- Email: [email protected]
MPPT and Unity Power Factor Achievement in Grid-Connected PV System Using Nonlinear control
MPPT and Unity Power Factor Achievement in Grid-Connected PV System Using Nonlinear control H. El Fadil*, F. Giri** *
National School of Applied Sciences (ENSA), Ibn Tofail University, Kénitra, Morocco (corresponding author; e-mail: [email protected]) **
GREYC Lab, UMR CNRS, University of Caen, 14032, Caen, France (e-mail: [email protected]).
Abstract: This work deals with the problem of controlling single-phase grid connected photovoltaic system. The system consists on a photovoltaic (PV) panel, a boost power converter and a single phase inverter connected to the grid. The control objective is fourfold: (i) asymptotic stability of the closed loop system, (ii) maximum power point tracking (MPPT) of PV module (iii) tight regulation of the DC bus voltage, and (iv) unity power factor (PF) in the grid. A nonlinear controller is developed using feedback linearizing technique based on an averaged nonlinear model of the whole controlled system. The model accounts, on the one hand, for the nonlinear dynamics of the underlying boost converter and inverter and, on the other, for the nonlinear characteristic of PV panel. It is formally shown, through theoretical analysis and simulation results, that the proposed controller does achieve its objectives. Keywords: Photovoltaic systems, Grid connection, Nonlinear control 1. INTRODUCTION With the deregulation of electricity markets and thrust to reduce greenhouse gas emissions from the traditional electric power generation systems, renewable energy resources such as wind turbines, photovoltaic panels, gas turbines and fuel cells, has gained a significant opportunity as new means of power generation to meet the growing demand for electric energy. Solar energy is considered to be one of the most useful natural energy sources because it is free, abundant, pollution-free, and most widely distributed. It can be used either at remote regions as standalone apparatus or in urban applications as grid interactive power source (Shimiza et al., 2003). On the other hand, a significant technology progress has been achieved over the few past years increasing efficiency of solar cells. The rapid growth of solar industry has expanded the importance of PV systems making them more reliable and efficient, especially for utility power in distributed generation (DG) (see Fig. 1) at medium and low voltages power systems. Implementing distributed energy resources (DER), into interconnected grids could be part of the solution to meet the rising electricity demand (Ortjohann et al. ,2008, Akorede et al., 2010). DG technologies are currently being improved through several research projects toward the development of smart grids. PV energy applications are divided into two categories: stand-alone systems and grid-connected systems. Stand-alone systems require a battery bank to store the PV energy; this is
suitable for low-power systems. On the other hand, gridconnected PV systems do not require battery banks; they are resorted generally in high power applications. The main purpose of the grid-connected system is to extract the maximal quantity possible of solar array energy and restitute it to grid with a unity power factor, despite changing atmospheric conditions (temperature and radiation). PV grid-connected systems represent the most important field applications of solar energy (Libo et al., 2007, Abeyasekera et al., 2006, Barbosa el al., 2006, Mastromauro, 2009 et al.). In general, a photovoltaic grid-connected system can be seen as a two-stage grid-connected inverter (Fig.1). The first stage is a dc-dc converter controlled so that the photovoltaic system operates in optimal condition i.e. seeking maximum power point tracking (MPPT) (El Fadil et al., 2011). The second stage is a dc-ac converter that controlled in a way that allows a grid connection with a unity power factor (PF). To this end, the output current (entering the grid) must be sinusoidal and in phase with the grid voltage. The dc-dc and dc-ac converters operate independently making easier the whole system control. The present paper is focusing on the problem of controlling single phase grid-connected PV power generation systems (Fig. 1). The control objective is fourfold: (i) global asymptotic stability of the whole closed-loop control system; (ii) achievement of MPPT of the PV array; (iii) ensuring a grid connection with unity PF; and (iv) ensuring a tight regulation of the dc-bus voltage. These objectives must be
met despite changes of the climatic variables (temperature and radiation). To this end, a nonlinear controller is developed using the backstepping design technique based on large-signal nonlinear model of the whole system. A theoretical analysis is developed to show that the controller actually meets its objectives a fact that is confirmed by simulation. The paper is organized as follows: the single phase grid connected PV system is described and modelled in Section II. Section III is devoted to controller design and analysis. The controller tracking performances are illustrated by numerical simulation in Section IV.
2.1 PV array model Typical (Ip-Vp) characteristics of solar cells arranged in N p parallel and N s -series can be found in many places (see e.g. Enrique et al., 2007, Chen-Chi and Chieh-Li, 2009). The PV array module considered in this paper is the NU-183E1. The corresponding electrical characteristics are listed in Table 1. The associated maximum power points (MPP) M1 to M4 under climatic conditions (temperature and radiation) are shown in Table 2. The data in Table 1 and Table 2 will be used latter for simulation purpose.
