Backstepping Control for Maximum Power Tracking in Single- Phase Grid-Connected Photovoltaic Systems

11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing July 3-5, 2013. Caen, France

FrS7T1.2

Backstepping Control for Maximum Power Tracking in Single- Phase Grid-Connected Photovoltaic Systems Abderrahim El Fadili ∗ ; Fouad Giri; Abdelmounime El Magri. University of Caen Basse-Normandie, GRYC CNRS UMR 6072, Caen, France. ∗

Corresponding authors: elfadili [email protected]

Abstract: The problem of maximum power point tracking (MPPT) in photovoltaic (PV) arrays is addressed considering a PV system including a PV panel, a PWM DC/AC inverter connected to single-phase grid. The maximum power point (MPP) of PV generators varies with solar radiation and temperature. To reduce the PV system cost, the MPPT is presently achieved without, chopper, resorting to solar radiation and temperature sensors. The proposed strategy involves a multi-loop nonlinear controller designed, using the backstepping design technique, to meet the three main control objectives i.e. (i) voltage reference generator designed to meet maximum power point tracking (MPPT), (ii) tight DC Link voltage regulation for a wide range voltage-reference variation and power factor correction (PFC) requirement must be satisfactorily realized. A formal analysis based on Lyapunov stability is developed to describe the control system performances. It is formally shown that the developed strategy control actually meets the MPPT requirement. Keywords: Photovoltaic arrays, MPPT, nonlinear control, backstepping technique, Lyapunov stability. 1. INTRODUCTION Photovoltaic power generators have gained a great popularity in recent years, due to their increasing efficiency and decreasing costs. Indeed, PV systems produce electric power without harming the environment, transforming a free inexhaustible source of energy, solar radiation, into electricity. Furthermore, PV devices are now guaranteed to last longer than ever and manufacturer warranties go over 20 years. Also, governments encourage resorting to such energy solutions through significant tax credits. All these considerations assure a promising role for PV generation systems in the near future. Dependence of the power generated by a PV array and its MPP on atmospheric conditions is readily be seen in the power-voltage (P -V ) characteristics of PV arrays as shown in Fig. 3 and Fig. 4. These show in particular that the array power depends nonlinearly on the array terminal operating voltage. Moreover, the MPP, varies with changing radiation and temperature, necessitating continuous adjustment of the array terminal voltage if maximum power is to be transferred. Different techniques to maximize PV power transfer to various loads have been reported in the literature, including the constant voltage method, the open circuit voltage method, the short circuit method, perturb and observe (P&O) method [Bianconi et al. , 2013, Yong and FangPing. , 2011], the incremental conductance method (IncCond) [Panagiotis et al. , 2011, Lalili et al. , 2011, Tsengenes et al. , 2011] and the Ripple Correlation Control (RCC) method 978-3-902823-37-3/2013 © IFAC

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Trishan et al. [2006]. The constant voltage method is the simplest one but it has been commented that the method could only collect about 80% of the available maximum power under varying irradiance. An improvement on the constant voltage method uses the open circuit voltage to estimate the maximum power output voltage while the short circuit current method uses the short circuit current to estimate the maximum power output current. (P&O) method provokes oscillations about the maximum power point (MPP) which can be minimized by reducing the perturbation step size. However, a small perturbation size slows down the MPPT. (RCC) and (IncCond) methods require the power, current and voltage derivatives which can be a cause of divide-by-zero singularity problems. However, in all these works, the converter dynamics were neglected which was only modeled by its steady state gain. In this paper, we are addressing the problem of controlling PV systems consisting of PV panels, PWM DC/AC inverter connected to single phase grid. The aim is to ensure maximum power point tracking whatever the position of the PV panel. Furthermore, to reduce the PV system (development and maintenance) cost, we are seeking a solution not necessitating climatic variable sensors (solar radiation, temperature). In the present paper, we develop a new control strategy involving an optimal voltage reference generator and a voltage and power factor correction (PFC) controller. The voltage reference optimality is to be understood in the 10.3182/20130703-3-FR-4038.00150

