Improved double integral sliding mode MPPT controller based parameter estimation for a stand-alone photovoltaic system

Energy Conversion and Management 139 (2017) 97–109

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Improved double integral sliding mode MPPT controller based parameter estimation for a stand-alone photovoltaic system Nasrin Chatrenour, Hadi Razmi ⇑, Hasan Doagou-Mojarrad Department of Electrical Engineering, East Tehran Branch, Islamic Azad University, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 17 December 2016 Received in revised form 30 January 2017 Accepted 19 February 2017

Keywords: Improved Double Integral Sliding Mode Controller (IDISMC) Maximum Power Point Tracking (MPPT) Photovoltaic (PV) system Genetic Algorithm (GA)

a b s t r a c t In this paper, an Improved Double Integral Sliding Mode MPPT Controller (IDISMC) for a stand-alone photovoltaic (PV) system is proposed. Performance of a sliding mode controller (SMC) is greatly influenced by the choice of the sliding surface. Switching surface coefficients were selected by the use of Hurwitz stability theorem. The IDISMC not only is robust against parametric and non-parametric uncertainties, but also has a very small steady-state error, thanks to the use of double integral of tracking voltage error in the definition of its sliding surface. For realistic simulation, Genetic Algorithm (GA) method was used to estimate parameters of solar panels model. The validity of the proposed double integral SMC in maximum power point tracking was approved by comparing the simulation results obtained for a sample PV system with the results of other methods. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The global warming, air quality and the increasing fuel costs worldwide call for the need to operate Renewable Energy Resources (RESs) [1,2]. The photovoltaic (PV) generation system is one of the most important RES in modern power systems, due to their desirable features such as pollution-free, free and infinite availability, and it requires less maintenance [3]. But, their applications are usually limited by high cost, siting and sizing restrictions, and the power of PV array varies with solar irradiance and ambient temperature [4]. The variation in parameters of the PV cell for changing the irradiance levels and temperature conditions results in changing electrically characteristics. In this paper, a Genetic Algorithm (GA)-based optimization method is used to find the parameters of stand-alone PV system. Therefore, the necessity of track the maximum power point during changes in the weather conditions for increasing the efficiency is inevitable [5,6]. So, the main goal of this paper is to develop the optimal operation of the PV array based on Maximum Power Point Tracking (MPPT) technique. In the literature, there are different MPPT techniques such as offline, online and hybrid techniques. Offline MPPT techniques such as Open Circuit Voltage (OCV) method and Short Circuit Current (SCC) method are also employed ⇑ Corresponding author. E-mail addresses: [email protected] (N. Chatrenour), razmi.hadi@ gmail.com (H. Razmi), [email protected] (H. Doagou-Mojarrad). http://dx.doi.org/10.1016/j.enconman.2017.02.055 0196-8904/Ó 2017 Elsevier Ltd. All rights reserved.

in PV systems because of their lesser complexity and inexpensive implementation. The major drawback of these techniques is that they usually fail under rapidly changing irradiance and do not perform well under partial shading [7]. Several papers have been focused on online MPPT techniques include the Constant Voltage (CV), Incremental Conductance (IC) and the Perturb and Observe (P&O) techniques. The main disadvantage of these methods is that they oscillate around the maximum power point [8]. Many adaptive and hybrid techniques have been developed to enhance the aforementioned techniques. A comprehensive review of these MPPT techniques are given in [9]. Also, the application of soft computing methods for MPPT of PV system have been reviewed and analyzed in [10]. MPPT control for PV system is a typical nonlinear control problem. Therefore, many MPPT based sliding mode controller (SMC) have been proposed in the literature; in both stand-alone and grid-connected PV systems. The salient features of the SMC method are as follows [11]: – It is robust with respect to internal and external disturbances, as well as parameter uncertainties. – The SMC utilizes a high frequency switching control signal to enforce the system trajectories onto a surface, namely sliding surface or sliding manifold. In [12], authors are focused on control of a stand-alone PV system with battery storage using MPPT-SMC. A DC/DC boost converter operating in continuous conduction mode, beyond varying

