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Solar Energy

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Optimal parameter design of fractional order control based INC-MPPT forPV system

Mujahed Al-Dhaifallaha,b,⁎, Ahmed M. Nassefa,c, Hegazy Rezka,d, Kottakkaran Sooppy Nisare

a College of Engineering at Wadi Addawaser, Prince Sattam Bin Abdulaziz University, Saudi Arabiab Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabiac Department of Computers and Automatic Control Engineering, Faculty of Engineering, Tanta University, Egyptd Electrical Engineering Dept., Faculty of Engineering, Minia University, Egypte Department of Mathematics, College of Arts & Science-Wadi Aldawaser, Prince Sattam Bin Abdulaziz University, Saudi Arabia

A R T I C L E I N F O

Keywords:Fractional calculusMPPTPV systemEnergy efficiencyOptimization

A B S T R A C T

A comprehensive and straightforward methodology for optimal parameter design of Fractional Order controlbased Incremental Conductance (FOINC)-Maximum Power Point Tracking (MPPT) is developed in this paper.The main objective is to develop a more efficient, robust MPPT algorithm based on the integration between thefractional-order control and Incremental Conductance (INC) method. The integration between INC robustnessand the accuracy of fractional-order can enhance the overall tracking performance in comparison to the con-ventional tracking techniques. Such integration ensures fast dynamics and high tracking accuracy for theMaximum Power Point (MPP) under tremendous climate variations. A small signal model for the whole systemhas been built to design the most appropriate order and gain of the fractional integrator for variable step sizeFOINC-MPPT. The ultimate gain (upper limit) of the controller has been specified properly via root locus plotbefore starting the optimization process to avoid falling in instability region. Radial Movement Optimization, asan optimization tool, is used for obtaining the optimal parameters of the fractional controller. The feasibility andeffectiveness of the proposed FOINC-MPPT algorithm are validated under different climate conditions with slowand rapid changes in solar irradiance. Simulation results show that the proposed FOINC-MPPT algorithm is ableto track the MPP accurately and rapidly in comparison with the conventional INC-based tracker.

1. Introduction

Nowadays, looking for different but environmentally friendly re-sources of energy becomes a must. This is because of the dramaticdrawbacks of the conventional fossil fuel energy. Gases, such as CO andCO2, which emitted from the combustion of fossil fuel is negativelyaffecting the living organs. These gases also deteriorate the Ozone layer,which in turn changes the climate and increases the temperature of theEarth. Additionally and most importantly, its bad effect on the humanhealth, especially the pulmonary system. Because of the above, one ofthe national research projects in worldwide is the field of RenewableEnergy. Solar Energy is definitely one of this type. Recently, there hasbeen an increasing interest in using the Photovoltaic Power System(PVPS) as a new source of electrical energy. It has many merits, such as:pollution-free, little maintenance and noise-free. However, the outputpower of PVPS varies with the atmospheric conditions, e.g. solar irra-diance level and temperature (Rezk et al., 2017; Rezk and Fathy, 2016;Fathy and Rezk, 2016). To increase the output power of the PVPS, it is

crucial to let the operating point to be as close as the Maximum PowerPoint (MPP). Accordingly, MPP-Tracking (MPPT) system should beapplied in the power electronic interface between the PVPS and theload. Employing the MPPT system will definitely increase the efficiencyof the PVPS, consequently, decreases the total number of requiredPhotovoltaic (PV) panels, and hence decreases the total cost (Rezk andEltamaly, 2015).

Most of the conventional MPPTs usually use a fixed-amplitude of theperturbation step in the converter duty cycle for tracking MPP.Accordingly, the step amplitude is considered as the main factor in thetracking process. With a step of large amplitude, the tracking speed toreach the MPP will be high, but unfortunately, the oscillation aroundthe MPP at steady-state will also be high (Ahmed and Shoyama,2011a,b; Mei et al., 2011). On the other hand, with a small step, thetracking speed is decreased with the minimization of the oscillationaround MPP. Therefore, the trade-off between the tracking speed andthe efficiency is considered as an interesting research topic. Further-more, such conventional techniques fail to track the MPP under rapid

https://doi.org/10.1016/j.solener.2017.11.040Received 21 June 2017; Received in revised form 23 October 2017; Accepted 15 November 2017

⁎ Corresponding author at: Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.E-mail address: mujahed@kfupm.edu.sa (M. Al-Dhaifallah).

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climate changes. To sort out this problem, several techniques that usevariable step-size are developed. Their core idea is based on starting thetracking process with large amplitude step to quickly track the MPP,and then the amplitude is reduced gradually whenever the operatingpoint is close to the MPP to eliminate the steady-state oscillationsaround MPP (Rezk, 2016). The main drawback of such techniques is thedependency of the step amplitude on the PV module characteristics.These characteristics are often highly dependent on the climate con-ditions. Therefore, selecting a proper step-size for all climate conditionsis difficult.

In this paper, a straightforward methodology for optimal parameterdesign of Fractional Order control based Incremental Conductance(FOINC)-MPPT is proposed. To obtain the optimal parameters somesteps are carried out. First, the PVPS is linearized at MPP to a straightline for studying the small signal stability. Second, a small signal modelis built for the overall PVPS integrated with FOINC-MPPT. Third, theclosed loop transfer function of the PVPS is extracted. Forth, the ulti-mate gain (upper limit) of the controller has been specified properly viaroot locus plot before starting the optimization process to avoid fallingin instability region. Finally, Radial Movement Optimization (RMO) hasbeen used as a tool for obtaining the optimal parameters of the frac-tional integrator.

The tracking performance of FOINC-based tracker is tested withdifferent conditions with slow and rapid changes of solar irradiance viaMATLAB/Simulink software. The proposed system performance iscompared with the conventional INC-based tracker.

