research article a newton-based extremum seeking mppt...

Research ArticleA Newton-Based Extremum Seeking MPPT Method forPhotovoltaic Systems with Stochastic Perturbations

Heng Li12 Jun Peng12 Weirong Liu12 Zhiwu Huang12 and Kuo-Chi Lin3

1 School of Information Science and Engineering Central South University Changsha 410075 China2Hunan Engineering Laboratory for Advanced Control and Intelligent Automation Changsha 410075 China3Department of Mechanical and Aerospace Engineering University of Central Florida Orlando FL 32816 USA

Correspondence should be addressed to Jun Peng pengjcsueducn

Received 2 May 2014 Revised 6 July 2014 Accepted 6 July 2014 Published 24 July 2014

Academic Editor Dimitrios Karamanis

Copyright copy 2014 Heng Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Microcontroller basedmaximumpower point tracking (MPPT) has been themost popularMPPT approach in photovoltaic systemsdue to its high flexibility and efficiency in different photovoltaic systems It is well known that PV systems typically operate undera range of uncertain environmental parameters and disturbances which implies that MPPT controllers generally suffer from someunknown stochastic perturbations To address this issue a novel Newton-based stochastic extremum seeking MPPT method isproposed Treating stochastic perturbations as excitation signals the proposed MPPT controller has a good tolerance of stochasticperturbations in nature Different from conventional gradient-based extremum seeking MPPT algorithm the convergence rateof the proposed controller can be totally user-assignable rather than determined by unknown power map The stability andconvergence of the proposed controller are rigorously proved We further discuss the effects of partial shading and PV moduleageing on the proposed controller Numerical simulations and experiments are conducted to show the effectiveness of the proposedMPPT algorithm

1 Introduction

Recent years have seen a growing interest in the research ofsolar energy which is mainly due to the advantages solarenergy has over traditional fossil energies including safetysustainable source of energy less environment pollution andnumerousmarket potentials As themain available solar tech-nology photovoltaic array has been widely used in satellitesand spacecrafts solar vehicles and domestic power supplysee [1 2] and the references therein

The process of guiding a photovoltaic array to its maxi-mum power point is called maximum power point trackingDue to the inherent nature of photovoltaic cells the power-voltage curves of PV cells depend nonlinearly on temperatureand irradiation intensity see [3ndash5]This fact however meansthat the operating current or voltage that maximizes the out-put power will change with environmental conditions Tomaintain the maximum output power regardless of environ-mental parameters one common way is to design MPPT

control system to regulate operating current or voltage to themaximum output power point

With the rapid development of embedded technologymicrocontroller based MPPT control system has been thedominated approach in current photovoltaic systems [5]Thisapproach is favoured because researchers and engineers pre-fer to program elegant and advancedMPPT algorithms ratherthan design complicated and expensive MPPT control cir-cuits

There are a number of MPPT control algorithms thathave been proposed in the literature among which the twoprimary algorithms are incremental conductance (IncCond)algorithm [5] and perturb and observe (PO) algorithm [6] InPO algorithms a step perturbation is added into the controlsignal and used to monitor the direction of changes in powerPO has been a commercial algorithm because of its ease ofimplementation The main drawback of PO algorithm is thatit will oscillate at the maximum power point Also it hasbeen shown that PO fails to track the rapidly changing

Hindawi Publishing CorporationInternational Journal of PhotoenergyVolume 2014 Article ID 938526 13 pageshttpdxdoiorg1011552014938526

2 International Journal of Photoenergy

irradiance [7] The IncCond algorithm tracks the maxi-mum power point by comparing the instantaneous andincremental conductances of the PV array Thus it can trackthe rapidly changing irradiance However the harmoniccomponents of array voltage and current need to bemeasuredand used to adjust the reference voltage which implies thaterrors at the maximum power point occur due to the lowprecision sensors used [8]

A promising newMPPTalgorithm is themethod of extre-mum seeking (ES) control see [9ndash12] As a model-free real-time optimization method ES is well suited for applicationswith unknown or partly known dynamics and externalperturbations such as photovoltaic systems [11] ES uses per-turbation signals (either from external disturbances or fromconverter ripples) as probing signals to estimate the gradi-ent of the power map and then update the control signalaccording to the estimation [12] It has been shown that EShas the advantages of both rigorously provable convergenceand simplicity of hardware implementation [13] Howeverexisting extremum seekingMPPT controllers still suffer fromthe following two limitations

(i) In existing extremum seekingMPPT controllers per-turbation signals are generally assumed to be peri-odic On the one hand the assumption is rather idealsince external disturbances are typically unknownand stochastic on the other hand the requirement oforthogonality makes periodic ES difficult to extend tomultivariable cases

(ii) In existing extremum seeking MPPT controllers theconvergence speed is typically defined by the gradientof power map of PV systems which implies thatcontrol systemswill be highly influenced by unknownand changing environmental conditions To constructa practical MPPT control system the convergenceof MPPT controller should be user-assignable ratherthan being dependent on environmental conditions

Thus the aim of this paper is to provide an improvedextremum seeking MPPT control algorithm to solve theseproblems Recent progress in stochastic averaging theoryhas made it possible to consider more general stochasticperturbations when designing extremum seeking controllerssee [14 15] In [14] Liu and Krstic provide a systematic designof extremum seeking when stochastic perturbations existand prove the stability of the stochastic extremum seekingvia a developed stochastic averaging theory The theoreticalpart of this paper is based on the findings of [14 15] andwe further propose a Newton-based stochastic extremumseeking MPPT controller considering physical features ofphotovoltaic systems We conduct extensive simulations andexperiments to show the effectiveness of the proposed controlalgorithm

The remainder of the paper is organized as follows InSection 2 preliminary knowledge about photovoltaic mod-elling and maximum power point tracking is introduced InSection 3 a new Newton-based stochastic extremum seekingMPPT controller is designed The tracking performanceof the proposed controller is evaluated in Section 4 Weconclude the paper in Section 5

Iph

Rp

Rs

Load

Ipv

UpvPV

module

Id

Figure 1 The equivalent electrical circuit of a photovoltaic module

2 Photovoltaic Modelling and MPPT

In this section we will introduce some preliminary knowl-edge about PV array and MPPT A photovoltaic array typ-ically consists of several photovoltaic modules connected inseries or parallel to obtain the desired output voltage and cur-rent Thus we first introduce the electrical characteristics ofPV modules More detailed background information can befound in [16]

A photovoltaic module can be seen as a black box thatproduces a current 119868pv at a voltage119880pvThe inside of that blackbox can be described by an electric circuit with 4 componentsas shown in Figure 1

(i) Current source this is the source of the photo currentwhich can be denoted by

119868ph = 119878 sdot 119867 sdot 120585 (1)

where 119878 is the module area 119867 is the intensity ofincoming light and 120585 is the response factor

(ii) Diode this nonlinear element reflects the dependenceon the band gap and losses to recombination It ischaracterized by the reverse current 119868119889