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Nomenclature Abbreviations SMC Sliding Mode Controller DISMC Double Integral Sliding Mode Controller IDISMC Improved Double Integral Sliding Mode Controller MPPT Maximum Power Point Tracking PV Photovoltaic GA Genetic Algorithm RES Renewable Energy Resource OCV Open Circuit Voltage SCC Short Circuit Current CV Constant Voltage IC Incremental Conductance P&O Perturb and Observe PWM Pulse width Modulation List of symbols I pv photo-generated current iD diode current leakage current ish v pv PV output voltage ipv PV output current Rse series equivalent resistor of solar panel parallel equivalent resistor of solar panel Rsh I0 dark-saturation current

meteorological conditions has been considered. In [13], an efficient MPPT controller based on hybrid backstepping and sliding mode is used. Similarity to [13], a SMC based on the admittance of the PV system was proposed in [14]. In the paper, a reference will be selected by an external MPPT algorithm to mitigate the perturbations generated by the load. In [15], effect of the SMC on a system connected to a PV panel consisting of two cascaded dc-dc boost converters has been investigated. An adaptive SMC, in which the output voltage is not required to be known, were shown in [16]. Then, in proposed model, an adaptive H1 control algorithm was proposed to ensure the fast and robust tracking. In [17], a framework for optimal operation of a small scale PV system connected to a micro-grid has been presented and a hybrid modified sliding mode has been proposed, which uses fuzzy logic control as a useful technique. In [18], a new design of stable SMC to maximize power extraction from the PV system was proposed. To improve the response speed of the PV system, a dual surface SMC was presented for MPPT in [19]. In [11], a new algorithm combining sliding mode variable structure control strategy and P&O method was addressed for fast track the global maximum power point of PV system. In [20], a new MPPT was proposed for a stand-alone PV system using the concept of double integral sliding mode controller (DISMC). The DISMC uses double integral of tracking voltage error term in its sliding surface to eliminate steady-state error apart from providing robust control actions in face of system uncertainties. However, in the most of existing literatures and research efforts on the higher order SMC, for example in [20], the coefficients of the defined sliding manifolds are taken as special values and given directly in the simulations. Furthermore, this paper proposes MPPT controller based improved double integral sliding mode via selection of switching surface coefficients by using Hurwitz stability. In the paper, in addition to using the tracking voltage error term, its first and double integral, we consider the tracking current error term in the sliding surface for calculating of equivalent switching signal. The stability of subsystem is derived by using Lyapunov theory. In the sliding mode, the sliding motion lies on the four coefficients, but, in a com-

vD

a Vt Ns kb e T Ipv ;ref KI G I0;ref Eg

vo

r pv K oc KG n p pmin pmax d l

diode voltage ideality factor of the diode thermal voltage of the diode number of series cells Boltzmann’s constant absolute value of electron’s charge temperature of solar panel photo-generated current at standard conditions short-circuit coefficient of temperature at standard conditions solar radiation dark-saturation current at standard conditions band gap energy of the silicon based on electron volts load voltage dynamic resistance of solar panel substance coefficient substance coefficient sliding surface order bit string of one of the parameters Ipv ;ref , I0;ref , Rse , Rsh and a minimum value of the parameter p maximum value of the parameter p decimal value of the bit string length of the bit string

plex and highly nonlinear form. Therefore, it is difficult to directly choose the coefficients to obtain the desired sliding motion. To simplify the switching surface design, the nonlinear sliding manifold is linearized around the desired equilibrium points, the nonlinear coefficients are calculated by Hurwitz stability. The main contributions of the paper are (i) Determine the parameters of PV system based on GA optimization method, (ii) Design of an improved double integral sliding mode controller (IDISMC) by choosing a surface coefficients by using Hurwitz stability and (iii) Consider the tracking current error term in the sliding surface for calculating of equivalent switching signal. The paper is organized as follows: Section 2 presents the modeling of solar panels. Section 3 contains the double integral sliding mode MPPT controller for PV system. Section 4 determines the switching surface coefficients. In Section 5, simulation results are presented to show the effectiveness of the proposed controller. 2. Mathematical modeling of solar panels The simple electrical model of a solar system subjected to solar radiation and connected to a load is in the form shown in Fig. 1 [21]. This circuit combines a current source, a diode, a resistor in series and a resistor in parallel. A single diode equivalent circuit of the PV cell is applied to the model, considering its simple implementation, proper accuracy and low computational efforts needed [22]. Resistors in equivalent circuit represent the losses in cells. Losses in cell are due to factors such as sunlight reflection on the cell surface, absorption of photons without formation of electron and free hole, and recombination of electrons and free holes. According to Fig. 1, characteristic equation of solar cell is expressed by Eq. (1).

