2013 13th International Conference on Control, Automation and Systems (lCCAS 2013) Oct. 20-23, 2013 in Kimdaejung Convention Center,Gwangju, Korea

Novel Algorithm of MPPT for PV Array Based on Variable Step Newton-Raphson Method Through Model Predictive Control

Seyed Hossein Hosseinil*, Amir Farakhor2 and Saeideh Khadem Haghighian3

1 Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, 51666, Iran (Tel : +98-914-313-4864; E-mail: hosseini@tabrizu.ac.ir) * Corresponding author

2 Department of Electrical Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran (Tel : +98-914-108-6010; E-mail:amir farakhor@yahoo.com)

3 Faculty of Electrical and Computer Engineering, University of Tabriz , Tabriz, 51666,1ran (Tel : +98-914-756-2308; E-mail:reyhanehkhadem@yahoo.com)

Abstract: In this paper a new method of maximum power point tracking (MPPT) strategy based on Newton-Raphson algorithm is presented for photovoltaic (PV) power systems, which makes a high efficiency for PV power systems. In this approach variable step Newton-Raphson algorithm is applied to improve the dynamic response of PV array MPPT which is based on improved perturb and observe (lP&O) algorithm. Improved perturb and observe (lP&O) MPPT algorithm and model predictive control (MPC) are developed to control the output current of a boost converter in order to extract maximum solar power from the panel. MPPT tracker compels the system to operate at maximum power point through appropriate control of boost DCIDC converter. In the fmal stage a simple model predictive current control strategy is employed for two-level three-phase inverter to deliver power to a load. Descriptions of the proposed system along with detailed simulation results which verify its feasibility are given using MATLAB/Simulink software.

Keywords: Photovoltaic array, Improved perturb and observe algorithm, Variable step Newton-Raphson method, Model predictive control.

1. INTRODUCTION

Demand for electrical energy has remarkably increased during the recent years with growing population and industrial progress. Since long time ago, fossil fuels have served as the major source of generating electrical energy, environmental consequences of these resources have made it necessary to benefit from new energy sources such as wind and solar. On the other hand, generation, transmission and distribution of electrical energy in the current manner cannot meet energy supply requirements of consumers. Transmission line losses, adjustment of improper voltage, and low power quality are among the problems of electrical energy consumers in the conventional methods. Photovoltaic systems (PV) could act as suitable choice for alleviating the aforementioned problems regarding provision of the local electrical energy for consumers, thanks to their low current expenditure and absence of transmission losses. Hence, tracking the maximum power of the PV arrays at real time is very important to increase the whole system performance; the interest in this area is significantly growing focusing mainly on the MPPT efficiency improvement. Recently, a large number of methods have been proposed for MPPT PV system [1], which can be classified into two main groups, first the methods which are based on fixed steps and the second group are based on variable steps methods. Both groups are employing conventional proposed algorithms for MPPT such as incremental conductance method (INC), perturbation and observation method (P&O), advanced intelligent algorithms [2], but the variable steps algorithms are

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improved algorithms of conventional methods which are based on numeral calculations and mainly employ new control approaches. These new control techniques feature advantages of simplicity and high flexibility, and less fluctuation around the maximum power point which increase efficiency of the PV system [3]-[6]. Model predictive control (MPC) is an attractive alternative to the classical control methods, due to its fast dynamic response, simple concept, and ability to include nonlinearities and constraints in the design of a controller [7]. MPC presents several advantages over the conventional control Techniques such as easy implementation which is expected to improve PV system utilization efficiency under continuous changes in solar irradiation overcoming disturbances and uncertainties [8]. The implementation of a PV array MPPT using MPC combines two keys of vital importance, speed and reliability, avoiding unacceptable oscillations despite the increased speed. Improved perturb and observe (IP&O) MPPT algorithm and MPC model is developed to control the output current of the boost converter in order to extract maximum solar power from the solar panel [9]. In the present paper, maximum power point tracking (MPPT) algorithm is proposed using improved perturbation and observation (IP&O) method. Variable steps were used for determining reference signal in this method in order to improve MPPT performance and also to enhance convergence speed and system precision. Model predictive control is used to apply the proposed MPPT scheme using a boost converter. Simulation results verify the proposed MPPT algorithm.

2. PROPOSED SYSTEM CONFIGURATION

Block diagram of the proposed system is illustrated in Fig. l.The system consists of a PV array, a boost converter and an inverter connected to a three phase load. Control strategy in this study is based on the DC step-up converter boosting the level of the PV system output voltage, as well as determining the factors of maximum power exploitation. PV system output voltage (Vpv) and current (lev) measurements are formed as inputs for the MPPT and the predictive controller. The MPPT reference output current (i*) is input data for the MPC in order to obtain information in one sampling time and setting duty cycle of boost converter. The discrete-time model of the inverter is used to predict the future behavior of the load currents for each of the 8 possible switching states. Load current (io) and DC link capacitor vo Itage (V c) are as input data for inverter controller. This inverter is applied to provide energy to the three phase load. The system components will be described in following sections respectively.

