Research Article MPPT Algorithm for Photovoltaic Panel...

Research ArticleMPPT Algorithm for Photovoltaic Panel Based onAugmented Takagi-Sugeno Fuzzy Model

Hafedh Abid, Ahmed Toumi, and Mohamed Chaabane

Laboratory of Sciences and Techniques of Automatic Control & Computer Engineering (Lab-STA),National School of Engineering of Sfax, University of Sfax, P.O. Box 1173, 3038 Sfax, Tunisia

Correspondence should be addressed to Ahmed Toumi; [email protected]

Received 30 October 2013; Accepted 19 December 2013; Published 17 February 2014

Academic Editors: T.-H. Meen and R. Miles

Copyright © 2014 Hafedh Abid et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper deals with the Maximum Power Point Tracking (MPPT) for photovoltaic energy system. It includes photovoltaic arraypanel, DC/DC converter, and load. The operating point for photovoltaic energy system depends on climatic parameters and load.For each temperature and irradiation pair, there exists only one optimal operating point which corresponds to themaximum powertransmitted to the load. The photovoltaic energy system is described by nonlinear equations. It is transformed into an augmentedsystem which is described with a Takagi-Sugeno (T-S) fuzzy model. The proposed MPPT algorithm which permits transfering themaximum power from the panel to the load is based on Parallel Distributed Compensationmethod (PDC).The control parametershave been computed based on Linear Matrix Inequalities tools (LMI). The Lyapunov approach has been used to prove the stabilityof the system. Some reliable simulation results are provided to check the efficiency of the proposed algorithm.

1. Introduction

In the most recent years, photovoltaic (PV) energy has beenthe subject of several research projects. It is well known thatthe PV array power panel depends on climatic variables suchas temperature and irradiation as shown in Figures 2 and 3.Actually, the operating point of the PV array panel dependson three parameters such as temperature, irradiation, and theload. In fact, the operating point results from the intersectionof the I-V characteristic and the load characteristic as shownin Figure 4. Inmost cases, the value of load is constant and theclimatic parameters vary in the day, so the load characteristicremains fixed and the characteristic of the panel variesaccording to climatic variables. Consequently, the operatingpoint is variable and the load cannot extractmaximumpowerfrom the panel. To overcome this disadvantage, a DC/DCconverter is inserted between the panel and the load. In thisway, the load value seen by the PV panel can be changed byvarying the duty cycle. In this context, several studies havebeen developed.

Most of papers dealingwith theMPPT control algorithmsare based on perturb and observe (P&O) [1–4], IncrementalConductance [3, 5], Mamdani type fuzzy logic controller

(FLC) [4, 6], and somedifferent approaches as neural networkcontroller (NNC) [7]. In this paper, the PV array panelhas been modelled by fuzzy system approach. At everytime, the desired state variables have been computed basedon the measurement of temperature and irradiation. Also,the MPP tracker algorithm has been developed based onParallelDistributedCompensation (PDC)methodwhichwasdesigned for fuzzy system. Most of papers which have usedfuzzy system applied Mamdani method. Whereas in thiswork, the main contribution which deals with MaximumPower Point Tracking for photovoltaic panel, consists ondeveloping a new algorithm based on augmented T-S fuzzysystem. It is well known that the Mamdani fuzzy systemincludes three blocks which are fuzzification, fuzzy inferencerules and defuzzification, whereas the T-S fuzzy system needsonly two blocks such as fuzzification and fuzzy inferencerules. The block number has been reduced. However, wedecided to choose the T-S type fuzzy system. In this paper,the T-S fuzzy system has been used in modelling stageof PV system and in the control stage. The optimal dutycycle, which permits extracting the maximum power fromthe photovoltaic array panel, is computed based on PDCtechniques. This paper is organised as follows. In Section 2,

Hindawi Publishing CorporationISRN Renewable EnergyVolume 2014, Article ID 253146, 10 pageshttp://dx.doi.org/10.1155/2014/253146

2 ISRN Renewable Energy

DC-DCconverterPV RL

Figure 1: Photovoltaic system.

we describe the problem statement andwe show the influenceof climatic parameters such as temperature and irradiationon the electrical characteristic of the PV array panel. Then,we recall the model of photovoltaic panel. At the end of thesecond section, we describe the photovoltaic energy systemby a state model. The third section is reserved to presentthe control strategy. In the first part, we recall the T-S fuzzysystem; then we describe the photovoltaic energy system byan augmentedT-S fuzzymodel. In the second part of the thirdsection, we describe the T-S fuzzy reference model and theMPPT algorithm which is based on T-S fuzzy system.

Section four is devoted to the stability analysis of theclosed loop system. The feedback gains have been com-puted by solving LMIs expressions. The simulation resultsof photovoltaic energy system showing performances of theproposedMPPT algorithm tracker are discussed in Section 5.Conclusions are drawn in the final section.