ipv ¼ Ipv  iD  ish

ð1Þ

where Ipv is photo-generated current, iD is diode current, ish ¼ ðv pv þ ipv Rse Þ=Rsh is leakage current, v pv and ipv are PV output

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99

Fig. 1. The equivalent circuit of a solar cell.

voltage and current, respectively, Rse and Rsh are series and parallel equivalent resistors of solar panel. Diode current is expressed based on I-V Characteristic of a Shockley diode by Eq. (2).


vD  iD ¼ I0 eaV t  1

ð2Þ

where I0 is dark-saturation current, v D ¼ v pv þ ipv Rse is diode voltage, a is ideality factor of the diode, V t ¼ ðN s  kb =eÞT is thermal voltage of the diode, N s is the number of series cells, kb is Boltzmann’s constant, e is the absolute value of electron’s charge, and T is temperature of solar panel. By substituting Eq. (2) in (1), a characteristic equation of solar panel, can be obtained as Eq. (3).

 v þi R  pv pv se v pv þ ipv Rse ipv ¼ Ipv  I0 e aV t  1  Rsh

ð3Þ

In Eq. (3), the value of photo-generated current Ipv , varies with the temperature and solar radiation based on Eq. (4) [12].

Ipv ¼ ½Ipv ;ref þ K I ðT  T n Þ

G Gn

ð4Þ

where Ipv ;ref is photo-generated current at standard conditions, K I is Short-Circuit coefficient of temperature at standard conditions, and G is solar radiation. Gn ¼ 1000 W=m2 and T n ¼ 25  C are standard test conditions of solar panel. In Eq. (3), dark-saturation current value I0 varies with the temperature based on Eq. (5).

 I0 ¼ I0;ref

T Tn

3

 exp

  e  Eg 1 1  kb  a  V t T n T

ð5Þ

where I0;ref is dark-saturation current at standard conditions, and Eg ¼ 1:121½1  0:0002677ðT  T n Þ is the band gap energy of the silicon based on electron volts [23]. Validity and accuracy of model presented in Fig. 1 depends heavily on the estimation of the parameters Ipv ;ref , I0;ref , Rse , Rsh and a. In this paper, these parameters were estimated using a GA optimization method. 2.1. Estimation of the parameters of solar panel Solar panel manufacturers present a catalog where N s , K I and output voltage and current parameters of the produced solar panel at standard test conditions in the following modes, are specified. – Open-circuit conditions: v pv ¼ V oc and ipv ¼ 0. – Short-circuit conditions: v pv ¼ 0, ipv ¼ Isc . – Maximum power conditions: v pv ¼ V mpp and ipv ¼ Impp . By plotting a smooth curve that connects these three points to each other, the current-voltage variation curve of a solar panel at standard test conditions can be obtained. I-V Characteristic of a sample solar panel is shown in Fig. 2.

Model parameters should be estimated in such a way that the solar panel model would be able to recreate the above modes. The following cost function is considered to achieve this purpose. 
V oc  V  oc cost function ¼ Ipv ;ref þ I0;ref eaV t  1 þ Rsh 
Isc Rse  I R  sc se þ Isc  Ipv ;ref þ I0;ref e aV t  1 þ Rsh  V þI R    mpp mpp se V mpp þ Impp Rse 1 þ þ Impp  Ipv ;ref þ I0;ref e aV t Rsh

