Use of Control Based on Passivity to Mitigate the ...

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Use of Control Based on Passivity to Mitigate the Harmonic Distortion Level of Inverters A. F. Cupertino, L. P. Carlette, F. Perez, J. T. Resende, S. I. Seleme Júnior, Member IEEE and H. A. Pereira, Member IEEE

Abstract – The insertion of distributed generation systems has growth in the last years. The use of switching converters can produce voltage and current harmonics in the point of common coupling (PCC). In this context, this work presents the use of the passivity-based control in a single-phase inverter connected to the grid. The performance of this technique is compared with the conventional proportional integral controllers and with the multi-resonant controller when there is a high distortion in the PCC voltage. Simulation results show that the passivity-based control is a good technique in this situation, due to its high capability to reject disturbances. Index Terms – Distributed Generation Systems, Harmonic Mitigation, Passivity-Based Control.

I. INTRODUCTION

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OWADAYS it can be observed a growth in the number of distributed generation (DG) systems. This technology is one of many factors that base the smart-grid philosophy. The insertion of these systems has some good points like an increase in the power system reliability, a reduction in transmission investments and an on-demand power quality of supply [1]. However, the DG systems which use switching converters are responsible to produce harmonics on the grid. The study of the harmonic distortion in the injected current by inverters is studied in many of works in literature [2], [3], [4]. Most of them propose design of filters, studies of new synchronism or new PWM techniques. In [5], is presented the effect of the control strategy applied in a single phase inverter using a LC filter, when this is connected to distorted grid. This reference proposes a multiresonant controller to reduce the current harmonics generated by the inverter. This technique showed good results but it is necessary to know what harmonics exist on the grid. Other works propose the use of non-linear techniques due to their high capability to reject disturbances [3]. One interesting non-linear technique is the passivity-based control that has been presented good results for DC/DC converters and three phase inverters [4], [6], [7]. The authors would like to thank FAPEMIG, CAPES and CNPQ by financial support. Allan Fagner Cupertino, Luan Peterle Carlette, Filipe Perez, José Tarcísio de Resende and Heverton Augusto Pereira are with the Department of Electrical Engineering, Universidade Federal de Viçosa, Viçosa, Brazil (emails: {allan.cupertino, luan.carlette, filipe.perez, resende, heverton.pereira}@ufv.br. Seleme Isaac Seleme Júnior is with the Department of Electrical Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil (email: [email protected]).

In this context, this work presents the application of the passivity-based control in a single-phase inverter connected to the grid. It will be compared the performance of the passivity based control with the conventional proportional integral controllers and with the multi-resonant controller when there is a high distortion level in the PCC voltage. II. METHODOLOGY The topology of the controlled single phase inverter studied in this work is presented in Fig. 1. Generally, the control of this structure uses two loops: the external loop, slower, controls the DC bus voltage and calculates the current reference. The control uses the variables in the natural
 reference frame. In this case, the DC bus voltage calculates the current wave pick. This value is multiplied by an unitary sinusoidal wave in phase with the grid voltage (obtained by the synchronism technique). It can be observed that (due to the objective of this work) the dc bus was considered constant for simplification. Only the current loop is analyzed. The parameters of the studied system are presented in Table I. The filter inductance was designed using the methodology proposed by [8].

Fig. 1. Single phase inverter connected to the grid. TABLE I

Parameters of the studied system. Parameter V (DC bus voltage) V (RMS grid voltage) V (magnitude of the carrier waveform) L (filter inductance) r (filter resistance) L (grid inductance) r (grid resistance) P (rated power)

Value 300 V 127 V 5V 7.3 mH 0.1 Ω 16.84 µH 10.6 µΩ 900 W

2 1.5 grid voltage (p.u.) PLL response 1

Amplitude (V)

A. Synchronism technique To obtain the current reference in phase with the grid voltage, it is used a single-phase PLL (phase locked loop) based on second order generalized integrator (SOGI), proposed by [9]. In this PLL, the SOGI filters and generates signals s in quadrature. After, it is used the Park’s transform to obtain the stationary dq reference frame components of the voltage. On these components the SRF-PLL is applied to obtain the voltage angle. The block diagram of the SOGI is shown in Fig. 2.

