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Dynamics and Control of Three-Phase Four-Leg Inverter
JENICA ILEANA CORCAU
Avionics Division
University of Craiova, Faculty of Electrical Engineering
107 Decebal Blvd., 200440 Craiova, Dolj
ROMANIA
jcorcau@elth.ucv.ro, jcorcau@yahoo.com
http://www.elth.ucv.ro
TEODOR LUCIAN GRIGORIE
Avionics Division
University of Craiova, Faculty of Electrical Engineering
107 Decebal Blvd., 20440 Craiova, Dolj
ROMANIA
lgrigore@elth.ucv.ro
NICOLAE JULA
Military Technical Academy
81-83 G. Cosbuc Blvd., Sector 5, Bucharest
ROMANIA
nicolae.jula@gmail.com
http://www.mta.ro
COSTIN CEPISCA
Electrical Engineering Faculty
University POLITEHNICA of Bucharest
313 Splaiul Independentei, Sector 6, Bucharest
ROMANIA
costin@wing. ro
www.electro.pub.ro
NICOLAE POPOVICIU
Hyperion University
169 Calea Calarasilor, Sector 3, Bucharest
ROMANIA
nicolae.popoviciu@yahoo.com
Abstract: In this paper the modelling and analysis of three-phase four-leg inverter are presented. Also, this paper
presents two strategies of control and their limitations. The frequency characteristics of the system are useful for
determining its control strategies. Initially a PI compensator is used. Finally, the control design strategy using
4zeros/5poles is chosen as optimum strategy. The system operation is verified for load variation from 1% to 100%
rated load. The closed loop performances of the inverter are verified using the Matlab/Simulink software package.
Key-Words: - Three-phase four-leg, Uninterruptible Power Supply, Power distribution system, control strategy.
The goal of the three-phase four-leg inverter is to
maintain the desired sinusoidal output voltage waveform
over all loading conditions and transients.
This is ideal for applications like data communication,
1 Introduction
The main feature of a three phase inverter, with an
additional neutral leg, is the ability to deal with
unbalance load in a standalone power supply system [1].
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industrial automation, military equipment, which require
high performance uninterruptible power supply (UPS).
Figure 1 presents the scheme of the switched inverter
model with four conductors [2].
The fourth conductor (N) permits the control of the
neutral current. For the three-phase inverters, if the load
requires us to connect the null, we utilize the null point
of the capacitor filter of the dc connection [2], [3].
In this case, the unbalanced loads or the non-linear
single phase loads generate null currents and distortions
when passing through zero. When using inverters with
four terminals (A, B, C and N), the control over the null
is obtained and the zero distortions caused by the
inverter control system are reduced.
2 Power stage modelling of the threephase four-leg inverter
Generally, in the literature, three levels of models are
available for inverter system: detailed model, behavioral
model and reduced order model.
The detailed model is based on equations derived from
the system structure and electrical circuit [4].
An example of a detailed model for a three-phase fourleg inverter would be a model that exactly describes all
voltage and current waveforms in circuit produced by the
switching action of the semiconductor switches. A
detailed model requires a very long time for numerical
simulation.
So, a detailed model often includes discrete and
discontinuous functions and it cannot be used for control
loop design based on linearization and frequency domain
analysis techniques. For this reason, a detailed model is
used only in simulations when a detailed study of the
system operation is required.
The behavioral model is derived from the detailed model
by time averaging of high frequency periodical
waveforms such as switching waveforms [4].
The behavioral model of inverter is called “average
model”.
The reduced order model is obtained from the behavioral
model by its linearization at an equilibrium point. So, it
is often called “linearized model”.
A large signal model is reduced to a small signal model,
which is available in the vicinity of this operating point
only.
The Matlab/Simulink software package can performs
automatically this linearization using the “linmod”
command. The average model of the three-phase fourleg inverter in steady-states coordinates is shown in
figure 3 ([5]).
+
Lfilter
A
B
C
Vin
Cfilter
-
Ln
N
Fig.1 The scheme of the switched inverter model with
four conductors
In figure 2 ([2]) is presented the three-phase four-leg
inverter used in the shipboard dc PDS (Power
Distribution System) to provide secondary AC power
distribution. It can be used to supply utility power for
combat equipments, radar and other critical electronic
loads [2].
Fig. 2 The diagram of Power Distribution System (PDS)
Fig. 3 The average model of inverter
The PDS comprises of a front-end boost rectifier feeding
the dc link and various loads connected to the dc link
which includes a three-phase four-leg inverter and a dc
to dc power converter.
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The model is obtained replacing the switches with
controlled voltage and current sources.
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The inverter output voltage and input current can be
represented as [5]
Va n   d an 