(iii) Shunt resistor 119877119901 it represents losses incurred byconductors

(iv) Serial resistor 119877119904 it also represents losses incurred bynonideal conductors

The relationship between current 119868pv and voltage 119880pv of aphotovoltaic module is then expressed by

119868pv = 119868ph minus 119868119889 [exp(

119880pv + 119868pv119877119904

119880119879

) minus 1] minus

119880pv + 119868pv119877119904

119877119901

(2)

where 119880119879 = 119902119896119879119890 with ideality factor 119902 temperature 119879Boltzmann constant 119896 = 138119890

minus23 and the elementary charge119890 = 1602119890

minus19Then the output power 119875pv of a photovoltaic module is

denoted by

119875pv = 119880pv119868pv = 119891 (119880pv) (3)

where function 119891(sdot) is determined by manufacturing andenvironmental parameters such as module area 119878 irradiance

International Journal of Photoenergy 3

0 5 10 15 20 250

05

1

15

2

25

3

35

4

Voltage (V)

Curr

ent (

A)

Maximum power pointH = 1000Wm2 T = 25∘CH = 800Wm2 T = 25∘C

H = 600Wm2 T = 25∘C

Figure 2 I-V characteristics of KC65T PV module at various irra-diance levels

119867 response factor 120585 and temperature119879 and thus is generallyunknown or partially known a priori

Figure 2 shows the I-V characteristics of the KC65Tphotovoltaic module at various irradiance levels see thedatasheet of KC65T [18] With light varying its intensitythroughout the day the maximum power point moves todifferent voltages and currents Thus a MPPT control systemis typically adopted between the photovoltaic module and theload to adjust the output voltage or current and maintain themaximum power output

The schematic of a photovoltaic MPPT control systemwith stochastic perturbations is shown in Figure 3 The pho-tovoltaic panel converts solar energy to electrical energy viathe photovoltaic effect A MPPT control system is designedbetween the photovoltaic panel and load to optimize the con-version efficiencyThe adoptedMPPT control system consistsof three components MPPT controller DC-DC driver andDC-DC converter The proposed MPPT system can be eithervoltage control or current control depending on which oneis used as the control variable of the MPPT controller Inthis paper we design the MPPT control system with voltagecontrol but the proposed method is also suited to currentcontrol cases

As the DC-DC driver and converter are relatively maturein electrical industry the main work in MPPT controlsystem is the design of MPPT control algorithm Such aresearch interest has continued for decades since even smallimprovements in performance can lead to great economicbenefitsThough considerable progress has beenmade in thisfield the following two problems have not been effectivelysolved

(i) how to optimize the output power with unknownpower map when stochastic external disturbancesexist

(ii) how to assign the convergence speed by designersrather than unknown power map

3 MPPT Controller Design

In this section we propose a new Newton-based stochasticextremum seeking MPPT controller to handle the abovetwo difficulties The proposed controller not only inheritsthe advantages of classical extremum seeking in model-freeoptimization but also shows its distinctive features in dealingwith stochastic perturbations and assigning convergencerates In what follows we will introduce the control struc-ture implementation and stability analysis of the proposedcontroller Some further discussions will be provided inSection 35

31 Controller Structure In order to illustrate the motivationof designing Newton-based extremum seeking MPPT con-troller we first introduce the application of classical gradient-based extremum seeking in MPPT of photovoltaic systemsFigure 4 shows the control structure of a gradient-basedextremum seeking MPPT controller

As shown in Figure 4 120591 is the duty cycle of PWM waves120578 is the stochastic perturbation and pv is the estimation ofdesired voltage that optimizes the power map The model ofDC-DC driver and converter can be simplified as follows

120591 = 119892 (119880pv pv) (4)

119880pv = ℓ (120591) (5)

The maximum power point (MPP) of the photovoltaicsystem is defined as

119875lowast

pv = 119891 (119880lowast

pv) = 119891 (ℓ (120591lowast)) ≜ max119875pv (6)

where 119880lowast

pv 120591lowast are the desired voltage and duty cycle at the

MPPWe then denote the voltage estimation error pv by

pv = pv minus 119880lowast

pv (7)

For the purpose of illustration assume that the powermap 119891(sdot) is of the quadratic form then the averaged systemis obtained by

119880pv = 119896Hpv (8)

where H is the second derivative (Hessian) of the power mapEquation (8) shows that the convergence rate is governed bythe unknown Hessian H which is highly influenced by envi-ronmental conditions such as irradiance and temperature

A practical MPPT control system often requires that theconvergence of the controller should be designer-assignableWe next show that this goal can be realized with the Newton-based extremum seeking

If the power map is known the following Newton opti-mization algorithm can be used to find the maximum powerpoint

119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv (9)


Sun irradiation

Photovoltaic panel MPPTcontroller

DC-DC converter

LoadC

LSW

DUpv

Ipv

Upv Ipv

Upv

PI DC-DC driverUlowast

pv

Stochasticperturbations

+minus+minus

Figure 3 The schematic of a photovoltaic MPPT control system with stochastic perturbations

DC-DCconverter

120591 DC-DCdriver

Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)

Figure 4The structure of gradient-based extremum seekingMPPTcontroller

If119891(sdot) is unknown then an estimator is needed to approx-imate 119889119891(119909)119889119909 and 119889

2119891(119909)119889119909

2 Then the purpose of thecontroller design is to combine the Newton optimizationalgorithm (9) with estimators of the first and second deriva-tives of power map to achieve the maximum power pointtracking

Let 119866 be the estimation of the first-order derivative Hthe estimation of the second-order derivative and Γ theestimation of inverse of second derivative Then we pro-pose a Newton-based stochastic extremum seeking MPPTcontroller in Figure 5 As shown in Figure 5 the first-orderderivative of 119891 is estimated by 119866 = 119875pv119872(120578) and the second-order derivative of 119891 is estimated by H = 119875pv119873(120578) It istypically difficult to derive Γ = 1H directly when H isclose to zero Instead a dynamic estimator Ξ = minusℎΞ + ℎHis employed to approximate Γ

We rewrite the proposed control algorithm in state spaceas

119880pv = 119892 (120591)

120591 = ℓ (119880pv pv)

119880pv = minus119896Γ119872(120578) 119875pv

Γ = ℎΓ minus ℎΓ2119873(120578) 119875pv

(10)

where 120578 is a general stochastic signal and 119878(120578) = 119886 sin(120578)119872(120578) and119873(120578) are the output of signal generatorΨ with theoriginal signal 120578

Remark 1 The signal generator outputs 119872(sdot) 119873(sdot) play animportant role in the Newton-based stochastic extremumseeking since they are used to estimate the first-order andsecond-order derivative of power map 119891(sdot) respectivelyTheoretically they can be any bounded and odd continuousfunctions However considering the practical stability of thephotovoltaic system we choose 119872(sdot) 119873(sdot) by following thedesign principle in [19] as

119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(11)

where variables 1198820(119902) and 1198821(119902) are defined as follows

1198822

0(radic2119902) = 2 (1198821 (119902) minus 119882

2

0(119902))

