Integral fuzzy control for photovoltaic power systems A. El Hajjaji, M. BenAmmar, J. Bosche, M. Chaabene, A. Rabhi Laboratoire de Modélisation, Information et Systèmes Université de Picardie Jules Verne 7, rue Moulin Neuf, 80000, Amiens, France
[email protected]
Abstract This paper addresses the control problem for a photovoltaic generator coupled to a boost DC-DC converter taking into consideration the nonlinearities and the parametric uncertainties of the model. After the analysis and modelling of the photovoltaic power system by a Takagi-Sugeno (TS) fuzzy model, the design strategy of the control law is proposed and the stabilization conditions are formulated in terms of Linear Matrix Inequalities (LMIs) which can be solved very efficiently using convex optimisation techniques. Simulation and experimental results will be given to demonstrate the performance of the proposed methods. Keywords : Photovoltaic energy, Fuzzy control, LMI, Stabilisation I. Introduction Photovoltaic energy is considered as an alternative energy to meet the ever increasing demand for energy in the world because it comes from solar energy which is abundant, renewable, clean and free. This energy is not only widely used in space applications but also in terrestrial applications. To obtain this energy we have essentially need of a photovoltaic generator and a Buck DC-DC converter, a Boost DC-DC converter or a combination of the two. Despite the research developed on photovoltaic systems in the least years, the problem of the maximum power continues to be studied because photovoltaic conversion systems are nonlinear and uncertain. To extract the maximum power from PV generators, control algorithms are usually used [1, 2 , 3, 4, 5, 6 ,7].. Many methods have been developed to track the maximum power point. Thus, in [5], a perturbed and observe (PO) method is proposed and an incremental conductance algorithm is used in [6]. Recently, the robust control like sliding mode, Youla parametrization [2], and Neural Networks [3,4,5] have also been proposed. In this work, we propose a design method to ensure the maximum power point, using integral TS fuzzy controllers, largely developed in literature in control and diagnosis fields of nonlinear systems [8,9,10,11]. It is well know that the I-V photovoltaic characteristic changes depending on the insulation and temperature as illustrated in figures 1 and 2. The purpose of this study is to develop an MPPT to extract the maximum power panel by developing a integral TS fuzzy controller for a photovoltaic system taking into account the weather conditions.
2
This paper is organized as follows; in section 2, we present the main characteristics of a photovoltaic panel and the influence of different weather and material factors on these features. Section 3 is devoted to fuzzy control design based on TS fuzzy representation of PV systems. Simulation and experiment results to illustrate the feasibility of the proposed algorithm and to demonstrate its high performances are presented in section 4. Finally, we conclude this work in section 5. II. Modelling and characterization of PV Systems The PV generator consists in combination of many PV cells connected in series and parallel modes to provide the desired value of output voltage and current. I-V characteristics of the photovoltaic generator can be obtained using its equivalent circuit. Atmospheric variables dramatically affect the available insulation for photovoltaic generators. Consequently, the current–voltage (I–V) curves and maximum power points (MPPs) of photovoltaic modules change with the solar radiation, as illustrated in Figure 1. Besides insulation, another important factor that influences the characteristics of a photovoltaic module is the cell temperature, as shown in Figure 2. The variation of cell temperature greatly changes the MPP along the x-axis. Fig. 3 illustrates the insulation effect based on the relationship of photovoltaic power and current. I-V Characteristic 6 1000 W/m 2
Current I[A]
5 800 W/m2
4
600 W/m2
3
400 W/m2
2
200 W/m2
1
0
0
2
4
6
8
10 12 Voltage V[V]
14
16
18
20
22
Figure 1 : I-V characteristics with insulation variations and constant cell temperature. I-V Characteristic 6
5
CurrentI[A]
4
3
4
6
8
10 12 Voltage V[V]
14
16
18
C 5° Ta=3
2
C 5° Ta=2
0
C 5° Ta=5
C 5° Ta=7
0
C 5° Ta=6
1
C 5° Ta=4
2
20
22
Figure 2: I-V characteristic with cell temperature variation and constant insulation.
