1

Abstract-- A three phase grid connected photovoltaic system is

simulated. The grid connected photovoltaic (PV) interface system consists of a sine PWM inverter and its control. The control algorithm is based on state feedback linearization technique. The Matlab/SimPowerSystems simulation of the grid connected PV interface under a sudden change of environmental conditions shows the validity of the state feedback linearization technique for maximum point tracking, power factor control and total harmonic distortion in compliance with IEEE519-1992.

Index Terms-- grid connected photovoltaic interface, maximum power point tracking, power factor control, state feedback linearization control.

I. INTRODUCTION HE. World Bank estimated in 2006 that 1.6 billion people in the world do not have access to electricity power. The

majority of this population does live in sub-Saharan Africa and South-Asia. This situation is mainly due to the lack of power generation infrastructure and the fast increase of oil prices. In fact, the growth of the population in these parts of the world is so high that, the existing power facilities cannot meet the energy demand. Moreover, because of the higher oil prices, the gross domestic product in these countries has dropped. Therefore, the population cannot afford the cost of electricity. Again, higher oil prices resulting from the high demand and the international conflicts can weaken the energy security of oil or gas importing countries. For example, the gas crisis between the Ukraine and Russia in 2006 has shown the weakness of European Union energy security. Another example is the effect of tensions in the Middle Eastern on the rise in pump prices. With all the afore-mentioned issues, how can oil or gas importing countries build a sustainable energy system for meeting their own energy demand without harming the environment nor affecting their energy safety?

In this case, renewable energy sources, nuclear energy or less pollutant fossil sources are good candidates for both protection of environment and energy safety. But nuclear energy contains some drawbacks such as radioactive waste and after Chernobyl, nuclear power plants are ruled by very strict international policy. More, nuclear power plants are very

The authors would like to thank Natural Sciences and Engineering Research Council of Canada (N.S.E.R.C.) for providing financial support for this research work.

A. Ovono Zué and A. Chandra are with the Department of Electrical Engineering, Ecole de Technologie Supérieure, Université du Québec, Montréal QC H3C-1K3, Canada (email : aslain.ovono-zue.1@ens.etsmtl.ca; chandra@ele.etsmtl.ca )

costly in comparison with fossil power plants. Therefore developing countries cannot afford the nuclear energy. The only solution viable for both poor countries and oil importing developed countries are renewable energy sources. It is important to mention that renewable energy such as solar energy, wind energy and hydro energy are greenhouse gases free. Also, the sun and the wind are available almost all over the planet. The renewable energy sources can assist the existing power systems which are presently on the verge of saturation. On the other hand, renewable sources can supply remote villages and hamlet. For this purpose, renewable sources can reduce not only the cost associated with the construction of power transmission line, but also the dependency of remote villages upon diesel fuel for power generation.

Our goal is to simulate and control a grid connected photovoltaic system. In fact photovoltaic is the term used to describe a device capable of converting directly the energy contained in photons of light into an electrical voltage. Photovoltaic systems have the advantage of being maintenance free and the fuel (Sun) is inexhaustible. When connected to the grid, PV systems require a power conditioning unit based on an inverter, which ensures the perfect compliance of PV systems operation with IEEE standard 519-1992. Because of the intermittence of the sun, power quality issues such as harmonic distortion, frequency fluctuation and reactive power injection into the grid can occur. Therefore the aim of the PV system and its control is to solve all the afore-mentioned power quality issues. Many linear control techniques such as the cascade structure based on PI controllers have been proposed. The cascade structure consists of an outer voltage loop and an inner current loop. The cascade structure is simple to implement. But the cascade structure can present poor dynamic when there are uncertainties in the power system model, leading the system to instability. PI controllers are designed with the approximation model of the PV system around an operating point. However when the system does function in the neighborhood of the latter operating point, PI controllers will fail to operate. In this paper, the state feedback linearization technique is used for controlling the grid connected PV interface current and for maximum power point tracking. The main advantage of state feedback linearization control technique is its insensitivity to operating point. A linearization control law is designed. After applying this nonlinear control law to the system, this one becomes linear. Therefore, a linear control law can be designed with pole placement for stabilizing the system. A