1Io&l 5

Predictive Cost Coo"o! fooction Boost

Optimiztion Converter R�i!!� J(K+l) -Ci1eu1ation

(MPl'I)

+ Two Le'-el Three e Ve Three leg Phase

In"erter Load

- - -S;. � � S. S, S�

Fi.nit� Cootrol Set

(Fes) Mo&.l Predictive

Coo"o!

Fig. 1 Block diagram of the proposed MPPT system.

3. PV ARRAY AND MPPT 3.1 Modeling PV Array and its Specifications

A PV array is defined as a group of several modules electrically connected in series-parallel combinations to generate the required current and voltage. An equivalent circuit of PV array is shown in Fig. 2.

Ip" Rs �

Idt +

Rp vp"

Fig. 2 Equivalent circuit of PV solar cell.

The nonlinear behavior of a PV array according to PV equivalent circuit can be described by these equations:

i h= (JHKi(T -D))(�) (1) p 1000

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PV cell is modeled as an ideal current source of value iph which depends on irradiation and ambient temperature, the diode is in parallel with this current source to show nonlinear nature of PV cell and Id is the diode current can be described by Eq. (2) and other information about PV are listed in Table 1 derived from the manufacturer's datasheet [10].

Ve, Jd = io(exp(-) -I) NVe

. T Eg T To = Jo( -)exp[-( - -1)] T, NV, T,

(2)

(3)

So that the terminal I-V relationship for the PV module is given by

(4)

Table 1 PV array parameters

10 Diode Reference Reverse

Saturation Current l. 13 x 10-6(A) Eg Semiconductor Band gap

Voltage 1.16 (e.v)

N Emission Coefficient 1.81

Ipv Reference Short-circuit Current 5.61 (A)

ki Short-circuit Temperature

Coefficient l.96 (mAIK) K Boltzmann's Constant 1.38* 1 0-23(J/K)

Tr Nominal Temperature 298. I 5 (K)

q Charge on an Electron l.6 * 10-19 (C)

Rs Series Resistance 2.48 (0)

Rp Parallel Resistance 8.7(0)

Ns Number of Series Cells 72

The nonlinear P-V, I-V and P-I characteristics of solar cells are well known and the simulation result are illustrated in Fig. 3. As the voltage and current of a PV array vary in respect to irradiation and temperature levels, the Eq. (4) has been computed for several irradiation levels in the same temperature level (T=25C). These figures illustrate the nonlinear variations of the PV maximum power point respect to irradiation levels. In this paper the PV array is modeled in the Matlab/Simulink based on Eq. (4) [7].According to Fig. 3(b) there is only one operating point on every Ppv-Ipv characteristic that the maximum PV power can be extracted. In this paper providing this operating point for PV array by using new method of Maximum power point tracking strategy based on Newton-Raphson algorithm is presented for photovoltaic system, which makes a high efficiency PV power system.

1S 20 25 30 3S 40 50 FV�V�M

(a) 2�r- ---'----�---P'�����· -- .-----�--�

! i ... _( .. - ... _ .. j ... -i

... _ ..... ,-! ,-",,,,1-,,,,,, -""'"

(b)

! i ! .""_. i·-.-",,.i_

! i i i i i i l .... _.,! . ... _ .... ,!._!

6 ·"""'--t"""'-1·"'-"'1"-"""r-' "'r"""--r"""'--l ! ! "'i """ ·······-i·······-l····_····!··_······f- ... +- .... + .... -i ... t-.....

�

..... _ ....... - ... 1--i i ···

·r 1

r===:1 .... _ .. 1..- ... � .... i- ... �

10 15 20 25 30 35 4G py"'*"v ..... M

(c) Fig. 3 Characteristic (a) P-V, (b) P-I and (c) I-V for

varying irradiance, T=25"C.

