A new control structure for grid-connected LCL PV ...

A New Control Structure for Grid-Connected LCL PV Inverters with Zero Steady-State Error and Selective Harmonic Compensation R. Teodorescu, F.Blaabjerg

U. Borup

M. Liserre

Institute of Energy Technology Aalborg University Pontoppidanstraede 101 DK-9220, Aalborg, Denmark [email protected]

PowerLynx A/S Ellegaardvej 36 DK-6400, Sønderborg Denmark [email protected]

Dept. of Electrotechnical and Electronic Eng. Polytechnic of Bari 70125-Bari, Italy [email protected]

1 Abstract— The PI current control of a single-phase inverter has well known drawbacks: steady-state magnitude and phase error and limited disturbance rejection capability. When the current controlled inverter is connected to the grid, the phase error results in a power factor decrement and the limited disturbance rejection capability leads to the need of grid feed-forward compensation. However the imperfect compensation action of the feed-forward control results in high harmonic distortion of the current and consequently non-compliance with international standards. In this paper a new control strategy aimed to mitigate these problems is proposed. Stationary-frame generalized integrators are used to control the fundamental current and to compensate the grid harmonics providing disturbance rejection capability without the need of feed-forward grid compensation. Moreover the use of a grid LCL-filter is investigated with the proposed controller. The current control strategy has been experimentally tested with success on a 3 kW PV inverter.

error and poor disturbance rejection capability. This is due to the poor performance of the integral action. In order to alleviate these problems, a second order generalized integrator (GI) as reported in [3] can be used. The GI is a double integrator that achieves an infinite gain at a certain frequency, also called resonance frequency, and almost no attenuation exists outside this frequency. Thus, it can be used as a notch filter in order to compensate the harmonics in a very selective way. This technique has been primarily used in three-phase active filter applications as reported in [3] and also in [4] where closed-loop harmonic control is introduced. Another approach reported in [5] where a new type of stationary-frame regulators called P+Resonant is introduced and applied to three-phase PWM inverter control. In this approach the PI dc-compensator is transformed into an equivalent ac-compensator, so that it has the same frequency response characteristics in the bandwidth of concern.

Keywords: single-phase PV inverter, current controller.

I.

In this paper generalized integrators are used in a combined control strategy for a single-phase PV grid-connected inverter to achieve both zero-steady-state error and selective harmonic compensation. Basically a P+Resonant (PR) controller is used to control the fundamental current and several GI that resonate for each harmonic frequency of interest (3rd, 5th and 7th) are used in a harmonic compensator in order to reduce the current THD for compliance with IEEE 929 standard.

INTRODUCTION

Due to the latest developments in power and digital electronics, the market for small distributed power generation systems like photovoltaic (PV) systems connected to the domestic grid is increasing rapidly. PV inverters in the range of 1-5 kW are currently available from several manufacturers. Harmonics level is still a controversial issue for PV inverters. The IEEE 929 standard from 2000 allows a limit of 5% for the current total harmonic distortion (THD) factor with individual limits of 4% for each odd harmonic from 3rd to 9th and 2% for 11th to 15th while a recent draft of European IEC61727 suggests something similar. These levels are far more stringent than other domestic appliances such as IEC61000-3-2 as PV systems are viewed as generation sources and so are subject to higher standards than load systems.

The current-controlled PV inverter is connected to the grid through a LCL filter, as depicted in Fig.1. Zs

us

ug

Lg

Li

ii ui

Cf

Ud

H-VSI

PI control with grid voltage feed-forward [1],[2] is commonly used for current-controlled PV inverters, but this solution exhibits two well known drawbacks: inability of the PI controller to track a sinusoidal reference without steady-state

CURRENT CONTROL

PWM

Fig.1. The H-bridge PV inverter connected to the grid through an LCL filter

The authors want to acknowledge the financial support of PSO-Eltra contract 4524

0-7803-8269-2/04/$17.00 (C) 2004 IEEE.

ig

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The design of the LCL filter has a very serious impact over the stability of the system, as it is a resonant circuit that should always be damped. As the grid impedance can vary substantially with the stiffness of the grid, the resonance frequency of the LCL filter changes and special care should be taken in the design of controller and LCL filter in order to solve the stability problem. Also, the low cost trend is challenging the LCL filter design in the direction of using lower inductance, which is expensive and higher capacitance, which is cheaper [6].