3
P-V Characteristic 70
60
10 00
40
0 80
30
2 /m W
0 60
20
2 /m W
0 40
/m 2 W
W /m 20 0
10
0
2
W /m
2
Power P[W /m ]
50
0
2
4
6
8
2
10 12 Voltage V[V]
14
16
18
20
22
Figure 3 : P-V characteristics with insulation variations and constant cell temperature. Now, we develop a dynamic model of our laboratory benchmark shown in figure 4.
Figure 4 : Laboratory benchmark This PV system is composed of a photovoltaic generator (100 Watts), a DC-DC boost converter and a battery (24 Volts). The equivalent circuit is presented in figure 5
L
ipv rpvi
vpv C1
α
RL
vFW
C2
Figure 5 : PV equivalent circuit
vbatt
4
The differential equations of the circuit can be described as follows [2]: d dt
− iL v = pv −
y =[ 0
1 L 1 r pvi C 1
RL L 1 C1
iL + v pv
V b at + V FW − L 0
α
iL 1] vpv
Where rpvi denotes the dynamic resistance of PV panel defined by
rpv =
dv pv
.
di pv
For fixed cells temperature and insulation, the I-V characteristics of the PV (T=25°C, G=1000W) system is represented in figure 6. To obtain the fuzzy model, we have approximated this curve by a fuzzy model with 4 Takagi Sugeno (TS) rules using the Levenberg-Marquardt optimisation algorithm: The obtained fuzzy rules are: If VPV is F1 Then
I pv = 0.43V pv + 8.54
If VPV is F2 Then
I pv = 0.61V pv
If VPV is F3 Then
I pv = 0.29V pv
If VPV is F4 Then
I pv = −0.35V pv
Where membership functions Fi (i=1,2,3,4) are represented in figure 3. 1
6
0.9
F1 F2 F3 F4
0.8
4
0.6 Current [A]
Membership functions
0.7
measured current estimated current
5
0.5 0.4 0.3
3
2
1
0.2 0
0.1 0
0
5
10
15 Voltage [V]
20
25
Figure 5 : Membership functions
30
-1
0
5
10
15
20
25
Voltage [V]
Figure 6 : I-V photovoltaic charactistic
5
∫
Fuzzy Controller
α
PV System
VPV IPV ,
Figure 7: Block diagram Using this idea, state model (1) can be described by the following TS Fuzzy model : .
If VPV is Fi then
x = Ai x + Biu + w (i=1,2,3,4)
where RL − L Ai = − 1 C1
1 L , ai C1
Vbat + V FW − Bi = L 0
, x = iL , u = α V PV
a1 = 0.43 , a2 = 0.61, a3 = 0.29 , a4 = −0.35 ,
T
w = [ 0 8.54]
The final output of the fuzzy system is inferred as follows 4
x& (t ) = ∑ hi (vpv(t )) { Ai ( x(t )) + Bi u (t ) + w}
(1)
i =1
with
hi =
Fi (VPV ) 4
∑ Fi (VPV )
i =1
It is easy to check that
4
hi (VPV ) ≥ 0, ∑ hi (VPV ) = 1. i =1
In order to guarantee zero steady state regulation error, we develop an integral T-S fuzzy control. Let r be a constant reference, the objective is to achieve that V → r when t → ∞ . To this end, we introduce an added state variable to acpv
6
count for the integral of output regulation error. Let us define the new state variable as: .
e = r − VPV
(2)
The TS fuzzy model given in (1) augmented by the error dynamics given in (2) becomes: .
4
x = ∑ hi (V pv ) { Ai ( x(t )) + Bi u (t ) + W }
(3)
i =1
Where
A Ai = i −C
0 Bi w , B = W = C = [ 0 1] i 0 0 r III Control strategies
In this work two control strategies are studied: 1) Incremental conductance method. This method consists in looking for the MPP. During the accurate tracking cycle, the output voltage of the photovoltaic array (PV) is adjusted by comparing the values of the incremental conductance and its instantaneous value.