1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(12)

32 Controller Implementation In this subsection we pro-vide a detailed flow chart of the implementation of theproposed Newton-based extremum seeking MPPT methodas shown in Figure 6 The basic idea of the controller imple-mentation is that the controller measures the output voltageand current and then computes the output power Then thefirst derivative and second derivative of the power map areestimated respectively based on the product of output powerand stochastic signals If the termination criterion is satisfiedgo to the next iteration If not regulate the duty cycle andoutput voltage to update the output power and go to the nextiteration


DC-DCconverter 120591

Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)

S(120578(t))M(120578(t))

N(120578(t))Signalgenerator Ψ

G

H

minusΓG

DC-DCdriver

times

Figure 5 The structure of Newton-based extremum seeking MPPT controller

Initialization

MeasureUpv (k) and Ipv (k)

Calculate powerPpv (k) = Upv (k) middot Ipv (k)

Estimate the first derivative of power map

Estimate the second derivative of power map

G(k) = 0 H(k) lt 0

Compute the estimation voltage

Yes

No

Upv (k + 1) = minuskΓ(k)M(120578(k))Ppv (k)

Γ(k + 1) = hΓ(k) minus hΓ2(k)N(120578(k))Ppv (k)

Update the duty cycle of DC-DC driver120591(k) = 120001(Upv (k) Upv (k))

Update the output voltage of DC-DC converterUpv (k) = g(120591(k))

k larr k + 1

(k) = Ppv (k) middot N(120578(k))H

(k) = Ppv (k) middot M(120578(k))G

Figure 6 Flow chart of theNewton-based extremumseekingMPPTmethod

33 A Heuristic Analysis In this subsection we provide aheuristic analysis of the Newton-based extremum seekingMPPT controller This illustrative analysis will be helpful

Voltage

A B C

Pow

er

UpvUpv

120591

120591

Figure 7 Three different regions of an illustrative P-V curve

in understanding the proposed MPPT algorithm A morerigorous stability analysis is provided in Section 34

In the I-V characteristics of a PV module there are threedifferent regions namely current source regionMPP regionand voltage source region [20] These three regions corre-spond to the regions A B and C in the P-V curve as shownin Figure 7 We will introduce the actions of the MPPT con-troller in these three regions and explain how the MPPT isachieved

The update of control signal 120591 in (10) can be furthersimplified as

120591 (119896 + 1) = 120591 (119896) + Δ120591 (119896) (13)

where Δ120591(119896) is the update of the control signal 120591 in one itera-tion

In what follows we will introduce the update of controlinput 120591 and output voltage119880pv in the following three differentregions


Region A In this region the first derivative of the power map119889119875pv119889119880pv gt 0 and the second derivative 119889

2119875pv119889119880

2

pv lt 0Then the Newton optimization algorithm (9) turns to be

119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvgt 0 (14)

which means that the update of the control signal Δ120591 gt 0Then the duty cycle 120591 in (10) is increasing The larger 120591 willdrive the DC-DC converter to produce larger voltage 119880pvand then 119880pv will move in the direction of converging to thedesired voltage 119880

lowast

pv

Region B At the MPP the first derivative 119889119875pv119889119880pv = 0 andthe second derivative 119889

2119875pv119889119880

2

pv lt 0Then the Newton opti-mization algorithm (9) is

119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv= 0 (15)

which means that the change of control signal Δ120591 = 0 Thenthe control input 120591 and output voltage 119880pv keep stable at thedesired 120591

lowast and 119880lowast

pv which implies that the MPPT is realized

Region C In this region the first derivative of the power map119889119875pv119889119880pv lt 0 and the second derivative 119889

2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvlt 0 (16)

which means that the change of control signal Δ120591 lt 0 Thenthe duty cycle 120591 in (10) is decreasing The smaller 120591 willdrive the DC-DC converter to produce smaller voltage 119880pvand then 119880pv will move in the direction of converging to thedesired voltage 119880

lowast

pv

34 Stability Analysis Stability is a basic requirement of apractical control system In this subsection we analyze thestability of the proposed MPPT algorithm using stochasticaveraging theory Then the main theoretical contribution ofthis paper is concluded as follows

Theorem 2 Consider the photovoltaic MPPT control systemshown in Figure 3 with DC-DC driver (4) DC-DC converter(5) and unknown power map (3) Assume that the outputpower119875119901V reaches its maximum119875

lowast

119901V at the desired control input120591lowast and desired voltage119880

lowast

119901V Then under the control law (10) forthe update of 120591 the stability of the closed-loop system is guaran-teed Moreover the output power 119875119901V exponentially convergesto the maximum power 119875

lowast

119901V which implies that the maximumpower point tracking is achieved

Proof We assume that the stochastic perturbation 120578(119905) con-sidered in this paper has invariant distribution 120583(119889119909) =

(1radic120587119902)119890minus11990921199022

119889119909 Denote the estimation error 120591 = 120591 minus 120591lowast

Γ = Γ minus Hminus1 Then the error system can be denoted by

120591 = minus119896 (Γ + Hminus1)119872 (120578) 119891 (119892 (120591lowast

+ 120591 + 119886 sin (120578)))

Γ = ℎ (Γ + Hminus1)

minus ℎ(Γ + Hminus1)2

119873(120578)119891 (119892 (120591lowast

+ 120591 + 119886 sin (120578)))

(17)

For simplicity a quadratic power map is considered

119891 (ℓ (120591)) = 119891lowast

+11989110158401015840

(ℓ (120591lowast))

2(120591 minus 120591

lowast)2= 119891lowast

+H2

(120591 minus 120591lowast)2

(18)

Then the error system can be simplified as

120591 = minus119896 (Γ + Hminus1)119872 (120578) (119891lowast

+H2

(120591 + 119886 sin (120578))2)



119873(120578) (119891lowast

+H2

(120591 + 119886 sin (120578))2)

(19)

To guarantee the stability of closed-loop system thegenerated stochastic signals 119872(120578) and 119873(120578) are chosen as

119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(20)

where 1198822

0(radic2119902) = 2(1198821(119902) minus 119882

2

0(119902)) variables 1198820(119902) and

1198821(119902) are defined as follows

1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(21)

Then we obtain the average system

120591ave

= minus119896 (120591ave

minus ΓaveH120591

ave)

Γ

ave= minusℎ (Γ

aveminus ℎ(Γ

ave)2

H)

(22)

Thus according to the stochastic averaging theory [14]system (22) has a locally exponentially stable equilibrium at(120591

ave Γ

ave) = (0 0) When 120591 rarr 0 we have 120591 rarr 120591

lowast and then119875pv rarr 119875

lowast

pv Thus the maximum power point tracking isrealized This completes the proof


Remark 3 From the equilibrium equation (22) we can findthat the convergence of closed-loop system is totally deter-mined by parameters 119896 and ℎ which implies that in thedesign of control algorithm (10) the convergence rate canbe arbitrarily assigned by the designer with an appropriatechoice of 119896 and ℎ Generally speaking relatively large 119896 and ℎ

will lead to a good convergence rate but too large 119896 and ℎwillalso result in oscillations during the convergence Hence 119896and ℎ should be chosen by fully considering both the dynamicand the steady responses