3.2 Maximum Power Point Tracker of Variable Step

Power extracted from PV can be expressed as a function of PV output current and solar irradiation. There exists an optimum Ipv which maximizes output power of PV for every irradiation. To get maximum power point, optimal value of Ipv is immediately probed using perturbation and observation technique. P&O algorithm [5] works based on the following procedures: if increment of Ipv causes an increase of P pv in the former step, in this state, the search for fmding optimal Ipv continues in the same direction, the opposite direction is followed otherwise. Increment of PV power (P pv) is approximated in terms of increase of power in dc linle For improving performance of the algorithm as shown in Fig. 4 a variable step for Ipv variations is assumed in improved perturbation and observation (IP&O) method [5]. This leads to improvement of convergence speed and system accuracy. Variable step technique operates according to Newton-Raphson method. In this method, root of function is estimated through the following equations:

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x =X _ F(X,,) 1/+1 1/ F·(Xn)

(5)

Xn is initial value of X, F (Xn) is value of function in point Xn, F'(Xn) is derivative of function in point Xn' The function F(Xn) can be written as below:

F(X,,+I)= F(lp)n+ 1)) = :: pv

P(n+l)-P(n) =

1 =grad(n +l)

pJn + 1) -1pJn)

. ( ) . ( ) d2 P F X,,+I = F Id,(n+ I) = dl2 pv

= grad(n + I) -grad(n)

11,,(n + 1) -11,,(n)

(6)

(7)

Using the previous equations, step of Ipv variations (L'lIpv) is expressed as:

M = F(Xn+1) = P(n+l)-P(n) (8) pv F' (Xn+l) grad(n + 1) -grad(n)

Therefore, reference voltage is calculated through the following equation:

1 pvref(n + 1) = 1 pvref(n) + M pl' (9)

Using variable steps, maximum power point tracker converges to the maximum power value more rapidly an� �ower fluctuations will decrease. Range of Mpv vanatIOns has been limited. Mpv limit changes depending on PV size and system design parameters. All simulation results for this controlling method will be presented in part (6) .

,-----'N�O __ --<1'(k» P(k+Ji:!:) �----=-Y"'=----_

Fig. 4 Flow chart of improved perturb and observe (IP&O) algorithm.

4. IMPLEMENTATION OF MODEL PREDICTIVE CONTROLLER

The main purpose of model predictive controller is the estimation of the future behavior of the controlled variables so that a cost function could be minimized [11] - [14]. A conventional boost converter is used to control the output current which is drawn from PV arrays. The reference output current of the PV arrays is also obtained by the MPPT algorithm. Fig. 1 indicates the equivalent circuit of the boost converter. There are only

two switching states for this converter. When the switch is considered as open, the operation of boost converter can be described as follows: dJpv = � TPv-� Vc (10)

dt L L In the case of the closed switch, the output PV current is calculated as follows: dJpv = � /1JV (11)

dt L The Euler approximation is then used to obtain the discrete model of the system:

Ts , Ts , Jpv(k + I) = -Tpv (k) --vc(k)+ lpv(k)

L L (12)

Jpv(k+l) = TSTpv(k)+Tpv(k) (13)

L Where, Ts is the sampling frequency. The Eqs. (12, 13) predict the future behavior of the PV array output current for both switching states. It is also clear that V Py, Ipy and Vc at time instant K is required for the prediction of the future behavior of PV output current. Therefore, one step predictive controller inputs V P" 'pv and V c estimating the future behavior of PV output current. Then a cost function must be minimized to determine the switching state at next time instant. Definition of the cost function plays a major role in MPC constraining the deviations of the controlled variables from their references so the cost function is given as follows:

J�:�,1 =1 Jpv, � n(k + I) -Jpv Rcr 1 (14)

The cost function assures the tracking of PV output current from the reference current provided by MPPT algorithm. For each sampling sequence the cost function is evaluated twice for each switching state. Comparison of cost function for different switching states determines the control actions for the following time instant. Fig. 5 shows the implemented control scheme to the boost converter.

Fig. 5 Flow chart of model predictive controller.

Variations may appear in the output voltage of the boost converter because the first priority of the boost converter is MPP tracking. Therefore, a two-level three-leg inverter is employed to feed a three-phase load with a regulated current [\5].

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5. MODEL PREDICTIVE CONTROL APPLIED TO A THREE-LEG INVERTER

Model predictive controller is also applied to the inverter. Since the switching states are limited, finite control set MPC is an efficient scheme to control the inverter. There are eight possible switching states for a two-level inverter as follows:

s � {[�H�H�H:m m m H:l} According to Fig. 6, inverter output phase voltages can be obtained as follows:

Ve V�m = -(2Su-S,,-Sw) 3

Vc Vvn = -(2Sv-Su - S .. )

3 Vc

V .. o = -(2Sw-Sc-Su) 3 (\5)

Where, Su, Sy and Sw are equal to 1 if the corresponding device is on, 0 if the device is off. The dynamics of the inverter currents can be expressed as:

, , dinn Vun = (R£+ Rl)lou+ (Lf+ LI)­

dt

Vv� = (R£+ Rl)iov+ (Lf+ LI) div" dt

Vwn = (Rr+ RI) iow+ (Lr+ LI) diuw dt

Discrete time model of the system is also given as: , T,(R/+RI) , T" lnn(k+ I) = (1 )lnn(k)+--Tnn(k)