11th IFAC ALCOSP July 3-5, 2013. Caen, France

sense to extract the maximum power from photovoltaic generator regardless of solar radiation. Therefore, the obtained optimal voltage reference law involves the PV delivered power. The voltage and PFC controllers are designed by backstepping designe technique. This new controllers enforce the voltage to perfectly track its varying reference trajectory, despite the solar radiation, and the grid current remains (almost) always in phase with the supply net voltage complying with the PFC requirement. The paper is organized as follows: in Section 2, the system modelling is presented; Sections 3 is devoted to the controller design; the controller tracking performances are illustrated through numerical simulations in Section 4. A conclusion and a reference list end the paper. 2. SYSTEM MODELLING Fig.1 shows the PV grid system. It consists of a 30 series photovoltaic modules SM55, a DC/AC inverter which is used to, achieve the maximum power point tracking (MPPT), and interface the PV array output to the single phase grid. PV generator Ig

DC/AC inverter s ie S

S' Lo

Rs

Iph

Where Iph is the photocurrent (generated current under a given radiation); Io is the cell reverse saturation current; Ior is the cell saturation current at Tr ; ISCR is the short circuit current at 298.15K and 1kW/m2 ; KI is the short circuit current temperature coefficient at ISCR ; λ is the solar radiation; EGO is the band gap for silicon; γ is the ideality factor; Tr is the reference temperature; T is the cell temperature; K is the Boltzmans constant and q is the electron charge. The analytical expressions of Iph and Io can be found in places, see e.g. Tan et al. [2004]. Here, let us just note that these only depend on the temperature T and radiation λ. The PVG is composed of many strings of PV modules in series, connected in parallel, in order to provide the desired values of output voltage and current. This PVG exhibits a non linear (Ig Vg ) characteristics given, approximately and ideally, by the following equation:

AC grid ie ve

S'

Vp

Rsh

Fig. 2. PV module equivalent circuit.

Ig = Iphg − Io {exp(Ag Vg ) − 1}

2C vg

ID

Ip

~

(2)

where Vg is the PVG voltage, Ig is the PVG current, Ag = A/Ns is the PVG constant, Iphg = Np Iph is the photocurrent of the PVG, Iog = Np Io is the saturation current of the PVG, Ns is the number of PV connected in series and Np is the number of parallel paths.

S

Power Voltage charctrestic (T=45°C,λ var) λ=800 W/m2

u

2000

2

Fig. 1. General diagram of the PV single-phase grid system.

λ=1200 W/m Power P(W)

2.1 Photovoltaic generator model The direct conversion of the solar energy into electrical power is obtained by solar cells. The equivalent circuit of PV module is shown in Fig.2 (Luque and Hegedus [2003]; Gow and Manning [1999]; Tan et al. [2004]). The traditional (Ip -Vp ) ideal characteristics (i.e. Rs = 0, Rsh = ∞) of a solar array are given by the following equation: Ip = Iph − Io {exp(AVp ) − 1} (1) where q A= γKT λ Iph = [ISCR + KI (T − Tr )] 1000  3    T qEGO 1 1 Io = Ior exp − Tr γK Tr T 660

2

λ=1000 W/m

1500

λ=1400 W/m2

1000

500

0 0

200 400 Voltage Vg (V))

600

Fig. 3. (P -V ) characteristics of the PVG, with constant temperature and varying radiation A PV array module considered in this paper is the SM 55. It has 36 series connected mono-crystalline cells. The arrays electrical characteristics are assembled in Table.1. 2.2 Modeling ‘DC/AC inverter’ The power supply net is connected to a H-bridge converter which consisting of four IGBT’s with anti-parallel diodes

11th IFAC ALCOSP July 3-5, 2013. Caen, France

where:

Power P(W)

Power Voltage charctrestic (T var ,λ=1000 W/m2) 1800 T=35°C 1600 T=45°C 1400 T=55°C

x1 = ie

x2 = v g

u=s

(8)

are the average values over cutting periods of ie , vg and s, respectively.