State Feedback Linearization Control of a Grid Connected Photovoltaic Interface with MPPT

Aslain Ovono Zué, Student Member, IEEE, and Ambrish Chandra, Senior Member, IEEE

T

2009 IEEE Electrical Power & Energy Conference 978-1-4244-4509-7/09/$25.00 ©2009 IEEE

2

brief review of photovoltaic cell mathematical model is presented in section IIA. Then maximum power point of PV array is defined is presented in section IIB. Finally the control strategy and simulation results are presented and discussed in sections III and IV respectively.

II. POWER SYSTEM DESCRIPTION The power system consists of a PV array, the dc bus

capacitor, a three phase sine wave PWM voltage source inverter and an L filter. The voltage source inverter is responsible for both maximum power point tracking and power displacement control whatever the environmental conditions.

A. PV array model PV module consists of PV cells connected in a given way

in parallel or series depending on the PV module ratings. A single module rating is limited to few hundreds watts. When higher power is required PV modules are connected in series and in parallel to obtain PV array. The theoretical model of a PV array is deducted from that of a single PV cell. This model is useful to simulate the PV array behavior with Matlab/Simulink. The well known model [1] of a PV cell is as follow:

[ ]{ }exp ( ) 1 ( ) /LGI I I V IR V IR Ros s s shα= − + − − + (1)

[ ] [ ]3/ exp (1/ 1/ )OS ORI I T T T Tr rβ= − (1) [ ]( 25) ( /1000)LG SCR II I K T λ= + − (2)

( /( ))q AkTα = (3)

0( ) /( )GqE Akβ = (4) The current delivered by a PV array made of Np modules in parallel and Ns cells in series per module can be derived from (1) as follow:

{ }/

/

exp ( / ) 1

( / ) /PV p LG p OS s s p

p s s p sh

I N I N I V N IR N

N V N IR N R

α⎡ ⎤= − + −⎣ ⎦− +

(5)

Where IOS cell reverse saturation current T cell temperature in K k Boltzmann’s constant q electronic charge KI=0.0017 A/˚C

short circuit current temperature coefficient at ISCR

λ solar irradiance in W/m2 ISCR short circuit current at 25˚C and

1kW/m2 ILG light generated current EG0=1.1 eV band gap of silicon A=1.92 ideality factor Tr=301.18˚K reference temperature IOR cell saturation current at Tr Rsh cell shunt resistance Rs cell series resistance I-V cell output current and voltage IPV PV array current

One can notice in (1) that there is a shunt resistance and a series resistance associated with the PV cell model. The

former resistance takes into account the output voltage drop caused by the shading effect on some cells in a PV string [2]. The latter resistance represents the contact resistance associated with the bond between the cell and its wire leads and the resistance of the semiconductor itself.

B. Maximum power point tracking The P-V characteristic of a photovoltaic exhibits a

maximum power point which is function of atmospheric conditions mostly the temperature and the irradiance. The maximum power point corresponds to a voltage and current Vmpp and Impp. Since the power efficiency of PV modules available in the market is low, we must ensure that the PV module do operate on the maximum power point. Before tracking the maximum power point of a PV array or module we should first locate it. But the P-V characteristic of a photovoltaic module is nonlinear (exponential) Fig.1-2, it may be difficult to locate the maximum of such a function. Therefore, we are using numerical techniques for computing the maximum power point. It is important to remind that the maximum of a given continuous function is the point at which its derivative is null. Then, to locate the maximum power point, one should solve the following equation:

0dP I dIdV V dV

= + = (6)

As in [3], the Newton algorithm will be used for solving the previous equation. The dc bus voltage will be controlled so as to track the voltage at the maximum power point.