Lf+LI LI+LI

, T,(RI+RI)),

(k) T,

T/ (k) lvv(k+l) = (1- Iv", +--Vv� L/+LI L/+LI

, T,(R/+RI) , T, T

(\6)

lnw(k + I) = (1- )low(k) +-- v wo(k) Lf + Ii LI + Ii (17)

To control the output current of the load, a cost function is defined as:

J =1 iou(k+ 1) -iOURd(k+ 1) 1 + 1 iov(k+ 1) -iOVRd(k+ 1) 1 + (18) 1 iow(k+ 1) -iOWRd(k + 1) 1 For each sampling time, the cost function will be calculated for all switching states. Then the proper control actions will be chosen for the next time instant.

Vc

p,------,----,----,

K L-� ______ � ____ _

Output ACFuter

I I I I 1 ______ -----1

Fig. 6 Two-level three-leg inverter.

Three Phase

6. SIMULATION RESULTS

In order to verify the feasibility of the proposed MPPT algorithm and applied predictive control, simulation results are provided using Matlab Simulink. The system parameters are given in Table 2.

Table 2 System parameters

L 20mH

C 1000uF

Lr 15mH

Rf 0.070

RLoad 100

Ts 50us

System behavior is studied under abrupt solar irradiation variations. Fig. 7 shows the MPPT current references for irradiations 800, 900 and 1000 W/m2• These changes occur at time instants 0.6s and 0.8s. The dynamic response of the proposed MPPT algorithm to sudden solar irradiation changes has improved due to the usage of variable steps. Also, oscillations around MPP point are decreased.

5 .• ,-------,----,---------,-----,-------,-----,

5,2

g s�··················· + ........................ j ............ ! ······················· H ············ ;························1 � au . , , n,6 � • �uf-.. · ............ ····· .. ·;····· .......... · .......... ···+ .. · · .......... ·· .. ···· .. ; ............................ ; ............................ ; ....................... ··1

4,2

!,4

Fig. 7 PV reference current under irradiation changes from 800W/m2 to 900W/m2 and then tol000W/m2•

Fig. 8 depicts the PV array output current which accurately tracks the reference value calculated by MPPT algorithm. Boost converter and applied predictive controller are able to extract the maximum power from PV array.

5.5,--------,-------,-------,------,------,-------,

:!: � 45 , 0 0; � 0; 4 0 > �

3,5

1,4 0,5 0,6 0,7 Tim. (I)

0,8 0,9

Fig. 8 PV output current under irradiation changes from 800W/m2 to 900W/m2 and then tol000W/m2•

158 1

As a result of increase in irradiation and effective control of MPPT system, output power PV array increase as illustrated in curve of Fig. 9. PV output power in steady state fails to have any fluctuation around the new operating value.

22 ° :

,

1- ...

°

....... . .......

,

.. .......... , . .. .. . .... ... ......... .

.. , ........ . .. .. ! " .,

. .. . -

01-...... , .. .. · .. .. .. ··,f .. · .. , .. .. , '-

12 0 • •

111ji,4 0,5 0,6

,

0.7 Time I')

•

0,8

•

0,9

Fig. 9 PV output power under irradiation changes from 800W/m2 to 900W/m2 and then tol000W/m2•

The output voltage of the boost converter is shown in Fig. 10. Some variations appear in boost converter output voltage because the first priority of the converter is devoted to MPP tracking. Therefore, an inverter is employed to feed the load with a constant regulated current.

0

60

50

0

8,4

".

! 0,5

(

,,-- ".

•

0,6

'"

! ,

0.7 lime(s)

V

i :

0,8

...

. ,.

,.. -

,

0,9

Fig. 10 Boost converter output voltage.

Despite the variation in the output voltage of the boost converter, the proposed inverter and applied predictive controller succeeds to regulate the load current. As it is shown in Fig. 11, reference values for inverter output current are well tracked.

0,6\ 0.7 0.75 0,8 08S TIme/5)

0,9

Fig. 11 Inverter output current.

0,95

7. CONCLUSION

In the current paper, a new strategy of maximum power point tracking was proposed for solar power systems; the algorithm used improved perturbation and observation (IP&O) method with variable steps. For promoting reliability of the system, the proposed maximum power point tracking system is implemented through model predictive control method. MPC benefits from advantages of simplicity, high flexibility and less fluctuation around the maximum power point, which increases efficiency of the PV system. Simulation results indicated that MPPT strategy can automatically probe optimal operational values and yield to maximum output power from PV panel despite variations of weather conditions including irradiation changes.

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