0.5

0

-0.5

-1

-1.5

0

0.005

0.01

0.015 Time (s)

0.02

0.025

0.03

Fig. 4. Transient response of the PLL.

Fig. 2.. SOGI blocks diagram [10].

1.5 grid voltage (p.u.) PLL response

The transfer functions of SOGI are:

Vs 

v  s k ω s   vs s  k ω s  ω  

Qs 

(1)



qv s kω   vs s  k ω s  ω 

(2)

Reference [10] showed that a critically-damped critically response is achieved when k  √2 . This value of gain results an interesting selection in terms of stabilization time and overshoot limitation. The frequency ω is equal to the grid fundamental frequency ω . The complete block diagram of the PLL is presented in Fig. 3.

Amplitude (V)

1

0.5

0

-0.5

-1

-1.5

0.08

0.09

0.1 Time (s)

0.11

0.12

0.13

Fig. 5.. Performance of the PLL for a distorted grid condition.

integral control B. Proportional-integral The first control technique analyzed in this work is the Proportional-integral integral controller (PI). The block diagram of a PI controller is presented in Fig. 6. The transfer function %&' is given by: %&' (  )# 

) (

(3)

Fig. 3. Synchronism system [9]. [9]

In this work was used: k  √2; ω  2" 60; k #  6.3 and k   3600. Using these values,, the PLL was simulated to analyze the transient response and its performance when there is harmonic distortion in the measured voltage. The results are presented in Fig. 4 and Fig. 5. 5 It can be observed that the PLL spends less than one cycle to synchronize with the grid voltage. Besides, when en there is a high distortion on the grid voltage (limits of the Brazilian standards) the SOGI is able to filter the signal and obtain the reference signal with a smaller harmonic distortion.

Fig. 6.. PI control strategy.

It must be observed that the reference value *+∗ is a sinusoidal wave. In this situation, the conventional techniques of design generally do not give good results. Fig. 7 shows the amplitude and phase relative errors in steady state, in function of the proportional and integral gains. It can be observed that the va values )# - 2 and ) - 7000 guarantee that the amplitude error is less than 1 % and the

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phase error is less than 0.1 %. For this reason, it was choose )#  3 and )  7000.

20

iL

15

iL

*

Current (A)

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

0.01

0.02

0.03

0.04 0.05 Time (s)

0.06

0.07

0.08

(a) 0.115 kp = 2

0.105 phase error (%)

Fig. 9.. System response for variation in the reference wave using PI.

kp = 1

0.11

C. Proportional-resonant resonant control modeling and design The second control technique presented in this work is the Proportional-multi-resonant resonant controller (PR) proposed by [5]. The block diagram of a PR controller is presented in Fig. 10.

kp = 3

0.1

kp = 4

0.095

kp = 5

0.09 0.085 0.08 0.075 0.07 0.065

0

2000

4000

6000

8000

10000

ki

Fig. 10.. PR control strategy.

(b) Fig. 7. Amplitude (a) and phase (b) relative errors in function of the PI controller gains.

The transfer function %&0 is given by:

Phase (deg)

Magnitude (dB)

2 ) 12 ( 2 )6 12 ( The bode diagram of the plant and the compensate plant is % (  )#    presented in Fig. 8.. It can be observed that tha the phase margin is &0 (  2 12 (  13 (   2 12 (  513  2 )44 12 ( positive and the gain margin is infinity, ensuring the stability   in closed loop. (4) (  2 12 (  713  2 )33 Using these gains, the response of the system was tested as 33 12 (   (  2 12 ((  1113  shown in Fig. 9.. The amplitude of the reference wave was changed in 30 ms. It is possible to see a fast response of the controller and a small error in steady state. where 13 is the fundamental frequency in ;
12 > 15). The last three terms of (44) are resonant controllers at the 100 plant 5th, 7th, andd 11th harmonic and they were ad added to mitigate plant+compensator harmonics generated by grid distortion. 50 The he parameters of the PR controller were chosen using a 0 similar ilar methodology of the PI. It was defined ω?  10 rad/s and k 56  k54  k 533  0.5 k k 5 [5]. Using these considerations, -50 it was possible to obtain the curves of Fig. 11. The values 0 k8  2 and k 5  40 guarantee that the amplitude error is less -45 than 1 % and the phase error is less than 0.1 %. The bode diagram of the open loop transfer function is presented in Fig. -90 12. -135 -180 -1 10

