U2013 Proceedings of Intemational Conference on Modelling, Identification & Control (ICMIC)

Cairo, Egypt, 31 stAug._ 2nd Sept. 2013

Closed-Loop Control of Single Phase Selective Harmonic Elimination PWM Inverter Using Proportional-Resonant

Controller I. Abdel-Qaweel, N. Abdel-Rahiml, H. A. MansourI, T. Dakrorl

I EE Department, Faculty of Engineering at Shoubra, Benha University, Egypt

2 EE Department, Faculty of Engineering, Benha University, Egypt

Abstract- This paper deals with the application of the

selective harmonic elimination technique of a closed-loop

control scheme of single-phase PWM inverter employing

proportional resonant controller. Selective harmonics

elimination (SHE) technique is used to eliminate low order

harmonics with relatively low inverter switching frequency.

The major challenge which faces this technique is solving large

numbers of nonlinear transcendental equations at each

modulation index to generate the power inverter switching

patterns. Proportional resonant controller is employed to

reduce both magnitude and phase errors of the load voltage.

Simulation of the proposed system is reported. The simulation

results show that this system is capable of producing load

voltage with low total harmonic distortion (THD).

Index Terms-Power Inverter, Proportional-Resonant

Controller, Selective Harmonic Elimination, PWM.

I. INTRODUCTION

Selective harmonic elimination pulse width modulation (SHE-PWM) technique is used to generate a predefined PWM pattern at the output of power inverters. A nearly sinusoidal load voltage is obtained from the inverter switching pattern using a second order low pass filter [1], [2]. The switching angles of the inverter are calculated off-line by solving a set of nonlinear transcendental equations. The number of equations depends on the number of harmonics that need to be eliminated. These switching angles are stored in a lookup table which is used for determining ON/OFF intervals of the power switches.

Several research papers have dealt with SHE-PWM technique for single phase and three phase inverters but most of them focused on algorithms used to solve the nonlinear equations only as it is considered the main challenge that faces this technique [3]-[8], [9]. In order to obtain high quality output waveforms at reasonable switching frequency, closed loop operation of the single-phase inverter employing proportional-resonant (PR) controller in the feed-forward path is investigated in this paper. Along with the SHE technique, a PR-controller is used, instead of the traditional PI controller to overcome drawbacks commonly associated

I. Abdel-Qawee is with the Electrical Engineering Department, Benha university (e-mail:islam.ahmed@feng.bu.edu.eg)

N. Abdel-Rahim is with the Electrical Engineering Department, Benha university (e-mail: nasser.abdelrehim@feng.bu.edu.eg).

H. A. Mansour is with the Electrical Engineering Department, Benha university (e-mail: h.abdelaziz@feng.bu.edu.eg).

169

with PI-controllers when the reference signal is sinusoidal. PR-controllers provide theoretical infinite gain in a narrow bandwidth that is centered at a predefined resonance frequency, hence eliminating the steady state error at that frequency and allowing outstanding tracking behavior with sinusoidal reference signals [10]-[ll]. Another advantage associated with the PR controllers is the possibility of implementing certain harmonics compensation without requiring excessive computational resources [12].

The objective of this paper is to achieve high quality output waveform of the single-phase inverter with a reasonable switching frequency. This is achieved using the SHE-PWM technique and the PR- controller in a closed loop control scheme of the single-phase inverter.

MATLAB/SIMULINK package is used to simulate the system. First, the mathematical equations of SHE technique are presented for bipolar (two-level) waveform and then the switching angles are determined. The design of the LC load filter and PR controller are provided. Finally, various waveforms of the proposed system are given.

II. EQUATION OF SHE TECHNIQUE

The output voltage waveform of the single-phase voltage source PWM inverter can be either bipolar, two-level, or unipolar, three level. This paper focuses on investigating the performance of the single-phase bridge inverter using bipolar switching pattern.