1200 1000

3. CONTROLLER DESIGN

800

3.1 Control Objectives

600 400

There are two operational control objectives:

200 0 0

200 400 Voltage Vg (V))

600

Fig. 4. (P -V ) characteristics of the PVG, with constant radiation and varying temperature Maximum Power Short circuit current Open circuit voltage Voltage at max power point Current at max power point KI (A/K)

Pm (W ) ISCR (A) Voc (V ) Vm (V ) Im (A)

55 3.45 21.7 17.4 3.15 4 10−4

(i) Continuous voltage optimization: DC link voltage vg must track as accurately as possible a state-dependent voltage reference vgref = F (P ) where P denotes the output power PVG and the function F (.) has yet to be determined so that vg = vgref entails a operating of PVG corresponding to maximum power point. (ii) PFC requirement: the inverter output current ie must be sinusoidal and in phase with the AC supply voltage ve . 3.2 DC link voltage reference optimization

Table 1. Electrical specifications for the solar module SM55 for bidirectional power flow mode (see Fig. 1). The converter should be controlled so that two main tasks are accomplished: (i) providing an optimal DC link voltage vg , in order to extract a maximum power from PVG; (ii) ensuring an almost unitary power factor connection with the power net. Applying Kirchhoffs laws, this subsystem is described by the following set of differential equations: die = svg − ve (3) Lo dt dvg 2C = Ig − sie (4) dt where ie is the current in inductor Lo , vg denotes the voltage in capacitor 2C (output of PV), Ig designates the √ PV output current, ve = 2E cos(ωe t) is the sinusoidal net voltage (with known constants E, ωe ) and s is the switch position function taking values in the discrete set {-1,1}. Specifically:  0 1 if S On and S Of f s= (5) 0 −1 if S Of f and S On The above (instantaneous) model describes accurately the physical inverter. Then, it is based upon to build up converter simulators. However, it is not suitable for control design due to the switched nature of the control input s. As a matter of fact, most existing nonlinear control approaches apply to systems with continuous control inputs. Therefore, control design for the above converter will be performed using the following average version of (3-4) Ortega et al. [1996]: dx1 = ux2 − ve (6) Lo dt dx2 2C = Ig − ux1 (7) dt 661

In this subsection, we seek the construction of a voltagereference optimizer that meets the MPPT requirement. Specifically, the optimizer is expected to compute on-line the optimal voltage value vgopt so that, if the voltage vg is made equal to vgopt then, maximal power is captured, and transmitted to the grid through the DC/AC inverter. Presently, the voltage-reference optimizer design is based on the power-voltage (P -Vg ) characteristic (Fig. 3) and feature the fact that it does not require radiation λ measurement. The used characteristic is that obtained at a temperature which the photovoltaic generator is operated at its neighborhood the most of the time. In this paper, we chose 45o C. The summits of these curves give the maximum extractable power Popt and so represent the optimal points. Each one of these points is characterized by the optimal voltage vgopt . It is readily seen from (Fig. 3) that for any radiation λ value, say λi , there is a unique couple (Vgi , Pi ) that involves the largest extractable power. The set of all such optimal couples (Vgi , Pi ) is represented by the red curve in (Fig. 3). A number of such couples have been collected from (Fig. 3) and interpolated to get a polynomial function (Vgopt = F (Popt )). The polynomial thus constructed is denoted: F (P ) = hn P n + hn−1 P n−1 + + h1 P + h0 (9) where the coefficients hi have a numerical values corresponding to characteristic of used PVG. The interpolation polynomial of function Vgref = F (P ) is shown by Fig.5. Remark 1. The polynomial interpolation yielding the function has been obtained using the Matlab functions POLYVAL, SPLINE, and POLYFIT. 3.3 DC/AC inverter control design Controlling inverter output current to meet PFC: The PFC objective means that the grid current should be sinusoidal and in phase with the AC supply voltage.