P =V ⋅I 1 dP I dI = + ,V > 0 V dV V dV
dP dV > 0 if G > ∆G dP dV = 0 if G = ∆G dP dV < 0
if G < ∆G
G=I V ∆G = − dI dV where I V is the conductance of the PV array, and cremental conductance.
dI dV is defined as the in-
7
The test procedure is shown by the following algorithm. This algorithm tracks the operating voltage point at which the instantaneous conductance is equal to the incremental conductance. Read V(k),
Read the voltage and current
dV = V(k) V(k-1)
Compute voltage and current changes Y
dV = 0? N Y
Y dI/dV= -
dI = N
N Y
Y dI/dV> -
dI > 0? N
N Vref = Vref +
Vref = Vref +
Vref = Vref +
V(k-1) = V(k)
Vref = Vref +
Modify references
Store voltage and current values
Return
Figure 8 : Flow chart of the MPPT Incremental Conductance Method 2) Integral TS fuzzy controller method We consider the well known PDC structure defined as: If VPV is Fi then
u = − Ki x
(i=1,2,3,4).
The designed fuzzy controller uses the same fuzzy sets as in the premise parts with the plant and has local linear controllers in the consequence parts. The global output of the fuzzy state feedback controller is given by : 4
u (t ) = −∑ hi (V pv )K i x (t )
(4)
i =1
By substituting (4) into (3), the closed loop fuzzy system can be represented as
8
.
4
4
4
4
j =1
j =1 i< j
x (t ) = ∑∑ hi h j Gij x (t ) = ∑ hi 2Gii x (t ) + ∑ hi h j ( Gij + G ji ) x (t ) + W i =1 j =1
A Gij = Ai − Bi K j = i −C
0 Bi − [ Ki 0 0
With
K Ii ]
The objective is to find K so that the closed loop system described by (3) is i
quadratically stable. The diagram block of integral TS fuzzy control is shown in figure 7. Using the relaxed stabilisation conditions proposed in [9], we can say that the equilibrium point of the PV system described by TS fuzzy model (2) is quadratically stabilisable via fuzzy controller 4) if there exist matrices Q>0, Yi,(i=1,2,3,4), Yiii,(i=1,2,3,4),Yjii=YiijT and Yiji ,(i=1,2,3,4, j≠i,j=1,2,3,4) and Yijl=YljiT, Yilj=YjjiT, Yjil=YlijT,(i=1,2,j=i+1 …3,l=j+1,…,4) satisfying the following LMI:
QAiT + AQ − Yi T BiT − BiYi < Yiii i T
2 QAiT + QA Tj + 2 Ai Q + A j Q − (Yi + Y j ) BiT − Yi T B Tj − Bi (Yi + Y j ) − B jYi ≤ Yiij + Yiji + Y iijT T
T
2Q ( Ai + Aj + Al )T − (Y j + Yl ) BiT − (Yi + Y j ) BlT T
− (Yi + Yl ) BTj + 2( Ai + Aj + Al )Q − Bi (Y j + Yl ) − Bl (Yi + Y j ) − B j (Yi + Yl ) ≤ Yijl + Yilj + Y jil YijlT + YiljT + Y jilT
Y1i1 Y 2 i1 Y3i1 Y4i1
Y1i 2
Y3i1
Y2i 2 Y2i 3 Y3i 2 Y3i 3 Y4i 2 Y4i 3
Y1i 4 Y2i 4 ≤0 Y3i 4 Y4i 4
Moreover, in this case, the fuzzy local state feedback gains are Proof : see [9]
K i = YiQ −1
9
VI- Simulation and experiments results To validate the proposed algorithm, simulation and experiment results are carried out. The simulations are developed in the matlab/simulink software and the experiment tests are achieved on a laboratory benchmark composed of the UNISOLAR PV generator (Pmax= 65 W, Voc =21.1 V, Isc=5.1 A, 124 Wp), a Boost DC-DC converter, Two Batteries (12 Volts, 40 Ah), two flash lights, a fan and a PIC18F-4520 microcontroller to receive the control laws as shown in figures 9, 10 and 11.