where: Ts – switching period, D – duty cycle. Neglecting the voltage drop across the LCL filter, the grid voltage can be expressed as u g = U dc ⋅ D , where the duty cycle

D = m ⋅ sin(ω t ) , (0 < ωt < π) is expressed as a function of the modulation index m = u$ g / U , where u$ g is the peak grid d

voltage and ω is the grid frequency in [rad/s]. Thus, a nondimensional expression of current ripple can be written as:

∆ii '(ω t ) =

An LCL design and stability analysis study using Matlab is proposed in paragraph II.Then the current controller with selective harmonic compensator design is discussed. The discretization of the current controller for digital implementation is described in IV and finally experimental results with a 3 kW DSP-controlled PV inverter are presented. Here a comparison with the classical PI feed-forward control is highlighted and compliance with IEEE 929 standard is discussed.

∆ii (ω t ) = (1 − m ⋅ sin(ω t ) ) ⋅ sin(ω t ) U dc ⋅ Ts 2 Li

(2)

The variable ∆ii´ is plotted in Fig. 3 for a half period of the grid voltage and it is a good way to estimate the maximum ripple knowing the dc voltage, switching period, modulation index for a given inductance. D i, m a x

II.

LCL FILTER DESIGN

0 .2 5

A step-by-step procedure in order to design the LCL-filter for grid-connected inverters described in [7] is applied. Though, the following special considerations need to be applied:

0.2

0 .1 5

0.1

• the converter side inductance design needs particular care since in single-phase application the ripple varies substantially within one period of the current

0 .0 5

0

• PV equipment have stringent requirements in terms of space, thus the overall inductors installed should be limited well below the 10 % of the base impedance

D/2

0.00 5 t im e [ s ]

0.00 6

0.0 07

0 .0 0 8

0 .0 0 9

0.0 1

(3)

(4)

where $i g is the peak of the fundamental current, usually equal with the filtered grid current. The expression (4) is very useful in the design of the filter inductance with the respect to the saturation level of the magnetic core. Next, in order to further attenuate the current ripple to a desired level, an LC second order filter is used as part of the LCL filter.

∆ii

Ts/4

Ts/4

The attenuation at the switching frequency can be determined by plotting the bode plot of the transfer function:

Ts/4

Fig 2 - The voltage accros Li and the inverter current ii

ig ( s )

The inverter current ripple can be expressed as:

(U

0.00 4

∆i (ωt )   ii ,max (ωt ) = max  $i g ⋅ sin(ωt ) + i 2  

ii

∆ii =

0.003

Knowing the ripple current, the peak current through the inverter can be calculated as:

uLi

Ts/4

0 .002

 1  1  ∆ii ,max ' = ∆ii ,max '  arcsin   = 2 4 m ⋅m   

The inverter current ii ripple can be seen in Fig. 2 for unipolar switching PWM strategy, where there are two half pulses in each switching period centred at Ts/4 and 3Ts/4, respectively.

Udc-ug

0 .0 0 1

Fig 3. Time variation of the non-dimensional current ripple ∆ii ‘ for a half period of the grid voltage

First the inverter side inductance Li is designed in order to limit the current ripple generated by the VSI.

D/2

0

− ug ) D Ts Li 2

dc

ii ( s )

=

1 1 + C f ⋅ Lg ⋅ s 2

where the grid impedance Zs has been neglected.

(1)

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(5)

The trend is to minimize the filter volume by using lower inductances and higher capacitances [6], knowing that inductors are more costly and bulky than capacitors.

III.

The classical PI current control strategy with voltage feedforward is depicted in Fig.5 and the new proposed control strategy in Fig. 6.

In turn, the control has to be more complex having to cope with stability problems when operating on a weak grid. The relative high current-ripple calls for the use of high-quality magnetic cores like the amorphous ones in order to keep the volume and losses on a low level.

i i*

u i*

Gd(s) Gf(s)

ii

ug

Fig. 5. The current loop of PV inverter with PI controller.

i i*

40

u i*

Gc(s) ii

LgC filter attenuation

60

Magnitude (dB)

GPI(s) ii

In the case of the 3 kW inverter having Ud = 350 V, m = 0.93, an LCL filter having a resonance frequency of 4.25 kHz has been designed. The Lg Cf fitler was designed to get around –20dB attenuation of the inverter current (5) at the switching frequency, as it can be seen in Fig. 4. where 19 dB attenuation is obtained. Thus the grid-side current ripple is highly attenuated.