10

0

1

2

10 10 Frequency (Hz)

10

3

Fig. 8. Open loop bode diagram for a PI compensator. compensator

10

4

4 20

iL

15

iL

*

Current (A)

10 5 0 -5 -10 -15 -20

(a) 0.076

0

0.04 0.05 0.06 0.07 0.08 Time (s) Fig. 13.. System response for variation in the reference wave using PR.

0.074 0.072 phase error (%)

0.068 kp = 1

0.064

kp = 2

0.062

kp = 3

0.06

kp = 4

0.058

kp = 5

0.056

0

20

40

60

80

v5 A B LC zEE B rC z  VIJJ

(b) Fig. 11.. Amplitude (a) and phase (b) relative errors in function of the PR controller gains. Bode Diagram plant plant+compensator

Magnitude (dB)

20

(5)

where z  iG . The vector of average dynamic error is defined by zH  z B z , where z  i∗G . So, doing z  zH  z it can be obtained that: LC zHE  rC zH  v5 A B VIJJ B LC zE   rC z 

0

(6)

The design of the PBC consists in modifying the system energy by adding damping through the dissipative structure. This modification is accomplished through the addition, in closed loop, of a dissipative term that emulates a resistor connected in series with the inductor, ind denoted by R3 [11]. The addition of the dissipative term is:

-20 -40 -60 0

Phase (deg)

0.03

100

ki

40

0.02

D. Passivity based control modeling and design The control by passivity applied to the dynamic system is based on energy functions. From this technique derives a control law allowing ing the plant to store less energy than it absorbs. This approach is valid for a wide range of operation and large signal stability is assured [11]. The Euler-Lagrange Lagrange model of the single-phase single inverter is given as:

0.07

0.066

0.01

-45

R ? zH  LrC  R3 MMzH ⇔ rC  R ? B R3

(7)

-90

Using (7), it is possible to verify the following followin change in the dynamic average error equation:

-135 -180 -1 10

10

0

1

10 10 Frequency (Hz)

2

3

10

10

4

Fig. 12.. Open loop bode diagram for a PR compensator.

The response of the system was simulated and the results are shown in Fig. 13. It is possible to see a fast response of the controller and a small error in steady state. The overshoot for this controller was smaller if compared with the PI controller.

LC zHE  R ? zH  v5 A B VIJJ B LC zE   rC z B R3 zH

(8)

The energy adjustment of the system is obtained doing: LC zHE  R ? zH  v5 A B VIJJ B LLC zE   rC z B R3 zHM  0

(9)

The desired energy in terms of the error can be modeled by H : 1 H  L L zH  2 C

(10)

H is a Lyapunov candidate for (9). The time derivative of (10) along the paths (9) results in: 1 H  LC zH  ⇒ HE  LC zH zHE  Bα H R 0 2 where α is strictly positive and constant.

(11)

5

With the Lyapunov’s theorem satisfied we can guarantee (10). Accomplishing some algebraic manipulations, the result is:

200 150 100

(12)

V TV .  Fig. 14 shows the result for the PBC controller using R3  400. It can be observed a fast response and a smaller overshoot if compared with the other controllers. where G5 A 

20

iL

15

iL

50

Voltage (V)

VIJJ  LLC zE   rC z B R3 zHM v∗  G5 A

0 -50 -100 -150 -200 0.08

*

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

0.11

0.115

0.12

PI controller 15

10

5

0 Current (A)

Current (A)

10

5

-5 -10

0

-5

-15 -10

-20 0

0.01

0.02

0.03

0.04 0.05 Time (s)