Main Supervisory Control
Table 1. Electrical specifications for the solar module NU183E1 Unit Control
GRID
Unit Control
PV Panels
System under study Unit Control BUS
Unit Control
MPPT
Wind Turbine
DC
Fuel Cell Generator
Unit Control
Parameter Maximum Power
Symbol Pm
Value 183.1W
Short circuit current
I scr
8.48 A
Open circuit voltage
Voc
30.1V
Maximum power voltage
Vm
23.9 V
Maximum power current
Im
7.66A
Number of parallel modules
Np
1
Number of series modules
Ns
48
Unit Control
Table 2. Maximum Power Points (MPP) Diesel Generator
Storage Batteries
MPP M1(λ=1000W/m2; T=25°C) M2(λ=400W/m2; T=25°C) M3(λ=1000W/m2; T=25°C) M4(λ=1000W/m2; T=60°C)
Fig.1: Control architecture in distributed energy resources
2. PRESENTATION AND MODELING OF GRID CONNECTED PV SYSTEM A typical configuration of a single-phase grid connected photovoltaic system is shown in Fig. 2. It consists of a solar array, an input capacitor Ci , a boost dc-dc converter which is used for boosting the array voltage and achieving MPPT for PV array, a DC link capacitor Cdc , a single-phase full-bridge inverter (including four power semiconductors) that is based upon to ensure a DC-AC power conversion a unity PF, a filter inductor Lg , and an isolation transformer.
+
idc u2
vp PV -
D1
Ci
u1
u2
ig
+
Cdc u2
Lg eg
u2
GRID
Li
Pm
23.82 V 22.94 V 23.82 V 20.38 V
183.1 W 70.8 W 183.1 W 153.4 W
2.2 Boost converter modeling The control input u1 of the boost converter is a PWM signal taking values in the set {0,1} . Applying the Kirchoff’s laws, successively with u1 = 1 and u1 = 0 , to the boost converter circuit of Fig. 2 one obtains the following instantaneous model dv p dt
ip
Vm
=
1 (i p − iLi ) Ci
(1a)
vp diLi v R = −(1 − u1 ) dc − i iLi + dt Li Li Li
(1b)
dvdc i 1 = (1 − u1 ) Li − idc dt Cdc Cdc
(1c)
-
where Ri is the equivalent series resistance (ESR) of input Fig.2: Single phase grid connected PV system
inductance Li .
2.3 Single-phase full-bridge inverter modeling The control input u 2 of the single-phase full-bridge inverter is also a PWM signal taking values in the set {0,1} . Applying the Kirchoff’s laws, successively with u 2 = 1 and u 2 = 0 , to the inverter circuit of Fig. 2 one obtains the following instantaneous model di Lg dt
= (2u 2 − 1)
eg vdc Rg − i Lg − L g Lg Lg
idc = (2u 2 − 1)iLg
(2a) (2b)
3.2 MPPT and PFC objectives The system of Fig.2 can be viewed as a MIMO system with two inputs µ1 and µ 2 an two outputs : y1 = x1 y 2 = x4
Bearing in mind the second control objective which is to enforce the output y1 to track the optimal point Vm , one can seek the relationship between the output y1 and the input µ1 . To this end let us differentiate the output y1
where Rg is the ESR of the filter inductance Lg .
y&1 =
2.4 Overall system model The instantaneous model (1a-c, 2a-b) of the single-phase grid connected photovoltaic system is useful for simulator design. But, it is not suitable for controller design because it involves binary control inputs u1 and u 2 . For control design purpose, it is more convenient to consider the following averaged model (Krein et al, 1990) dx1 1 = (i p − x2 ) dt Ci x R dx2 x = −(1 − µ1 ) 3 − i x2 + 1 dt Li Li Li dx3 x x = (1 − µ1 ) 2 − (2µ 2 − 1) 4 dt Cdc Cdc Rg eg x dx4 x4 − = (2 µ 2 − 1) 3 − dt Lg L g Lg
(4a) (4b)
1 (i p − x2 ) Ci
(5)
Since y&1 is still not directly related to the input µ1 , let us differentiate again. We now obtain &y&1 =
1 Ci
⎛& x R x ⎞ ⎜ i p + (1 − µ1 ) 3 + i x2 − 1 ⎟ ⎜ ⎟ L L L i i i ⎠ ⎝
(6)
The unity PF objective means that the grid current iLg should be sinusoidal and in phase with the AC grid voltage e g . We
(3a) (3b) (3c) (3d)
therefore seek a regulator that enforces the current y 2 to tack a reference signal y 2 ref of the form:
y 2 ref = γeg
At this point γ is any real positive parameter that is allowed to be time-varying. To generate a direct relationship between the output y 2 and the second input , let us differentiate the output y 2
where x1 , x2 , x3 , x4 , i p , µ1 and µ 2 denote the average
y& 2 = (2 µ 2 − 1)
values, respectively of, v p , iLi , vdc , iLg , i p , u1 and u 2 . All variables averaging are performed over switching periods. Consequently, the quantities µ1 and µ 2 , commonly called duty ratios, vary continuously in the interval [0 1] and act as the input control signals. 3. CONTROLLER DESIGN 3.1 Control objectives In this section, we aim at designing a controller that will be able to ensure: (i) a global stability of closed loop system, (ii) a perfect MPPT (whatever the position of the PV panel). Specifically, the controller must enforce the voltage x1 to track, as accurately as possible, the unknown (and slowly varying) voltage Vm . Note that Vm is unknown because it depends on the temperature T and solar variation λ and these are not supposed to be measured; (iii) a unity PF in the grid; and (iv) a tight regulation of the dc bus voltage x3 .