35 Further Discussions Besides the stability property dis-cussed above there are some more issues that should beconsidered in the implementation of MPPT controllers suchas tracking efficiency effects of partial shading and PVmodule ageing effects In what follows we will discuss thesetopics that may be helpful in the application of the proposedMPPT method

(1) Tracking Efficiency Tracking efficiency has been the mostimportant consideration of the MPPT method in manycommercial applications The tracking efficiency of a MPPTalgorithm can be calculated by the following equation [6]

120578119879 =

int119905

0119875pv (119905) 119889119905

int119905

0119875max (119905) 119889119905

(23)

where 119875pv(119905) is the measured power produced by the PVarray under the control of MPPT algorithm and 119875max(119905) is thetheoretical maximum power that the array can produce

The discrete definition of the tracking efficiency can bedenoted by [21]

120578119879 =1

119899

119899

sum

119896=0

119875pv (119896)

119875max (119896) (24)

where 119899 is the number of samplesTracking efficiency evaluates the overall performance

of a MPPT algorithm In the next section we will verifythe tracking efficiency of the proposed MPPT method withnumerical results

(2) Effects of Partial Shading The partial shading effect hasreceived much attention recently in the implementation ofMPPT controller which is partly due to the rapid develop-ment of building integrated photovoltaics (BIPV) [22] As theBIPV is commonly installed on the rooftop it typically suffersfrom partial shading due to the space limitation clouds andso forth

When a PV system is subjected to partial shading the P-V curve often exhibits a global extremum and several localextremums as shown in Figure 8 Hence in order to trackthe global MPP of power map the adopted MPPT algorithmshould have a global convergence However from (22) wefind that the proposed controller is locally stable at the MPPwhich implies that it may get trapped at local extremumsand may not be a good choice for PV systems under partialshading Fortunately several global or semiglobal extremum

Voltage

Pow

er

Global MPPLocal MPPs

0

Figure 8 An illustrative P-V curve for PV systems under partialshading

0 20 40 60 80 100 120 140 160 180

1

2

3

4

5

Voltage (V)

Curr

ent (

A)

I-V characteristics at BOLI-V characteristics 4 years later

Figure 9 Degradation of 20 a-Si modules after 4 years of operationin [17]

seeking control designs have been available see [23ndash25] formore information

(3) Effects of PV Module Ageing Reliability and lifetime ofPV modules are key factors to PV system performance andare mainly dominated by the PV module ageing effect [26]In order to separate the ageing effect from the irradianceand temperature it is necessary to evaluate the Beginningof Life (BOL) performance Then the deviations between theexperimental data and BOL performance explain the ageingof the PV system [17]

Figure 9 shows the degradation of 20 a-Si modules after 4years of operation in the literature [17] We can find that theMPP declines slowly during the 4-year time Mathematicallythe maximum power point tracking for an ageing PV systemis essentially an optimization process for a slowly varyingunknown cost function [13] The proposed Newton-basedextremum seeking MPPT controller optimizes the powermapwith real-timemeasurements and calls for no knowledgeof model information Thus it can deal with the PV moduleageing effect well

4 Tracking Performance Evaluation

In this section we provide several simulation and experimentresults to show the effectiveness of the proposed MPPT


Light source

PV panel

Multimeters

Upper computerDC-DC

Light sourcecontrol box

converterS7-200 PLC

Auxiliary power supply (APS)

Figure 10 The experiment setup of the light source-driven photo-voltaic MPPT system

algorithm In order to evaluate the tracking performanceof the proposed MPPT algorithm under arbitrary desiredirradiance we adopt the light source-driven photovoltaicMPPT system as shown in Figure 10

In Figure 10 there are two incandescents working as theirradiation source The KC65T-PV panel converts the solarenergy to electrical energy via the photovoltaic effect TheSiemens S7-200 PLC works as the MPPT controller drivingthe DC-DC converter and regulating the output voltage ofthe converter The upper computer is used to collect andstore experimental data and multimeters are used to displaythe measurements The light source control box is used toregulate the irradiation intensity of light sources

In what follows we will provide simulation examples andexperiment results to illustrate the tracking performance ofthe proposedMPPT algorithmunder uniform irradiance andnonuniform irradiance cases respectively The data sheetsof KC65T photovoltaic module under different irradianceconditions are shown in Tables 1 and 2 see [18] for detailedinformation

41 Tracking under Uniform Irradiance In this subsectionwe consider the tracking performance evaluation of theproposed MPPT algorithm under the uniform irradiance Asimulation and an experiment are provided respectively toillustrate the implementation of the algorithm The simula-tion is conducted in MATLABSimulink and the parametersetting of the proposed control algorithm is shown in Table 3For simplicity we adopt only one KC65T PV module in thesimulation

Figure 11 shows the simulation results of the proposedMPPT algorithm under uniform irradiance of 800Wm2 at25∘CThe convergence of themaximumpower point is shownin Figure 11(a) The output power oscillates in the directionof MPP which is detected by the product of the power andperturbations We can see that the output power convergesto the MPP within 5 s The output power in the steady stateis about 461W whereas the maximum power of the PV

Table 1 Data sheet of KC65T at 800Wm2 25∘C

Maximum power (119875max) 477WMaximum power voltage (119880mpp) 155 VMaximum power current (119868mpp) 308A


Maximum power (119875max) 653WMaximum power voltage (119880mpp) 174VMaximum power current (119868mpp) 375 A

Table 3 Parameter setting of the proposed MPPT controller

Parameter Value119886 01119896 1ℎ 008119902 40

module is 477W Then the simulated tracking efficiency ofthe proposed MPPT algorithm is about 966

Figure 12 shows the comparison between the proposedNewton-based extremum seeking MPPT method and clas-sical extremum seeking MPPT method We compare thetracking performance of the two MPPT methods in termsof tracking efficiency and convergence time The numericalcomparison of the two MPPT methods is shown in Table 4

In addition to the simulations experiments on the lightsource-driven photovoltaic MPPT system are conducted forthe purpose of verification The irradiance is set to be800Wm2 and the temperature is about 23sim27∘C The PVpanel comprises four KC65T PVmodules with two modulesconnected in parallel and two in series

The experiment results of the proposed MPPT algorithmare shown in Figures 13 and 14 From Figure 13(a) we cansee that the output power converges to the MPP within 6 sThe average power at steady state is about 178W whereas themaximum power is 1908W then the tracking efficiency isabout 932

42 Tracking under Nonuniform Irradiance In this subsec-tion we will consider the tracking performance of the pro-posed MPPT algorithm under nonuniform irradiance withan abrupt change from 800Wm2 to 1000Wm2 at 10 s Forthe purpose of comparison we also provide the experimentresults of the classical gradient-based extremum seekingMPPT method under the same experimental conditions