Gd(s)

Gf(s)

ii

Gh(s)

20

Fig. 6. The current loop of PV inverter with PR and HC

-19 dB at 17 kHz

0

-20 System: Zfil Frequency (Hz): 1.7e+004 Magnitude (dB): -19.1

-40

where ui* is the inverter voltage reference and ii* is the inverter current reference.

-60 180

The PI current controller GPI(s) is defined as:

135

Phase (deg)

CONTROL STRATEGY

GPI ( s ) = K P +

90

45

0

1

10

2

3

10

4

10

10

5

10

Fig. 4 The frequency response of the LgCf filter at the output of the inverter

Here it should be mentioned that the grid impedance has a positive effect by further increasing the ripple attenuation at the switching frequency. So this attenuation analysis could be considered worst-case.

The P+Resonant (PR) current controller Gc(s) is defined as [5], [10]:

Gc ( s ) = K P + K I

The transfer function of the LCL filter in terms of inverter current and voltage neglecting the resistances is [7], [9]: G f (s) =

( (

2 2 ii ( s ) 1 s + z LC = 2 ui ( s ) Li s s 2 + ω res

) )

(7)

The PI controller is not able to track a sinusoidal reference without steady-state error and in order to get a good dynamic response, a grid voltage feed-forward is used, as depicted in Fig. 5. This leads in turn to the presence of the grid-voltage background harmonics in the current waveform. Thus, a poor THD of the current will typically be obtained.

0 10

KI s

s s + ωo2

(8)

2

The harmonic compensator (HC) Gh(s) as defined in [3]:

(6)

Gh ( s ) =

∑

h =3,5,7

where z 2 =  L C  −1 and ω 2 = ( Li + Lg ) ⋅ z LC . LC res  g f L 2

K Ih

s s + (ωo h ) 2

2

(9)

is designed to compensate the selected harmonics 3rd, 5th and 7th as they are the most prominent harmonics in the current spectrum .

i

The grid side inductance Lg will be connected in series with the grid inductance and thus the attenuation and resonant frequency (6) can vary significantly leading to instability.

A processing delay typical equal to Ts for the PWM inverters [8] is introduced in Gd ( s) . The filter transfer function Gf(s) is expressed in (6).

Special attention has to be paid when designing the current controller in order to ensure stability over grid inductance variation.

The current error - disturbance ratio rejection capability at null reference is defined as:

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u g ( s ) i* i

= =0

o p e n lo o p

60

G f (s)

1 + ( Gc ( s ) + Gc ( s ) ) ⋅ Gd ( s ) ⋅ G f ( s )

(10)

where: ε is current error and the grid voltage ug is considered as the disturbance for the system.

5 th

fund

w i tho ut ha rm . c o m p . w i th ha rm . c o m p .

50

Magnitude [db]

ε (s)

3 rd

40

7 th

30 20 10 0

The Bode plots of disturbance rejection for the PI and PR controllers are shown in Fig 7. As it can be observed, around the fundamental frequency the PR provides 140 dB attenuation while the PI provides only 17 dB. Moreover around the 5th and 7th harmonics the situation is even worst, the PR attenuation being 125 dB and the PI attenuation only 8 dB. Moreover from Fig. 7 it is clear that the PI rejection capability at 5th and 7th harmonic is comparable with that one of a simple proportional controller, the integral action being irrelevant.

c ro s s -o ve r fre q =4 6 0 H z

-1 0 1 10

10

2

10

0

w itho ut ha rm . c o m p . w ith ha rm . c o m p .

-2 0 -4 0

Phase [Grad]

-6 0

P M = 7 2 g rd

-8 0 -1 0 0 -1 2 0 -1 4 0 -1 6 0 -1 8 0 1 10

2

10 F re q ue nc y [H z]

10

Fig. 8.Bode plot of open-loop PR current control system with and without harmonic compensator

0

c lo s e d lo o p -50 PR+HC PI P

4

Magnitude [db]

-150

7 th

5 th

3 rd

fun d

-100

2 0

B W =6 50H z

-2

-3 d B

-4

w i th ha rm . c o m p . w i th o u t h a rm . c o m p .