C. The power circuit The power circuit depicted in Fig.3 consists of a

photovoltaic array with a dc link capacitor on the dc side. The dc capacitor is used to steady the dc link voltage. On the ac side are a three leg two level inverter sine wave PWM controlled, a passive filter L with its internal resistance R at each phase and the grid. The L filter attenuates the inverter output current high frequency component due to the switching of the inverter. The nonlinear controlled three phase PWM inverter is responsible for both maximum power point tracking and displacement factor correction. The parameters of the power circuit simulated are summarized in table I.

III. CONTROL SYSTEM DESCRIPTION

A. State model of the PV system For deriving the dynamic model of our three phase grid

connected PV interface over a switching period, we will assume the system balanced without the neutral connection. More, we will neglect the internal losses of the power converter. We will also assume that the switching frequency of the PV inverter is as high as the parameters of the system can be considered constant over a switching period. It has been shown in [4] that the model of the inverter in abc frame is:

3

Fig. 1. PV module P-V characteristic

1

31( ) )

3 1

31( ) )2 3 1

31( ) )3 3 1

(

(

(

Sa a

Sb b

Sc c

L di dt C vSa i dci

L di dt C vSb i dci

L di dt CSc ii

vdc

Ri e

Ri e

Ri e

c

c

c

∑−=

∑−=

∑−=

⎧ −⎪⎪⎪ −⎨⎪⎪

−⎪⎩

= −

= −

= −

(8)

1 2 3dc PV sa sb scC dv dt i c i c i c i= − − − (9) Where

1c 2c 3c are the PWM inverter switching functions iSa,b,c are the grid currents vdc is the dc link voltage L, R are the inverter output filter inductance and

resistance C is the dc link capacitor iPV is the current generated by the PV array If we replace the switching functions defined in (1) with the

following switching functions d defined as followed: 31

)3 1

(k k Cii

d c ∑−=

= (10)

The new inverter state model in abc frame becomes:

1

2

3

( )

( )

( )

Sa a

Sb b

Sc c

L di dt vSa dc

L di dt vSb dc

L di dtSc vdc

Ri e

Ri e

Ri e

ddd

⎧ −⎪⎪ −⎨⎪

−⎪⎩

= −= −= −

(11)

TABLE I SIMULATION PARAMETERS

Utility voltage Vga 120VrmsUtility frequency fn 60Hz Inverter power Pn 1.7kVA Switching frequency fs 6.3kHz Output inductance L 8mH Dc bus capacitor Cdc 1.5mF PV array

- PV module power - No of modules in series - No of modules in parallel

NS

Np

170P

10 1

The above state space model (11) is nonlinear since it contains the switching functions which can have the values 1 or zero depending of the state of the upper and lower switches of each arm of the inverter bridge.

Fig. 2 PV module I-V characteristic The power system depicted in Fig.3 is assumed balanced without neutral. Therefore, the equivalent state model of (11) in the synchronous reference frame dq is:

( )

( )

d

q

d q d

d q q

L di dt vd dcL di dt vq dc

Ri i ei Ri e

dd

ωω−

+ + −⎧⎪⎨ + −⎪⎩

= −= − (12)

With the same way, we can derive the equivalent dynamic model of (9) in the synchronous reference frame. The dc link capacitor dynamic can also be derived by doing the active power balance between the dc and the ac sides of the inverter as in [5]-[6]. Since our goal is also to achieve unit displacement factor we will assume the q-axis component of grid voltage equal to zero. Therefore the active power delivered to the grid by the PV interface in synchronous d-q frame is:

(3 2) d dP e i= (13) The power in the dc link voltage is equal to the power

supplied by the PV array minus the power delivered by the interface in the grid. Then the dynamic of the dc bus voltage can be written as follow:

(3 2)dc PV d d dcC dv dt i e i v= − (14) We can notice that equations (12) and (14) are nonlinear

because they contain respectively the switching functions and division between state variables. To control such a nonlinear system many techniques do exist. Most of them consist of first, making the system linear and then controlling it with the linear techniques. We can use the linearization around the equilibrium point with Taylor series as in [7]. That technique contains many drawbacks. First of all, it is a linear approximation of a nonlinear system. That approximation is only valuable around the equilibrium point. Second, this technique may lead to control designs with poor performance. More, this approximation can be uninformative about the dynamic of the system. Another family of linearization technique contains the state space feedback linearization and the input output linearization. Both linearization techniques are also known as respectively full and partial linearization techniques. In the following section we will explain briefly the input-output linearization technique and then the full state feedback linearization technique.