2. Problem Statement

Thephotovoltaic power depends on climatic parameters suchas temperature and irradiation as shown in Figures 2 and3. In fact, the photovoltaic power, which is transmitted tothe load, is function of the impedance of the load and theclimatic parameters as shown in Figure 4.However, to changethe impedance seen by the panel, it is necessary to inserta DC/DC converter. The photovoltaic system consists of aphotovoltaic array panel connected to a DC-DC converterwhich provides energy to the load, as shown in Figure 1.

Figures 2 and 3 show the evolution of the generatedpower curves as a function of voltage, respectively, for a givenconstant irradiation and different values of temperature andthen for a given constant temperature and different values ofirradiation.

In conclusion, we can say that the PV array panel isnonlinear and time-variant system. From Figures 2 and 3,it is clear that the temperature affects essentially the voltageand the irradiation affects fundamentally the intensity of thePV array panel. Also, we can conclude that the output powergenerated by the PV array panel depends on the climaticparameters “𝐺 and 𝑇.” In fact, the power increases with anincrease in solar radiation and decreases with an increasein temperature. For each given pair of parameters (𝐺, 𝑇),there exists only one Maximum Power Point (MPP). Theoperating point is determined by the intersection of the panelcurrent-voltage characteristic and the load current-voltagecharacteristic.

0 5 10 15 20 250

10

20

30

40

50

60

70PV curve

V (V)

P(W

)

25∘

55∘

75∘

G = 1000W/m2

Figure 2: PV power curves with different 𝑇.

0 5 10 15 20 250

10

20

30

40

50

60

70Power curve

V (V)

P(W

)

G = 1000

G = 700

G = 400

T = 25∘C

Figure 3: PV power curves with different 𝐺.

0 5 10 15 20 250

1

2

3

4

5

6

Load characteristic

Panel characteristic

Vpv (V)

I(A

)

1/R op

1/R

1/R

Figure 4: Operating point of the PV array panel.

ISRN Renewable Energy 3

+

VV

IshRs

Ipv Ipv

IoIph(G, T)

Rsh−

Figure 5: Equivalent circuit for PV cell.

However, a specific algorithm tracker should be used tosearch the optimal operating point which permits to extractthe maximum power from the PV array panel.

In the next part of this section, we recall themost popularmodel which is developed by Singer et al. [8]; then wedescribe the modeling of the overall photovoltaic systemenergy.

The electrical equivalent circuit of the PV cell is given byFigure 5.

It consists of a current generator which depends onirradiation (𝐺) and temperature (𝑇), in parallel with a diode,and connected to an internal parallel and series resistor,namely, respectively, 𝑅sh and 𝑅

𝑠.

ThePVcellmodel is described by the following equations:

𝐼sh =𝑉 + 𝑅

𝑠𝐼pv

𝑅sh,

𝐼pv = 𝐼ph − 𝐼𝑜[exp(

𝑉 + 𝑅𝑠𝐼

𝑉𝑡

) − 1] −(𝑉 + 𝑅

𝑠𝐼)

𝑅sh,

𝑉𝑡=

𝑛𝑠𝑘𝑇

𝑞.

(1)

The generated current by the photovoltaic panel varies withtemperature and irradiation; its expression is given by thefollowing equation:

𝐼ph = (𝐼ph,𝑛 + 𝐾𝐼Δ𝑇)

𝐺

𝐺𝑛

. (2)

𝐼ph,𝑛 is the rated current generated by the PV panel understandard condition of temperature and irradiation (𝑇 = 25

∘Cand 𝐺 = 1000w/m2):

𝐼𝑜=

(𝐼ph + 𝐾𝐼Δ𝑇)

exp ((𝑉oc + 𝐾VΔ𝑇) /𝑉𝑡) − 1

, (3)

where 𝐼𝑜is a reverse saturation current:

𝑉oc = 𝑛𝑠

𝐾𝑇

𝑞log(

𝐼sc + 𝐼𝑜

𝐼𝑜

) ,

𝑉𝑐= 𝑛𝑠

𝐾𝑇

𝑞log(

𝐼sc + 𝐼𝑜− 𝐼pv

𝐼𝑜

) ,

(4)

where𝑉oc is the open circuit voltage and 𝐼sc is the short circuitcurrent.

The overall photovoltaic system energy can be repre-sented by the scheme illustrated in Figure 6.

The average dynamic model of the photovoltaic systemgiven by Figure 6 can be expressed in continuous conductionby the following equations:

𝑑𝑉pv

𝑑𝑡=

1

𝐶1

(𝐼pv − 𝐼𝐿) ,

𝑑𝐼𝐿

𝑑𝑡=

1

𝐿[𝑉pv − 𝑉

𝑐2(1 − 𝜇)] ,

𝑑𝑉𝑐2

𝑑𝑡=

1

𝐶2

[𝐼𝐿(1 − 𝜇) −

𝑉𝑐2

𝑅𝐿

] .

(5)

In the continuous conduction, the average value of 𝐼PVcurrent is equal to the average value of 𝐼

𝐿current.

It is very clear that the system can be described as the formof

�� (𝑡) = 𝐴 (𝑥, 𝑡) 𝑥 (𝑡) + 𝐵𝑢 (𝑡) , (6)

where 𝑥(𝑡) = [𝑉pv 𝐼𝐿

𝑉𝑐2]𝑇 is the state vector, 𝐴(𝑥, 𝑡) is the

state matrix, 𝐵 is the input vector, and 𝜇 is the duty ratio.However, the state matrix is nonlinear.

