Novel GSSA Modeling and Control of High Power
Inverters for Modern Aircraft Electric Power Systems
Hadi Ebrahimi, Hassan El-Kishky*, Justin Erdle, Mohammed Biswass, and Melvin Robinson
The University of Texas at Tyler
3900 University Blvd.
Tyler, TX75799
Abstract— Growing complexity and nonlinear structure of
modern aircraft electric power systems with extensive nonlinear
loading made available state-space averaging models inadequate
for accurate analysis and characterization of such systems. In
this paper, the Generalized State-Space Averaging (GSSA) model
recently developed by the authors is applied for the modeling,
control, and characterization of high power inverters in
advanced aircraft electric power systems. The proposed GSSA
model has been applied to derive the corresponding averaged
model of the reduced-order system for the voltage source
inverter, while taking into consideration the interaction with the
entire aircraft electric power system. An accurate model of the
switching function approximations has been derived by
developing average duty cycle intervals over several regions of
the PWM switching pulses. The first order approximations of the
modulated signals are subsequently generated and the inverter’s
input and output profiles have been captured according to the
GSSA model. Moreover, the proposed approximated model can
be employed to generate real-time control signals using available
hardware. Furthermore, the developed model and the results
obtained are presented and discussed.
Keywords— SPWM, Switching Function, Control Signal
I. INTRODUCTION
Growing complexity and the extensive nonlinear structure of
modern aircraft electric power systems with extensive
nonlinear loading made available state-space averaging
models inadequate for accurate analysis and characterization
of such systems [1-9, 13-18]. Switched-mode power suppliers
including 6 and 12-pulse inverters are widely used in a long
list of applications, including the more electric aircrafts
(MEA). Sinusoidal pulse-width modulation (SPWM)
switching method is widely used particularly in the case of
AC-side control. In a conventional hardware modulator, the
SPWM waveforms are generated by comparing the sine
reference wave with the triangular carrier wave through the
"natural sampling" process. As the linear SPWM region is
exceeded into the transition region, the harmonic quality of the
waves significantly deteriorates with the introduction of the
lower-order harmonics [4,5]. Both output voltage and
operating frequency of the high power inverters in advanced
aircraft electric power systems (AAEPS) must be accurately
regulated to satisfy all applicable standards [13-17].
A few models can be used to generate the switching function
in an attempt to reduce the complexity of real-time analysis
and characterization of the switching function associated with
the inverter. Yet, it is highly desirable to develop an
approximation model that can be used to simplify the control
and performance characterization of such a system. In this
study, a new method of switching function generation and
inverter control has been proposed based on the GSSA model
developed by the authors. In this method, the SPWM discrete
signal is approximated with clock pulse modulated signals.
This assumption can be realized based on linearization of the
sine function over several time intervals.
In this paper, the Generalized State-Space Averaging
(GSSA) model recently developed by the authors is applied
for the modeling, control, and characterization of high power
inverters in advanced aircraft electric power systems. The
proposed GSSA model has been applied to derive the
corresponding averaged model of the reduced-order system
for the voltage source inverter, while taking into consideration
the interaction with the entire aircraft electric power system.
An accurate model of the switching function approximations
has been derived by developing average duty cycle intervals
over several regions of the PWM switching pulses. The
developed models as well as results and discussion and
general conclusions are presented in subsequent sections.
II. SYSTEM CONFIGURATION AND MODELING
Figure 1 below shows the model used in this study for a 12pulse high power voltage source inverter (VSI) with a static
load and an output filter.
12-pulse VSI
CF Main AC Bus 115Vrms/400Hz
Iia-Y
IInv
Iib-Y
Iic-Y
270 VDC
Iia-Δ
Iib-Δ
Iic-Δ
Y
Δ
Y
Iia Lf
Iib Lf
Lf
I
ic
Rf
Vcb
Ilb
Vcc
Cf
LL
Ila
Vca
Rf
Rf
RL
RL
RL
LL
LL
Ilc
Cf
Cf
PWM Control
Signal
Fig.1: PWM 12-pulse voltage source inverter (VSI) scheme.
II.1 Switching Function Generation: Figure 2 presents the
exact model of the PWM signal generation method which is
commonly used for regulation of the output voltage and
frequency of a switching voltage source inverter. The
operating frequency at the 208/115V main AC Bus is set equal
to the frequency of the sinusoidal function. Also, the
amplitude of the output voltage at the main AC bus is related
to the modulation ratio 𝑀𝑎 , which is equal to (𝐴𝑚 /𝑉𝑡𝑟𝑖 ). This
can be achieved by choosing appropriate values for the
system’s PI control parameters.