(7)
eg x3 Rg − x4 − Lg Lg Lg
(8)
Equations (6) and (8) can be gathered as follows ⎛ &y&1 ⎞ ⎛µ ⎞ ⎜⎜ ⎟⎟ = A( x)⎜⎜ 1 ⎟⎟ + b( x) & ⎝ y2 ⎠ ⎝ µ2 ⎠
(9a)
⎛ − x3 ⎜ LC A( x) = ⎜⎜ i i ⎜⎜ 0 ⎝
(9b)
where
⎛ 1 ⎜ ⎜C b( x) = ⎜ i ⎜ ⎜ ⎝
⎞ 0 ⎟ ⎟ 2 x3 ⎟ ⎟ Lg ⎟⎠
⎛& x R x ⎞⎞ ⎜ i p + 3 + i x2 − 1 ⎟ ⎟ ⎜ Li Li Li ⎟⎠ ⎟ ⎝ ⎟ Rg eg x ⎟ − 3 − x4 − ⎟ Lg Lg Lg ⎠
(9c)
Using linearizing feedback technique (Slotine, 1991), one can choose the following control signals ⎞ ⎛ ⎛ν ⎞ ⎛ µ1 ⎞ ⎜⎜ ⎟⎟ = A −1 ( x)⎜ ⎜⎜ 1 ⎟⎟ − b( x) ⎟ ⎟ ⎜ ⎝ µ2 ⎠ ⎠ ⎝ ⎝ν 2 ⎠
(10)
where ν 1 andν 2 being a new inputs to be determined. The nonlinearity in (9a) is cancelled, and we obtain a simple linear integrators relationship between the outputs and the new inputs &y&1 = ν 1 y& 2 = ν 2
e1 = y1 − Vm e2 = y 2 − y 2 ref
(12a) (12b)
be the tracking errors, and choosing the new inputs as
ν 1 = V&&m − α 0 e1 − α1e&1 ν 2 = y& 2 ref − β 0 e2
and the temperature T , by acting on the duty cycle µ1 . Then, the optimal voltage Vm can be generated using a PI regulator as shown in Fig. 3. This regulator (denoted PI-1 in Fig. 3) is defined as follows: Vm = G1 ( s ) ε p
(13a) (13b)
e&2 + β 0 e2 = 0
and
(14)
Which represent an exponentially stable error dynamics. Therefore the errors e1 and e2 converge to zero exponentially.
3.3 DC bus voltage regulation objective Now, the aim is to design a variation law for the ratio γ in (7) so that the dc voltage x3 =< vdc > is steered to a given constant reference Vd > 0 . To this end, the following PI control law (called PI-2) is used:
γ = G2 ( s )ε dc
(15a)
where (1 + τ 2 s ) τ 2s
ε dc = x3 − Vd
(15b) (15c)
(17a)
where G1 ( s ) = k1
εp =
(1 + τ 1s ) τ 1s
(17b)
dP dv p
(17c)
where s denotes the Laplace variable. ip vp
P
d dt
d dt
with α 0 , α1 and β 0 being positive constants. The tracking errors of closed loop system are given by
G2 ( s ) = k2
must keep dP / dv p equals zero, whatever the radiation λ
(11a) (11b)
The design of a tracking controller for (11a-b) is simple because of the availability of linear control technique. For instance, letting
e&&1 + α1e&1 + α 0 e1 = 0
with P = v p i p being the PV power. That is, the controller
dP / dt
dP dv p
εp +
dv p / dt
-
PI-1
Vm
0
Fig.3: Optimal voltage generation diagram
The performances of the controller, consisting of the control laws (26a-c), (23) and (16) are described in the following proposition. Proposition 1 (main result). Consider the single-phase gridconnected PV system shown in Fig. 2, represented by its average model (3a-d), together with the controller consisting of the control laws (10) and (13a-b). Then, one has the following results: i) The closed loop system is globally asymptotically stable. ii) The tracking error e1 = y1 − Vm vanishes exponentially, implying MPPT achievement (because Vm is the optimal PV voltage corresponding to maximum power i.e. dP / dv p = 0 ). Interestingly, Vm is computed online according to the law (17a-c). iii) The error e2 = y2 − y 2 ref converges to zero ensuring a unity PF (because y 2 ref = γeg ). iv) The tracking error ε dc = x3 − Vd converges to zero guaranteeing a tight regulation of the dc bus voltage.
3.4 Generation of the optimal voltage Vm 4. SIMULATION RESULTS
The MPP ( Vm , Pm ) is reached when: ∂P ∂v p
=0 v p =V m
(16)
The theoretical performances, described by Proposition 1, of the proposed nonlinear controller are now illustrated by simulation. The experimental setup, described by Fig. 4, is simulated using MATLAB. The characteristics of the controlled system are listed in Table 3. Let us emphasize that the controlled system is simulated using the instantaneous
model (2a-e). The averaged model (3a-d) is in effect used only in controller design. The resulting closed-loop control performances are illustrated by Fig 5 to Fig 8. Lg PV
Boost converter
Single phase inverter
Cdc
u1
vp ip Diagram Fig.3
Vm
u2
u2
PWM µ1 Regulator (10), (13a)
PWM µ2
vdc + Vd -
PI-2
Regulator ig (10), (13b) eg
Fig.4: Experimental bench for single phase grid connected system control
between 298.15K and 333.15 K (i.e. between 25°C and 60°C), while the radiation λ is constant equal to 1000 W/m 2 . It is seen that the controller keeps the whole system at the optimal operation conditions. Indeed, the captured PV power P achieves the values 183.1W or 153.4W corresponding (see Table 2) to the maximum points (M3 and M4) associated to the temperatures 333.15 K and 298.15 K , respectively. The figure also shows that the DC bus voltage vdc is regulated to its desired value Vd=48V. Fig. 8 illustrates the grid current iLg and the grid voltage eg. This figure also shows that the current iLg is sinusoidal and in phase with the voltage eg, which proves the unity PF achievement.