4

Sai

Sbi

Sci

ae

be

ce

,RL

dcVC

PVI

2(.) 0.5

abc

dq↓

ddt

Sabci

abc

dq↓

dqi 123x

dcv

123z12v 12u

0

mppv

0

*123z

Fig. 3. PV utility interface and its control system It is important to mention that both techniques are feedback linearization techniques.

B. Feedback linearization Let consider the following nonlinear SISO system:

( ) ( )( )

x f x g x uy h x

= +=

(15)

Let use the change of coordinates (16) to transform the nonlinear state equation (15) from the x coordinates into z coordinates with T an invertible function:

( )z T x= (16)

Where nx R∈ is the state vector, u is the control input,

: nf D R→ and : n pg D R ×→ are two smooth functions on a domain nD R⊂ . This system is feedback linearizable if there exists a diffeomorphism : nT D R→ such that the

( )zD T D= contains the origin and the change of coordinates (9) transforms the nonlinear state space model (15) into (17)[8].

[ ]( ) ( )z Az B x u xγ α= + − (17)

With (A,B) controllable and ( )xγ nonsingular for all x D⊂ . Where the state feedback control u:

( ) ( )u x x vα β= + (18) Such that the original model (15) in the new coordinates z is controllable linear system: z Az Bv= + (19)

C. State feedback linearization control design of the PV system

To design the state feedback linearization control of the PV system, the non linear multivariable dynamic model (12) and (14) should be transformed in one canonical form compatible

with state feedback linearization. For that purpose, the change of variable (20) is done.

Then if we replace (20) in (12) and (14) we can derive the following canonical model as in [9]:

1

22

3 ( ) / 2

d

q

dc

x ix i

x v

=⎧⎪ =⎨⎪ =⎩

(20)

1 1 1 2 2 3 1 4

2 2 1 1 2 3 2 5

3 6 1 7 3

x x x ux x x u

x x x

θ θ θ θθ θ θ θ

θ θ

⎧ = − + + −⎪

= − − + −⎨⎪ = − +⎩

(21)

Where 1

2

d

q

vdc

vdc

u d

u d

=

= (22)

θ is the vector of the PV system parameters while u1 and u2 are respectively the d and q components of the signal control of the three phase PWM inverter. We assume that all the parameters of the system are known.

[ ]1 2 3 4 5 6 7Tθ θ θ θ θ θ θ θ= (23)

/ 1/ / / (3/ 2) / /T

d q d PVR L L e L e L e C i Cθ ω⎡ ⎤= ⎣ ⎦ f(x,θ) and g1,2(x,θ) as in (25) and(26) respectively.

2

1

1 1 2

2 2 3

( , ) ( , )

( )( )

i ii

x f x g x u

y h x xy h x x

θ θ=

= +

= == =

∑ (24)

1 1 2 2 4

2 1 1 2 5

6 1 7 3

( , )x x

f x x x

x x

θ θ θθ θ θ θ

θ θ

⎡ ⎤− + −⎢ ⎥

= − − −⎢ ⎥⎢ ⎥− +⎣ ⎦

(25)

5

[ ]3

1 2 3

0( , ) ( , ) 0

0 0g x g x

θθ θ θ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(26)

In addition, one can notice from (21) that both the d component of the grid current x1 and the dc bus voltage x3 are linked. So, by controlling the dc bus voltage, it is no more necessary of controlling the d component of the grid current. Note that controlling vdc square instead of vdc does accelerate the dynamic of dc link voltage. It is a way of controlling the energy stored by the dc link capacitor. As shown in (24), the sum of relative degrees of h1 and h2 is 3 which is the number of states of the system. Therefore, there will be no internal dynamic.