In the bipolar PWM technique, the power circuit is shown in Fig.2 and the output voltage is either +Vde or -V de as shown in Fig.l. There are two mathematical models for representing the bipolar SHE-PWM technique depending on whether the number of eliminated harmonics is odd or even [8], [13]. In this work, odd-numbered harmonics (41 harmonics) are eliminated. Hence, the Fourier representation of the inverter output waveform is given by [\3]

Vol!: r-- r-- - . . - I I I I I I I I I I I I I

-Vole - .. I .. - - - - � Q] 0.1 u}_ Fig.!. Bipolar SHE-PWM output voltage waveform

U20l3 Proceedings oflnternational Conference on Modelling, Identification & Control (lCMIC)

Cairo, Egypt, 3lstAug.- 2ndSept. 2013

where: Uk : the switching angles V de : the input supply voltage Vn : nth harmonic of the inverter output voltage N number of switching angles in the first

period.

And

(1 ),

quarter

(2)

In order to eliminate the first 4 I odd harmonics in the inverter output pattern, 42 non-linear equations need to be simultaneously solved. There are several techniques to solve these nonlinear transcendental equations such as Newton -Raphson, Optimization, and Genetic algorithm techniques [1]-[8], [9]. In this paper, Matlab Optimization toolbox using "fsolve" function is used to solve the equations. The magnitude of the fundamental component of the inverter output voltage is determined from (1) by substituting n=l and the equation itself is set to the desired magnitude of the fundamental component of the inverter output voltage. Odd harmonic orders starting from the 3rd and up-to the 8Sth harmonic order are set to zero.

III. DESIGN OF FILTER PARAMETERS

The power circuit of the single phase inverter with an LC load output filter is shown in Fig.2 . In order to produce a load voltage with low total harmonic distortion (THD), a low pass filter must be used to eliminate the low order harmonics and improve the quality of the output voltage. Second order low pass filter is used to achieve this purpose. Fig.2 shows the power inverter with the second order filter (LrCf) interfacing between the inverter and the load. Fig.3 shows the equivalent circuit of that shown in Fig.2.

Fig.2. Power circuit of single phase inverter

Fig.3. Equivalent circuit of single phase inverter

170

Where:

Cf Lf rf RL Vinv: VOUI:

fi Iter capacitor. filter inductor. resistance of the filter inductor. load resistor. inverter output voltage. load output voltage.

Referring to Fig.3, the transfer function of the open-loop voltage gain is expressed as:

v'mJs) v'nv (s)

(3)

rf I As - « -- , and RL» rf. Therefore, (3) can be Lf RIC I

rewritten to be:

Vow (s) v'nv(s)

LrCf

From (4), the magnitude voltage gain is expressed as

Where

(4)

(S)

(6).

The relationship between the magnitude of the inverter output voltage and load voltage is given by (S). The equation shows that the magnitude of the output voltage is affected by two factors: (1) the filter parameters (Lfi Cf), and (2) the load resistance RL. The plot of (S) is shown in FigA. As can be seen from the figure, Eq. (S) is maximum at wf (see (6)) and, for fixed filter parameters, whose magnitude varies with the value of the load resistance. The magnitude of (S) is equal to unity when the frequency of the harmonic components of the inverter output voltage is much lower than wf' The magnitude of (S) is very much reduced at frequencies much higher than wi Typically, the magnitude of (S) at three times wlis -20dB and at large value of (J) the filter gain is nearly equal to zero. Thus, the filter parameters have to be carefully selected such that wI of the selected filter parameters is located far below the first dominant harmonic that appears in the inverter output voltage.

U20l3 Proceedings of International Conference on Modelling, Identification & Control (ICMIC)

Cairo, Egypt, 3lstAug.- 2ndSept. 2013

------I �--: I���;;I I I ------

1 -r ------ :----r- ---------� : J� •••••• ••••...