  
(1)
Vb n  =  d bn  ⋅ V g ,
V   d cn 
 cn 
The Bode diagrams for system open loop control, in the
light load case, are presented in figure 4. Figure 5
depicts the same characteristics for heavy load case.
0 0
0 0
0 0
I a 
(2)
I p = [d an d bn d cn ] ⋅  I b ,
 I c 
where Vin (i=a0, b0, c0) are the inverter outputs voltages
(line-neutral), Ia, Ib, Ic are line currents, and din (i=a, b,
c) are the line-to-neutral duty cycles.
The duty cycles din (i=a, b, c) are controlled to produce
sinusoidal voltages at the filter output as a function of
the load; the load could be light, heavy, unbalanced or
non-linear.
The system requirements can be represented as
 cos(ωt ) 
Van 
V  = V cos(ωt − 120 o ) ,
(3)
m

 bn 
o
Vcn 
cos(ωt + 120 )
Fig. 4 The Bode diagrams for system open loop control
in the light load case
where Vin (i=a, b, c) is the output load voltage and Vm
the rated output voltage.
To produce the desired sinusoidal output voltages, the
steady-state duty cycles are time-varying and sinusoidal.
But, to apply classical control techniques a DC operating
point is necessary. As a consequence, to get the power
stage model in rotating coordinates the transformation T
is applied.
In this way, the steady-state output load voltages and the
duty cycles are dc quantities being described as follows
Vd  Vm 
V  =  0  ,
(4)
 q  
V0   0 
and
(1 − ω 2 LC )Vm / V g 
 Dd  