11th IFAC ALCOSP July 3-5, 2013. Caen, France

3.4 DC link voltage regulation 550

reference voltage(V)

500 450 400 350 300 250 0

500

1000 Power (W)

1500

2000

Fig. 5. Optimal power-voltage characteristic obtained from the polynomial interpolation of points (Vgi , Pi ) for 30 series SM55 module Therefore, one seeks a regulator that enforces the current x1 to tack a reference signal x∗1 of the form: x∗1 = kve (10) At this point, k is any real parameter that is allowed to be time-varying. The sign of this parameter depends on the direction of power transfer. Indeed, k is positive when the PV system feeds into the network, k is negative when the network charges the capacitor 2C. Introduce the current tracking error: z1 = x1 − x∗1 (11) In view of (6), the above error undergoes the following equation: 1 1 z˙1 = ux2 − ve − x˙ ∗1 (12) Lo Lo To get a stabilizing control law for this first-order system, consider the quadratic Lyapunov function V1 = 0.5z12 . It can be easily checked that the time-derivative V˙ 1 is a negative definite function of z1 if the control input is chosen as follows: ve Lo (−c1 z1 + + x˙ ∗1 ) (13) u= x2 Lo with c1 > 0 is a design parameter. The properties of such control law are summarized in the following proposition, the proof of which is straightforward. Proposition 1. Consider the system, next called current (or inner) loop, composed of the current equation (6) and the control law (13) where c1 > 0 is arbitrarily chosen by the user. If the reference x∗1 = kve and its first time derivative are available then one has the following properties: 1) The current loop undergoes the equation z˙1 = −c1 z1 with z1 = x1 − x∗1 . As c1 is positive this equation is globally exponentially stable i.e. z1 vanishes exponentially, whatever the initial conditions. 2) If in addition k converges (to a finite value), then the PFC requirement is asymptotically fulfilled in average i.e. the (average) input current x1 tends (exponentially fast) to its reference kve as t → ∞

662

The aim is now to design a tuning law for the ratio k in (10) so that the inverter input voltage x2 = v g is steered to a given reference value vgref . The first step in designing such a tuning law is to establish the relation between the ratio k (control input) and the output voltage x2 . This is the subject of the following proposition. Proposition 2. Consider the power inverter described by (6-7) together with the control law (13). Under the same assumptions as in Proposition 1, one has the following properties: 1) The output voltage x2 varies, in response to the tuning ratio k, according to the equation: dx2 1 2C = Ig − (kve2 + z1 ve ) (14) dt x2 2) The squared voltage (y = x22 ) varies, in response to the tuning ratio k, according to the equation: 1 1 dy = Ig x2 − (kve2 + z1 ve ) (15) dt C C The ratio k stands up as a control signal in the firstorder system defined by (15). As previously mentioned, 2 the reference signal yref = vgref (of the squared DClink voltage x2 = vg ). Then, it follows from (15) that the tracking error z2 = y − yref undergoes the following equation: √ 2Ez1 E2k E2k − cos(2ωe t) − cos(ωe t) z˙2 = − C C C Ig x2 + − y˙ ref (16) C √ where one has used the fact that ve = 2E cos(ωe t) and ve2 = E 2 (1 + cos(2ωe t). To get a stabilizing control law for the system (16), consider the following quadratic Lyapunov function: V2 = 0.5z22 (17) ˙ It is easily checked that the time-derivative V2 can be made negative definite in the state z2 by letting: √ kE 2 + kE 2 cos(2ωe t) + 2Ez1 cos(ωe t) Ig x2 = Cc2 z2 + − C y˙ ref (18) C The point is that such equation involves a periodic singularity due to the mutual neutralization of the first two terms on the left side of (18). To get off this singularity and to avoid an excessive chattering in the solution, the two terms in cos(.) on the left side of (18) are ignored and bearing in mind the fact that the first derivative of the control ratio k must be available (Proposition 1), the following filtered version of the above solution is proposed: C Ig x2 k˙ + dk = d 2 {−c2 z2 + − y˙ ref } (19) E C The regulator parameters (d, c2 ) are any positive real constants. Let us summarize the main findings in the following proposition. Proposition 3. Consider the system control consisting of the AC/DC inverter described by (6-7) together with the control laws (13) and (19). Using Proposition 1 (Part 1), it