-270

-6 1 10

10

2

10

3

-360

0 -450

Phase [Grad]

-2 0

-540 10

1

10

2

10

3

3

Fig. 7. Bode plot of disturbance rejection (current error ratio disturbance) of the PR+HC, P and PR current controllers.

-4 0 -6 0 -8 0

w i th h a rm . c o m p . w i th o u t ha rm . c o m p .

-1 0 0

2

10 F re q ue nc y [H z]

Thus it is demonstrated the superiority of the PR controller respect to the PI in terms of harmonic current rejection.

10

Fig.9. Bode plot of closed loop PR current control system with and without harmonic compensator

Then the tuning for the PR controller can be adressed considering first the system without harmonic compensator. The open loop and closed loop frequency response of the system can be seen in Fig.8 and Fig 9 respectively.

closed loop poles/zeros

1

LCL resonance poles 0.6π /T

0.8

The size of the proportional gain Kp from PR controller determines the bandwidth and stability phase margin [3], in the same way as for the classical PI controller.

0.5π /T

0.4π /T 0.10.3π /T

0.7π /T

0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.8π /T

0.4 0.9π /T

A gain equal to Kp = 2 leads to a bandwidth of about 650Hz as it can be seen on the closed-loop current loop bode plot in Fig. 9. This was considered satisfactory with respect to the sampling frequency of 8.5 kHz.

0.2

0

dominant poles

0.2π /T

0.1π /T

π /T π /T

-0.2 0.9π /T

From the open-loop bode plot depicted in Fig. 8, the phase margin (PM) is determined to be equal with 72 deg. indicating a high stability.

0.1π /T

-0.4 0.8π /T

-0.6

0.7π /T

-0.8

The dominant poles of the controller are well damped as it can be observed in Fig. 10 exhibiting a damping factor higher than 0.9 that is not the technical optimum as recommended in [1].

0.2π /T

0.3π /T 0.6π /T

-1

-1

-0.8

-0.6

-0.4

-0.2

0.5π /T

0

0.4π /T

0.2

0.4

0.6

0.8

1

Fig.10 Pole-zeros placement of closed-loop current control system

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3

3

This is because a more conservative approach has been adopted leading to a smooth transient response but the system is stable even if the grid impedance varies considerably.

s = 2 u (s) s + ω2

Also the high-frequency poles from the LCL-filter are inside the unity-circle indicating stability even without any kind of damping. The one-sample delay of the system also increases the stability by pushing the LCL-filter poles toward the unity-circle centrum.

(11)

Fig. 12 shows the block diagram of the GI equivalent form. y

u

The integral constant KI acts to eliminate the steady-state error [5]. Another aspect is that KI determines the bandwidth centred at the resonance frequency, in this case the grid frequency, where the attenuation is positive. Usually, the grid frequency is stiff and is only allowed to vary in a narrow range, typically ± 1%.

∫ v ω2

∫

Fig. 12 GI integrator decomposed in two simple integrators.

In order to avoid an algebraic loop during discrete implementation it is suggested that the direct integrator is discretized using forward method while the feedback one is discretized using the backwards method giving:

A value of KI = 100 was determined by simulations in order to eliminate the steady error at the grid frequency as it can be seen in Fig. 11 being able to cope with the ± 1% frequency variations. 1 .5

1   y ( s ) = s u ( s ) − v ( s )  ⇔  v ( s ) = 1 ⋅ ω 2 ⋅ y ( s )  s

y ( s)

 yk = yk −1 + Ts ⋅ (uk −1 − vk −1 )  2 vk = vk −1 + Ts ⋅ ω ⋅ yk

ig ig *

(12)

1

Amplitude

0 .5

Thus, the PR controller transfer function expressed in terms of inverter voltage reference ui* and current error ε is:

0

K   ui* ( s ) = ε ( s ) ⋅  K p + 2 I 2  s +ω  

- 0 .5

-1

(12)

and can be written in discrete form as follows:

- 1 .5 0

0 .0 1

0 .0 2

Kp =2

K i =1 00

0 .0 3

L g r id [u H ] = 0

0 .0 4

Rg r id [o h m ] = 0

0 .0 5

0 .0 6

0 .0 7

Po u t[W ] = 3 0 0 ( s e c )

 yk = yk −1 + Ts ⋅ K I ⋅ ε k −1 − Ts ⋅ vk −1  * ui ,k = K p ⋅ ε k + yk v = v + T ⋅ ω 2 ⋅ y k −1 s k  k ε k −1 = ε k y = y k  k −1 vk −1 = vk

