzzy rules are: If v v is Ff Then p
E
f X2
X3
=
¢
=
fs with s
{�:
=
xref - x2
=
v
If
v
If
v
pv
pv I'
P
isF2Then
= -0.003521
ipl' =-0.03369 vpl' i
V',.ef -vpv
-F3
L
OO���--���O��'�5������� I-V Characteristic
L
With
XT
=
L u+
o
o
-1
i.e.:
X = AX +B.u+W
[ (] x,
Voltage IVI
Fig. 7 : Membership functions
o o
F4
(4)
x2
It leads to the following augmented state-space model: L
8' -F2
I I
AX+_B.u+W
; - vref 1
+5.147
PI' = -0.1007 vI'I' + 6.144 is F4 Then if}\' =-1.224 vI'I' +24.6 is F3 Then
The closed loop state space model is deducted from the two equations below:
R{
PI' +5.006
V
The membership functions Fi (i=1,2,3,4) are represented in fig.7.
Fig.6. Closed-loop system architecture We define
If
PI'
i
(5)
(I)
represent the augmented state vector. °O���--��--�'O--�'2��� Voltago V[V)
B.
T-S Fuzzy Model
The model presented in (5) is nonlinear in the state. Indeed, the dynamic matrix of the system depends on the resistance of PV-Module. In electrical circuits, resistance is defined as the ratio of the voltage across a circuit element by the current
Fig.S. Linear approximation of the I-V photovoltaic characteristic.
The consequence of this piecewise linearization, four models are defmed which correspond to the four regions of the I-V curve approximation.
Using this idea, the state model (7) can be described by the following TS Fuzzy model : 4
X = Ih, (A/X+B.u+W;)
=-:;-
(7)
i=1
-
Fi(Vpv) are the activation functions and: -'-'-t:..:. -' _ .:.... ! It=1 Fi(Vpv) - (vba/ +vFW ) Vbal + VPW _R, 0
With: h.
-_
L
Ai=
L
C1
0
i =1,
...
!!L C1 -1
0
B=
The sufficient condition to satisfy in this case is the following:
L
L 0
; W;
0
0
=
V(X)+yTy- rWTW O
!l C]
{
v,'eJ
,4
At the present, the Boost parameters vanatlOns are considered. Indeed, it is assumed that these parameters vary in a defined domain, as result; the structured uncertainties are taken into account in this study. These parameters are R/; Vbat and a(
+
R, = R,o M,; Vbal So it comes:
=
0
f>.a;
0
C1
0
0
0 0
L'l.B
P+
+P ( IlA + llBK; ) + CTC
r
+ SoK, )+ ( IlA + IlSK; p
x
(13)
+wrT PX +xTiw _y2i.f/W .A+f>.BK, ) x +w, }
1111 Ai 112 < I;il11B 112 < 1
So the following LMIs are obtained after substituting
4
=
OK
Such as: DAi, DB, EAiand EB are predefined matrix. llA and llB
0
X = Ihd(A/o+L'l.A)X+(Bo+L'l.B).u+W}
{X j;hl
;=1
+
L'l.B= DBL'l.BEB
L
Therefore the T-S fuzzy model can be written in the following fonn:
C.
t h,XT
The structured uncertainties can be rewritten as [18]:
L'l. vbal
0
)
(12)
L'l.A = DAiL'l.AEAI
Such as:
MI L
Ai( O ! X P(AiO I
(11)
The development of the condition (11) leads to [17]:
VbalO + tl �al; a; = a I D + l1a;
A; = AID + tlA; B = Bo + I1B
f>.A;=
minimize the impact of exogenous inputs on the output system. • Disturbance Attenuation: design a controller such that the influence of the disturbance W on the output Y (W" norm) of the system should be minimal as: 1IYliz < yllWllz·
(10)
y=c.x
W( represents the disturbance vector. Y: represents the output. .. In this study, two objectives are covered by the control: the first is to place the poles in a region of the left half- complex plane to ensure certain performance; the second is used to
f...B:
¢i+f.1.1-IDAIDAiT T T +/ii1DBDTB +CTC QE�i Ri EB Q 1 0 0