Figure 15 shows the experiment results of the proposedNewton-based extremum seeking MPPT method under thenonuniform irradiance It is shown in Figure 15(a) that thenew MPP is located within 2 s The new steady power isabout 243W while the maximum power is 2612W then thetracking efficiency is about 93

Figure 16 shows the experiment results of the classicalgradient-based extremum seeking MPPT method under the


0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle

Figure 11 Simulation results of the proposed MPPT method under uniform irradiance of 800Wm2 at 25∘C

Table 4 The simulated comparison of the two methodsMPPT method Efficiency ConvergenceNewton-based ES MPPT 966 lt4 sGradient-based ES MPPT 921 gt6 s

0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

Newton-based ES MPPT methodGradient-based ES MPPT method

Figure 12 The comparison between the proposed MPPT methodand classical extremum seeking MPPT method

nonuniform irradiance It is shown in Figure 16(a) that thenew MPP is located within 4 s The new steady power isabout 231W while the maximum power is 2612W then thetracking efficiency is about 884

In order to illustrate the effectiveness of the proposedMPPT method more intuitively we provide the numericalcomparison between the proposed algorithm and classicalextremum seeking MPPT algorithm under different irradi-ance Figures 17 and 18 show the comparison of the twometh-ods under irradiance 800Wm2 and irradiance 1000Wm2respectively We can see that the proposed method shows itssuperiority in both tracking efficiency and convergence rate

5 Concluding Remark

In this paper we propose a new Newton-based extremumseeking MPPT algorithm for photovoltaic systems The pro-posed algorithm benefits the following advantages it callsfor no knowledge of the power map it has a good toleranceof stochastic perturbations and its convergence rate can betotally designer-assignable The stability and convergence ofthe proposed MPPT controller are rigorously proved withstochastic averaging theory Some topics related to applica-tions are discussed such as partial shading effects and PV


020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) CurrentFigure 13 Experiment results of output power and current collected in the upper computer

2 s

30V3V

(a) Voltage

4V

20 ms

(b) PWM signalFigure 14 Experiment results of output voltage and PWM signal measured in the oscilloscope

0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal

Figure 15 Tracking performance of the Newton-based extremum seeking MPPT algorithm under nonuniform irradiance


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal

Figure 16 Tracking performance of the gradient-based extremum seeking MPPT algorithm under nonuniform irradiance

87

88

89

90

91

92

93

94

Newton-based ES MPPT Gradient-based ES MPPT

()

Tracking efficiency

(a) Tracking efficiency

0

2

4

6

8

10

12


(s)

Convergence time

(b) Convergence time

Figure 17 The numerical comparison between the proposed MPPT algorithm and classical extremum seeking MPPT algorithm underirradiance 800Wm2

module ageing effects Simulation and experiment results areprovided to show the effectiveness of the proposed method

It is also worth mentioning that both the Newton-basedES and gradient-based ES belong to the so-called static

extremum seeking which requires that the power map ofphotovoltaic systems is rather slower than extremum seekingWhen irradiance changes quickly the tracking performanceof ES MPPT algorithms will be worse In order to propose


86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time



a commercial ES MPPT algorithm we will further investi-gate the dynamic extremum seeking and its applications inmaximum power point tracking in future work

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their valuable comments This work is supported by theNational Natural Science Foundation of China (Grant nos61071096 and 61379111) the Fundamental Research Fundsfor the Central Universities of Central South University andHunan Provincial Innovation Foundation for Postgraduate

References

[1] E F Camacho M Berenguel F R Rubio and D MartinezControl of Solar Energy Systems Springer 2012

[2] E F CamachoM Berenguel and F R RubioAdvanced Controlof Solar Plants Springer 1997

[3] T Esram andP L Chapman ldquoComparison of photovoltaic arraymaximum power point tracking techniquesrdquo IEEE Transactionson Energy Conversion vol 22 no 2 pp 439ndash449 2007

[4] V Salas E Olıas A Barrado and A Lazaro ldquoReview of themaximum power point tracking algorithms for stand-alonephotovoltaic systemsrdquo Solar Energy Materials and Solar Cellsvol 90 no 11 pp 1555ndash1578 2006

[5] K H Hussein I Muta T Hoshino and M Osakada ldquoMax-imum photovoltaic power tracking an algorithm for rapidlychanging atmospheric conditionsrdquo IEE Proceedings GenerationTransmission and Distribution vol 142 no 1 pp 59ndash64 1995

[6] D P Hohm and M E Ropp ldquoComparative study of maximumpower point tracking algorithmsrdquo Progress in PhotovoltaicsResearch and Applications vol 11 no 1 pp 47ndash62 2003

[7] T Kawamura K Harada Y Ishihara et al ldquoAnalysis of MPPTcharacteristics in photovoltaic power systemrdquo Solar EnergyMaterials and Solar Cells vol 47 no 1ndash4 pp 155ndash165 1997

[8] C Hua J Lin and C Shen ldquoImplementation of a DSP-controlled photovoltaic systemwith peak power trackingrdquo IEEETransactions on Industrial Electronics vol 45 no 1 pp 99ndash1071998

[9] R Leyva C Alonso I Queinnec A Cid-Pastor D Lagrangeand L Martınez-Salamero ldquoMPPT of photovoltaic systemsusing extremummdashseeking controlrdquo IEEE Transactions onAerospace and Electronic Systems vol 42 no 1 pp 249ndash2582006

[10] S L Brunton C W Rowley S R Kulkarni and C ClarksonldquoMaximum power point tracking for photovoltaic optimizationusing ripple-based extremum seeking controlrdquo IEEE Transac-tions on Power Electronics vol 25 no 10 pp 2531ndash2540 2010

[11] H Yau and C Wu ldquoComparison of extremum-seeking controltechniques for maximum power point tracking in photovoltaicsystemsrdquo Energies vol 4 no 12 pp 2180ndash2195 2011

[12] A Ghaffari M Krstic and S Seshagiri ldquoExtremum seeking forwind and solar energy applicationsrdquo ASME Dynamic Systems ampControl Magazine pp 13ndash21 2014

[13] D Dochain M Perrier and M Guay ldquoExtremum seekingcontrol and its application to process and reaction systems asurveyrdquoMathematics and Computers in Simulation vol 82 no3 pp 369ndash380 2011

[14] S Liu and M Krstic ldquoStochastic averaging in continuous timeand its applications to extremum seekingrdquo IEEETransactions onAutomatic Control vol 55 no 10 pp 2235ndash2250 2010

[15] S Liu and M Krstic ldquoNewton-based stochastic extremumseekingrdquo Automatica vol 50 no 3 pp 952ndash961 2014

[16] httpwwwgreenrhinoenergycomsolar[17] A Abete F Scapino F Spertino and R Tommasini ldquoAgeing

effect on the performance of a-Si photovoltaic modules ina grid connected system experimental data and simulation


resultsrdquo in Proceedings of the Conference Record of the 28th IEEEPhotovoltaic Specialists Conference pp 1587ndash1590 AnchorageAlaska USA 2000