For designing the control laws of our system we will first find a change of coordinates as (16) which transforms the system state space model into Brunovsky canonical form.

31

2 6 1 7 3

3 2

( )

xzz z T x x x

z xθ θ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥= = = − +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(27)

The function T(x) is invertible for all x є R3. Both the inverse T-1(z) of T(x) and T(x) are smooth functions for x3=0. It is easy to show that the rank of the jacobian matrix ∂T(x)/∂x is 3. Therefore T is a diffeomorphism.

7 1 261

12 3

3 1

1 [ ]

( )

z zx

x x T z zx z

θθ

−

⎡ ⎤−⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

(28)

Assuming the system parameters known and constant the following state space model of the PV system in the new coordinates is derived:

1 6 1 7 3

22 6 1 1 6 2 2 6 3 1 6 4 7 7 6 1 3

3 2 1 1 2 3 2 5

/ 2 /(2 )

z x x

z x x u x xz x x u

θ θ

θ θ θ θ θ θ θ θ θ θ θθ θ θ θ

= − +

= − − + + −= − − + −

(29)

The above state space model is also equivalent to: 1 2

7 22 1 7 1 2 2 6 3 3 6 1 4 6

1

23 1 3 5 7 1 2 3 2

6

2

ˆ

z zzz z z z uz

z z z z u

θθ θ θ θ θ θ θ θ

θθ θ θ θθ

=

⎡ ⎤= − − − + +⎣ ⎦

⎡ ⎤= − − − − +⎣ ⎦

(30)

From (29) we can derive the two nonlinear control laws u1 and u2 which are useful for input state feedback linearization:

( )

27 7 6 1

1 1 1 6 1 2 6 2 4 63 6 3

2 2 2 1 1 2 53

1( , )2 2

1( , )

xu x v x x

x

u x v x x

θ θ θθ θ θ θ θ θ θθ θ

θ θ θ θθ

⎛ ⎞= − + − + + −⎜ ⎟⎜ ⎟

⎝ ⎠

= + + +

(31)

If we substitute the two nonlinear control laws of (31) in (30), the new linear equivalent state space model of the PV system is:

1 2

2 1

3 2

z zz vz v

===

(32)

Where v1 and v2 are the linear control laws. If our system is minimum phase the following state feedback stabilizing control law is:

1 1 1 2 2

2 1 3

v k z k zv k z

= − −= −

(33)

For trajectory tracking we will take the above linear control laws as functions of tracking errors:

* *1 1 1 1 2 2 2

*2 1 3 3

( ) ( )

( )

v k z z k z z

v k z z

= − − − −

= − − (34)

In this paper our interest is regulation of outputs (vdc and iq) rather than trajectory tracking.

IV. SIMULATION RESULTS The simulation has been performed under a step change of irradiance and a constant cell temperature of 25˚C. At 0.5 second irradiance is decreased from 1kW/m2 to .8kW/m2 as shown in Fig.5. The dc bus voltage reference is the voltage at the MPP computed with MPPT algorithm. The Fig.5 shows that the maximum power point voltage is tracked for this change of irradiance. Grid current THD (Fig.4) is lower than 5% meaning compliance with IEEE-519-1992. Grid current of phases a, b,c is presented on Fig.5. One can notice in Fig.5 that the grid currents are 120 degrees from each other with a 60Hz frequency. As shown in Fig.6 phase a current is synchronized with phase a voltage. That means that the power factor is nearly equal to unity.