11th IFAC ALCOSP July 3-5, 2013. Caen, France

s ie

vg

2C

PVG

Lo

DC/AC Inverter

ve

~

1200

AC grid

u Equa. 13 +

1300

ie

Radiation(W/m2)

Ig

ie

-

ve

1100

1000

900

k Equa. 20

Ig vg

vgref Equa. 9

800 0

+

-

2

4 6 Time (s)

8

10

Fig. 7. Radiation λ (W/m2 ) Fig. 6. Simulation protocole 500 495 Voltage Vg (V)

turns out that the system control undergoes, in the (z1 , z2 , k) -coordinates, the following equation, where z1 = x1 − x∗1 and z2 = y − yref : Ig x2 Z˙ = AZ + g(x)( − y˙ ref ) + f (Z, t) (20) C with   " # −c1 0 0 z1 2 Z = z2 ; A= 0 0 − EC  dC k 0 E 2 c2 −d   0 √ 2 2E  f (Z, t) =  −E C k cos(2ωe t) − C z1 cos(ωe t) 0   0 g(x) =  1 

490 485 480 475 470 0

2

4 6 Time (s)

8

10

4 6 Time (s)

8

10

Fig. 8. Voltage Vg (V )

dC E2

4

4. SIMULATION RESULTS

The simulated system is given the following characteristics: . Supply network: is triphase 220V /50Hz . DC/AC inverter: Lo = 15mH; C = 1.5mF ; modulation frequency 10KHz. . 30 series photovoltaic module SM55, it is a 55W whose characteristics are summarized in Table 1. The radiation variation is shown by Fig.7, the simulation protocol is proceeded in first time at tempurature T = 45o C. The indicated values of design parameters (c1 , c2 , d) have been selected using a try-and-error search method and proved to be suitable. The experimental setup is simulated within the Matlab/Simulink environment with a calculation step of 5µs. This value is motivated by the fact that the inverter frequency commutation is 10kHz. Fig. 8 shows the resulting (state-dependent) optimal voltage reference (red curve) and measured photovoltaic voltage vg . It is clearly seen that the state-dependent voltage reference varies significantly in function of the radiation 663

3.5 Current Ig(A)

The experimental setup is described by Fig. 6 and the controller, developed in Section 3, including the optimizer (9) and the control laws (13) and (19), will now be evaluated by simulation.

3

2.5

2 0

2

Fig. 9. Current Ig (A) λ and measured photovoltaic voltage vg track quickly its reference after each change in λ. Fig.9 shows the photovoltaic current Ig . It is observed that the current amplitude changes whenever the radiation λ vary. The power extracted from the PV panel, as shown by Fig.11 is always maximal regardless of the value of radiation λ. Fig. 10(a) shows the measured output current ie response. The current frequency is constant and equal to voltage ve frequency. Specifically, the current remains most time either in phase or opposed phase (charging the capacitor) with the supply net voltage, complying with the PFC

11th IFAC ALCOSP July 3-5, 2013. Caen, France

5. CONCLUSIONS ie

15

0.05 ve

10

In this paper, the problem of achieving MPPT in singlephase grid-connected photovoltaic system has been addressed. A new control strategy has been developed using backstepping designe technique regulators, based on the system nonlinear model (6-7) that accounts for the DC/AC inverter dynamics and the single phase grid inductor. The voltage reference optimizer is resorted to cope with the changing operating conditions (temperature and radiation). The obtained MPPT controller consists of the optimizer voltage reference generator (9), the control laws (13) and (19). It is shown to meet its control objectives, despite the climatic conditions. This performance are checked by simulations.