[18] httpwwwkyocerasolarcomassets0015170pdf[19] S Liu and M Krstic Stochastic Averaging and Stochastic

Extremum Seeking Springer London UK 2012[20] P Bhatnagar and R K Nema ldquoMaximum power point tracking

control techniques state-of-the-art in photovoltaic applica-tionsrdquo Renewable and Sustainable Energy Reviews vol 23 pp224ndash241 2013

[21] A Reza Reisi M HassanMoradi and S Jamasb ldquoClassificationand comparison of maximum power point tracking techniquesfor photovoltaic system a reviewrdquo Renewable and SustainableEnergy Reviews vol 19 pp 433ndash443 2013

[22] K Ishaque and Z Salam ldquoA review of maximum power pointtracking techniques of PV system for uniform insolation andpartial shading conditionrdquo Renewable amp Sustainable EnergyReviews vol 19 pp 475ndash488 2013

[23] Y Tan D Nesic I M Y Mareels and A Astolfi ldquoOn globalextremum seeking in the presence of local extremardquo Automat-ica vol 45 no 1 pp 245ndash251 2009

[24] F Esmaeilzadeh Azar M Perrier and B Srinivasan ldquoA globaloptimization method based on multi-unit extremum-seekingfor scalar nonlinear systemsrdquo Computers and Chemical Engi-neering vol 35 no 3 pp 456ndash463 2011

[25] W H Moase and C Manzie ldquoSemi-global stability analysisof observer-based extremum-seeking for Hammerstein plantsrdquoIEEE Transactions on Automatic Control vol 57 no 7 pp 1685ndash1695 2012

[26] R Khatri S Agarwal I Saha S K Singh and B KumarldquoStudy on long term reliability of photo-voltaic modules andanalysis of power degradation using accelerated aging tests andelectroluminescence techniquerdquo Energy Procedia vol 8 pp396ndash401 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


irradiance [7] The IncCond algorithm tracks the maxi-mum power point by comparing the instantaneous andincremental conductances of the PV array Thus it can trackthe rapidly changing irradiance However the harmoniccomponents of array voltage and current need to bemeasuredand used to adjust the reference voltage which implies thaterrors at the maximum power point occur due to the lowprecision sensors used [8]

A promising newMPPTalgorithm is themethod of extre-mum seeking (ES) control see [9ndash12] As a model-free real-time optimization method ES is well suited for applicationswith unknown or partly known dynamics and externalperturbations such as photovoltaic systems [11] ES uses per-turbation signals (either from external disturbances or fromconverter ripples) as probing signals to estimate the gradi-ent of the power map and then update the control signalaccording to the estimation [12] It has been shown that EShas the advantages of both rigorously provable convergenceand simplicity of hardware implementation [13] Howeverexisting extremum seekingMPPT controllers still suffer fromthe following two limitations

(i) In existing extremum seekingMPPT controllers per-turbation signals are generally assumed to be peri-odic On the one hand the assumption is rather idealsince external disturbances are typically unknownand stochastic on the other hand the requirement oforthogonality makes periodic ES difficult to extend tomultivariable cases

(ii) In existing extremum seeking MPPT controllers theconvergence speed is typically defined by the gradientof power map of PV systems which implies thatcontrol systemswill be highly influenced by unknownand changing environmental conditions To constructa practical MPPT control system the convergenceof MPPT controller should be user-assignable ratherthan being dependent on environmental conditions

Thus the aim of this paper is to provide an improvedextremum seeking MPPT control algorithm to solve theseproblems Recent progress in stochastic averaging theoryhas made it possible to consider more general stochasticperturbations when designing extremum seeking controllerssee [14 15] In [14] Liu and Krstic provide a systematic designof extremum seeking when stochastic perturbations existand prove the stability of the stochastic extremum seekingvia a developed stochastic averaging theory The theoreticalpart of this paper is based on the findings of [14 15] andwe further propose a Newton-based stochastic extremumseeking MPPT controller considering physical features ofphotovoltaic systems We conduct extensive simulations andexperiments to show the effectiveness of the proposed controlalgorithm

The remainder of the paper is organized as follows InSection 2 preliminary knowledge about photovoltaic mod-elling and maximum power point tracking is introduced InSection 3 a new Newton-based stochastic extremum seekingMPPT controller is designed The tracking performanceof the proposed controller is evaluated in Section 4 Weconclude the paper in Section 5

Iph

Rp

Rs

Load

Ipv

UpvPV

module

Id

Figure 1 The equivalent electrical circuit of a photovoltaic module

2 Photovoltaic Modelling and MPPT

In this section we will introduce some preliminary knowl-edge about PV array and MPPT A photovoltaic array typ-ically consists of several photovoltaic modules connected inseries or parallel to obtain the desired output voltage and cur-rent Thus we first introduce the electrical characteristics ofPV modules More detailed background information can befound in [16]

A photovoltaic module can be seen as a black box thatproduces a current 119868pv at a voltage119880pvThe inside of that blackbox can be described by an electric circuit with 4 componentsas shown in Figure 1

(i) Current source this is the source of the photo currentwhich can be denoted by

119868ph = 119878 sdot 119867 sdot 120585 (1)

where 119878 is the module area 119867 is the intensity ofincoming light and 120585 is the response factor

(ii) Diode this nonlinear element reflects the dependenceon the band gap and losses to recombination It ischaracterized by the reverse current 119868119889

(iii) Shunt resistor 119877119901 it represents losses incurred byconductors

(iv) Serial resistor 119877119904 it also represents losses incurred bynonideal conductors

The relationship between current 119868pv and voltage 119880pv of aphotovoltaic module is then expressed by

119868pv = 119868ph minus 119868119889 [exp(

119880pv + 119868pv119877119904

119880119879

) minus 1] minus

119880pv + 119868pv119877119904

119877119901

(2)

where 119880119879 = 119902119896119879119890 with ideality factor 119902 temperature 119879Boltzmann constant 119896 = 138119890

minus23 and the elementary charge119890 = 1602119890

minus19Then the output power 119875pv of a photovoltaic module is

denoted by

119875pv = 119880pv119868pv = 119891 (119880pv) (3)

where function 119891(sdot) is determined by manufacturing andenvironmental parameters such as module area 119878 irradiance


0 5 10 15 20 250

05

1

15

2

25

3

35

4

Voltage (V)

Curr

ent (

A)


H = 600Wm2 T = 25∘C












120591 = 119892 (119880pv pv) (4)

119880pv = ℓ (120591) (5)


119875lowast

pv = 119891 (119880lowast

pv) = 119891 (ℓ (120591lowast)) ≜ max119875pv (6)

where 119880lowast




pv (7)


119880pv = 119896Hpv (8)




119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv (9)


Sun irradiation


DC-DC converter

LoadC

LSW

DUpv

Ipv

Upv Ipv

Upv


pv


+minus+minus


DC-DCconverter

120591 DC-DCdriver

Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)