PV Voltage Vp(V) 200
26
150
24
100
22
50
20
Table 3. Characteristics Of Controlled System Parameter
Symbol PV model PV array PV power Ci Boost converter Li Ri DC link capacitor Cdc Lg Grid filter inductor Rg PWM Switching frequency fs Transformer ratio Grid AC source
Value NU-183E1 183.1W 4700µF 1mH 0.65Ω 6800µF 2.2mH 0.47 Ω 25kHz
0
0.2
0.4
0.6
0.8
0
0
0.2
0.4
0.6
0.8
DC bus Voltage Vdc (V)
Radiation λ (W/m2) 1000
60 40
500 20 0
0
0.2
0.4 time (s)
0.6
0.8
0
0
0.2
0.4 time (s)
0.6
0.8
Fig.5 : MPPT and DC bus voltage behavior in presence of radiation changes.
22:220 220V
PV Power P(W)
28
Current i
and Grid voltage e
Lg
g
40
eg
30 20
4.1 Radiation change effect Fig. 5 illustrates the perfect MPPT in presence of radiation changes. Specifically, the radiation varies between 400 W/m 2 and 1000 W/m 2 , meanwhile the temperature is kept constant, equal to 298.15K (i.e. 25°C). The figure shows that the captured PV power varies between 70.8 W and 183.1 W . These values correspond (see Table 2) to the maximum points (M2 and M1) of the curves associated to the considered radiations, respectively. The figure, also shows that the DC bus voltage vdc is regulated to its desired value Vd=48V. Fig. 6 illustrates the grid current iLg and the grid voltage eg. This figure clearly shows that the current iLg is sinusoidal and in phase with the voltage eg, proving unity PF achievement. 4.2 Temperature variation effect Fig.7 illustrates the controller behavior when facing temperature changes. Specifically, the temperature T varies
i
10
Lg
0 -10 -20 -30 0.3
0.35
0.4
0.45
0.5
0.55
time (s)
Fig.6 : Unity PF behavior in presence of radiation changes
PV Voltage Vp(V)
PV Power P(W) 200 150
20
100 10
0
50 0
0.2
0.4
0.6
0.8
0
0
0.2
0.4
0.6
0.8
DC bus Voltage Vdc (V)
Temperature T (K) 340 60 320 40 300 20 280 0
0.2
0.4 time (s)
0.6
0.8
0
0
0.2
0.4 time (s)
0.6
0.8
Fig.7 : MPPT and DC bus voltage behavior in presence of temperature changes.
Current iLg and Grid voltage eg 40
eg
30 20 iLg 10 0 -10 -20 -30 0.3
0.35
0.4
0.45
0.5
0.55
time (s)
Fig.8 : Unity PF behavior in presence of temperature changes 5. CONCLUSION The problem of controlling a single-phase grid connected PV system has been considered. The controller is obtained from the nonlinear average model (3) using nonlinear control technique. Using both a theoretical analysis and simulation, it is proved that the controller does meet the performances for which it was designed, namely: (i) global asymptotic stability of the closed-loop system, (ii) perfect maximum power point tracking of PV array, (iii) good unity power factor in the grid, (iv) tight regulation of the DC bus voltage. REFERENCES Abeyasekera, T., Johnson CM, Atkinson D, Amstrong M. (2005). Suppression of line voltage related distortion in current controlled grid connected inverters. IEEE Trans Power Electron;20:1393–401. Akorede, M.F., H. Hizam, E. Pouresmail, (2010). Distributed energy resources and benefits to environment.