This paper is organized as follows: Section 2 presents the previousworks in the MPPT. Modeling of PVPS components, including themathematical representation of the PV panel, Boost converter, INC-basetracker and building the overall small signal model, is introduced inSection 3. An introduction to fractional calculus and RMO are presentedin Sections 4 and 5, respectively. Section 6 discusses the obtained re-sults. Finally, the conclusions are addressed in Section 7.

2. Literature review

Many researches tried to tackle the MPPT problem and the fol-lowing are their contributions: Rezk discussed the tradeoff betweensteady state and dynamic performance of incremental resistance basedMPPT for fuel cell system (Rezk, 2016). Fixed step and variable stepbased trackers are considered. Dealing the tradeoff between the time oftracking and steady-state distortions in the case of fixed step size isdone with a variable step size-based tracker. In such tracker, the in-tegrator gain is employed to adapt the error signal to be in a suitablerange before integration process. The main drawback of such work isonly five different gain values were employed for testing system per-formance without clear methodology for designing the optimal values(Rezk, 2016). Arulmurugan and Suthanthiravanitha proposed an im-proved fractional order variable step size incremental conductancebased tracker (Arulmurugan and Suthanthiravanitha, 2014). The pro-posed algorithm modifies the existing conventional Inc-Cond controllerbased on improved fractional order variable step size which differs fromthe existing. The new proposed controller combines the merits of bothimproved fractional order (FO) and variable step size (VSS) incrementalconductance which is well suitable for control and execution. The maindrawback of proposed method can be summarized by the following;there is no accurate methodology for designing the main two factors(the order of fractional and scaling factor). Also, the proposed system istested by only one pattern of radiation change (from 200 to 850W/m2)(Arulmurugan and Suthanthiravanitha, 2014). Chendi et. al. presented anew INC-based tracker with an adaptive variable step size (Li et al.,2016). Such system automatically updates the step size. Accordingly,the PVPS operates with a large step size when the operating point is farfrom the MPP and with a small step size when the operating point isclose to the MPP. A tuning method of initial step sizes is also presented,which is derived from the approximate linear relationship between the

open-circuit voltage and MPP voltage (Li et al., 2016). Chen et. al.proposed an auto-scaling variable step-size MPPT technique (Chenet al., 2014). Such technique eliminates the problems in the conven-tional variable step-size method. It uses the simple judgment criterionand auto-scaling variable step size to enable PVPS to achieve fast dy-namic response and stable steady-state output power (Chen et al.,2014). Tang et. al. integrated the fractional order with fuzzy MPPT(Tang et al., 2017). Such integration combined the robustness of fuzzylogic and the accuracy of fractional for enhancing the performance ofthe tracking efficiency. The order of the fractional system is adaptedwith large or small values according to the dynamic operating range ofthe fuzzy controller (Tang et al., 2017). Tey et. al. presented the con-fusing of the classical MPPT techniques during the increment of solarradiation (Tey and Mekhilef, 2014). Furthermore, they proposed amodified incremental conductance algorithm that responds accuratelywhen the solar irradiation level increases (Tey and Mekhilef, 2014).

Ramdan et. al. proposed a new MPPT technique for a PV systembased on the integration between Lagrange Interpolation (LI) andParticle Swarm Optimization (PSO) for eliminating the traditionalmethods initialization problem. Using LI allows the particles to migratevery close to the best position directly and hence avoids local minimatrapping. Such technique minimizes the PSO number of iterations forreaching the MPP. The results are compared with Perturb and Observe(P&O), INC, and the conventional PSO. The comparison indicated thatthe proposed technique enhances the search stability and the trackingspeed capability under any abrupt change (Koad et al., 2017). Rahmaniand Yusof proposed a new stochastic optimization technique calledRadial Movement Optimization (RMO) based on swarm populationwhich uses spherical boundaries in the search-space. This techniqueneeds only to adjust few numbers of the parameter which makes thesearch process robust and fast, accordingly a small memory storage isrequired (Rahmani and Yusof, 2014). Therefore, this technique isadopted in this work. Vanithasri et. al. introduced a modified version ofthe RMO (MRMO) for better results, fast convergence and preventingtrapping in local minima. The MRMO has also less memory storage, fewparameters to be tuned and is able to reach the global minimum effi-ciently (Vanithasri et al., 2016). Seyedmahmoudian et. al. proposed amethod to track the global maximum power point (GMPP) as fast aspossible for PV system under partial shading based on RMO technique.The speed of convergence, efficiency, stability and computationalcomplexity were considered in the comparison with other techniques(Seyedmahmoudian et al., 2016). Gharaveisi et. al. proposed an optimaldesign of MPPT controller for a stand-alone PV system. They used threemethods of optimization: Genetic Algorithm (GA), Discrete Action Re-inforcement Learning Automata (DARLA), and Vector Based SwarmOptimization (VBSO). These methods were applied to tune the para-meters of the PID controller. Their simulation results show that VBSOwas the best to adjust the controller’s gains (Gharaveisi et al., Sep.2014). Oshaba et. al. proposed a new MPPT for PV–DC motor pumpsystem by designing two PI controllers. One for adjusting the duty cycleof the DC/DC converter and the other for controlling the DC motorspeed. They used the artificial bee colony (ABC) to optimize the twocontrollers’ parameters. Their system performance was compared andproved competent with that of GA (Oshaba et al., 2017). Paul did acomparison study between the application of the conventional PI andthe fuzzy controllers in MPPT. He offline trained Artificial NeuralNetwork (ANN) using GA to get the reference voltage of the MPP. Thisvoltage is then compared with PV voltage and is fed to the (PI/Fuzzy)controller which gives the gating signal to the boost converter. TheFuzzy controller was superior to the traditional PI controller in terms oferror reduction (Paul, 2013). Cheng et. al. proposed an asymmetricalfuzzy logic control (FLC)-based MPPT. They used the power–voltage(P–V) curve of solar cells and the (PSO) to optimize the input mem-bership function (MF) setting values. The speed/tracking accuracyshowed an improvement relative to the traditional (P&O) and symme-trical FLC (Cheng et al., 2015). Kumar et. al. proposed a method for