1 l-k\�, : � � I o���_+��--�����==�==���� mr 0 0.2 0.4 0.6 0.11 Ol (ra�/sec)

FigA.The relationship between magnitude voltage gain and angular frequency at load resistances (RJ =5, R2= 15, and R3=30) Q

A. Harmonic Analysis of Inverter Output Voltage

The output of the SHE-PWM bipolar inverter consists of fundamental component and odd harmonics. The Fourier series representation of the inverter output voltage is given by

en �nJmt) = � sin(m)) + I Vn sin(nm)) 11=3,5

en

=VdcM, sin(mot) + Vdc I Cn sin(nm"t) n=3,5

Where

0)0: angular frequency (rad/sec) of the fundamental component,

Mj: modulation index,

(7)

Cn: normalized value of the nth harmonic component of the inverter output voltage and is given by V nIY dc, and

n : the harmonic order (only odd values).

The order of the first harmonic to appear in the inverter output voltage is the 85th harmonic order. Hence, (7) can be re-written as

�nv(mt) = VdC M i sin(m,,!) + VdcCX5 sin(85m,,!) +�kC87 sin(87m})

B. Selection of the filter Cut-off Frequency

(8)

The selection of the cut-off frequency of the filter depends on the order of the dominant harmonic to appear in the inverter output voltage. In this paper, switching frequency is given by [14].

fsw = (2N + l)f (9)

Where

Isw: inverter switching frequency, and f: fundamental frequency of the inverter output voltage.

The Cut-off frequency of the filter is chosen such that it is one-fifth the frequency of the dominant harmonic in the inverter output voltage. This is to ensure that all undesired harmonics are significantly attenuated at the load filter. Since the frequency of the dominant harmonic is 4250 Hz, hence the filter cut off frequency becomes 850 Hz (5340 rad/sec).

Having determined the cut-off frequency of the filter (see (6)), the filter parameters Lf and Cf can be determined. The optimized values of the filter parameters were found to be

171

Lf =600/lH and Cr60.O/lf.

c. Filter Rating

The selection of the filter inductor is determined using the the peak and RMS values of the inductor current, while the rating of the capacitor is determined by peak voltage and its current RMS value. 1. Inductor Rating

The nth harmonic of the filter input current is given by

Where

v _ xc V i _ = � = " �sin(nm t-() ) n.mv � Iz 1

11 n zn,mv n=l,3 n,inv

(l0).

in .i nv: nth harmonic of inverter output current magnitude,

Vn,inv: nth harmonics of inverter output voltage magnitude,

Zn.inv' nth harmonics input impedance, and

()n : nth harmonics phase shift.

The magnitude of the inverter output voltage (Vn,inv) at nth harmonic component is determined from (1) by setting N=42, and substituting the values of the switching angles read from Fig.8. The Bode plot of the impedances of the loaded filter at the nth harmonic component at various loading conditions is given in Fig.5.

Bode diagram of the input impedance of the loaded filter

I I I IIIII1 I II II § 40 I I I IIIII1 : :::: -� 3

0 -- T - I I T 111 11 ---- - 1- 1 1 11--­� 10 -----r--T T-rTl1n--- ------ --r---ITTT 111 -----+--++-++++-1-1------------+---++-++-

il., ,,' 111'

1 11 1 1111 I 1 1 1 1111

I I 1 1 1 1111 .. I I 1 1 1 1111 = I I 1 1 1 1111 e I I 1 1 1 1111

I I 1 1 1 1111 _11� ••

n'

I IIIII1 I I I IIIII1 I IIIII1 I I I IIIII1

,,'

co (rad/sec)

,,'

Fig.5. Inverter input impedance Zn)nv versus angular frequency at load resistances (Rl =10, R2= 20, R3=30, and R4=40) Q

The peak value of the filter input current can be obtained by maximizing (10) with respect to time as expressed [� Vn_inv . 1

IL,I1JJJX = � -1---

I sm(nm)-()I1 )

n-1,3 Zn inv , m�

And, the RMS value of filter current can be obtained by

2. Capacitor Rating

( I I )

(12)

The magnitude transfer characteristic of the filter can be concluded from (5) to be given by

(13)

So the peak voltage across the filter capacitor can be obtained

U2013 Proceedings of International Conference on Modelling, Identification & Control (ICMIC)

Cairo, Egypt, 3 jstAug.- 2nd Sept. 2013

by

(14)

Where IT!'; I: Filter transfer characteristic at the frequency of the

fundamental component of the inverter output voltage (000,) , and

VUnv: Peak fundamental component of inverter voltage

which can be obtained from (8).