ð6Þ If the value of above cost function is equal to zero, the parameters of solar panel are estimated correctly. Based on previous studies [21], I0;ref , in silicon cells is about 100 pA; Ipv ;ref has a value of about 3–4 amp; a, depends on the type and the physical structure of the diode and has a value between 1 and 2; Parallel resistance Rsh , represents the internal losses or leakage current of Shockley diode which usually has a value of about 200–400 X; and Series resistance Rse , which is connected to the solar panel terminal, depends on the cell fabrication quality and has a small value about 0.05–0.1 X. 3. Improved Double Integral Sliding Mode Controller (IDISMC) A PV system consists of a solar panel, a dc to dc converter, a MPPT controller and a load. A topology of a stand-alone PV system connected to a load is shown in Fig. 3. In these systems, converter is an intermediary between load and solar panel and tracks the maximum power point of solar system by its duty cycle variation. Changes in duty cycle of converter are made by a controller. Any simple, fast, accurate and reliable control method that can provide maximum power point for solar panel is desirable and thus a basic need of these systems. 3.1. State equations of boost converter Generally, converters used in PV systems are mainly of buck or boost type [24]. In this paper, a detailed model of the DC-DC boost converter is used for the simulation. Voltage stabilization on the output of PV systems by these converters is usually carried out through Pulse width Modulation (PWM) at a fixed frequency. Switching in this converter is performed by one of the elements of MOSFET, BJT or IGBT. Fig. 4(a) and (b), show equivalent circuit of boost converter in open and closed switch operations, respectively [20]. According to Fig. 4(a), dynamic equations of system in open switch operation can be expressed by Eqs. (7) and (8).

v_ pv ¼ 

1 1 iL þ v pv c1 c 1 r pv

_iL ¼ 1 v pv  1 v o L L

ð7Þ ð8Þ

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Fig. 2. I-V characteristic of a sample solar panel.

In the above equations, v o is load voltage and rpv is dynamic resistance of solar panel, which is defined by Eq. (11).

  @ v pv r pv ¼  @ipv

ð11Þ

Eqs. (7)–(10), in open and closed switch operations, are different in just one part. Assuming d as duty cycle of control signal u for controlling the switch, these equations can be rewritten into Eqs. (12) and (13) in state space form.

1 1 iL þ v pv c1 c 1 r pv

ð12Þ

_iL ¼ 1 v pv  ð1  dÞ 1 v o L L

ð13Þ

v_ pv ¼  Fig. 3. Topology of a stand-alone PV system.

In these equations, d ¼ 0 and d ¼ 1 indicates open and closed switch operations, respectively. In above equations, v pv and iL are state variables and d is input variable of the system.

According to Fig. 4(b), when the switch is closed, then

v_ pv ¼ 

1 1 iL þ v pv c1 c1 r pv

1 i_L ¼ v pv L

ð9Þ 3.2. MPPT design

ð10Þ

At the maximum power point, v pv is equal to V mpp and ipv is equal to Impp . It should be noted that in steady state condition,

Fig. 4. Boost converter schematic; (a) when switch is open, (b) when switch is closed.

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current flow from the capacitor c1 is very small and thus, iL also at the maximum power point is almost equal to Impp . So, in order to design maximum power point tracker, first values of V ref and Iref which are V mpp and Impp in different weather conditions, must be identified. When load is disconnected from circuit, the last term of Eq. (3) is ignored, giving Eq. (14).

    V oc 1 0 ¼ Ipv  Io exp Ns V t

ð14Þ

So, without load being disconnected from the system, open circuit voltage of V oc can be calculated by solving relation (14) as follows.

V oc ¼ Ns V t ln

  I pv þ I o Io

a1 ðV_ ref  v_ pv Þ þ a2 ðV ref  v pv Þ þ a3   s ¼0 þ Kjsja sat /

d¼

After calculating V oc , value of V ref is calculated as follows [20].

ð16Þ

where K oc is the substance coefficient and is related to structure and material of PV module. K oc usually varies with the age and condition of PV module. This coefficient change over significantly long durations. Various experiments on solar panels have shown that value of Iref is a function of radiation intensity G. It can be proved that relation between these two variables is approximately linear and can be expressed by the following equation.

Iref ¼ K G G

ð17Þ

where K G is the substance coefficient and is related to structure and substance of PV module. The first step in designing a maximum power point tracker based on SMC is to define the switching surface. Sliding surface s should be designed to provide optimal system performance and satisfy the desired control purposes. This means that optimal control system performance can be expected whenever system somehow has to move on above surface [19]. Switching surface is usually defined by a linear combination of state variables error and its derivatives, as follows:

 s¼

n1 d þb eðxÞ dt

ð18Þ

where n is sliding surface order and eðxÞ is state variable tracking error of x. If it is assumed that n ¼ 1, sliding surface will be: s ¼ eðxÞ. When designing a double integral sliding mode controller, tracking error is defined as follows:

eðxÞ ¼ a1 e1 þ a2 e2 þ a3 e3 þ a4 e4

ð19Þ

ð22Þ

 a1 a a1 V_ ref  v pv þ 1 iL þ a2 e1 þ a3 e2 þ a4 I_ref a4 v o c 1 r pv c1   a4 a4 s  v pv þ v o þ Kjsja sat / L L L