3. Control Strategy

The control strategy that we propose is given by Figure 7.The control strategy consists of three blocks: reference

model, controller, and plant, which is defined by PV system(see Figure 1).

3.1. T-S Fuzzy Model. Several studies have been proving thatthe Takagi-Sugeno fuzzy system can describe the behaviorof continuous nonlinear system. However, we use in thiswork the T-S fuzzy system to describe the nonlinear energyconversion system. The fuzzy model is described by fuzzyrules where each rule represents input-output relations oflinear local model. The 𝑖th rule of the fuzzy model has thefollowing form:

IF 𝑧1is 𝑀𝑖1and 𝑧

2is 𝑀𝑖2and, . . . , and 𝑧

𝑛is 𝑀𝑖𝑛

THEN{�� (𝑡) = 𝐴

𝑖𝑥 (𝑡) + 𝐵

𝑖𝑢 (𝑡) ,

𝑦 (𝑡) = 𝐶𝑖𝑥 (𝑡) , 𝑖 = 1, 2, . . . , 𝑐,

(7)

where {𝑀𝑖𝑗} are the fuzzy sets, 𝑥(𝑡) is the state vector, 𝑢(𝑡)

is the input vector, 𝐴𝑖is the state matrix, 𝐵

𝑖is the input

matrix, 𝑧1(𝑡), . . . , 𝑧

𝑛(𝑡) are the premise variables, 𝑥(𝑡) ∈ 𝑅

𝑛,𝐴𝑖∈ 𝑅𝑛×𝑛, 𝐵𝑖∈ 𝑅𝑛𝑥𝑚, 𝑦(𝑡) ∈ 𝑅

𝑚, and 𝑐 is the number of fuzzyrules. The global fuzzy model of the system has the followingform:

�� (𝑡) =∑𝑐

𝑖=1𝑤𝑖(𝑧 (𝑡)) [𝐴


𝑖𝑢 (𝑡)]

∑𝑐

𝑖=1𝑤𝑖(𝑧 (𝑡))

. (8)


Inductor Diode

Fuzzy MPPT

PV array

algorithm

Boost converter

Load

+

Ipv IL

LG

G

T

T

T

−

IC1IC2

Ich

VoutC1C2VPV RL

Figure 6: Photovoltaic system.

Photovoltaic energyconversion system

T-S fuzzy referencemodel

T-S fuzzycontroller (PDC)

𝜇 x(t)e(t)T

G −+

xr(t)

Figure 7: Control strategy.

For each rule 𝑅𝑖attributed a weight 𝑤

𝑖(𝑧(𝑡)) which depends

on grade of membership function of premise variables 𝑧𝑗(𝑡)

in fuzzy sets 𝑀𝑖𝑗:

𝑤𝑖(𝑧 (𝑡)) =

𝑛

∏

𝑗=1

𝑀𝑖𝑗(𝑧𝑗(𝑡)) , 𝑤

𝑖(𝑧 (𝑡)) ≥ 0,

𝑐

∑

𝑖=1

𝑤𝑖(𝑧 (𝑡)) > 0, for 𝑖 = 1, . . . , 𝑐.

(9)

𝑀𝑖𝑗(𝑧𝑗(𝑡)) is the grade of membership of 𝑧

𝑗(𝑡) to the fuzzy set

𝑀𝑖𝑗:

ℎ𝑗(𝑧 (𝑡)) =

𝑤𝑗(𝑧 (𝑡))

∑𝑐

𝑖=1𝑤𝑖(𝑧 (𝑡))

, 0 ≤ ℎ𝑖(𝑧 (𝑡)) ≤ 1,

𝑐

∑

𝑖=1

ℎ𝑖(𝑧 (𝑡)) = 1, 𝑖 = 1, . . . , 𝑐.

(10)

The polytopic form of state equation is

�� (𝑡) =

𝑐

∑

𝑖=1

ℎ𝑖(𝑧 (𝑡)) [𝐴


𝑖𝑢 (𝑡)] . (11)

3.2. Augmented T-S Fuzzy Model of Photovoltaic System. Inthe first stage of this subsection, we transform the averagedynamic model of the photovoltaic system described by (5)into an augmentedmodel. However, a new state variablemustbe added to the state vector. In other words an integrator isincluded previous to the real input 𝜇 and let �� = 𝑢.Therefore,𝜇 becomes new state variable and 𝑢 is the new control inputof the augmented system. Then, the system can be describedas a nonlinear system in the form of

�� (𝑡) = 𝐴 (𝑥, 𝑡) 𝑥 (𝑡) + 𝐵𝑢 (𝑡) , (12)

where 𝑥(𝑡) = [𝑉pv 𝐼𝐿

𝑉𝑐2

𝜇]𝑇 is the state vector, 𝐴(𝑥, 𝑡) is

the state matrix, and 𝐵 is the input matrix:

𝐴 =

[[[[[[[[[[

[

𝛼

𝐶1

−1

𝐶1

0 0

1

𝐿0 −

1

𝐿

𝛽

𝐿

01

𝐶2

−1

𝑅𝐿𝐶2

−𝛾

𝐶2

0 0 0 0

]]]]]]]]]]

]

, 𝐵 =[[[

[

0

0

0

1

]]]

]

. (13)

with 𝛼 = 𝐼pv/𝑉pv, 𝛽 = 𝑉𝑐2, and 𝛾 = 𝐼

𝐿.