ωLI d
D  = 
.
(5)
 q 
Vg

 D0  

0


The parameters on three-phase four-leg inverter are [2]:
VL-N = 277 Vrms, L = 333 µH, C = 100µF, VDC = 800
V, P0 = 150 kW, fs = 20 kHz, f = 60 Hz.
The load of the inverter is represented as a disturbance
and is considered resistive. The output varies from 100%
to 1% rated power, so load varies from 1.53 Ω to 153 Ω.
The average large signal model is simulated using the
Matlab/Simulink software package. In order to obtain
the small signal model, the average model is processed
with “linmod” Matlab command around an operating
point, in the small perturbations hypothesis; the small
signal model is necessary in the control design phase.
ISSN: 1790-5117
Fig. 5 The Bode diagrams for system open loop control
in heavy load case
Neglecting the coupling in the power stage, the open
loop transfer function is approximated as
Vg
V
(6)
H (s) = d ≈
d d 1 + (L / R ) ⋅ s + (L ⋅ C ) ⋅ s 2
This transfer function can be replaced with a seconddegree low-pass filter by the form
Vg
(7)
H (s) =
,
1 + s /(Q ⋅ ω 0 ) + (s / ω 0 ) 2
1
(8)
,
ω0 =
L⋅C
Q = R ⋅ C / L,
28
(9)
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ω0 =
1
The Bode diagrams for the system using the PI controller
in the heavy load case are presented in figure 7. Also, it
can be observed that the amplitude-frequency
characteristic, presents a peak between 103 rad/sec and
104 rad/sec.
Closing the loop gain below the resonance yields low
system bandwidth and this results in a poor transient
response. Another possibility is to close the system loop
after resonance [8], [9].
The finally chosen compensator architecture is
4zeros/5poles (design II), with zeros placed at z1 = 780
Hz, z2 = 800 Hz, z3 = 820 Hz, z4 = 840 Hz, and poles p1 =
1 kHz, p2 = 2.5 kHz, p3 = 10 kHz, p4 = 10 Hz, p5 = 0 Hz.
The Bode diagrams for this control architecture are
presented in figure 8. The Bode diagrams for the same
control architecture but with heavy load are shown in
figure 9.
(10)
.
L⋅C
ω0 is the resonant frequency (approximately 872 Hz),
and Q is the circuit quality factor. According with the
equation (9), Q is a function of the resistive load, so, Q is
larger when R is larger.
It can be observed that the amplitude-frequency
characteristic depicted in figure 5, presents a peak
between 103 rad/sec and 104 rad/sec. The controller is
designed to satisfy the system performance for both
heavy and light load cases.
3 Control loop design
In figure 6 is presented the closed loop model of the
inverter implemented in Matlab/Simulink. The output
filter capacitor voltages are sensed and transformed from
stationary to rotating coordinates. These are used as
feedback signals for d-q-0 channels voltage
compensators.
The voltage loop gain Tv is
V
(11)
Tv (s) = d ⋅ H v (s).
dd
The three-phase four-leg inverter is a multi loop and
highly coupled multi-variable system [7]. The d-q-0
channels compensators are designed independently with
the other 2 channels assumed as open (q and 0 channels).
In the first stage a PI compensator was used (design I)
([6])
K
K 
s 
H v (s) = K p + i = i 1 +
(12)
,
s
s  K i / K p 
where the coefficients are chosen with the numerical
values Kp=1.25·10-5 and Ki=0.07.
1
1
dd
1
iLd
Vg
Vd
L.s
Gain 2
Gain
Fig. 7 The Bode diagrams for the system using the PI
controller in the heavy load case
C.s
Gain 3
To Workspace
-K-
Gain 1
-K-
1
Out 1
-K-
dq0
Gain 9
Transfer Fcn
abc
sin_cos
-K-
numc (s)
Freq
Gain 6
denc (s)
-K-
Step
-K-
Gain 5
dq 0_to_abc
Transformation
Sin_Cos
-Kwt
Gain 8
Vg
2
dq
Vabc
To Workspace 3
1
Gain 4
L.s
1
2
Out 2
C.s
iLq
Gain 7
Discrete
Virtual PLL
To Workspace1
Vq
-K-
Constant
0
num (s)
3
d0
1
1
Vg
den (s)
(L+3*Ln )s
Gain 10
C.s
3
Out 3
Gain 11
-K-
V0
To Workspace2
Fig. 6 The closed loop model of the inverter
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Figure 10 shows the plots in time of Vd and Vq with light
load and figure 11 - with heavy load. The compensator is
designed in rotating coordinates and hence it controls Vd
and Vq. The final output voltage is derived from them
after applying the transformation T.
In figure 12 are presented the plots in time of Va, Vb and
Vc. The V0 turns out to be 0 in these simulations as the
load is taken as the balanced load.
Fig. 8 The Bode diagrams for compensator with
4zeros/5poles
Fig. 11 Vd and Vq for heavy load case
Fig. 9 The Bode diagrams for compensator with
4zeros/5poles and heavy load
Fig. 12 The plot in time of Va, Vb and Vc.
Control design II is superior to design I as the cross over
frequency in case II is much higher than in case I.
Dynamic simulations of inverter employing strategy I
shows that the settling time for an output load voltage is
higher.
4 Conclusions
In this paper, the modeling and control of three-phase
four-leg inverter is presented. Two control strategies are
presented in this paper. The performances of inverter are
verified through simulation using the Matlab/Simulink
software package.
As is shown in figure 5, for system open loop control in
heavy load case there is a large peak. Also, the presence
Fig. 10 Vd and Vq for light load case
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[4] K., Louganski, Modeling and analysis of a dc power
distribution system in 21st century air lifters. Master
of Science in Electrical Engineering, 1999.
[5] R., Zhang, High performance power converter
system for non-linear and unbalanced load source.
Dissertation, Virginia Tech, 1998.
[6] M.N., Rashid, Power Electronics Circuits, Devices
and Application. Second Edition, 1992.
[7] Y., Han, A.G., Yao, L.D., Zhou, C., Chen, Design
and Experimental Investigation of a Three-Phase
APF based on Feed-Forward plus Feedback
Control., WSEAS TRANSACTIONS on POWER
SYSTEMS, Issue 2, Volume 3, February 2008.
[8] N.Y.A., Shammas, S., Eio, D., Chamund,
Semiconductor Devices and Their Use in Power
Electronic Applications, WSEAS TRANSACTIONS
on POWER SYSTEMS, Issue 4, Volume 3, April
2008.
[9] Y., HAN, M.G., Yao, L.D., Zhou, C., Chen,
Harmonic Mitigation of Residential Distribution
System using a Novel Hybrid Active Power Filter,
WSEAS TRANSACTIONS on POWER SYSTEMS,
Issue 12, Volume 2, December 2007.
of a similar peak can be observed on the amplitudefrequency characteristic for the system using the PI
controller in the heavy load case (figure 7).
So, control design II (4zeros/5poles compensator) is
superior to design I (PI compensator) as the cross over
frequency in case II is higher than case I (see figures 8
and 9). Dynamic simulations of inverter employing
strategy I shows that the settling time for an output load
voltage is higher.
References:
[1] I., Aron, J., Corcau, Modeling and analysis of a dc
power distribution system. The 30th Session of
scientific presentation “Modern technologies in the
XXI Century”, Bucuresti, 06-07 November 2003, pp.
9-15, CD.
[2] G.S., Thandi, Modeling Control and Stability of a
PEBB based dc distribution power System. Master of
Science in Electrical Engineering, 1997.
[3] U. N., Jensen, A new control method for 400Hz
Ground Power Units for Aircraft. IEEE Transactions
on Industry Applications, vol. 36, no. 1, 2000, pp.
180-187.
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