2119891(119909)119889119909




119880pv = 119892 (120591)

120591 = ℓ (119880pv pv)

119880pv = minus119896Γ119872(120578) 119875pv

Γ = ℎΓ minus ℎΓ2119873(120578) 119875pv

(10)



119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(11)


1198822

0(radic2119902) = 2 (1198821 (119902) minus 119882

2

0(119902))

1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(12)




Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)

S(120578(t))M(120578(t))


G

H

minusΓG

DC-DCdriver

times


Initialization





G(k) = 0 H(k) lt 0


Yes

No





k larr k + 1





Voltage

A B C

Pow

er

UpvUpv

120591

120591





120591 (119896 + 1) = 120591 (119896) + Δ120591 (119896) (13)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvgt 0 (14)


lowast

pv


2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv= 0 (15)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvlt 0 (16)


lowast

pv



lowast


lowast


lowast



(1radic120587119902)119890minus11990921199022




+ 120591 + 119886 sin (120578)))



119873(120578)119891 (119892 (120591lowast

+ 120591 + 119886 sin (120578)))

(17)


119891 (ℓ (120591)) = 119891lowast

+11989110158401015840

(ℓ (120591lowast))

2(120591 minus 120591


+H2

(120591 minus 120591lowast)2

(18)



+H2

(120591 + 119886 sin (120578))2)



119873(120578) (119891lowast

+H2

(120591 + 119886 sin (120578))2)

(19)


119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(20)

where 1198822

0(radic2119902) = 2(1198821(119902) minus 119882

2

0(119902)) variables 1198820(119902) and


1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(21)


120591ave

= minus119896 (120591ave

minus ΓaveH120591

ave)

Γ

ave= minusℎ (Γ

aveminus ℎ(Γ

ave)2

H)

(22)


ave Γ



lowast







120578119879 =

int119905

0119875pv (119905) 119889119905

int119905

0119875max (119905) 119889119905

(23)



120578119879 =1

119899

119899

sum

119896=0

119875pv (119896)

119875max (119896) (24)





Voltage

Pow

er


0


0 20 40 60 80 100 120 140 160 180

1

2

3

4

5

Voltage (V)

Curr

ent (

A)









Light source

PV panel

Multimeters

Upper computerDC-DC


converterS7-200 PLC






















0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 5 10 15 20 250

05

1

15

2

25

3

35

4

Voltage (V)

Curr

ent (

A)


H = 600Wm2 T = 25∘C












120591 = 119892 (119880pv pv) (4)

119880pv = ℓ (120591) (5)


119875lowast

pv = 119891 (119880lowast

pv) = 119891 (ℓ (120591lowast)) ≜ max119875pv (6)

where 119880lowast




pv (7)


119880pv = 119896Hpv (8)




119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv (9)


Sun irradiation


DC-DC converter

LoadC

LSW

DUpv

Ipv

Upv Ipv

Upv


pv


+minus+minus


DC-DCconverter

120591 DC-DCdriver

Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)



2119891(119909)119889119909




119880pv = 119892 (120591)

120591 = ℓ (119880pv pv)

119880pv = minus119896Γ119872(120578) 119875pv

Γ = ℎΓ minus ℎΓ2119873(120578) 119875pv

(10)



119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(11)


1198822

0(radic2119902) = 2 (1198821 (119902) minus 119882

2

0(119902))

1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(12)




Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)

S(120578(t))M(120578(t))


G

H

minusΓG

DC-DCdriver

times


Initialization





G(k) = 0 H(k) lt 0


Yes

No





k larr k + 1





Voltage

A B C

Pow

er

UpvUpv

120591

120591





120591 (119896 + 1) = 120591 (119896) + Δ120591 (119896) (13)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvgt 0 (14)


lowast

pv


2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv= 0 (15)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvlt 0 (16)


lowast

pv



lowast


lowast


lowast



(1radic120587119902)119890minus11990921199022




+ 120591 + 119886 sin (120578)))



119873(120578)119891 (119892 (120591lowast

+ 120591 + 119886 sin (120578)))

(17)


119891 (ℓ (120591)) = 119891lowast

+11989110158401015840

(ℓ (120591lowast))

2(120591 minus 120591


+H2

(120591 minus 120591lowast)2

(18)



+H2

(120591 + 119886 sin (120578))2)



119873(120578) (119891lowast

+H2

(120591 + 119886 sin (120578))2)

(19)


119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(20)

where 1198822

0(radic2119902) = 2(1198821(119902) minus 119882

2

0(119902)) variables 1198820(119902) and


1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(21)


120591ave

= minus119896 (120591ave

minus ΓaveH120591

ave)

Γ

ave= minusℎ (Γ

aveminus ℎ(Γ

ave)2

H)

(22)


ave Γ



lowast







120578119879 =

int119905

0119875pv (119905) 119889119905

int119905

0119875max (119905) 119889119905

(23)



120578119879 =1

119899

119899

sum

119896=0

119875pv (119896)

119875max (119896) (24)





Voltage

Pow

er


0


0 20 40 60 80 100 120 140 160 180

1

2

3

4

5

Voltage (V)

Curr

ent (

A)









Light source

PV panel

Multimeters

Upper computerDC-DC


converterS7-200 PLC






















0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Sun irradiation


DC-DC converter

LoadC

LSW

DUpv

Ipv

Upv Ipv

Upv


pv


+minus+minus


DC-DCconverter

120591 DC-DCdriver

Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)



2119891(119909)119889119909




119880pv = 119892 (120591)

120591 = ℓ (119880pv pv)

119880pv = minus119896Γ119872(120578) 119875pv

Γ = ℎΓ minus ℎΓ2119873(120578) 119875pv

(10)



119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(11)


1198822

0(radic2119902) = 2 (1198821 (119902) minus 119882

2

0(119902))

1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(12)




Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)

S(120578(t))M(120578(t))


G

H

minusΓG

DC-DCdriver

times


Initialization





G(k) = 0 H(k) lt 0


Yes

No





k larr k + 1





Voltage

A B C

Pow

er

UpvUpv

120591

120591





120591 (119896 + 1) = 120591 (119896) + Δ120591 (119896) (13)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvgt 0 (14)


lowast

pv


2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv= 0 (15)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvlt 0 (16)


lowast

pv



lowast


lowast


lowast



(1radic120587119902)119890minus11990921199022




+ 120591 + 119886 sin (120578)))



119873(120578)119891 (119892 (120591lowast

+ 120591 + 119886 sin (120578)))

(17)


119891 (ℓ (120591)) = 119891lowast

+11989110158401015840

(ℓ (120591lowast))

2(120591 minus 120591


+H2

(120591 minus 120591lowast)2

(18)



+H2

(120591 + 119886 sin (120578))2)



119873(120578) (119891lowast

+H2

(120591 + 119886 sin (120578))2)

(19)


119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(20)

where 1198822

0(radic2119902) = 2(1198821(119902) minus 119882

2

0(119902)) variables 1198820(119902) and


1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(21)