Fig.11. 50Hz response of the PR current control system

Having the fundamental component current controller designed, the harmonic compensator is being added. In this case, the integral constant KIh has the same effect as for the fundamental component i.e. eliminating the steady-state error, just that the resonance frequencies are synchronous with the 3rd, 5th and 7th harmonics. Having added the harmonic compensator, the open-loop and closed-loop bode graphs changes as it can be observed in Fig. 8 and Fig. 9 with dashed line. The change consists in the appearance of gain peaks at the harmonic frequencies, but what is interesting to notice is that the dynamics of the controller, in terms of bandwidth and stability margin remains unaltered.

The block diagram of the PR control implementation is shown in the diagram from Fig. 12 kp

In this way a selective harmonic compensation can be achieved without affecting the fundamental controller dynamics. IV.

(14)

ii*

aw

u

ε ii

DISCRETE IMPLEMENTATION

ui*

Gf(s)

v ∫

The GI integrators expressed in (8) are actually double integrators and in order to ease the discretization they are decomposed in two simple integrators as it follows:

y

∫

kI

ω2

Fig.12 Control diagram of the PR controller implementation

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ii

In Fig. 12, the anti-windup function implemented as:

, y > ymax  ymax − y aw =  − ymax − y , y < − ymax

25

(15)

15 10

and has been added in order to avoid the well known winding up problems.

5 0

For the PR+HC control, three additional GI are added as described by (9) using the same implementation model. V.

Ig (exp) [5A/div] Ug (exp) [100/div]

20

-5 -10

EXPERIMENTAL RESULTS

-15

A test setup consisting of one 3 kW PV full-bridge (H) inverter with LCL filter having the resonance fequency at 4.25 kHz and 19 dB attenuation around the switching frequency for the inverter current has been built, as depicted in Fig. 13.

-20 -25

0

0.005

0.01

0.015

0.02 time[sec]

0.025

0.03

0.035

0.0

Fig.14 Experimental results at 3kW. Grid voltage and current with PI controller.

The following equipment is been used: 400V/10A regulated dc power supply to simulate PV panels string, digital scope Tektronix TDS3014B and power analyzer PM100.

25 Ig (exp) [5A/div] Ug (exp) [100/div]

20 15 10 5 0 -5 -10 -15 -20 -25

0

0.005

0.01

0.015

0.02 time[sec]

0.025

0.03

0.035

0.0

Fig.15 Experimental results at 3kW. Grid voltage and current with PR controller

Fig.13 Experimental test-setup of the 3kW PV inverter

The control strategy has been implemented and tested on a 16-bit fixed-point TMS320F24xx DSP platform. The execution time for the control loop was measured to about 40µs, for the PR including the harmonic compensator.

25 Ig (ex p) [5A /div] U g (ex p) [100/div]

20 15

The system was tested in the following conditions: dc voltage UD = 370 V, grid voltage Ug = 226 VRMS with a THD of 1.46 % background disttortion. The grid impedance was measured to 1.1 ohms with a series inductance of less then 30 µH.

10 5 0 -5

The grid voltage and current was measured using Tektronix differential probe TP5205 and current probe TCP202 respectively. The data from the scope was acquired on a pc and plotted using Matlab.

-10 -15 -20

The grid current and grid voltage at 3 kW for PI, PR and PR+HC controllers are shown in Fig.14, Fig.15 and Fig. 16.

-25

In Fig. 17, a comparison of the spectrum for PI, PR and PR+HC in the lower frequency region is shown. The levels for IEEE 929 standard are also shown for reference.

0

0.005

0.01

0.015

0.02 tim e[s ec ]

0.025

0.03

0.035

0.04

Fig.16 Experimental results at 3kW. Grid voltage and current with PR+HC controller

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to act as a notch filter at the resonance frequency and thus it can track a high frequency sinusoidal reference without having to increase the switching frequency or adopting a high gain, as it is the case for the classical PI controller.