It is clear that (12) is nonlinear. To obtain the T-S fuzzymodel, we choose the following three premises variables: 𝐼PV,𝑉PV, and𝑉

𝑐2. However, eight local models have been obtained

to describe the T-S fuzzymodel.The statematrix of each localmodel has the following structure:

𝐴𝑖=

[[[[[[[[[[

[

𝛼𝑖

𝐶1

−1

𝐶1

0 0

1

𝐿0 −

1

𝐿

𝛽𝑖

𝐿

01

𝐶2

−1

𝑅𝐿𝐶2

−𝛾𝑖

𝐶2

0 0 0 0

]]]]]]]]]]

]

, (14)

where each of the variables 𝛼𝑖

= 𝐼pv𝑖/𝑉pv𝑖, 𝛽𝑖 = 𝑉𝑐2𝑖, and

𝛾𝑖

= 𝐼𝐿𝑖

must be replaced, respectively, by the appropriatevalue according to fuzzy rule base (𝛼min or 𝛼max), (𝛽min or𝛽max), and (𝛾min or 𝛾max).

3.3. Reference Model. It is well known that the maximumpower produced by panel, also the corresponding optimal


Photovoltaic energyconversion system

T-S fuzzy reference

modelMPPT algorithm based on

T-S fuzzy controller

x(t)e(t)T

G

xr(t) 𝜇(t)−

c

∑i=1

hi(z)Ki−+

Figure 8: Control strategy of MPPT.

tension 𝑉MP and optimal current 𝐼MP, depends on temper-ature 𝑇 and irradiation 𝐺. However, these latest are used aspremise variables, 𝑍

𝑅1= 𝑇 and 𝑍

𝑅2= 𝐺, to compute the

reference model based on T-S fuzzy system. The 𝑖th rule ofthe fuzzy reference model has the following form:

IF 𝑧1𝑅

is 𝜆𝑖1and 𝑧

2𝑅is 𝜆𝑖2and, . . . , and 𝑧

𝑛𝑅is 𝜆𝑖𝑛

THEN{��𝑅(𝑡) = 𝐷

𝑖𝑥𝑅(𝑡) ,

𝑦𝑅(𝑡) = 𝐶

𝑅𝑖𝑥𝑅(𝑡) , 𝑖 = 1, 2, . . . , 𝑐𝑟,

(15)

where {𝜆𝑖𝑗} are the fuzzy sets, 𝑥

𝑅(𝑡) = [𝑉pv𝑟 𝐼

𝐿𝑟𝑉𝑐2𝑟

𝜇𝑟]𝑇

is the state reference variable vector, 𝐷𝑖∈ 𝑅𝑛×𝑛 is the local

reference statematrix, and {𝑧1𝑅

(𝑡), . . . , 𝑧𝑛𝑅

(𝑡)} are the premisevariables. 𝑦(𝑡) ∈ 𝑅

𝑚 is the output vector, 𝑐𝑟 is the number offuzzy rules. Then, the T-S fuzzy reference model is given bythe following equation:

��𝑅(𝑡) =

𝑐𝑟

∑

𝑖=1

𝜂𝑖(𝑧 (𝑡))𝐷

𝑖𝑥𝑅(𝑡) (16)

with ∑𝑐𝑟

𝑖=1𝜂𝑖(𝑧(𝑡)) = 1 and 𝐷

𝑖is the local reference state

matrix:

𝐷𝑖=

[[[[[[[[[[[[

[

𝛿𝑖

𝐶1

−1

𝐶1

0 0

1

𝐿0 −

1

𝐿

𝜎𝑖

𝐿

01

𝐶2

−1

𝑅𝐿𝐶2

−𝜁𝑖

𝐶2

1

1 − 𝜇op𝑖0 −1 0

]]]]]]]]]]]]

]

(17)

with 𝜁𝑖= 𝐼MPP𝑖, 𝛿𝑖 = 𝜁

𝑖/𝑉MPP𝑖, 𝜇op𝑖 = 1 − 1/√𝑅

𝐿𝛿𝑖, and 𝜎

𝑖=

𝑉MPP2𝑖 = 𝑉MPP𝑖/(1 − 𝜇op𝑖).

3.4. MPPT Algorithm Based on T-S Fuzzy Model. The crucialfunction of the MPPT algorithm is to search the coordinatesof the optimal operating point. In this work, the MPPTalgorithm provided at each time the appropriate duty ratio 𝜇

based on the desired and measured state variables. However,the MPPT algorithm which represents a controller is basedon Parallel Distributed Compensation (PDC) method. TheT-S fuzzy controller is designed from the T-S fuzzy modelof photovoltaic energy conversion system. The T-S fuzzycontroller is based on Parallel Distributed Compensation(PDC) technique, which is proposed by Wang et al. [9]. Theparticularity of the PDC controller is that it shares the same

fuzzy sets in the premise parts with the fuzzymodel. Based onthe T-S fuzzymodels, the PDC fuzzy controller is designed asfollows.