ð20Þ

ð23Þ

3.3. Stability demonstration To analysis system stability, positive definite Lyapunov function is considered as follows:

V¼

1 2 s 2

ð24Þ

To maintain stability condition of Lyapunov theorem; h
i V_ ¼ ss_ ¼ s Kjsja sat /s must always be negative definite and thus coefficient K should be selected as positive. This means that applied control signal d, according to Eq. (23), guarantees that system error in a limited time reaches zero. In [20] has been shown that to achieve the stable condition of Lyapunov theorem (ss_ < 0), the selected constant K must be sufficiently large. 4. Determination of switching surface coefficients To determine switching surface coefficients, first, values of s and s_ in Eqs. (19) and (20) must be zero. By doing this, Eqs. (25) and (26) are obtained.

Z a2 ðV ref  v pv Þdt a1 Z Z a3 a4  ðV ref  v pv Þdt dt  ðIref  i_L Þ a1 a1

V ref  v pv ¼ 

a1 a2 a3 I_ref  i_L ¼  ðV_ ref  v_ pv Þ  ðV ref  v pv Þ  a4 a4 a4

Z

ð25Þ

ðV ref  v pv Þdt ð26Þ

R R y1 ¼ f ðV ref  v pv Þdtgdt,

where

e1 ¼ V ref  v pv Z e2 ¼ ðV ref  v pv Þdt Z Z ðV ref  v pv Þdt dt e3 ¼

ðV ref  v pv Þdt þ a4 ðI_ref  _iL Þ

Due to presence of jsja in s_ , the speed of reaching to sliding surface increases with distance. In this equation, a is a number between zero and one. Also, / is a small number that is used to reduce the chattering. As a result, using Eqs. (12) and (13), the control input d will be obtained as Eq. (23).

ð15Þ

V ref ¼ K oc V oc

Z

R

Considering, y2 ¼ ðV ref  v pv Þdt _ y3 ¼ Iref  iL , the following equations are obtained.

and

y_ 1 ¼ y2

ð27Þ

y_ 2 ¼ 

a2 a3 a4 y  y  y a1 2 a1 1 a1 3

ð28Þ

y_ 3 ¼ 

 a1 _ 1 1 a a V ref  v pv þ iL  2 ðV ref  v pv Þ  3 y2 c 1 r pv c1 a4 a4 a4

ð29Þ

e4 ¼ Iref  iL And a1 to a4 coefficients are constants, whose values will be obtained by using Hurwitz Stability Theorem. To obtain control signal d, sliding surface and its derivative must be considered to be zero. By doing this, steady-state error will approach zero. Switching surface derivative is calculated by the following equation.

The set of linear Eq. (30) are obtained by linearization with Taylor expansion around the equilibrium point.

s_ ¼ a1 e_ 1 þ a2 e_ 2 þ a3 e_ 3 þ a4 e_ 4

2

Considering s_ ¼ Kjsja sat equation give Eq. (22).


 s /

ð21Þ [20] and substituting it into the above

3 2 3 y1 y_ 1 6_ 7 6 7 4 y2 5 ¼ A4 y2 5; y3 y_ 3

2

0 6 a A ¼ 4  a31

1

0

 aa21

 aa41

A31

A32

A33

3 7 5

ð30Þ

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where

A31 ¼

A32

chromosome has five parts: Ipv ;ref , I0;ref , Rse , Rsh and a. To represent each chromosome, the binary coding system was used. The bit strings of each parameters should be decoded by Eq. (31).