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designing an optimum control-based MPPT for a PV system using a PIDcontroller to obtain the duty cycle of the boost converter. The PIDparameters were tuned by Ziegler-Nicholas method. The results showedthat the time taken to obtain the optimal point is very less and theresulting power is maximized with the PID-MPPT controller (AshokKumar et al., 2015). Mahammad et. al. presented two controllers; tra-ditional PID and Fuzzy Logic, to control the MPPT under partiallyshaded conditions. Their simulated results showed that PID controllergives better performance. But their work does not show clearly how thegains of the PID controller are tuned (Mahammad et al., 2013). Yau et.al. proposed a two-stage system to implement MPPT in addition to theoptimal charge control of Li-ion battery. The variable step size incre-mental conductance method (VSINC) for tracking the MPP is obtainedby the optimum parameters of the PI charge controller. The controllerparameters were tuned by PSO and GA which let the system to providea constant charging voltage (Yau et al., 2013).

Due to its potential to improve the performance and efficiency ofmany engineering and scientific systems recent researches has beenattracted to fractional order calculus (FOC). However, one of thechallenges in the FOC is to find out the best order of the system and itsoptimal gain. The use of FOC in MPPT is one of the emerging researchfields. Some of the recent works investigated the fractional order con-trol of MPPT. In (Liu et al., 2017) authors discussed the role of frac-tional order controller in extremum seeking control (ESC). The con-troller is designed with integer order integrator (IO-I) and low passfilter (IO-LPF) together with fractional order high pass filter (FOHPF),by substituting the normal HPF in the original ESC system. A novel typeof variable FO incremental conductance algorithm (VFOINC) combinedwith extenics variable step size (EVSS) control into the MPPT schemefor photovoltaic (PV) power systems was proposed in Yu et al. (2015).The variable fractional order is selected when the maximum powerpoint is approached. The obtained results showed that the fractionalorder method has a better MPPT effect. The methods proposed in var-ious studies on the PV power generation system have addressed thedefect that tracking continuously near the maximum power pointcauses the power loss. To overcome this defect, a new method of FOChaos synchronization dynamic error detector for the MPPT scheme ofa PV power system was offered in Yu et al. (2015). The experimentalresult of this paper showed that about 4% of steady-state vibrationenergy can be saved. In Neçaibia et al. (2015), the authors used thecontrol scheme established on extremum-seeking (ES) combined withfractional order systems (FOS). They showed that the FO operators canimprove the plant dynamics with respect to time response and dis-turbance rejection. A new FO incremental conductance (FOINC) algo-rithm to plan and study the MPPT of small wind power systems wasapplied in Yu and Liao (2015). The benefit of the proposed FOINC isthat it can control the algorithm only by capturing the voltage andcurrent generated by the wind power system without adjusting theparameter setting for different systems.

It is noticed from the above review that the methodology for ob-taining the optimal parameters of the fractional order controller basedMPPT does not show a complete scenario. This motivated the authors tointroduce a clear and straightforward methodology based on the use ofroot locus and optimization techniques.

3. Modeling of PV system components

The PVPS is mainly consisting of PV panels, DC-DC boost converter,control system and the load as illustrated in Fig. 1. The PVPS is in-tegrated with the proposed FOINC-based MPPT technique. According tomaximum power transfer theory, maximum power is being transferredfrom source to load when source impedance is equal to the load im-pedance. To change the PV panel voltage to operate around the MPP, aDC-DC converter is required. Load matching can be obtained by ad-justing the converter duty-cycle. Thus, the converter must operate witha specific duty-cycle that hopefully yields the maximum power. In the

case of changing climate conditions, the converter duty-cycle has to beadjusted for extracting maximum power from PVPS, so as to improvethe efficiency (Rezk and Eltamaly, 2015; RezaReisi et al., 2013). Thereare several configurations from DC-DC converters. In this paper, theboost converter has been selected. It is very spread thanks to its highreliability with respect to other more complex configurations, to thereduced number of components and also to the high-minded experiencein its operation.

3.1. Mathematical representation of PV panel

PV solar panel generates DC electrical power from the solar radia-tion. It works under the phenomenon of the photoelectric effect, whereit directly converts sunlight into electrical power. Several PV cells areinterconnected in series for stepping up the rated voltage of PV panel.Fig. 2 illustrates the equivalent circuit model for a PV module. The solarcell can be regarded as a non-linear current source. Its generated cur-rent depends on the characteristic of material, irradiation and celltemperature. Equations (1)–(3) describe the I-V characteristics of the PVmodel (Fathy and Rezk, in press; Rezk and Hasaneen, 2015).

= − −I I I Ipv L D sh (1)

⎜ ⎟= ⎧⎨⎩

⎛⎝

⎞⎠− ⎫⎬⎭

I I VV

exp 1 ,D oD

t (2)

= + −I R I a T T[ ( )]L L ref c c ref, , (3)

where, =VtAKT

qc , = ×+V V I RD pv s, = ×V V Npv swhere

IC= Ipv Solar cell current (A)Vc Solar cell voltage (V)Vpv PV module voltage (V)IL Light generated current (A)Io Reverse saturation Currentq Charge of electron=1.6×10−19 (coulomb)K Boltzmann constant (J/K)A Ideality factorVt Thermal voltageRs Cell series resistance (Ω)Rsh Cell shunt resistance (Ω)a Temperature coefficient of the short circuit current A/°CTc PV cell temperatureTc,ref Reference Temperature 25 °CNs Number of solar cells connected in series per module

LA361K51S PV module is chosen to be used in MATLAB simulationmodel. The module is made of 36 multi-crystalline silicon solar cells inseries and provides 51W of nominal maximum power. Table 1 displaysthe LA361K51S PV module electrical specifications. The voltage, cur-rent and power curves under different irradiance levels are shown inFigs. 3 and 4.