Renewable and Sustainable Energy reviews, 14, pp. 724734. Barbosa, PG., Braga HAC, Barbosa MC, Teixeria EC. (2006) Boost current multilevel inverter and its application on single phase grid connected photovoltaic system. IEEE Trans Power Electron;21:1116–24. Chen-Chi, C., Chieh-Li, C. (2009). “Robust maximum power point tracking method for photovoltaic cells: A sliding mode control approach”. Solar Energy, 83, pp. 1370–1378. El Fadil, H. and Giri, F. (2011). Climatic sensorless maximum power point tracking in PV generation systems. ELSEVIER Trans on Control Engineering Practice ,19(5), pp. 513–521. Enrique, J.M., E. Duran, M. Sidrach-de-Cardona, J.M. Andujar. (2007). “Theoritical assessment of the maximum power point tracking efficiency of photovoltaic facilities with different converter topologies”. Solar Energy, 81, pp. 31-38, Krein, P.T., Bentsman, J., Bass, R. M., & Lesieutre, B. (1990). “On the use of averaging for analysis of power electronic system”. IEEE Transactions on Power Electronics, 5(2), pp. 182–190. Libo,W., Zhao Zhengming, Liu Jianzheng. (2007). “A singlestage three-phase grid-connected photovoltaic system with modified MPPT method and reactive power compensation” IEEE Transactions on Energy Conversion, v 22, n4, pp. 881-886, December. Mastromauro, R.A., Liserre, M. Kerekes, T. Dell'Aquila, A. (2009). “A Single-Phase Voltage-Controlled GridConnected Photovoltaic System With Power Quality Conditioner Functionality”. IEEE Transactions on Indust. Electronics, 56(11), pp. 4436–4444. Ortjohann, E., Lingemann M., Mohd A., Sinsukthavorn W., Schmelter A., Hamsic N. and Morton, D.. (2008). A General Architecture for Modular Smart Inverters, IEEE ISIE, Cambridge, England. Shimizu, T., Hashimoto O, Kimura G. (2003). A novel highperformance utilityinteractive photovoltaic inverter system. IEEE Trans Power Electron ;18:704–11. Slotine J.-J. E. and W. Li (1991). Applied nonlinear control. Prentice Hall.
National School of Applied Sciences (ENSA), Ibn Tofail University, Kénitra, Morocco (corresponding author; e-mail: [email protected]) **
GREYC Lab, UMR CNRS, University of Caen, 14032, Caen, France (e-mail: [email protected]).
Abstract: This work deals with the problem of controlling single-phase grid connected photovoltaic system. The system consists on a photovoltaic (PV) panel, a boost power converter and a single phase inverter connected to the grid. The control objective is fourfold: (i) asymptotic stability of the closed loop system, (ii) maximum power point tracking (MPPT) of PV module (iii) tight regulation of the DC bus voltage, and (iv) unity power factor (PF) in the grid. A nonlinear controller is developed using feedback linearizing technique based on an averaged nonlinear model of the whole controlled system. The model accounts, on the one hand, for the nonlinear dynamics of the underlying boost converter and inverter and, on the other, for the nonlinear characteristic of PV panel. It is formally shown, through theoretical analysis and simulation results, that the proposed controller does achieve its objectives. Keywords: Photovoltaic systems, Grid connection, Nonlinear control 1. INTRODUCTION With the deregulation of electricity markets and thrust to reduce greenhouse gas emissions from the traditional electric power generation systems, renewable energy resources such as wind turbines, photovoltaic panels, gas turbines and fuel cells, has gained a significant opportunity as new means of power generation to meet the growing demand for electric energy. Solar energy is considered to be one of the most useful natural energy sources because it is free, abundant, pollution-free, and most widely distributed. It can be used either at remote regions as standalone apparatus or in urban applications as grid interactive power source (Shimiza et al., 2003). On the other hand, a significant technology progress has been achieved over the few past years increasing efficiency of solar cells. The rapid growth of solar industry has expanded the importance of PV systems making them more reliable and efficient, especially for utility power in distributed generation (DG) (see Fig. 1) at medium and low voltages power systems. Implementing distributed energy resources (DER), into interconnected grids could be part of the solution to meet the rising electricity demand (Ortjohann et al. ,2008, Akorede et al., 2010). DG technologies are currently being improved through several research projects toward the development of smart grids. PV energy applications are divided into two categories: stand-alone systems and grid-connected systems. Stand-alone systems require a battery bank to store the PV energy; this is
suitable for low-power systems. On the other hand, gridconnected PV systems do not require battery banks; they are resorted generally in high power applications. The main purpose of the grid-connected system is to extract the maximal quantity possible of solar array energy and restitute it to grid with a unity power factor, despite changing atmospheric conditions (temperature and radiation). PV grid-connected systems represent the most important field applications of solar energy (Libo et al., 2007, Abeyasekera et al., 2006, Barbosa el al., 2006, Mastromauro, 2009 et al.). In general, a photovoltaic grid-connected system can be seen as a two-stage grid-connected inverter (Fig.1). The first stage is a dc-dc converter controlled so that the photovoltaic system operates in optimal condition i.e. seeking maximum power point tracking (MPPT) (El Fadil et al., 2011). The second stage is a dc-ac converter that controlled in a way that allows a grid connection with a unity power factor (PF). To this end, the output current (entering the grid) must be sinusoidal and in phase with the grid voltage. The dc-dc and dc-ac converters operate independently making easier the whole system control. The present paper is focusing on the problem of controlling single phase grid-connected PV power generation systems (Fig. 1). The control objective is fourfold: (i) global asymptotic stability of the whole closed-loop control system; (ii) achievement of MPPT of the PV array; (iii) ensuring a grid connection with unity PF; and (iv) ensuring a tight regulation of the dc-bus voltage. These objectives must be
met despite changes of the climatic variables (temperature and radiation). To this end, a nonlinear controller is developed using the backstepping design technique based on large-signal nonlinear model of the whole system. A theoretical analysis is developed to show that the controller actually meets its objectives a fact that is confirmed by simulation. The paper is organized as follows: the single phase grid connected PV system is described and modelled in Section II. Section III is devoted to controller design and analysis. The controller tracking performances are illustrated by numerical simulation in Section IV.