𝑖th controller rule:

IF 𝑧1is 𝑀𝑖1and 𝑧

2is 𝑀𝑖2and, . . . , and 𝑧

𝑛is 𝑀𝑖𝑛

THEN 𝑢 (𝑡) = −𝐾𝑖𝑥 (𝑡) , 𝑖 = 1, 2, . . . , 𝑐.

(18)

The global fuzzy controller is represented by

𝑢 (𝑡) = −

𝑐

∑

𝑖=1

ℎ𝑖(𝑧)𝐾𝑖𝑥 (𝑡) . (19)

4. Stability Analysis

In the previous section, we discussed the main blocks of thecontrol strategy and we have developed an MPPT algorithm(Figure 8). However, the system of energy conversion trans-mits the maximum PV generated power to the load and it iscrucial to ensure that the closed-loop system is stable.

Theorem 1. Consider the reference model (16) which is usedto compute the reference state variables, the nonlinear system(5) which can be modeled by the T-S fuzzy model (11), and theMPPT algorithm (19) based on the PDC techniques. If thereexist a common symmetric positive definite matrix 𝑄 and afeedback gains 𝐾

𝑖which satisfy the following LMIs (20), then

the closed loop system is asymptotically stable and the trackingerror converges toward zero:

[𝑄𝐴𝑇

𝑖+ 𝐴𝑖𝑄 − 𝐵

𝑖𝑀𝑖− 𝑀𝑇

𝑖𝐵𝑇

𝑖(𝐴𝑖− 𝐷𝑘) 𝑄

𝑄(𝐴𝑖− 𝐷𝑘)𝑇

−𝜌2

𝐼] < 0,

𝑓𝑜𝑟 𝑖 = 1, . . . , 𝑐, 𝑘 = 1, . . . , 𝑐𝑟,

[𝑄𝐴𝑇


𝑖𝑀𝑗− 𝑀𝑇

𝑗𝐵𝑇



−𝜌2

𝐼] < 0,

𝑓𝑜𝑟 𝑖 = 1, . . . , 𝑐, 𝑗 = 1, . . . , 𝑐, 𝑖 = 𝑗, 𝑘 = 1, . . . , 𝑐𝑟

(20)

with 𝜌 > 0.

Proof. The state tracking error is given by

𝑒 (𝑡) = 𝑥𝑅(𝑡) − 𝑥 (𝑡) . (21)


The following quadratic Lyapunov candidate function whichis positive definite, has been used to verify the system stabilityand compute the feedback gains 𝐾

𝑖:

𝑉 (𝑒) = 𝑒𝑇

(𝑡) 𝑃𝑒 (𝑡) +1

𝜌2∫

𝑡

0

𝑥𝑇

𝑅(𝜏) 𝑥𝑅(𝜏) 𝑑𝜏. (22)

The system is asymptotically stable if we prove that ��(𝑒) < 0:

�� (𝑒) = 𝑒𝑇

𝑃𝑒 + 𝑒𝑇

𝑃 𝑒 +1

𝜌2𝑥𝑇

𝑅𝑥𝑅,

𝑒 (𝑡) =

𝑐

∑

𝑖=1

𝑐𝑟

∑

𝑘=1

ℎ𝑖(𝑧) 𝜂𝑘(𝑧)

× [𝐴𝑖𝑒 (𝑡) + 𝐵

𝑖𝑢 (𝑡) + (𝐴

𝑖− 𝐷𝑘) 𝑥𝑅(𝑡)] ,

𝑒 (𝑡) =

𝑐

∑

𝑖=1

𝑐

∑

𝑗=1

𝑐𝑟

∑

𝑘=1

ℎ𝑖(𝑧) ℎ𝑗(𝑧) 𝜂𝑘(𝑧)

× [𝐴𝑖𝑒 (𝑡) − 𝐵

𝑖(𝐾𝑗𝑒) + (𝐴

𝑖− 𝐷𝑘) 𝑥𝑅(𝑡)] ,

�� (𝑒) =

𝑐

∑

𝑖=1

𝑐

∑

𝑗=1

𝑐𝑟

∑

𝑘=1


× [(𝐴𝑖− 𝐵𝑖𝐾𝑗) 𝑒 (𝑡)

+ (𝐴𝑖− 𝐷𝑘) 𝑥𝑅(𝑡) ]𝑇

𝑃𝑒

+ 𝑒𝑇

𝑃

𝑐

∑

𝑖=1

𝑐

∑

𝑗=1

𝑐𝑟

∑

𝑘=1


× [(𝐴𝑖− 𝐵𝑖𝐾𝑗) 𝑒 (𝑡)

+ (𝐴𝑖− 𝐷𝑘) 𝑥𝑅(𝑡) ]