3.2. DC/DC boost converter

Boost converter is often employed for stepping step up solar panelvoltage via varying the duty cycle of the MOSFET switch. The dutycycle (D) is defined as the ratio between the on state period (ton) ofMOSFET switch to the total switching period of the converter (ts) andtherefore (D= ton/ts). Duty cycle D of the boost converter is con-tinuously controlled in order to reach the MPPT of the PV system. Forvarious mission profiled of the PV system at different irradiance valuedand ambient temperature, the boost stage is operated at different dutycycles according to the control signal coming from the MPPT algorithm.For limiting the high frequency harmonic components, an input capa-citor is inserted in the input side of the PV panel. The duty cycle cor-responding to MPP can be estimated as following (Rezk and Eltamaly,2015; Kumar and Gupta, 2012);

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652

= −×

DV

P R1mpp

mpp

mpp load (4)

where Vmpp represents the voltage of PV panel voltage corresponding to

MPP, Pmpp is the power of panel at MPP, Rload represents the equivalentresistance of the boost converter output and Dmpp is the operating dutycycle of the boost converter at MPPT.

3.3. Incremental conductance based MPPT

One of the well-known algorithms developed to track MPP of PVPSis INC. Fig. 4 shows the relationship among PV module power and itsoutput voltage. There is a single point located at power curve. Thatpoint represents MPP at which PVPS produces its maximum power. Thecore idea of INC method is that PV power derivative w.r.t its voltage iszero at the MPP (Fig. 5, dark cyan line). The derivative of power againstvoltage can be represented by (5) and accordingly, the error signal maybe calculated as in (6) (Rezk and Eltamaly, 2015; Ahmed and Shoyama,2011a,b).

= ∗ = ∗ +dpdv

d v idv

didv

v i( )PV

PV

PV PV

PV

PV

PVPV PV (5)

= +e didv

iv

PV

PV

PV

PV (6)

Accordingly, tracking the MPP needs the following strategy:

Fig. 1. Schematic diagram of PVPS integrated withFOINC-MPPT.

Fig. 2. Equivalent circuit of solar cell.

Table 1Electrical specifications data of LA361K51S solar module.

Characteristics Specification

Maximum power 51WShort circuit current 3.25 AOpen circuit voltage 21.2 VCurrent at MPP 3.02 AVoltage at MPP 16.9 V

Fig. 3. P-V curve of LA361K51S PV module.

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⎧

⎨⎪

⎩⎪

= + ∗ >= == − ∗ <

d new d old K e when ed new d old when ed new d old K e when e

1) ( ) ( ) 02) ( ) ( ) 03) ( ) ( ) 0

Iλ

Iλ

(7)

where KI and λ are the parameters of the fractional order integrator(gain and order).

The variable step size FOINC-based tracker starts the tracking pro-cess with a large step size for quickly catching MPP. Then, it reduces thestep size when the operating point is near to MPP for mitigating theoscillation around MPP. FOINC-MPPT is implemented with the help of afractional order integrator (FOC) with gain KI and fed by the errorsignal.

3.4. Small signal model of PV system

The PVPS is linearized at MPP to a straight line for studying thesmall signal stability. Then, a small signal model is built for the overallPVPS equipped with FOINC-MPPT. After that, the closed loop transferfunction of the PVPS is extracted. Finally, the ultimate gain (upperlimit) of the controller has been specified properly via root locus plot to

be used in the optimization process to avoid falling in instability region.Based on the state space averaging model of the DC/DC boost

converter integrated with the PV panel around MPP, the transferfunction (PV module voltage to the converter duty ratio) of the PVmodule equipped with DC-DC boost converter and the load resistance isshown in the following equation (Ahmed and Shoyama, 2011b);

= ⎡⎣⎢

++ + +

⎤⎦⎥

Vd

z S zp S p S p S p

ΔΔ

PV 1 2

13

22

31

4 (8)

where

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

== − +== += −= − +

Z V R R CZ I R R d VP C LR R CP C LR LR CP d C R RP I R R d V

(1 )

(1 )(1 )

o L mpp out

L L mpp o

in L mpp out

in mpp L out

in L mpp

L L mpp o

1

2

1

2

32

4

The PV module performance can be linearized around the MPP by

Fig. 4. I-V curve of LA361K51S PV module.

Fig. 5. PV module output power and its derivativeagainst PV output voltage.

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(9) as illustrated in Fig. 6;

= + ∗ − ∗ = − ∗V V I R I R V I R( ) 2PV mpp mpp mpp PV mpp mpp PV mpp (9)

Linearizing the error signal around (Vmpp, Impp) by applyingTaylor series expansion to (6) and considering only the linear term(neglect higher order terms) with the help of the linearized equationwritten in (9), the error will be as in (10):

= +∂∂

↓ ∗ − +∂∂

↓ ∗

− + ⋯

e e V Ie v i

vv V

e v ii

i

I

( , )( , )

( )( , )

(

) .

m mpv pv

pvV I pv mpp

pv pv

pvV I pv

mpp

, ,m m m m

= −⎡

⎣⎢ + ⎤

⎦⎥e R

RI

VI

I2 mppmpp

mpp

mpp

mppPV2

(10)

Therefore, the linearized error will be:

= −⎡

⎣⎢ + ⎤

⎦⎥e

RI

VI

iΔ Δmpp

mpp

mpp

mppPV2

(11)

Then, the relation between Δd and Δe is formulated as follows;

=d kS

eΔ ΔIλ (12)

Based on the above equations, the overall small signal model of theFOINC-MPPT, including the fractional order integrator (FOI), is as il-lustrated in Fig. 7.