2.1 PV array model Typical (Ip-Vp) characteristics of solar cells arranged in N p parallel and N s -series can be found in many places (see e.g. Enrique et al., 2007, Chen-Chi and Chieh-Li, 2009). The PV array module considered in this paper is the NU-183E1. The corresponding electrical characteristics are listed in Table 1. The associated maximum power points (MPP) M1 to M4 under climatic conditions (temperature and radiation) are shown in Table 2. The data in Table 1 and Table 2 will be used latter for simulation purpose.
Main Supervisory Control
Table 1. Electrical specifications for the solar module NU183E1 Unit Control
GRID
Unit Control
PV Panels
System under study Unit Control BUS
Unit Control
MPPT
Wind Turbine
DC
Fuel Cell Generator
Unit Control
Parameter Maximum Power
Symbol Pm
Value 183.1W
Short circuit current
I scr
8.48 A
Open circuit voltage
Voc
30.1V
Maximum power voltage
Vm
23.9 V
Maximum power current
Im
7.66A
Number of parallel modules
Np
1
Number of series modules
Ns
48
Unit Control
Table 2. Maximum Power Points (MPP) Diesel Generator
Storage Batteries
MPP M1(λ=1000W/m2; T=25°C) M2(λ=400W/m2; T=25°C) M3(λ=1000W/m2; T=25°C) M4(λ=1000W/m2; T=60°C)
Fig.1: Control architecture in distributed energy resources
2. PRESENTATION AND MODELING OF GRID CONNECTED PV SYSTEM A typical configuration of a single-phase grid connected photovoltaic system is shown in Fig. 2. It consists of a solar array, an input capacitor Ci , a boost dc-dc converter which is used for boosting the array voltage and achieving MPPT for PV array, a DC link capacitor Cdc , a single-phase full-bridge inverter (including four power semiconductors) that is based upon to ensure a DC-AC power conversion a unity PF, a filter inductor Lg , and an isolation transformer.
+
idc u2
vp PV -
D1
Ci
u1
u2
ig
+
Cdc u2
Lg eg
u2
GRID
Li
Pm
23.82 V 22.94 V 23.82 V 20.38 V
183.1 W 70.8 W 183.1 W 153.4 W
2.2 Boost converter modeling The control input u1 of the boost converter is a PWM signal taking values in the set {0,1} . Applying the Kirchoff’s laws, successively with u1 = 1 and u1 = 0 , to the boost converter circuit of Fig. 2 one obtains the following instantaneous model dv p dt
ip
Vm
=
1 (i p − iLi ) Ci
(1a)
vp diLi v R = −(1 − u1 ) dc − i iLi + dt Li Li Li
(1b)
dvdc i 1 = (1 − u1 ) Li − idc dt Cdc Cdc
(1c)
-
where Ri is the equivalent series resistance (ESR) of input Fig.2: Single phase grid connected PV system
inductance Li .
2.3 Single-phase full-bridge inverter modeling The control input u 2 of the single-phase full-bridge inverter is also a PWM signal taking values in the set {0,1} . Applying the Kirchoff’s laws, successively with u 2 = 1 and u 2 = 0 , to the inverter circuit of Fig. 2 one obtains the following instantaneous model di Lg dt
= (2u 2 − 1)
eg vdc Rg − i Lg − L g Lg Lg
idc = (2u 2 − 1)iLg
(2a) (2b)
3.2 MPPT and PFC objectives The system of Fig.2 can be viewed as a MIMO system with two inputs µ1 and µ 2 an two outputs : y1 = x1 y 2 = x4
Bearing in mind the second control objective which is to enforce the output y1 to track the optimal point Vm , one can seek the relationship between the output y1 and the input µ1 . To this end let us differentiate the output y1
where Rg is the ESR of the filter inductance Lg .
y&1 =
2.4 Overall system model The instantaneous model (1a-c, 2a-b) of the single-phase grid connected photovoltaic system is useful for simulator design. But, it is not suitable for controller design because it involves binary control inputs u1 and u 2 . For control design purpose, it is more convenient to consider the following averaged model (Krein et al, 1990) dx1 1 = (i p − x2 ) dt Ci x R dx2 x = −(1 − µ1 ) 3 − i x2 + 1 dt Li Li Li dx3 x x = (1 − µ1 ) 2 − (2µ 2 − 1) 4 dt Cdc Cdc Rg eg x dx4 x4 − = (2 µ 2 − 1) 3 − dt Lg L g Lg
(4a) (4b)
1 (i p − x2 ) Ci
(5)
Since y&1 is still not directly related to the input µ1 , let us differentiate again. We now obtain &y&1 =
1 Ci
⎛& x R x ⎞ ⎜ i p + (1 − µ1 ) 3 + i x2 − 1 ⎟ ⎜ ⎟ L L L i i i ⎠ ⎝
(6)
The unity PF objective means that the grid current iLg should be sinusoidal and in phase with the AC grid voltage e g . We
(3a) (3b) (3c) (3d)
therefore seek a regulator that enforces the current y 2 to tack a reference signal y 2 ref of the form:
y 2 ref = γeg
At this point γ is any real positive parameter that is allowed to be time-varying. To generate a direct relationship between the output y 2 and the second input , let us differentiate the output y 2
where x1 , x2 , x3 , x4 , i p , µ1 and µ 2 denote the average
y& 2 = (2 µ 2 − 1)
values, respectively of, v p , iLi , vdc , iLg , i p , u1 and u 2 . All variables averaging are performed over switching periods. Consequently, the quantities µ1 and µ 2 , commonly called duty ratios, vary continuously in the interval [0 1] and act as the input control signals. 3. CONTROLLER DESIGN 3.1 Control objectives In this section, we aim at designing a controller that will be able to ensure: (i) a global stability of closed loop system, (ii) a perfect MPPT (whatever the position of the PV panel). Specifically, the controller must enforce the voltage x1 to track, as accurately as possible, the unknown (and slowly varying) voltage Vm . Note that Vm is unknown because it depends on the temperature T and solar variation λ and these are not supposed to be measured; (iii) a unity PF in the grid; and (iv) a tight regulation of the dc bus voltage x3 .