+1

𝜌2𝑥𝑇

𝑅𝑥𝑅,

�� (𝑒) =

𝑐

∑

𝑖=1

𝑐

∑

𝑗=1

𝑐𝑟

∑

𝑘=1


× (𝑒𝑇

(𝐴𝑖− 𝐵𝑖𝐾𝑗)𝑇

𝑃𝑒 + 𝑒𝑇

𝑃 (𝐴𝑖− 𝐵𝑖𝐾𝑗) 𝑒

+ 𝑥𝑇

𝑅(𝐴𝑖− 𝐷𝑘)𝑇

𝑃𝑒 + 𝑒𝑇

𝑃 (𝐴𝑖− 𝐷𝑘) 𝑥𝑅)

+1

𝜌2𝑥𝑇

𝑅𝑥𝑅,

�� (𝑒) =

𝑐

∑

𝑖=1

𝑐𝑟

∑

𝑘=1

ℎ2

𝑖(𝑧) 𝜂𝑘(𝑧)

× (𝑒𝑇

(𝐴𝑖− 𝐵𝑖𝐾𝑖)𝑇

𝑃𝑒 + 𝑒𝑇

𝑃 (𝐴𝑖− 𝐵𝑖𝐾𝑖) 𝑒

+ 𝑥𝑇

𝑅(𝐴𝑖− 𝐷𝑖)𝑇

𝑃𝑒 + 𝑒𝑇

𝑃 (𝐴𝑖− 𝐷𝑘) 𝑥𝑅

+1

𝜌2𝑥𝑇

𝑅𝑥𝑅)

+

𝑐

∑

𝑖=1

𝑖 = 𝑗

𝑐

∑

𝑗=1

𝑐𝑟

∑

𝑘=1


× (𝑒𝑇


𝑃𝑒

+ 𝑒𝑇

𝑃 (𝐴𝑖− 𝐵𝑖𝐾𝑗) 𝑒 + 𝑥

𝑇


𝑃𝑒

+ 𝑒𝑇

𝑃 (𝐴𝑖− 𝐷𝑘) 𝑥𝑅+

1

𝜌2𝑥𝑇

𝑅𝑥𝑅) .

(23)

We note in this analysis that the feedback gains and stabilityconditions will be transformed to an LMI problem. However,the inequality ��(𝑒) < 0 is guaranteed when the followinginequalities are satisfied:

𝑒𝑇

(𝐴𝑖− 𝐵𝑖𝐾𝑖)𝑇

𝑃𝑒 + 𝑒𝑇

𝑃 (𝐴𝑖− 𝐵𝑖𝐾𝑖) 𝑒 + 𝑥

𝑇


𝑃𝑒

+ 𝑒𝑇


1

𝜌2𝑥𝑇

𝑅𝑥𝑅

< 0,

𝑒𝑇


𝑃𝑒 + 𝑒𝑇

𝑃 (𝐴𝑖− 𝐵𝑖𝐾𝑗) 𝑒 + 𝑥

𝑇


𝑃𝑒

+ 𝑒𝑇


1

𝜌2𝑥𝑇

𝑅𝑥𝑅

< 0.

(24)

Using Schur complement [10], the inequalities given in (24)can be written as

[

[

𝐴𝑇

𝑖𝑃 + 𝑃𝐴

𝑖− 𝑃𝐵𝑖𝐾𝑖− 𝐾𝑇

𝑖𝐵𝑇

𝑖𝑃 𝑃 (𝐴

𝑖− 𝐷𝑘)

(𝐴𝑖− 𝐷𝑘)𝑇

𝑃 −𝜌2

𝐼

]

]

< 0,

[

[

𝐴𝑇

𝑖𝑃 + 𝑃𝐴

𝑖− 𝑃𝐵𝑖𝐾𝑗− 𝐾𝑇

𝑗𝐵𝑇

𝑖𝑃 𝑃 (𝐴

𝑖− 𝐷𝑘)

(𝐴𝑖− 𝐷𝑘)𝑇

𝑃 −𝜌2

𝐼

]

]

< 0,

for 𝑖 = 1, . . . , 𝑐, 𝑗 = 1, . . . , 𝑐, 𝑖 = 𝑗, 𝑘 = 1, . . . , 𝑐𝑟.

(25)

Since coupled elements, such as 𝑃𝐵𝑖𝐾𝑖, have been enclosed

in these inequalities, then we have BiLMIs inequalities.However, we must transform them to the LMIs using acongruence transformation by diag [𝑃

−1

𝐼 𝐼] to (29) andconsidering 𝑄 = 𝑃

−1, 𝑀𝑖= 𝐾𝑖𝑃−1, we obtain the following

matrices in the LMI form:

[𝑄𝐴𝑇


𝑖𝑀𝑖− 𝑀𝑇

𝑖𝐵𝑇



−𝜌2

𝐼] < 0,

[𝑄𝐴𝑇


𝑖𝑀𝑗− 𝑀𝑇

𝑗𝐵𝑇



−𝜌2

𝐼] < 0,

for 𝑖 = 1, . . . , 𝑐, 𝑗 = 1, . . . , 𝑐, 𝑖 = 𝑗, 𝑘 = 1, . . . , 𝑐𝑟.