The closed-loop characteristic equation of the small signal model isshown in (13)

⎜ ⎟⎜ ⎟+ ⎛⎝

+ ⎞⎠⎛⎝

⎞⎠⎡⎣⎢

− −+ + +

⎤⎦⎥=k

SVI

RI R

z S zp S p S p S p

1 1 0Iλ

m

m

m

m mpp2

1 2

13

22

31

4 (13)

4. Fractional order control

Presently, fractional order (FO) controllers are being widely used byseveral scientists in order to reach the most robust performance of thesystems (Monje et al., 2008; Chen et al., 2008; Podlubny, 1994; Daset al., 2012; Zamani et al., 2009; Zhao et al., 2005). The fractional orderPID controller (Podlubny, 1994) has higher flexibility and better- closedloop performance compared to classical PID controller because, in orderto tune the controller the PID controller has to change three parameters,while the fractional order PID (FOPID) controller has five. ThePI Dλ μ-controller, which involves an integrator of order λ and differ-entiator of order μ, is proposed in Podlubny (1994). A detailed surveyabout the different type of fractional controller is given in the recentwork of Shah and Agashe (2016).

The most commonly used definitions of fractional calculus in thecontrol systems (Oldham and Spanier, 1974) are:

(1) Grunwald-Letnikov (GL)

The GL definition is stated as

∑= − −→ =

⎡⎣− ⎤⎦ ( )D

hrk f t khlim 1 ( 1) ( )b t

β

h βk

t bh

k0 0 (14)

where ⎡⎣ ⎤⎦−t bh integer part b and t are the limits of operator. r is the

integer which satisfies − < <r β r1 , where ( )rk is the binomial coeffi-

cient defined by

=−( )r

kr

k r k!

! ( )!.

(2) Riemann-Liouville (RL)

Fig. 6. Linearization of PV panel performance at MPP.

Fig. 7. Block-diagram of small signal model for PVPS.

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The RL definition given in the following expression

∫=− − − +D g t

r kddt

g τg t τ

dτ( ) 1Γ( )

( )( )b t

kr

r b

t

k r 1 (15)

where −r kΓ( ) denotes the gamma function and − < <r k r1 .

(3) Caputo definition

The Caputo definition is given by

∫=− − − +D h t

k rh τ

h t τdτ( ) 1

Γ( )( )

( )b tk

b

t r

k r 1 (16)

where − < <r k r1 .The FOPID is the generalized version of classical PID. The FO con-

trollers are more robust to change of parameters of a controlled system.A FO controller can achieve the property of iso-damping easily (Chaoet al., 2010). The generalized transfer function of this FOPID is givenby:

= = + + ⩾C s U sE s

K KS

K s λ μ( ) ( )( )

, ( , 0)pIλ D

μ(17)

where C(s) is the controller output, U(S) is the control signal, E(s) is theerror signal, KP is the proportional constant gain, KI is the integrationconstant gain, KD is the derivative constant gain, λ is the order of in-tegration, and μ is the order of differentiator. In FOPID, we have to setfive parameters, namely proportional, derivative, integral constants, KP,KI, KD in addition to the fractional integration and differentiation λ, μ.To set these parameters according to given specifications, it is needed toperform parameter optimizations. Fig. 8 shows that the closed loopconfiguration of FOPID controller is cascade with a plant. The two extraparameters in the FOPID give more flexibility to improve the generalperformance of the system. However, tuning of FOPID parameters ischallenging task. Many algorithms have been proposed for tuning suchas neural network, particle swarm optimization, and Ziegler-Nicholsmodified method (Li et al., 2010; Luo et al., 2010; Padula and Visioli,2011).FOPID was introduced by Podlubny in 1994 to control FOsystem. The first work to investigate stability given by I. Petras based ona Riemann surface (Petras, 2009). Further studies about stability foundin the work of many authors (Tavazoei and Haeri, 2009; Rivero et al.,2013; Hamamci, 2008; Cheng and Hwang, 2006). The real realizationor implementation of the fractional-order systems usually has two folds;software and hardware implementations. In the software implementa-tion, a Fractional-order Modeling and Control (FOMCON) toolbox,which works under Matlab© platform, is suitable for this task and it isadopted in this work (http://fomcon.net/fomcon-toolbox/download/).However, for the hardware implementation, Carlson and Halijak (1964)and Roy (1967) gave the realization of fractional-order differentiators.In Carlson and Halijak (1964), authors attempted to create a “fractionalcapacitor” having a transfer function of 1/s1/n where n is a positive realnumber. The electronic realization of the FO controlled object and FOcontroller, whose mathematical model is fractional order differentialequation is given in Dorčák et al. (2013).

In summary, the digital implementation of FO controller can be asfellows; firstly, the controller transfer function is discretized using ei-ther direct or indirect discretization techniques. Then, using the sam-pling time Ts, the discretized function is transformed to the z-domain.Lastly, the resulting z-domain transfer function is implemented to a

discrete function controller.

5. Radial movement optimizer

If a function f is dependent on a vector of unknown variables,X=[x1, x2, ….…. , xn], of dimension n, then the process of finding thebest solution Xbest, that produces the minimum/maximum of the func-tion f, is called an optimization process. In engineering, most of theoptimization problems are minimization because of the objectivefunction, which should be minimized, is often the error function.Swarm-based optimization algorithms such as PSO, Ant ColonyOptimization (ACO), and RMO, are examples of stochastic optimizationtechniques and have many things in common. At the beginning of theprocess, they usually propose a population of solutions (initialization).Each nominated solution represents a point in the d-dimensional space(d-unknown variables) and always should be within the upper and thelower limits of the search space. Then, the positions of these solutionsare updated iteratively until it reaches the maximum number of itera-tions or a stop criterion is met.

In this paper, RMO is used to optimize the parameters of the frac-tional-order controller. This optimizer differs from the other swarmoptimization algorithms in the way of updating the solutions’ positions.That is the particles are scattered from a center point and the location ofthis center is always updated every iteration. As an illustration, in a 3Doptimization problem, the solutions are randomly distributed along theradii of a sphere with different velocities. The location of each solutionis then evaluated by the objective function and the best (minimum/maximum) solution is obtained.