Vcont
Selecting the line current and the output voltage shown in
Fig. 1 as state variables, the differential equations describing
inverter’s dynamics are given as follow:
𝑑𝐼𝑖𝑎
𝑑𝑡
=−
𝐼
𝐿𝑓 𝑖𝑎
𝑑𝐼𝑙𝑎
=−
𝑑𝑡
𝑑𝑣𝑐𝑎
Vtri
1
{
ωt
0
-1
Fig. 2: SPWM function generator.
For real-time system control function generation, Fig. 2, the
SPWM switching signal 𝑆𝑖𝑗 can be given by;
1
𝑆𝑖𝑗 = 𝑈[𝑡𝑟𝑖(𝑡, 𝑇) − 𝐴𝑚 𝑆𝑖𝑛(𝜔𝑡)] = {
0
𝑅𝑓
𝑡𝑟𝑖 > 𝑆𝑖𝑛
𝑡𝑟𝑖 ≤ 𝑆𝑖𝑛
(1)
where 𝑈 represents the communication function that generates
either logic “1” or “0”. Fig.3 shows the PWM discrete function
used for controlling the conduction state of the inverter, as
predicted by equation (1).
Sij
1
T
2
Fig. 3: SPWM control signal
𝑅𝐿
=
1
𝑣
𝐿𝑓 𝑐𝑎
𝐼 +
+
1
𝐿𝑓
𝑣𝑖𝑎
𝑣𝑐𝑎
𝐿𝐿 𝑙𝑎
𝐿𝐿
𝐼𝑖𝑎
𝐼𝑙𝑎
𝐶𝑓
−
(3)
𝐶𝑓
where 𝑣𝑖𝑎 and 𝐼𝑖𝑎 represent the ac voltage and line current at the
output of the VSI terminals considered for a 3-phase balanced
system, respectively. 𝑅𝑓 , 𝐿𝑓 , 𝐶𝑓 correspond to the harmonics
filter, 𝑅𝑙 , 𝐿𝑙 and 𝐼𝑙𝑎 model the series RL passive load and the
current through the introduced passive load. Also,
𝑣𝑐𝑎 represents the ac voltage at the harmonics filter’s
terminals. Furthermore, the inverter’s output voltage for a 3phase system can be obtained by [10]:
𝑉𝑖𝑎
2
𝑉
[𝑉𝑖𝑏 ] = 𝑑𝑐 [−1
3
𝑉𝑖𝑐
−1
−1
2
−1
−1 𝑠11
−1] [𝑠12 ]
2 𝑠13
(4)
where 𝑠11 , 𝑠12 , 𝑠13 introduce SPWM switching function for
output voltage control and frequency regulation purposes of
the VSI, and 𝑉𝑑𝑐 represents the input dc voltage. By applying
30° phase shift to the described switching functions, the
control signals corresponding to the upper row of the ∆ − 𝑌
rectifying bridge can be introduced as 𝑠31 , 𝑠32 , 𝑠33 . Using this
set of switching functions, the relationship between input
current and inverter’s output ac currents can be expressed by
the following equation
𝐼𝑖𝑛𝑣 = 𝑆11 𝐼𝑖𝑎−𝑌 + 𝑆12 𝐼𝑖𝑏−𝑌 + 𝑆13 𝐼𝑖𝑐−𝑌 + 𝑆31 𝐼𝑖𝑎−∆ + 𝑆32 𝐼𝑖𝑏−∆ +
A total of 12 switching functions for the upper and lower
bridge have been defined to express the inverter’s line voltage
and current profiles at the input and output terminals. Each
phase is controlled by two switches. The switching operation
can be given by;
𝑆𝑖𝑗 = ∑ 𝐴𝑛 sin(𝑛𝜔𝑡 + 𝑞𝑖𝑗 )
𝑑𝑡
−
(2)
where q11=0˚, q12=120˚, q13=240˚, q31=30˚, q32=150˚, q33=270˚.
The upper and lower switches corresponding to each phase
cannot be on and off simultaneously.