120591ave

= minus119896 (120591ave

minus ΓaveH120591

ave)

Γ

ave= minusℎ (Γ

aveminus ℎ(Γ

ave)2

H)

(22)


ave Γ



lowast







120578119879 =

int119905

0119875pv (119905) 119889119905

int119905

0119875max (119905) 119889119905

(23)



120578119879 =1

119899

119899

sum

119896=0

119875pv (119896)

119875max (119896) (24)





Voltage

Pow

er


0


0 20 40 60 80 100 120 140 160 180

1

2

3

4

5

Voltage (V)

Curr

ent (

A)









Light source

PV panel

Multimeters

Upper computerDC-DC


converterS7-200 PLC






















0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



Upv

Upvtimes+

Ppv = f(Upv )

k

s

120578(t)

S(120578(t))M(120578(t))


G

H

minusΓG

DC-DCdriver

times


Initialization





G(k) = 0 H(k) lt 0


Yes

No





k larr k + 1





Voltage

A B C

Pow

er

UpvUpv

120591

120591





120591 (119896 + 1) = 120591 (119896) + Δ120591 (119896) (13)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvgt 0 (14)


lowast

pv


2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv= 0 (15)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvlt 0 (16)


lowast

pv



lowast


lowast


lowast



(1radic120587119902)119890minus11990921199022




+ 120591 + 119886 sin (120578)))



119873(120578)119891 (119892 (120591lowast

+ 120591 + 119886 sin (120578)))

(17)


119891 (ℓ (120591)) = 119891lowast

+11989110158401015840

(ℓ (120591lowast))

2(120591 minus 120591


+H2

(120591 minus 120591lowast)2

(18)



+H2

(120591 + 119886 sin (120578))2)



119873(120578) (119891lowast

+H2

(120591 + 119886 sin (120578))2)

(19)


119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(20)

where 1198822

0(radic2119902) = 2(1198821(119902) minus 119882

2

0(119902)) variables 1198820(119902) and


1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(21)


120591ave

= minus119896 (120591ave

minus ΓaveH120591

ave)

Γ

ave= minusℎ (Γ

aveminus ℎ(Γ

ave)2

H)

(22)


ave Γ



lowast







120578119879 =

int119905

0119875pv (119905) 119889119905

int119905

0119875max (119905) 119889119905

(23)



120578119879 =1

119899

119899

sum

119896=0

119875pv (119896)

119875max (119896) (24)





Voltage

Pow

er


0


0 20 40 60 80 100 120 140 160 180

1

2

3

4

5

Voltage (V)

Curr

ent (

A)









Light source

PV panel

Multimeters

Upper computerDC-DC


converterS7-200 PLC






















0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvgt 0 (14)


lowast

pv


2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pv= 0 (15)





2119875pv119889119880

2


119889119875pv

119889119905= minus(

1198892119891(119880pv)

1198891198802pv)

minus1119889119891 (119880pv)

119889119880pvlt 0 (16)


lowast

pv



lowast


lowast


lowast



(1radic120587119902)119890minus11990921199022




+ 120591 + 119886 sin (120578)))



119873(120578)119891 (119892 (120591lowast

+ 120591 + 119886 sin (120578)))

(17)


119891 (ℓ (120591)) = 119891lowast

+11989110158401015840

(ℓ (120591lowast))

2(120591 minus 120591


+H2

(120591 minus 120591lowast)2

(18)



+H2

(120591 + 119886 sin (120578))2)



119873(120578) (119891lowast

+H2

(120591 + 119886 sin (120578))2)

(19)


119872(120578) =1

1198861198820 (119902)sin (120578)

119873 (120578) =1

11988621198822

0(radic2119902)

(sin2 (120578) minus 1198820 (119902))

(20)

where 1198822

0(radic2119902) = 2(1198821(119902) minus 119882

2

0(119902)) variables 1198820(119902) and


1198820 (119902) =1

2(1 minus 119890

minus1199022

)

1198821 (119902) =3

8minus

1

2119890minus1199022

+1

8119890minus41199022

(21)


120591ave

= minus119896 (120591ave

minus ΓaveH120591

ave)

Γ

ave= minusℎ (Γ

aveminus ℎ(Γ

ave)2

H)

(22)


ave Γ



lowast







120578119879 =

int119905

0119875pv (119905) 119889119905

int119905

0119875max (119905) 119889119905

(23)



120578119879 =1

119899

119899

sum

119896=0

119875pv (119896)

119875max (119896) (24)





Voltage

Pow

er


0


0 20 40 60 80 100 120 140 160 180

1

2

3

4

5

Voltage (V)

Curr

ent (

A)









Light source

PV panel

Multimeters

Upper computerDC-DC


converterS7-200 PLC






















0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of






120578119879 =

int119905

0119875pv (119905) 119889119905

int119905

0119875max (119905) 119889119905

(23)



120578119879 =1

119899

119899

sum

119896=0

119875pv (119896)

119875max (119896) (24)





Voltage

Pow

er


0


0 20 40 60 80 100 120 140 160 180

1

2

3

4

5

Voltage (V)

Curr

ent (

A)









Light source

PV panel

Multimeters

Upper computerDC-DC


converterS7-200 PLC






















0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Light source

PV panel

Multimeters

Upper computerDC-DC


converterS7-200 PLC






















0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)

0 02 04 06 08 10

10

20

30

40

(a) Power

0 5 10 15 200

2

4

6

8

10

12

14

16

18

t (s)

U(V

)

0 02 04 06 08 115

16

17

18

(b) Voltage

0 5 10 15 200

05

1

15

2

25

3

35

t (s)

I(A

)

0 02 04 06 08 1

3

2

1

0

(c) Current

0 5 10 15 2005

06

07

08

09

1

t (s)

120591

0 02 04 06 08 106

07

08

09

1

(d) Duty cycle



0 5 10 15 200

10

20

30

40

50

t (s)

P(W

)





5 Concluding Remark



020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


020406080

100120140160180200

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)


2 s

30V3V

(a) Voltage

4V

20 ms


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

30V3V

332V

(c) Voltage

4V

20ms

(d) PWM signal



0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0

50

100

150

200

250

300

00

981

962

943

92 49

588

686

784

882 9

810

78

117

612

74

137

214

715

68

166

617

64

186

219

6

Pow

er (W

)

Time (s)

(a) Power

0

1

2

3

4

5

6

7

8

9

00

941

882

823

76 47

564

658

752

846 9

410

34

112

812

22

131

614

115

04

159

816

92

178

618

819

74

Curr

ent (

A)

Time (s)

(b) Current

2 s

295V3V

324V

(c) Voltage

4V

20ms

(d) PWM signal


87

88

89

90

91

92

93

94


()

Tracking efficiency


0

2

4

6

8

10

12


(s)

Convergence time







86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


86

87

88

89

90

91

92

93

94


()

Tracking efficiency


0

05

1

15

2

25

3

35

4


(s)

Convergence time






Acknowledgments


References






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

research article a newton-based extremum seeking mppt...

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