The tradeoff between exploration and exploitation is still a chal-lenging dilemma. Because up to the knowledge of the authors, till nowno one knows exactly when to switch between exploration and ex-ploitation. So, updating the particles’ positions radially can improve theexploration of new search areas and also the exploitation of the localarea.

Another advantage of using RMO is that it needs small size ofmemory for computations because updating the particles’ locationsdoes not need any information about their previous locations and/orvelocities. The information needed to update the center are only theglobal best, Gb, and the radial best, Rb. Therefore, the center is movediteratively to an area attracted by the Gb to prevent sticking to a localoptima. The whole optimization process is illustrated in Fig. 9(Rahmani and Yusof, 2014; Vanithasri et al., 2016).

At each iteration step, the objective function is evaluated for thewhole scattered particles around the center. The location of the bestfitness value is considered as the best radial solution, Rb, and the bestRb found so far is considered as the global best, Gb. So, Rb will be thebest solution found at the current iteration and Gb is the best solutionall over the whole iterations. The process stops as soon as the maximumnumber of iterations is reached or the change in the objective functionvalue is very small for a long time.

The stages of the RMO algorithm as in the flowchart are summar-ized as follows (Rahmani and Yusof, 2014; Vanithasri et al., 2016):

5.1. Initialization

The first step in the optimization process is to assign values ran-domly to the positions and velocities of the particles to represent theinitial solutions. To cover the whole search space, these initial valuesare obtained from a normally distributed random generator. The loca-tions of the particles are defined by an nPop x nDim matrix called Xij,where nPop and nDim refer to the number of particles and the numberof dimensions, respectively. The nPop is usually chosen by the user andshould be appropriate to the nDim as well as the range of search space.While, the nDim has to be typically equal to the number of unknownvariables to be optimized. The location matrix for the algorithm isshown below (Rahmani and Yusof, 2014; Vanithasri et al., 2016):Fig. 8. General block diagram of fractional PID controller.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

656

=⎡

⎣⎢⎢

⋯⋮ ⋱ ⋮

⋯

⎤

⎦⎥⎥

xx x

x xij

nDim

nPop nPop nDim

1,1 1,

,1 ,

where i=1; 2; 3; …; nPop and j=1; 2; 3; …; nDim. The initializationprocess is accomplished according to (18):

= + −x Lb r Ub Lb( )ij (18)

where Lb and Ub are the lower and upper bounds of the searchspace, respectively; r is a normally distributed random number in therange [0 1].

5.2. Updating the particles’ locations

As soon as the center is obtained the proposed solutions are scat-tered from the center along the radii. This would make the solutionschange their locations in straight lines from the center along the radiibased on velocity vector, Vij. The velocity vector Vij is obtained

according to (19) Rahmani and Yusof, 2014; Vanithasri et al., 2016.

= ⇒⎧

⎨⎪

⎩⎪

== …= …

−

V rVVi nPopj nDim

1,2, ,1,2, ,

ij max j

max jx x

k( )

( )j jmax( ) min( )

(19)

The parameter k must be an integer number which can be chosenadequately. This parameter shows how many divisions the search spaceis divided into. In this paper, k is set to 5. Same as PSO, the velocityvector is associated with an inertia weight as shown in Eq. (17), whichcontrols the convergence rate of the algorithm.

The value of inertia weight, W, is decreasing linearly throughout theiterations as in (20). As the velocity vector, Vij

k , is a function of Wk, thevalues of Wmax and Wmin controls the influence of the velocity vectorson the movement

⎜ ⎟= −⎛⎝

− ⎞⎠

W W W WIteration

Iterationk maxmax min

maxk

(20)

Start

Set the RMO algorithm parameters andInitialize the places of nPop solutions

randomly

Scatter the solutions around the center along the radii

Evaluate the fitness value of each solution

K = 1

Center k+1=Center k+C3Rbk

Gb > Rb

End

Evaluate the fitness value of each solution

no

yes

Yes No

K+1

K = max

Yes

No

Choose the best as a Center

Center k+1= Center k+ C1(Gb - Center k)+C2(Rb k-Center k)

Gb = Rb

Fig. 9. The RMO algorithm flowchart.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

657

=V rW Vijk

k jmax( ) (21)

where Wmax, Wmin are the maximum and minimum values of the inertiaweight and are set to 1 and 0, respectively. Iterationmax, is the maximumnumber of iterations, and r is a random number between 0 and 1.

Eq. (21) calculates the values of the velocity vector of the particlesin space. This velocity vector represents in which extend the particleswill move around the center. This movement should be controlled ef-ficiently to switch properly between the exploration and exploitationstages and hence avoid stochastic jumps. These random and long jumpswill definitely prevent from reaching the global minimum which is themain goal of the optimization. On the other hand, Eq. (20) representsthe linear decay of the velocity vector’s weight. At the very beginning ofthe optimization process, the values of the weight vector are big (ex-ploration stage) and iteratively these values become smaller andsmaller (exploitation stage). So, Eq. (20) calculates the values of theweight at each iteration and ensures that these values starts from(Wmax, = 1) and iteratively it ends up to (Wmin=0).

Fig. 10 illustrates the radial movement of the solutions from thecenter-point and the boundaries of the sphere where the solutions arescattered. These boundaries are limited with the Vmax.

After evaluating the fitness function of all scattered solutions, theone with the best fitness value is selected as the radial best (Rb). Thelocations of the Gb and Rb solutions, found in the current iteration, areused to calculate the new center-point location as in (22) and (23).Fig. 11 illustrates how the new center-point is obtained using Gb andRb.

= ++Center Center Updatek k1 (22)

= − + −Update C Gb Center C Rb Center( ) ( )k k k1 2 (23)

where C1 and C2 are two constants and in this works, their valuesare set to 0.7 and 0.8, respectively.