a3 1 a2 a3  þ a4 c1 r pv a1 a4

p ¼ pmin þ

a2 1 a2 a3 ¼  þ 2  a4 c1 r pv a1 a4 a4

A33 ¼

1 1 a1 a2 þ  þ c1 r pv c1 a4 a1

According to Hurwitz stability theorem, matrix A is stable if and only if all its Eigen values have negative real part. By finding characteristic equation of matrix A by using equation jkI  Aj and equating it to a stable desirable characteristic equation such as ðk þ 1Þðk þ 2Þðk þ 3Þ, the coefficients of switching surface s will be obtained. By doing this and assuming a4 ¼ 1, other coefficients are obtained as follows:

a1 ¼ 6c1 

1 ; r pv

a2 ¼ 11c1 ;

and a3 ¼ 6c1

According to the above values, it can be seen that value of a1 is a function of r pv and thus varies dynamically with conditions. 5. The simulation results MSX-60 PV module made by Solarex is selected for simulation. Information about this module under standard test conditions are available on the datasheet provided by its manufacturer and are given in Table 1. Information about the elements in the boost converter used in this study are presented in Table 2 [20]. In Section 5.1, the results of the use of GA optimization method to estimate parameters of PV modules model are presented. The simulation results of MPPT in different weather conditions are shown in Section 5.2. Finally, in Section 5.3, a comparative analysis with other methods are presented to show the effectiveness of the proposed controller. 5.1. Estimation of the parameters of solar panel by GA There are five parameters in the single diode model of PV system including the photo-generated current at standard conditions (Ipv ;ref ), the dark-saturation current at standard conditions (I0;ref ), the series equivalent resistor of solar panel (Rse ), the parallel equivalent resistor of solar panel (Rsh ) and the ideality factor of the diode (a) which must be estimated by the GA. Therefore, each

pmax  pmin 2l  1

d

ð31Þ

where p is the bit string of one of the parameters Ipv ;ref , I0;ref , Rse , Rsh and a; pmin and pmax are the minimum and maximum values of the parameter p, respectively; d is the decimal value of the bit string; and l is the length of the bit string. Note that the values of l, pmin and pmax is determined by user. The numbers of bits of each part, based on the required accuracy and according to engineering experience, were selected equal to 20. Minimum and maximum values of these parameters, according to the description provided in Section 2.1, are considered in accordance with Eq. (32).

1:0e  7 6 I0;ref 6 1:5e  7 3:7 6 Ipv ;ref 6 4 1 6 a 6 1:5

ð32Þ

0:01 6 Rse 6 0:5 300 6 Rsh 6 400 In the GA, the first generation begins to search for the optimal response by a series of random solutions called the initial population. The initial population has 3000 chromosomes and later, these solutions and genetic operators of selection, crossover and mutation are used to build the next new population. Usually, the new population is fitter, which means that the next generation is better than the older one. This process will continue when the numbers of generations reaches to 2000. The most important operator in GAs is crossover operator. Crossover in GA eliminates dispersion or genetic variation. In this study, Single point crossover is used. Mutation is another operator and produces other possible solution. For simulations of this section, mutation probability is assumed to be 0.15. Roulette wheel method is used to select parents. Finally, the best solution found in the search process, must be decoded to obtain their actual form by Eq. (31). The results obtained from the use of GA to estimate the parameters of studied solar panel are as follows:

I0;ref ¼ 1:60623e  7 Ipv ;ref ¼ 3:802346 a ¼ 1:319639 Rse ¼ 0:1936763

Table 1 MSX-60 PV module information [20]. Parameter

Value

Isc Voc Impp Vmpp KI Ns

2.8 A 21.6 V 2.2 A 18.2 V 0.06%/K 36

Table 2 Boost converter information. Parameter

Value

RL L C1 C2 Switching frequency of IGBT

20 X 5 mH 380 lF 330 lF 10 kHz

Rsh ¼ 313:3311 In Figs. 5–10, examples of I-V, P-V and P-I characteristics are plotted by values of estimated parameters in the previous section for tested solar panel in the states of, (1) Solar radiation is 1000 W=m2 and temperatures are 0, 25, 50, 75 and 100  C (2) Temperature is 25  C and Solar radiations are 200, 400, 600, 800 and 1000 W=m2 In these curves, effects of temperature change and solar radiation change on these characteristics are quite clear and evident. According to this fact that the solar panel output voltage and current values at maximum power point (V ref and Iref to design IDISMC) are obtained by values of these curves. V oc and V ref values to calculate the K oc and G and Iref to obtain K G are given in Tables 3 and 4, respectively.