The rated peak voltage of the selected capacitor is chosen to be the next higher standard value which is calculated from (14).

The RMS value of the filter capacitor current is given by

where

(15),

Peak value of nth harmonics capacitor voltage that

can be obtained from (5) at w=nwo .. and Vn,inv= VdcCn , Reactance at nth harmonics which can be given by

(16) where XCI is reactance at fundamental frequency.

(16).

IV. PROPORTIONAL RESONANT CONTROLLER

Proportional resonant (PR) controller is employed in the closed-loop operation of the single phase inverter system when the reference signal is sinusoidal [10], [11]. In this work, the PR controller is employed along with SHE-PWM to construct high quality power supply with reasonable inverter switching frequency. The transfer function of the PR controller is given by [12]:

Where

Kp: Proportional controller gain. KI: Integration controller gain.

(17)

Equation (17) shows that the PR-controller provides very large gain at the resonant frequency; therefore, steady state error is eliminated at this frequency. The proportional controller gain, K1" is chosen such that good transient response and stability margins are achieved. Kp also determines the bandwidth of the outer feedback voltage loop of the system. Kr is chosen such that the magnitude and phase steady-state error are eliminated [10], [12]. The Bode plot of the PR controller at Kp=O, KF10 and 000 =21(50 rad/sec is shown in Fig.6.

172

,� __________ � ___ � __ ��m ____________ ___

is i 0 � !-O i ---------------­,�

, ��. --��--����======----======= � J � � Fr!C!liI'ncy Ir.Ko'�l

Fig.6. Frequency response of PR controller

V. CLOSED LOOP OPERATION

In open-loop operation, the output voltage magnitude is affected by various loading conditions as indicated in FigA.

In some applications, like UPS, the output voltage must maintain a predetermined value over a wide range of loading conditions [15]. Thus, to achieve good load-voltage regulation, closed loop operation of the power inverter is employed. The L C filter and PR controller which were designed in sections III and IV, respectively will be used to obtain sinusoidal load voltage and load current with THO less than 5% and output voltage steady state error less than 1%.

The closed loop system is shown in Fig.7. The principle of operation is as follows. The load voltage is compared with its reference sinusoidal signal to produce the error signal, ev• the ey is multiplied by internal proportional gain (Ky=woxCf) and the output is compared with the actual capacitor current to produce, ec. ec is then fed into the PR-controller to produce the modulation index, M;. The modulation index, Mi , is used to select a predefined switching pattern stored in a look-up table. The switching pattern is then processed by the gate drive circuit (please see Fig.7) and is then used to control the inverter power switches. The inverter switching devices are turned ON/OFF such that the steady state error between the actual load voltage and its reference signal is reduced.

Fig.7. Closed loop control scheme of single phase inverter

VI. SIMULATION RESULTS

In this work, 41 odd-numbered harmonics are eliminated. Hence, there are 42 switching angles. MATLAB optimization toolbox is used to solve these 42 nonlinear

U2013 Proceedings ofTntemational Conference on Modelling, Identitication & Control (JCMIC)

Cairo, Egypt, 3 PtAug.- 2ndSept. 2013

equations. Fig.8 shows the relationship between the modulation index, Mi, and the resulting inverter switching angles.

Inverter output pattern switching angles

Modulation index (Mj)

Fig.S. Switching angles for different modulation indices at N=42

The simulation results of the closed loop operation of the proposed system are presented in this section. The steady state inverter output voltage, load voltage, and load current with resistive load (R=10 Q) are shown in Fig.9. The figure shows that the load voltage is a nearly perfect sinusoidal waveform with low THD. Fast Fourier Transform (FFT) of the inverter output voltage is shown in Fig. IO. The figure confirms that 41 harmonics have been eliminated from the inverter output voltage. The figure also shows the first harmonic component that appears in the inverter output voltage is the 85th harmonic. Fig.11 shows the FFT of the load voltage where the THD of the load voltage was calculated to be 4.54%.