After obtaining the new center-point, the particles are scattered

again and the process is repeated. The optimization process is continueduntil a stopping criterion is met. Fig. 12 shows the new center-pointupdate process in two consecutive iterations based on Eqs. (5) and (6),where the location of the new center-point is obtained from the updatevector.

6. Results and discussion

In control systems, stability test is mandatory before starting thedesign of the controller. Therefore, the ultimate gain, KI, of the con-troller has to be specified properly before starting the optimizationprocess to avoid falling in instability region. Because of this, the ulti-mate gain will be the upper limit of the search space. Root locus is animportant tool used to plot the roots of the characteristic equation(system poles) when the gain K is changed from 0 to ∞. As soon as theroot locus plot is available, the ultimate gain, Ku, can be specified at theintersection point of the root locus with the imaginary axis. In thispaper, the toolbox of plotting the root locus of fractional order systemsdesigned by Machado is used (Machado, 2011). The characteristicequation of the fractional order system is shown in (24):

+ + + + +

=

− −S e S e S S K S(1.066 2.447 0.01476 11.15) (0.3239 196.8)

0

λI

9 3 6 2

(24)

The root locus is plotted for different values of λ changing from 0.1to 0.9 with a step of 0.1. Fig. 13 shows the root locus at λ=0.1, 0.9,1.5 and 2. It is worth mentioning that the system will be unstable whenλ≥ 2.

At the imaginary axis, the values of system’s natural frequency, ω,and ultimate gain, Ku, can be calculated from the characteristic equa-tion when the real and imaginary parts are equal to zero as in (25) and(26), respectively.

⎛⎝

⎞⎠

− − ⎛⎝

⎞⎠

−

+ =

− + + − +π λ ω e ω π λ ω e βω

K

cos2

(11.15 2.447 ) sin2

(0.01476 1.066 )

196.8 0

λ λ λ λ

u

6 2 1 9 3

(25)

⎛⎝

⎞⎠

− + ⎛⎝

⎞⎠

−

+ =

+ − + − +π λ ω e ω π λ ω e ω

ωK

cos2

(0.01476 1.066 ) sin2

(11.15 2.447 )

0.3239 0

λ λ λ λ

u

1 9 3 6 2

(26)

Table 2 illustrates the obtained values of ω and Ku, at differentvalues of λ in the range [0.1 0.9]

To find out the relationship between the upper limit of the in-tegrator gain KI and the system’s ultimate gain Ku, the characteristicequation in (13) can be put in the form: 1+ KI G(S)= 0. When thesystem operates at the point of intersection of the root locus and theimaginary axis, the DC loop gain at steady state (i.e. at S= 0) Ku will be|KI G(0)|, accordingly:

= ⎛⎝

+ ⎞⎠

=

−

⎛

⎝⎜ + ⎞

⎠⎟

( )( )K K

K KOr,

u IV

I

RI R

ZP

IP R

Zu

1mpp

mpp

mpp

mpp mpp

mpp

VmppImpp

RmppImpp

22

4

4

2 2 (27)

So, based on Table 2 and Eq. (27), the upper limit of the integrator gain,at each λ, is calculated.

The parameters of the RMO optimizer and their values are shown inTable 3.

The RMO algorithm is run to obtain the optimal values of KI and λthat maximizes the output energy of the PV system. The RMO problemis formulated as follows:

Fig. 10. Scattering the particles along the radii; Vmax is the largest radius of the sphere.

Fig. 11. Updating the center-point through the update vector.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

658

Fig. 12. Updating center location of a three-dimen-sion optimization problem.

Fig. 13. Root locus at λ=0.1, 0.9, 1.5 and 2.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

659

∑= < < <

<⎛⎝

+ ⎞⎠

=E P K λ λ K

P R

ZK

Maximize (t, , ); subject to 0 1 and 0

,

t

T

PV I I

mpp

V

I

RI

u

0

4

2mpp

mpp

mpp

mpp2

where E, PPV, and T are the total energy, the PV output power, and thesimulation time, respectively.

The resulting optimal values of the FOI parameters, KI and λ, are0.3688 and 0.9, respectively. A comparison including the calculatedparameters of the iterated process at iterations 30 and 50 is given inTable. 4.

The optimization cost function E, the center-point (CP) locations ofKI and λ and the best values of KI and λ at every iteration are shown in

Fig. 14.These optimal values are used to test the FOINC-based tracker

performance with different patterns of solar irradiance. The Simulinkmodel of the PVPS is illustrated in Fig. 15. This system consists of onesolar panel with 51W output power, DC/DC boost converter operatingin continuous conducted current mode with switching frequency of30 kHz, an input inductance of 1 mH, an output capacitor of 47 μF and40Ω resistive loads.

Several solar irradiance patterns are designed to evaluate the pro-posed FOINC system performance. Fig. 16(a) illustrates three differentpatterns. Every pattern includes several possible changing conditionsfor irradiance. These changing conditions simulate a step and rampchange. The first pattern simulates small step change (at seconds 1, 4and 4.5) and a slow ramp (starts at second 2). However, the secondpattern includes rapid step change from 1000W/m2 (at second 1). TheThird pattern includes steep ramp changes from 600W/m2 to 900W/m2 in only one second (starts at second 3).

Fig. 16(b) shows the corresponding output PV power under everypattern. It is clear from the figure that the proposed FOINC-basedtracker is fast enough and successfully reaching the MPP under allstudied climate conditions as mentioned above. The detailed analysissuch as power, current, voltage and boost converter duty-cycle for eachpattern are shown in Figs. 17–19, respectively.

Fig. 20 shows the comparison between PV output power of INC andFOINC based trackers. It can be noted that the INC-based tracker takes0.24 s to fully reach MPP. On the other hand, the FOINC-MPPT takes0.14 s only. This means that employing fractional order integrator in-creases the tracking speed by 41.67% compared with the conventionalINC-based tracker.