(7)
eg x3 Rg − x4 − Lg Lg Lg
(8)
Equations (6) and (8) can be gathered as follows ⎛ &y&1 ⎞ ⎛µ ⎞ ⎜⎜ ⎟⎟ = A( x)⎜⎜ 1 ⎟⎟ + b( x) & ⎝ y2 ⎠ ⎝ µ2 ⎠
(9a)
⎛ − x3 ⎜ LC A( x) = ⎜⎜ i i ⎜⎜ 0 ⎝
(9b)
where
⎛ 1 ⎜ ⎜C b( x) = ⎜ i ⎜ ⎜ ⎝
⎞ 0 ⎟ ⎟ 2 x3 ⎟ ⎟ Lg ⎟⎠
⎛& x R x ⎞⎞ ⎜ i p + 3 + i x2 − 1 ⎟ ⎟ ⎜ Li Li Li ⎟⎠ ⎟ ⎝ ⎟ Rg eg x ⎟ − 3 − x4 − ⎟ Lg Lg Lg ⎠
(9c)
Using linearizing feedback technique (Slotine, 1991), one can choose the following control signals ⎞ ⎛ ⎛ν ⎞ ⎛ µ1 ⎞ ⎜⎜ ⎟⎟ = A −1 ( x)⎜ ⎜⎜ 1 ⎟⎟ − b( x) ⎟ ⎟ ⎜ ⎝ µ2 ⎠ ⎠ ⎝ ⎝ν 2 ⎠
(10)
where ν 1 andν 2 being a new inputs to be determined. The nonlinearity in (9a) is cancelled, and we obtain a simple linear integrators relationship between the outputs and the new inputs &y&1 = ν 1 y& 2 = ν 2
e1 = y1 − Vm e2 = y 2 − y 2 ref
(12a) (12b)
be the tracking errors, and choosing the new inputs as
ν 1 = V&&m − α 0 e1 − α1e&1 ν 2 = y& 2 ref − β 0 e2
and the temperature T , by acting on the duty cycle µ1 . Then, the optimal voltage Vm can be generated using a PI regulator as shown in Fig. 3. This regulator (denoted PI-1 in Fig. 3) is defined as follows: Vm = G1 ( s ) ε p
(13a) (13b)
e&2 + β 0 e2 = 0
and
(14)
Which represent an exponentially stable error dynamics. Therefore the errors e1 and e2 converge to zero exponentially.
3.3 DC bus voltage regulation objective Now, the aim is to design a variation law for the ratio γ in (7) so that the dc voltage x3 =< vdc > is steered to a given constant reference Vd > 0 . To this end, the following PI control law (called PI-2) is used:
γ = G2 ( s )ε dc
(15a)
where (1 + τ 2 s ) τ 2s
ε dc = x3 − Vd
(15b) (15c)
(17a)
where G1 ( s ) = k1
εp =
(1 + τ 1s ) τ 1s
(17b)
dP dv p
(17c)
where s denotes the Laplace variable. ip vp
P
d dt
d dt
with α 0 , α1 and β 0 being positive constants. The tracking errors of closed loop system are given by
G2 ( s ) = k2
must keep dP / dv p equals zero, whatever the radiation λ
(11a) (11b)
The design of a tracking controller for (11a-b) is simple because of the availability of linear control technique. For instance, letting
e&&1 + α1e&1 + α 0 e1 = 0
with P = v p i p being the PV power. That is, the controller
dP / dt
dP dv p
εp +
dv p / dt
-
PI-1
Vm
0
Fig.3: Optimal voltage generation diagram
The performances of the controller, consisting of the control laws (26a-c), (23) and (16) are described in the following proposition. Proposition 1 (main result). Consider the single-phase gridconnected PV system shown in Fig. 2, represented by its average model (3a-d), together with the controller consisting of the control laws (10) and (13a-b). Then, one has the following results: i) The closed loop system is globally asymptotically stable. ii) The tracking error e1 = y1 − Vm vanishes exponentially, implying MPPT achievement (because Vm is the optimal PV voltage corresponding to maximum power i.e. dP / dv p = 0 ). Interestingly, Vm is computed online according to the law (17a-c). iii) The error e2 = y2 − y 2 ref converges to zero ensuring a unity PF (because y 2 ref = γeg ). iv) The tracking error ε dc = x3 − Vd converges to zero guaranteeing a tight regulation of the dc bus voltage.