2 PI P +RE S P +RE S +HC IE E E 92 9

1.8

- [% ]THD =1 2.6 6 5 - [% ]THD =8 .1 6 55 - [% ]THD =5.4028 - [% ]THD =5

1.6

Current harmonic [A]

1.4

Several resonant controllers have been added in order to compensate selcted harmonics without influencing the controller’ s dynamics.

1.2

1

A modest computaion burden of around 40 µs was achieved for PR+HC on a 16-bit fixed-point TMS320F24xx DSP platform due mainly to an original discretization method.

0.8

0.6

0.4

Considerable lower current THD in comparison with the classical stationary PI controller is obtained and compliance with IEEE 929 standard can be achieved.

0.2

0

50

150

250

3 50 F re quency [Hz]

4 50

550

650

REFERENCES

Fig. 17. Experimental results at 3kW (a) grid voltage and current with PR controller; b) The measured grid current harmonic spectrum for PI, PR and PR+HC control strategies and the limits for IEEE 929

[1]

M. Kazmierkowski, R.Krishnan, F.Blaabjerg, Control in Power Electronics. Selected Problems, Academic Press 2002, ISBN 0-12402772-5. [2] C. Cecati, A. Dell'Aquila, M. Liserre and V. G. Monopoli, "Design of Hbridge multilevel active rectifier for traction systems", IEEE Trans. on Ind. Applicat., vol. 39, Sept./Oct. 2003, pp. 1541-1550. [3] X. Yuan, W. Merk, H. Stemmler, J. Allmeling – “Stationary-Frame Generalized Integrators for Current Control of Active Power Filters with Zero Steady-State Error for Current Harmonics of Concern Under Unbalanced and Distorted Operating Conditions” IEEE Trans. on Ind. App., vol. 38, no. 2, Mar./Apr. 2002, pp.523 – 532. [4] P. Mattavelli “A Closed-Loop Selective Harmonic Compensation for Active Filters” IEEE Trans. on Ind. App., vol. 37, no. 1, january/february, 2001, pp. 81 – 89. [5] D. N. Zmood, D. G. Holmes, “Stationary Frame Current Regulation of PWM Inverters with Zero Steady-State Error” IEEE Trans. on Power Electr., vol. 18, no. 3, May 2003, pp. 814 – 822. [6] J. H. R. Enslin and P. J. M. Heskes, “Harmonic interaction between a large number of distributed power inverters and the distributed network”, PESC 2003, pp. 1742-1747. [7] M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an LCLfilter based active rectifier,” IEEE Trans. on Ind. App., vol. 38, no. 2 Sept./Oct. 2001, pp. 299-307. [8] D.G. Holmes and T. Lipo, Pulse Width Modulation for Power Converters : Principles and Practice, 2003, ISBN 0471208140. [9] R. Teodorescu, F. Blaabjerg, M. Liserre and A. Dell’Aquila “A stable three-phase LCL-filter based active rectifier without damping” IAS 2003, pp. 1552 – 1557. [10] H-S.S. Song, R.Keil, P.Mutschler, J.Weem, K.Nam – “Advanced Control Scheme for a Single-Phase PWM Rectifier in Traction Applications” IAS 2003, pp. 1558 –1565.

As it can be seen in Fig. 17, the PI controller fails to comply with the IEEE 929 standard, exhibiting very large harmonics content, especially the 3rd and the 5th with a THD of 12.65%. The steady-state error in amplitude was about 20 % with a phase shift of about 7 deg. Using PR controller without harmonic compensation, a noticeable improvement is being obtained, as it can be seen in Fig. 15 and Fig. 17. The THD was 8.16 % and the steady-state error was negligible. Enabling now also the harmonic compensator (PR+HC), as depicted in Fig. 16 and Fig. 17 the 5-th and the 7-th harmonics are more attenuated and the current THD decreases to 5 %, the limit imposed by the IEEE 929 standard. As the 3rd harmonic is still relative high, accounting for about 4 % of the funemental, it is considered that implementing a deadtime compensation technique can further reduce it. The used deadtime was 500 ns. VI.

CONCLUSIONS

A new control strategy based on P+Resonant (PR) controller and harmonic compensator (HC) has been developed and successfully tested on a DSP-controlled 3kW PV inverter. As it was shown, the gain of PR becomes infinity in a narrow bandwith centered on the resonant frequency and almost null outside the bandwith. This makes the PR controller

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