7. Conclusions

A comprehensive methodology for optimal parameter design offractional order control based incremental conductance (FOINC)-MPPThas been provided. A small signal model for the whole system has beenbuilt to design the most appropriate order and gain of the fractionalorder integrator (FOI). Using the root locus technique and the radialmovement optimizer, the appropriate values of the order and gain forFOI, which ensures a good dynamic tracking performance for severaldifferent patterns of climate changes, are determined as 0.9 and 0.3866,respectively. The combination between conventional INC method andthe fractional-order integrator ensures fast dynamics as well as hightracking accuracy for the MPP under tremendous climate variations.

Table 2Values of ω and Ku, at different values of λ.

λ ω Ku

0.1 11619.49 1.03910.2 6878.03 0.69890.3 5478.51 0.83920.4 4827.66 1.26400.5 4447.38 2.14280.6 4191.12 3.90490.7 4000.20 7.47550.8 3846.60 14.8390.9 3715.10 30.295

Table 3Parameters of the RMO optimizer and their values.

Parameter Value Parameter Value

K 5 Number of particles 10C1 0.7 Number of iterations 50C2 0.8

Table 4The calculated parameters of the iterated process at iterations 30 and 50.

Iterations Best KI Best

30 0.3312 0.950 0.3688 0.9

Fig. 14. The optimization cost function, the center-point (CP) locations of KI and λ and the best values of KI and λ at every iteration.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

660

The results indicated that the FOINC-based tracker successfully andaccurately tracked MPP with better performance and employing FOIincreased the tracking speed by 41.67% compared with the conven-tional INC-based tracker. Future work will focus on experimental im-plementation and realization of the algorithm to verify the results. Also,

application of other optimization techniques, and exploring otherfractional order control strategies such as full fractional PID controller,fractional sliding mode control, fractional order model referenceadaptive control, etc. needs to be considered for the future research.

Fig. 15. MATLAB model of PVPS integrated with FOINC-MPPT.

Fig. 16. (a) Three different patterns of solar irradiances; and (b) the corresponding PV module output power.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

661

Appendix A.

Derivation of Eqs. (25) and (26):In the characteristic equation of the fractional order system shown below, Let Ku=KI (Eq. (24) in the paper):

+ + + + + =− −S e S e S S K S(1.066 2.447 0.01476 11.15) (0.3239 196.8) 0λu

9 3 6 2 (A1)

At the intersection of the root locus with the imaginary axis, the closed loop pole S is equal to jω. Substituting in Eq. (A1):

+ + + + + =− −jω e jω e jω jω K jω( ) [1.066 ( ) 2.447 ( ) 0.01476( ) 11.15] (0.3239( ) 196.8) 0λu

9 3 6 2 (A2)

− − + + + + =− −jω j e ω e ω j ω j ωK K( ) [ 1.066 2.447 0.01476 11.15] 0.3239 196.8 0λu u

9 3 6 2

− + − + + =− −jω e ω j ω e ω j ωK K( ) [(11.15 2.447 ) (0.01476 1.066 )] 0.3239 196.8 0λu u

6 2 9 3 (A3)

Fig. 17. The detailed analysis of FOINC-based tracker for first pattern of solar irradiance.

Fig. 18. The detailed analysis of FOINC-based tracker for second pattern of solar irradiance.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

662

=

=

=

= ⎡⎣+ ⎤⎦

( )

( ) ( )

jω ωe

jω ωe

ω e

jω ω λ j λ

And

( )

( ) cos sin (from Euler formula)

j

λ j λ

λ j λ

λ λ π π2 2

π

π

π

2

2

2

(A4)

Substituting from (A4) into (A3):

⎡⎣

⎛⎝

⎞⎠+ ⎛

⎝⎞⎠⎤⎦

− + − + + =− −ω π λ j π λ e ω j ω e ω j ωK Kcos2

sin2

[(11.15 2.447 ) (0.01476 1.066 )] 0.3239 196.8 0λu u

6 2 9 3

Or,

⎡⎣

⎛⎝

⎞⎠+ ⎛

⎝⎞⎠⎤⎦

− + − + + =− + + − +π λ j π λ ω e ω j ω e ω j ωK Kcos2

sin2

[(11.15 2.447 ) (0.01476 1.066 )] 0.3239 196.8 0λ λ λ λu u

6 2 1 9 3

Multiplying the two brackets it gives:

⎛⎝

⎞⎠

− − ⎛⎝

⎞⎠

− + + ⎛⎝

⎞⎠

−

+ ⎛⎝

⎞⎠

− + =

− + + − + + − +

− +

π λ ω e ω π λ ωω e ω K j π λ ω e ω

j π λ ω e ω j ωK

cos2

(11.15 2.447 ) sin2

(0.01476 1.066 ) 196.8 cos2

(0.01476 1.066 )

sin2

(11.15 2.447 ) 0.3239 0

λ λ λ λu

λ λ

λ λu

6 2 1 9 3 1 9 3

6 2(A5)

Taking the real and imaginary parts equal to zero

Fig. 19. The detailed analysis of FOINC-based tracker for third pattern of solar irradiance.

Fig. 20. Comparison between PV output power of INC and FOINC based trackers.

M. Al-Dhaifallah et al. Solar Energy 159 (2018) 650–664

663

⎛⎝

⎞⎠

− − ⎛⎝

⎞⎠

− + =− + + − +π λ ω e ω π λ ω e ω Kcos2

(11.15 2.447 ) sin2

(0.01476 1.066 ) 196.8 0λ λ λ λu

6 2 1 9 3(A6)

And,

⎛⎝

⎞⎠

− + ⎛⎝

⎞⎠

− + =+ − + − +π λ ω e ω π λ ω e ω ωKcos2

(0.01476 1.066 ) sin2

(11.15 2.447 ) 0.3239 0λ λ λ λu

1 9 3 6 2(A7)

So, Eqs. (A6) and (A7) are the same as Eqs. (24) and (25) in the paper, respectively.

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