N. Chatrenour et al. / Energy Conversion and Management 139 (2017) 97–109

Fig. 5. Effect of temperature change on the I-V characteristic.

Fig. 6. Effect of temperature change on the P-V characteristic.

Fig. 7. Effect of temperature change on the P-I characteristic.

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Fig. 8. Effect of solar radiation change on the I-V characteristic.

Fig. 9. Effect of solar radiation change on the P-V characteristic.

Fig. 10. Effect of solar radiation change on the P-I characteristic.

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N. Chatrenour et al. / Energy Conversion and Management 139 (2017) 97–109 Table 3 V oc , V ref and Iref for proposed controller at different temperatures and solar radiation of 1000 W=m2 . Temperature (°C)

Voc (V) Vref (V) Iref (A)

0

25

50

75

100

23.62 19.70 3.48

21.10 17.20 3.48

19.22 15.18 3.51

17.84 13.90 3.48

16.86 12.81 3.50

Table 4 V oc , V ref and Iref for proposed controller at 25  C and different solar radiation. Irradiance (W/m2)

Voc (V) Vref (V) Iref (A)

200

400

600

800

1000

19.06 15.71 0.66

19.95 16.39 1.37

20.46 16.75 2.08

20.82 16.90 2.80

21.10 17.20 3.48

5.2. MPPT by the proposed controller In s_ ¼ Kjsja sat


 s /

, a, / and K are assumed equal to 0.01, 0.97

and 10e10, respectively. First, it is assumed that the temperature is 25  C and solar radiation is 1000 W=m2 . According to Table 3, in these conditions, the values of 17.20 V and 3.48 A, should be tracked. In Figs. 11–14, changes of the solar panel output voltage, the solar panel output current, tracking voltage error and tracking current error after applying the IDISMC are shown, respectively. The performance of the proposed controller was also tested in other weather conditions. For example, Figs. 15–18 show the changes of solar panel output voltage, solar panel output current and voltage tracking error, at 75  C and solar radiation of 400 W=m2 . In this case, the values of 12.99 V and 1.37 A must be tracked. As can be seen in these figures, the desirable performance of IDISMC in various conditions has been maintained and tracking operations have been carried out as well.

Assuming h1 and h2 as upper and lower limits of value of output voltage of the PV cell at steady-state, respectively, chattering magnitude can be obtained by Eq. (33).

h ¼ h1  h2

ð33Þ

The steady-state error value of output voltage of PV cell is also calculated by Eq. (34).

SSE ¼ maxðjV ref  h1 j; jV ref  h2 jÞ

ð34Þ

In Table 5, the results of the proposed method in this paper and other methods provided in [20], in standard conditions, are compared in terms of chattering magnitude (33) and steady-state error (34). As can be seen in the results of this table, the proposed controller outperforms the five controllers provided in [20]. The reasons for this superiority is not only the use of double integral of voltage error in the definition of switching surface, but also the two following factors:

5.3. Discussion Chattering and steady-state error are two important factors that must be investigated to evaluate the performance of the IDISMC.

(1) Using current error in the definition of switching surface; (2) Using Hurwitz stability theorem to calculate accurately the switching surface.

Fig. 11. The solar panel output voltage at 25  C and solar radiation of 1000 W=m2 .

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Fig. 12. The solar panel output current at 25  C and solar radiation of 1000 W=m2 .

Fig. 13. Voltage error at 25  C and solar radiation of 1000 W=m2 .

Fig. 14. Current error at 25  C and solar radiation of 1000 W=m2 .

N. Chatrenour et al. / Energy Conversion and Management 139 (2017) 97–109

Fig. 15. Solar panel output voltage at 75  C and solar radiation of 400 W=m2 .

Fig. 16. Solar panel output current at 75  C and solar radiation of 400 W=m2 .

Fig. 17. Voltage error at 75  C and solar radiation of 400 W=m2 .

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Fig. 18. Current error at 75  C and solar radiation of 400 W=m2 .

Table 5 Comparison of the performance of control methods presented in [20] and the proposed controller in standard conditions. MPPT controller

h (V) SSE (V)

P&O

Adaptive P&O

SMC

ISMC

DISMC

IDISMC

6 2.865

4 1.865

0.03 0.015

0.0025 0.01

0.01 0.005

6.9541e4 6.5754e5

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