To examine the system dynamic performance, a ±50% step change in the load was simulated. The point at which the step change occurs in the load was chosen such that it corresponds to the point where a peak load current took place. This corresponds to a worst case condition of such step change in the load. Fig.12 shows various waveforms of the proposed system for a 50% step change in the load (i.e. load current changed from 10.7 A to 21.4 A). The Figure shows that the system has the capability of resuming its sinusoidal output load voltage in two msecs. The figure also shows that the load voltage experiences a dip of 25% of the peak value of the load voltage.

Fig.13 shows the load voltage and current waveforms of the proposed system when -50% step change in the load was performed. It can be seen that there is a 13% dip in the load voltage and the system takes O.17msec to regain its steady state conditions.

173

!l �11"11111111111111.�I�I��11 � m m � � � m m w w m

time(see) Output voltage

�l:� � � m � m g � m a a �

time(see)

Fig.9. Output inverter voltage, output load voltage, and output load current for steady state operation

FFT analysis----------------------.

!l 100 c E 80 .. -g 60 = u.. '0 40 C 20 tJ) .. :;

Fundamental (50Hz) = 303.6 , THD= 146.04%

50 100 Harmonic order

150

Fig.IO. FFT of inverter output voltage

11 200

- FFT analysis---------------------;

.:- 100 .. .. c Q) 80 E .. 'tI 60 c :::I

u.. ... 40 0

C 20 tJ) .. ::

0

Fundamental 150Hz) = 304.1 , THD= 4.54%

50 100 Harmonic order

Fig.ll. FFT of load voltage

150 200

U2013 Proceedings of International Conference on Modelling, Identification & Control (ICMIC)

Cairo, Egypt, 31 s'Aug.- 2nd Sept. 2013

(a)

�§ 5"�11111 i 111111111111111111,:I}IIII,lllllllllil�1Il1111111-1III�11111�UI111J-rlrlllll,I,IIII:::: II ... -5W!194 O.�96 O.�98 0:2 O.�02 O.�04 O.�06 O.�08 O.�l O.�12 0.214

time (sec) 350 (b) ---�

1 300 , ,250 �� 0 -- : -----

�5

(l :; , , 0.2 0.205 I 0.21 .... -soo I , , , ,

0.19 0.2 0.21 0.22 0.23 0.24 50 r � time (sec)

(c)

50,---,------,--�����=--,-------rl

� ._� 0 - 21

0.23 0.24 -50 �-(c-).1'c:9------(�1.2=------c:-.:-=------::-:-:----,-:-'.:-::-----�·

Fig.12. 50% step change in the load: (a) output inverter voltage when the load is changed (b) output load voltage (c) Output load current

'"

��18 ------�0.1�9 ----�G*j------�O j�'------G�n�----�o�� �� oo.r-____

-, ______ �----�_,I\b=J====u_'

_n'=(_�

__ -)======�:��

i�1$------��1 �9 ----�Q*2------�02�, ----�Q�n�----�Q��-J lime I""')

Fig.13. -50% step change in the load: (a) output load voltage, and (b) load current

VII. CONCLUSION

In this paper, closed loop operation of a single-phase voltage-source inverter has been investigated with SHE-PWM technique. To achieve low harmonic content in the load voltage, a low pass filter has been employed to interface between the power inverter and load. Methodology of designing the load filter has been presented in the paper. To achieve very low steady-state error and successful operation of the feedback closed loop control of the system, a proportional resonant controller was employed as a regulator in the outer voltage feedback loop. Simulation results of the proposed system were reported. It was shown that the system was capable of producing sinusoidal load voltage with low total harmonic distortion at reasonable switching frequency. The results show that the system has good capability of coping with load dynamics.

174

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