0.06

0.07

0.08 -15 0.08

0.085

0.09

0.095

0.1

0.105

Fig. 14. System response for variation in the reference wave using PBC. PR controller 15

III. RESULTS

10

Current (A)

5

0

-5

-10

-15 0.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.11

0.115

0.12

PBC controller 15

10

5 Current (A)

It was simulated in the software Matlab/Simulink the single phase inverter using the parameters of the TABLE I. The controllers performance was tested in a grid including 7.5% 5th, 6.5 % 7th, 4.5 % 11th, 2.5 % 17th and 2% 19th harmonics. This is the worst case provided by the Brazilian standards [12]. Fig. 15 presents the results for the PI, the PR and the passivity based controller. The TABLE II shows the current THD in the steady state and the THD when there is distortion on the grid. Fig. 16 presents the current harmonic spectrum for the three presented techniques. It can be observed that PR controller had a better response than the PI controller for the 5th, 7th and 11th harmonics and a smaller THD. The multi-resonances reduce significantly the amplitude of the harmonics. The passivity based control presented a better response than the PI controller. This fact can be justified by the non-linear characteristic of the controller that reduces the influence of the harmonics. It can be seen that the THD for this technique change a little but the range is smaller than the other controllers. Although the PR and PBC harmonic spectra were similar, the PBC presents some advantages because it is not tuned in a specific frequency. In a real power system, the harmonic spectrum change during the day, so the passivity based control will have a better performance.

0

-5

-10

-15 0.08

0.085

0.09

0.095

0.1 0.105 Time (s)

0.12

Fig. 15. Effect of the PCC voltage harmonic distortion in the injected current for the 3 studied controllers.

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IV. CONCLUSIONS TABLE II VALUE OF CURRENT THD IN FUNCTION OF THE GRID SITUATION

Harmonic Amplitude (%)

Harmonic Amplitude(%)

Controller PI PR PBC

Non-distorted grid 0.18 % 0.22 % 0.11 %

Distorted grid 1.91 % 1.35 % 1.13 %

Non distorted grid 1.5 PI PR PBC

1 0.5 0

0

5

10

15

20

Distorted grid 1.5 PI PR PBC

1 0.5 0

V. REFERENCES [1] SIMÕES, M. et al. A Comparison of Smart Grid Technologies and Progresses in Europe and the U.S. IEEE Transactions on Industry Applications, v. 48, p. 1154-1162, July 2012. [2] CHEN, C.-L. et al. Design of Parallel Inverters for Smooth Mode Transfer Microgrid Applications. IEEE Transactions on Power Electronics, v. 25, p. 6-15, January 2010. [3] MEZA, C. et al. Lyapunov-Based Control Scheme for Single-Phase Grid-Connected PV Central Inverters. IEEE Transactions on Control Systems Technology, v. 20, p. 520-529, March 2012.

10 15 20 Harmonic order Fig. 16. Comparison of the current harmonic spectrum for the three studied controllers.

0

5

Fig. 16 shows that when the grid is distorted harmonic components appear in the injected current which order is different from the voltage harmonics. To understand this, it is shown in Fig. 17 the harmonic spectrum of the reference current calculated by the PLL. It can be observed that appears components of 3th, 9th, 13th and 15th harmonics in the reference current due to the SOGI. The passivity based controller had a better performance for this harmonics, which contributed for its less value of current THD. 1.5 non distorted grid distorted grid Harmonic Amplitude (%)

In this paper, three techniques to mitigate the grid current harmonics injected by inverters have been explained. The presence of voltage harmonics on the grid affects the performance of these three controllers. In this situation the passivity-based control has a better response than the PI controller, reducing the grid current harmonics. In a real power system, the use of the passivity based control will have a better performance than the multi-resonant controller because it is not necessary to tune the PBC control in harmonics that exists on the grid. It was also observed that the performance of the PLL has an influence in the results because sometimes it produces harmonics in the current reference signal.