Fig.4 Grid current Spectrum

V. CONCLUSION A state feedback linearization control technique was applied to a three phase PWM grid connected PV inverter. The system was modeled in dq coordinates after assuming a balance of power between dc and ac sides of the inverter. A change of coordinates and two control laws were proposed to derive state equation in the normal form. A proper change of coordinates is important to apply state feedback linearization technique which requires the same number of controlled states and inputs. An indirect control of inverter output current was done

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Fig. 5 Simulation results of the PV utility interface by controlling the dc link voltage. By doing so, we are not dealing with cross coupling terms in d-q currents. The simulation results such as maximum power point tracking and unity displacement factor under irradiance variation show the validity of the state feedback linearization technique used in this paper.

Fig.6 Grid current and voltage

VI. REFERENCES [1] E. Koutroulis, Kalaitzakis, K., Voulgaris, N.C., "Development of a

Microcontroller-Based Photovoltaic Maximum Power Point Tracking Control System," IEEE Transactions on Power Electronics, vol. 16, pp. 46-54, 2001.

[2] G. M. Masters, Renewable and efficient electric power systems. Hoboken, N.J.: John Wiley & Sons, 2004.

[3] A. O. Zue and A. Chandra, "Grid Connected Photovoltaic Interface with VAR Compensation and Active Filtering Functions," in Proc. 2006 IEEE Power Electronics, Drives and Energy Systems Conf., pp. 1-6.

[4] V. Blasko, Kaura, V., "A Novel Control to Actively Damp Resonance in Input LC Filter of a Three-Phase Voltage Source Converter," IEEE Transactions on Industry Applications, vol. 33, pp. 542-550, Mar-Apr 1997.

[5] I.-S. Kim, "Robust maximum power point tracker using sliding mode controller for the three-phase grid-connected photovoltaic system," Solar Energy, vol. 81, pp. 405-414, 2007.

[6] L. Yacoubi, "Contribution à l'étude, la modélisation et la commande des redresseurs triphasés non-polluants : application au convertisseur trois-niveaux à point neutre calé," Ph.D. thesis, Dept. Elect. Eng: École de technologie supérieure, Montréal, 2004.

[7] K.-K. Shyu, M.-J. Yang, Y.-M. Chen, and Y.-F. Lin, "Model Reference Adaptive Control Design for a Shunt Active Power Filter System," in Proc 2006 IEEE Industrial Electronics Conference, pp. 73-78.

[8] H. K. Khalil, Nonlinear systems, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002.

[9] L. Yacoubi, K. Al-Haddad, L. A. Dessaint, and F. Fnaiech, "A DSP-based implementation of a nonlinear model reference adaptive control for a three-phase three-level NPC boost rectifier prototype," IEEE Transactions on Power Electronics, vol. 20, pp. 1084-1092, 2005.

[10] A. Isidori, Nonlinear control systems, 3rd ed. Berlin: Springer, 1995.

VII. BIOGRAPHIES

Aslain Ovono Zué graduated as an electromechanical engineer from Ecole Polytechnique de Masuku, Gabon in 2001 and is presently pursuing his M.ing degree in the department of electrical engineering, Ecole de Technologie Supérieure, Université du Québec, Montréal Québec. His employment experience includes oilfield mechanical engineering and oilfield operation engineering from 2002 to 2004. His research interest includes grid connected photovoltaic interfaces, power quality and distributed generation. Ambrish Chandra (SM’99) received the B.E. degree from the University of Roorkee, Roorkee, India, the M.Tech. degree from the Indian Institute of Technology (I.I.T.), New Delhi, India, and the Ph.D. degree from University of Calgary, Calgary, AB, Canada, in 1977, 1980, and 1987, respectively.He worked as a Lecturer and later as a Reader at University of Roorkee, presently I.I.T. Roorkee. Since 1994, he has been a professor with the Electrical Engineering Department, Ecole de Technologie Supérieure, Université du Québec, Montréal, QC, Canada. His main research interests are power quality, active filters, static reactive power compensation and flexible AC transmission systems (FACTS) and control of renewable energy sources. Dr Chandra is member of Ordre des Ingénieurs du Québec.