0.02

5

0 ratio k

0.05*ve (V)/ie

0.04

0 −5

−0.02 −0.04

−10 −0.06 −15 4

4.02

4.04 4.06 Time (s)

4.08

−0.08 0

4.1

2

(a) Current ie (A) / 0.05ve (V)

4 6 Time (s)

8

10

(b) ratio k

Fig. 10. Power factor correction cheking T=45°C 2000

Power P (W)

REFERENCES 1500

1000

500 0

2

4

6

8

10

8

10

Fig. 11. Power P (W ) at T = 45o C T=55°C 2000 1800

Power P (W)

1600 1400 1200 1000 800 600 400 0

2

4 6 Time (s)

Fig. 12. Power P (W ) at T = 55o C requirement. This is further demonstrated by Fig. 10(b) which shows that the ratio k takes a constant value after short transient periods following changes in the radiation λ. Note that the ratio k is negative when the network gets to back energy to charge the capacitor 2C. The second part of simulation is performed at temperature T = 55o C different than 45o C, (the latter is the temperature used when designing the optimizer F (P )). Fig. 12 shows,with red curve, the maximum power point at temperature T = 55o C function of the the radiation λ as varying in Fig.7. The power extract with our strategy control is shown by black curve in Fig. 12. It is clear that the error made in a temperature variation of 10o C is less than 2.5% In the light of the closed-loop responses (see Figs 7-12), it is seen that the multiloop controller meets all its objectives and enjoy quite satisfactory transient performances. 664

Gow J. A and C. D. Manning. Development of a photovoltaic array model for use in powerelectronics simulation studies. IEE Proceedings on Electric Power Applications, 146(2),(1999), pp:193-200. Enrico Bianconi, Javier Calvente, Roberto Giral, Emilio Mamarelis, Giovanni Petrone, Carlos Andres RamosPaja, Giovanni Spagnuolo, Massimo Vitelli. (2013) Perturb and Observe MPPT algorithm with a current controller based on the sliding mode. International Journal of Electrical Power and Energy Systems (IJEPES), Vol. 44, pp: 346–356. Yong Yang, Fang Ping Zhao. (2011) Adaptive perturb and observe MPPT technique for Grid connected Photovoltaic Inverters. Procedia Engineering, Vol. 23, pp: 468–473. Panagiotis E. Kakosimos, Antonios G. Kladas. (2011) Implementation of photovoltaic array MPPT through fixed step predictive control technique.Renewable Energy, Elsevier, Vol. 36, pp: 2508–2514. Lalili D, A. Mellit, N. Lourci, B. Medjahed, E.M. Berkouk. (2011) Input output feedback linearization control and variable step size MPPT algorithm of a grid-connected photovoltaic inverter. Renewable Energy, Elsevier, Vol. 36, pp: 3282–3291. Georgios Tsengenes, Georgios Adamidis. (2011) Investigation of the behavior of a three phase grid-connected photovoltaic system to control active and reactive power. Electric Power Systems Research, Vol. 81, pp: 177–184. Trishan Esram, Jonathan W. Kimball, Philip T. Krein, Patrick L. Chapman, Pallab Midya. (2006) Dynamic Maximum Power Point Tracking of Photovoltaic Arrays Using Ripple Correlation Control. IEEE Transactions On Power Electronics, Vol. 21, No. 5, pp: 1282–1291. Khalil H Nonlinear Systems. Prentice Hall, NJ, USA. Krstic M, I. Kanellakopoulos and P. Kokotovic. (1995) Nonlinear and Adaptive Control Design. John Willey & Sons, Inc. Ortega R., P.J. Nicklasson , G. Espinosa-Perez. (1996) On Speed Control of Induction Motors. Automatica, Vol. 32, No. 3, pp 455-460. Luque A and S. Hegedus. (2003) Handbook of Photovoltaic Science and Engineering. John Wiley & Sons,Ltd. Tan Y. T, D.S. Kirschen, and N. Jenkins. (2004) A model of PV generation suitable for stability analysis. IEEE Trans. Energy Convers., Vol. 19, No.4, pp: 748-755.