3.4 Generation of the optimal voltage Vm 4. SIMULATION RESULTS
The MPP ( Vm , Pm ) is reached when: ∂P ∂v p
=0 v p =V m
(16)
The theoretical performances, described by Proposition 1, of the proposed nonlinear controller are now illustrated by simulation. The experimental setup, described by Fig. 4, is simulated using MATLAB. The characteristics of the controlled system are listed in Table 3. Let us emphasize that the controlled system is simulated using the instantaneous
model (2a-e). The averaged model (3a-d) is in effect used only in controller design. The resulting closed-loop control performances are illustrated by Fig 5 to Fig 8. Lg PV
Boost converter
Single phase inverter
Cdc
u1
vp ip Diagram Fig.3
Vm
u2
u2
PWM µ1 Regulator (10), (13a)
PWM µ2
vdc + Vd -
PI-2
Regulator ig (10), (13b) eg
Fig.4: Experimental bench for single phase grid connected system control
between 298.15K and 333.15 K (i.e. between 25°C and 60°C), while the radiation λ is constant equal to 1000 W/m 2 . It is seen that the controller keeps the whole system at the optimal operation conditions. Indeed, the captured PV power P achieves the values 183.1W or 153.4W corresponding (see Table 2) to the maximum points (M3 and M4) associated to the temperatures 333.15 K and 298.15 K , respectively. The figure also shows that the DC bus voltage vdc is regulated to its desired value Vd=48V. Fig. 8 illustrates the grid current iLg and the grid voltage eg. This figure also shows that the current iLg is sinusoidal and in phase with the voltage eg, which proves the unity PF achievement.
PV Voltage Vp(V) 200
26
150
24
100
22
50
20
Table 3. Characteristics Of Controlled System Parameter
Symbol PV model PV array PV power Ci Boost converter Li Ri DC link capacitor Cdc Lg Grid filter inductor Rg PWM Switching frequency fs Transformer ratio Grid AC source
Value NU-183E1 183.1W 4700µF 1mH 0.65Ω 6800µF 2.2mH 0.47 Ω 25kHz
0
0.2
0.4
0.6
0.8
0
0
0.2
0.4
0.6
0.8
DC bus Voltage Vdc (V)
Radiation λ (W/m2) 1000
60 40
500 20 0
0
0.2
0.4 time (s)
0.6
0.8
0
0
0.2
0.4 time (s)
0.6
0.8
Fig.5 : MPPT and DC bus voltage behavior in presence of radiation changes.
22:220 220V
PV Power P(W)
28
Current i
and Grid voltage e
Lg
g
40
eg
30 20
4.1 Radiation change effect Fig. 5 illustrates the perfect MPPT in presence of radiation changes. Specifically, the radiation varies between 400 W/m 2 and 1000 W/m 2 , meanwhile the temperature is kept constant, equal to 298.15K (i.e. 25°C). The figure shows that the captured PV power varies between 70.8 W and 183.1 W . These values correspond (see Table 2) to the maximum points (M2 and M1) of the curves associated to the considered radiations, respectively. The figure, also shows that the DC bus voltage vdc is regulated to its desired value Vd=48V. Fig. 6 illustrates the grid current iLg and the grid voltage eg. This figure clearly shows that the current iLg is sinusoidal and in phase with the voltage eg, proving unity PF achievement. 4.2 Temperature variation effect Fig.7 illustrates the controller behavior when facing temperature changes. Specifically, the temperature T varies
i
10
Lg
0 -10 -20 -30 0.3
0.35
0.4
0.45
0.5
0.55
time (s)
Fig.6 : Unity PF behavior in presence of radiation changes
PV Voltage Vp(V)
PV Power P(W) 200 150
20
100 10
0
50 0
0.2
0.4
0.6
0.8
0
0
0.2
0.4
0.6
0.8
DC bus Voltage Vdc (V)
Temperature T (K) 340 60 320 40 300 20 280 0
0.2
0.4 time (s)
0.6
0.8
0
0
0.2
0.4 time (s)
0.6
0.8
Fig.7 : MPPT and DC bus voltage behavior in presence of temperature changes.
Current iLg and Grid voltage eg 40
eg
30 20 iLg 10 0 -10 -20 -30 0.3
0.35
0.4
0.45
0.5
0.55
time (s)
Fig.8 : Unity PF behavior in presence of temperature changes 5. CONCLUSION The problem of controlling a single-phase grid connected PV system has been considered. The controller is obtained from the nonlinear average model (3) using nonlinear control technique. Using both a theoretical analysis and simulation, it is proved that the controller does meet the performances for which it was designed, namely: (i) global asymptotic stability of the closed-loop system, (ii) perfect maximum power point tracking of PV array, (iii) good unity power factor in the grid, (iv) tight regulation of the DC bus voltage. REFERENCES Abeyasekera, T., Johnson CM, Atkinson D, Amstrong M. (2005). Suppression of line voltage related distortion in current controlled grid connected inverters. IEEE Trans Power Electron;20:1393–401. Akorede, M.F., H. Hizam, E. Pouresmail, (2010). Distributed energy resources and benefits to environment.
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