[5] WANG, F. et al. Modeling and Analysis of Grid Harmonic Distortion Impact of Aggregated DG Inverters. IEEE Transactions on Power Electronics, v. 26, p. 786-797, March 2011. [6] BECHERIF, M.; AYAD, M. Y.; ABOUBOU, A. Hybridization of Solar Panel and Batteries for Street Lighting by Passity Based Control. IEEE International Energy Conference, Al Manamah, p. 664-669, 2010. [7] ORTEGA, R. et al. Passivity based Control of Euler Lagrange Systems: Mechanical, Electrical and Electromechanical Applications. [S.l.]: Springer-Verlag, 1998. [8] PONNALURI, S.; KRISHNAMURTHY, V.; KANETKAR, V. Generalized System Design & Analysis of PWM based Power Electronic Converters. Industry Applications Conference, v. 3, p. 1972-1979, 2000. [9] CIOBORATU, M.; TEODORESCU, R.; BLAABJERG, F. A new single-phase PLL structure base on second order generalized integrator. PESC, 2006. [10] RODRÍGUEZ, P. et al. New Positive-sequence Voltage Detector for Grid Synchronization of Power Converters under Faulty Grid Conditions. Proceedings of Power Electronics, June 2006. 1 - 7. [11] CUPERTINO, A. F. et al. A Grid-Connected Photovoltaic System with a Maximum Power Point Tracker using Passivity-Based Control applied in a Boost Converter. INDUSCON, Fortaleza, November 2012.

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[4] PING, Q.; BING, X. Passivity-Based Control Strategies of Doubly Fed Induction Wind Power Generator Systems. 2nd International Conference on Information Science and Engineering, Hangzhou, Dezembro 2010.

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10 15 20 Harmonic order Fig. 17. Harmonic spectrum of the reference calculated by the PLL.

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BIOGRAPHIES Allan Fagner Cupertino was born in Visconde do Rio Branco, Brazil. He is student of Electrical Engineering at Federal University of Viçosa, Viçosa, Brazil. Currently is integrant of GESEP, where develop works about power electronics applied in renewable energy systems. His research interests include solar photovoltaic, wind energy, control applied on power electronics and grid integration of dispersed generation. Luan Peterle Carlette was born in Cachoeiro de Itapemirim, Brazil. He is student of Electrical Engineering at Federal University of Viçosa, Viçosa, Brazil. He works with Power Systems, especially with photovoltaic energy and control applied to converters.

Filipe Perez was born in Uberaba-MG, Brazil. He joined the electrical engineering course at the Federal University of Viçosa in 2008. He is a member of GESEP, working in the area of power electronics system with emphasis on solar photovoltaic. Interests in the area of control in power electronics as converters and inverters.

José Tarcísio de Resende received M.S. degrees in electrical engineering from the Federal University of Itajubá (UNIFEI), Itajubá, Brazil, in 1994, and P.H. degree in electrical engineering from the Federal University of Uberlândia (UFU), in 1999. He is currently Professor at Federal University of Viçosa, Brazil. His research interests include modeling of electric machines, power systems and renewable energy. Seleme Isaac Seleme Jr. received the B.S. degree in electrical engineering from the Escola Politecnica (USP), Sao Paulo, Brazil, in 1977, the M.S. degree in electrical engineering from the Federal University of Santa Catarina, Florianópolis, Brazil, in 1985, and the Ph.D. degree in control and automation from the Institut National Polytechnique de Grenoble (INPG),Grenoble, France, in 1994. He spent a sabbatical leave with the Power Electronics Group, University of California, Berkeley, in 2002. He is currently an Associate Professor with the Department of Electronic Engineering, Federal University of Minas Gerais, Belo Horizonte, Brazil. His research interests are electrical drives, control applied to power electrics and electromechanic systems. Heverton Augusto Pereira (M’12) received the B.S. degree in electrical engineering from the Universidade Federal de Viçosa (UFV), Brazil, in 2007, the M.S. degree in electrical engineering from the Universidade Estadual de Campinas (UNICAMP), Brazil, in 2009, and currently is Ph.D. student from the Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, Brazil.

Since 2009 he has been with the Department of Electric Engineering, UFV, Brazil. His research interests are wind power, solar energy and power quality.