(26)


Table 1: Characteristics of the PV array panel.

𝑁𝑝= 1

𝑁𝑠= 36

𝑞 = 1.6𝑒 − 19C𝐴 = 1.92

𝐸𝑔= 1.1

𝑇𝑟= 298.18

∘K𝑇𝑟= 25∘C

𝐼or = 9.579𝑒 − 6A𝑉co = 27.4V𝑅𝑠= 0.09Ω

𝑅sh = 100Ω

𝑅load = 𝑅𝐿= 30Ω

𝑃max = 61W𝐼sc = 4.8A𝐾𝐼= 0.00171A/∘C

𝐹 = 10KHz𝐶2= 68 𝜇F

𝐾 = 1.38.10−23 J/K (Boltzmann’s constant)

5. Simulation Results

This section is reserved for presenting the main results.We use MATLAB to simulate the behavior of the energyconversion system. The main characteristics of the PV arraypanel are given by Table 1.

The resolution of the LMIs gives the following matricesand feedback gains, respectively, 𝑃,𝑄,𝐾

1,𝐾2,𝐾3,𝐾4,𝐾5,𝐾6,

𝐾7, and 𝐾

8.

𝑃 = 100[[[

[

78.9262 −0.9022 0.5037 −0.0002

−0.9022 0.1748 −0.0088 −0.0000

0.5037 −0.0088 0.0056 −0.0000

−0.0002 −0.0000 −0.0000 0.0008

]]]

]

,

𝑄 =[[[

[

0.0003 0.0002 −0.0266 0.0000

0.0002 0.0623 0.0802 0.0004

−0.0266 0.0802 4.3183 −0.0007

0.0000 0.0004 −0.0007 12.1668

]]]

]

,

𝐾1= −1.0𝑒 + 004 [−0.3951 −0.0579 −0.0008 −4.5760] ,

𝐾2

= −1.0𝑒 + 004 [−0.3661 −0.0492 −0.0035 −4.5715] ,

𝐾3= −1.0𝑒 + 004 [−0.4047 −0.0608 −0.0009 −4.5780] ,

𝐾4= −1.0𝑒 + 004 [−0.3983 −0.0543 −0.0010 −4.5786] ,

𝐾5= −1.0𝑒 + 004 [−0.3757 −0.0522 −0.0036 −4.5735] ,

𝐾6= −1.0𝑒 + 004 [−0.3693 −0.0457 −0.0037 −4.5741] ,

𝐾7= −1.0𝑒 + 004 [−0.4015 −0.0644 −0.0007 −4.5754] ,

𝐾8= −1.0𝑒 + 004 [−0.3725 −0.0557 −0.0035 −4.5710] .

(27)

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

35

40

45

50

55

Times

Tem

pera

ture

(∘C)

×104

Figure 9: Evolution of temperature.

500

600

700

800

900

1000

1100

1200Ir

radi

atio

n

0 0.5 1 1.5 2 2.5Times ×104

Figure 10: Evolution of irradiation.

To demonstrate the performance of the proposed MPPTcontrol approach, we apply a sudden variation of temperatureor solar irradiation as shown in Figures 9 and 10.

In Figure 9, we have applied a sudden change of tem-perature, although it is impossible to have a really dramaticchange.

In this test, we have chosen four pairs of irradiation andtemperature. We know that for each pair there exists onlyone optimal operating point which can be determined fromthe power-voltage characteristics of the PV array panel whichis not always available for each pair (𝐺, 𝑇). It is importantto mention that it is not possible to know the appropriatecoordinates of the ideal optimal operating point (𝑉MPP, 𝐼MPP)for all pairs (𝐺, 𝑇) as there are infinitely of pairs (𝐺, 𝑇).

In Table 2, we give the ideal corresponding values(𝑉MPPR, 𝐼MPPR) of operating point for each pair of temperature


Table 2: Coordinates for each operating point.

Temperature in ∘C Irradiation (wm−2) 𝑉MPPR (V) 𝐼MPPR (A) 𝑉MPP (V) 𝐼MPP (A)35 800 14.445 3.343 14.2494 3.287535 700 14.519 2.940 14.2226 2.880525 1000 14.92 4.16 14.9846 4.145745 700 13.794 2.917 13.4918 2.8504

10

15

20

25

30

35

40

45

50

0 0.5 1 1.5 2 2.5Times ×104

Vc2Vc2r

Vs

(V)

Figure 11: Evolution of output voltage.

Erro

r (V

)

−30

−25

−20

−15

−10

−5

10

5

0

15

0 0.5 1 1.5 2 2.5Times ×104

Figure 12: Evolution of error output voltage.

and irradiation and the computed values (𝑉MPP, 𝐼MPP) by ouralgorithm.

Figures 11, 12, 13, 14, 15, 16, 17, 18, and 19 show, respectively,the evolution of 𝑉MPP voltage,error of 𝑉MPP voltage, output

13

13.5

14

14.5

15

15.5

0 0.5 1 1.5 2 2.5Times ×104

VMPVMPR

Evolution of operating panel voltage

Vpv

(V)

Figure 13: Evolution of the 𝑉MPP voltage.