Figure 9 : Power block of PV system
Figure 10 : PIC 18F-4520 Microcntroller
10
Figure 11 : generated Power Based on the Incremental conductance method, we have charged a 24V battery, powered a fan and flash light with a power efficiency of 94.89% as shown in figure 11. In simulation, we consider that the MPP is obtained with considering a reference voltage equal to 15,5 volts. The simulation results using the integral TS fuzzy control given in figure 12 show the regulation performance of the algorithm proposed.
V P V oltage V olts
20
10
VP voltage reference signal
0
-10
0
0.05
0.1
0.15 0.2 Time (s)
0.25
0.3
0.35
0
0.05
0.1
0.15 0.2 Time (s)
0.25
0.3
0.35
1
Duty c ycle
0.8 0.6 0.4 0.2 0
Figre 12 : The response of output voltage and the control input
11
V. Conclusion. In this paper, a method based on the TS fuzzy system has been used to modelize, analyse and control the PV systems. The controller design conditions are formulated in LMI terms and an MPPT was developed to implement the proposed MPPT Methods. [1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
References G. Petrone, G. Spagnulo, M. Vitteli, “Analytical model of mismatched photovoltaic fields by means of Lambert W-function”, Solar Energy Materials & solar Cells 2007 pp 1652-1657. W. Xiao, W. G. Dunford, P. R. Palmer, and A. Capel, Regulation of Photovoltaic Voltage, IEEE Transactions on Industrial Electronics, Vol. 54, No. 3, June 2007. A. B. G. Bahgat, N. H. Helwa, G.E. Ahmad, E.T.El Shenawy « Maximum power point tracking controller for PV systems using neural networks » Renewable energy 30 (2005) pp 1257-1268. R Leyva, C. Alonso, I. Queinnec, A. Cid-Pastor, D. Lagrange, L. Martinez “MPPT of photovoltaic systems using Extremum- Seeking Control », IEEE Transaction on aerospace and electronic systems vol 42, n° 1 January 2006 pp 249-258 N. Femia, G. Petrone, G. Spagnuolo, and M. Vitelli, “Optimization of perturb and observe maximum power point tracking method,” IEEE Trans. Power Electron., vol. 20, no. 4, pp. 963–973, Jul. 2005. T. Tafticht, K. Agbossou, M. L. Doumbia, A.Chériti : « An improved maximum power point tracking method for photovoltaic systems », Renew Energy (2007), pp 1-9 K. Hussein, I. Muta, T. Hoshino, and M. Osakada, « Maximum photovoltaic power tracking : an algorithm for rapidly changing atmosphere conditions » IEEE, Proc. Inst. Electr. Eng, vol 142 n°1 pp59-64 Kazuo Tanaka, Takayuki Ikeda, and Hua O. Wang, « Fuzzy Regulators and Fuzzy Observers: Relaxed Stability Conditions and LMI-Based Designs », IEEE Trans. On Fuzzy Systems Journal, Vol. 6, N°2, May 1998 pp 250265. C. Fang, Y. Liu, S. Kau, L. Hong, and C. Lee « A New LMI-Based Approach to Relaxed Quadratic Stabilization of T–S Fuzzy Control Systems, IEEE Transactions On Fuzzy Systems, Vol. 14, No. 3, June 2006. M. Oudghiri, M. Chadli and A. El Hajjaji, One-Step Procedure For Robust Output H∞ Fuzzy Control, CD-ROM of the 15th Mediterranean Conference on Control and Automation, IEEE-Med’07, June 27-29, Athens, Greece. W. El Messoussi, J. Bosche, O. Pages, A. El Hajjaji, Non-Fragile ObserverBased Control of Vehicle Dynamics Using T-S Fuzzy Approach, proceeding of the 17th IFAC World Congress, July 6-11, Seoul Korea (2008).