Erro

r

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5

−3

−3.50 0.5 1 1.5 2 2.5

Times ×104

Evolution of operating panel voltage error

Figure 14: Evolution of error 𝑉MPPvoltage.

voltage of converter, error of output voltage of converter,panel current, error of panel current, delivered power, errorof delivered power, and the duty cycle.

In Figures 12, 13, 14, 15, 16, 17, and 18, we observe momen-tary peaks; they are due to sudden and significant change intemperature and irradiation.The changes in temperature andirradiation are notmade as that way given in Figures 9 and 10,but we have used it to show the performance of the proposed


0 0.5 1 1.5 2 2.51

1.5

2

2.5

3

3.5

4

4.5

Times ×104

I pv

Ipv

(A)

Ipvr

Evolution of PV current

Figure 15: Evolution of the inductance current.

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5Times ×104

Erro

r (A

)

Evolution of error current operating panel

Figure 16: Evolution of the error inductance current.

algorithm. It is clear that at the steady state, the errorstend toward zero and the state variables reach the referenceone. Also, it is visible that the computed coordinates, ofoptimal operating point, based on the proposed algorithm,are almost the same as the ideal optimal operating point.This analysis allows demonstrating the performance of theproposed algorithm.

6. Conclusion

In this paper, a new algorithm strategy based on the aug-mented Takagi-Sugeno type fuzzy system has been proposed

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5Times ×104

P

Pr

Figure 17: Evolution of the power.Po

wer

erro

r (W

)

−40

−30

−20

−10

0

10

20

30

0 0.5 1 1.5 2 2.5Times ×104

Figure 18: Evolution of the error power.

for the MPPT of a PV energy system. All the PV systemhas been modeled by T-S fuzzy system. Based on themeasurement of temperature and irradiation, we deduce thecoordinates of the desired optimal operating point whichcorresponds to the maximum power.

The MPPT algorithm is based on an augmented T-Sfuzzy model and PDC method. The controller parametershave been computed based on the LMI tools. The stability ofsystem has been proved based on Lyapunov approach. Thesimulation results show that the proposed algorithm tracksquickly the optimal operating point despite sudden variationsof temperature and irradiation.


0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 0.5 1 1.5 2 2.5Times ×104

Figure 19: Evolution of the duty cycle.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

[1] A. Chermitti, O. Boukli-Hacene, and B. Mohamed, “Improve-ment of the “perturb and observe” MPPT algorithm in a pho-tovoltaic system under rapidly changing climatic conditions,”International Journal of Computer Applications, vol. 56, no. 12,pp. 5–10, 2012.

[2] J. J.Nedumgatt, K. B. Jayakrishnan, S.Umashankar,D.Vijayaku-mar, and D. P. Kothari, “Perturb and observe MPPT algorithmfor solar PV systems-modeling and simulation,” in Proceedingsof the Annual IEEE India Conference: Engineering SustainableSolutions (INDICON ’11), Hyderabad, India, December 2011.

[3] A. P. K. Yadav, S. Thirumaliah, and G. Haritha, “Comparisonof MPPTalgorithms for DC-DC converters based PV systems,”International Journal of Advanced Research in Electrical, Elec-tronics and Instrumentation Engineering, vol. 1, no. 1, pp. 18–23,2012.

[4] M. Azzouzi, “Comparaison between MPPT P&O and MPPTfuzzy controls in optimizing the photovoltaic generator,” Inter-national Journal of Advanced Computer Science and Applica-tions, vol. 3, no. 12, pp. 57–62, 2012.

[5] S. Gomathy, S. Saravanan, and S. Dr. Thangave, “Design andimplementation of maximum power point tracking (MPPT)algorithm for a standalone PV system,” International Journal ofScientific & Engineering Research, vol. 3, no. 3, pp. 2229–5518,2012.

[6] G. Balasubramanian and S. Singaravelu, “Fuzzy logic controllerfor themaximum power point tracking in photovoltaic system,”International Journal of Computer Applications, vol. 41, no. 12,pp. 22–28, 2012.

[7] L. Jie and C. Ziran, “Research on the MPPT algorithms of pho-tovoltaic system based on PV neural network,” in Proceedingsof the Chinese Control and Decision Conference (CCDC ’11), pp.1851–1854, Mianyang, China, May 2011.

[8] S. Singer, B. Rozenshtein, and S. Surazi, “Characterization of PVarray output using a small number of measured parameters,”Solar Energy, vol. 32, no. 5, pp. 603–607, 1984.

[9] H. O. Wang, K. Tanaka, and M. Griffin, “Parallel distributedcompensation of nonlinear systems by Takagi-Sugeno fuzzymodel,” in Proceedings of the IEEE International Conference onFuzzy Systems, pp. 531–538, March 1995.

[10] L. El Ghaoui andG. Scorletti, “Control of rational systems usinglinear-fractional representations and linearmatrix inequalities,”Automatica, vol. 32, no. 9, pp. 1273–1284, 1996.

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