II.2 Proposed GSSA Modeling and Control of Inverters
The main concept of the GSSA modeling is to replace the
real-time domain variables with their complex Fourier
coefficient over a desired time interval [11]. In fact, a signal
𝑥(𝜏) can be approximated, using finite number of the
corresponding Fourier transform coefficients in a finite time
interval 𝜏𝜖 [𝑡 − 𝑇, 𝑡] . Detailed description and rigorous
derivation of the GSSA model can be found in previous work
by the authors [9-11].
𝑆33 𝐼𝑖𝑐−∆
(5)
where 𝐼𝑖𝑛𝑣 represents the input current of the12-pulse PWM
inverter.
III. THE PROPOSED SWITCHING AND CONTROL
MODEL
In this section, we’re presenting a new method that has
been developed to wield an approximated model of the
switching function derived from the GSSA representation of
multi-pulse modulated signals. In the proposed approach, the
SPWM non-linear control signals with discrete waveforms
and desired frequencies can be generated with square waves
and regulated frequencies and duty cycles over desired
intervals. The required degree of accuracy can be obtained by
choosing appropriate frequency and duty cycle. This can be
simply achieved by adding the products of time modulated
signals while employing appropriate phase synchronization to
the corresponding clock pulses.
As depicted in Fig. 4, the SPWM control signal for the 12pulse switching inverter is constructed using a specific
combination of square-wave signals. In order to produce the
desired constant duty cycle over two regions of time domain,
two clock pulses with identical frequencies and two separate
duty cycles are considered. The clock pulse 𝑓1 with a duty
cycle of 𝑑1 , is multiplied by the clock signal 𝐶𝐿𝐾1 with a fifty
percent duty cycle and a frequency of 800Hz. The clock
signal 𝑓2 with the larger duty cycle of 𝑑2 is also multiplied by
the signal 𝐶𝐿𝐾1 with a phase shift of 180°. The resulting
signals are added and subsequently multiplied by a final clock
signal 𝐶𝐿𝐾2 with a fifty percent duty cycle, a frequency of
400Hz and a phase shift of 90°. Further investigation into Fig.
5, one may conclude that deriving the Fourier representation
of the approximated SPWM signal is quite feasible by
applying the convolution property of the time-domain
modulated signals.
f1
CLK-1
𝐶𝐿𝐾1 = 𝑝0 + 2𝑝2 sin(𝜔2 𝑡)
(7)
𝑔1 = 𝐴0 + 𝐹(𝜔1 , 𝜔2 , 𝜔1 − 𝜔2 , 𝜔1 + 𝜔2 )
(8)
𝑓2 = 𝑚0 + 2𝑚1 cos(𝜔1 𝑡) − 2𝑚2 sin(𝜔1 𝑡)
𝑔2 = 𝐴′0 + 𝐹(𝜔1 , 𝜔2 , 𝜔1 − 𝜔2 , 𝜔1 + 𝜔2 )
(10)
𝐶𝐿𝐾2 = 𝑝0 + 2𝑝2 cos(𝜔0 𝑡)
(11)
𝑆11 = (𝑔1 + 𝑔2 ) ∗ 𝐶𝐿𝐾2 = 𝐵0 + 𝑔(𝜔0 , 2𝜔0 , 3𝜔0, … , 𝜔1 , 𝜔1 +
2𝜔0 , 𝜔1 − 2𝜔0 )
(12)
In equation (12) Bo is the dc value of the switching function
S11. Also, the spectral density regarding dominant components
of S11 has been shown in the graph below. In this study,
15kHz has been used as the frequency of the saw-tooth
function, and the inverter’s operating frequency is set to
400Hz. The Fourier coefficients representing SPWM
switching signal are presented in Fig. 5.
0
Sprectral Density
Amplitude
g1
f2
CLK-1
-180
Fig. 5: Spectral density of the modeled switching function.
g1+g2
Estimated PWM signal function
-90
t
Fig. 4: Control signal generation scheme.
The first-order Fourier representation of the modulating
signal𝑓1 can be derived as;
𝑓1 = 𝑧0 + 2𝑧1 cos(𝜔1 𝑡) − 2𝑧2 sin(𝜔1 𝑡)
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
Frequency, Hz
g2
CLK-2
(9)
(6)
where 𝑧0 introduces the dc value of the clock signal 𝑓1 and
𝑧1 𝑎𝑛𝑑 𝑧2 represent the real part and imaginary part of the
first-order Fourier coefficient corresponding to 𝑓1 . As detailed
in Table. 1, d1 appearing in 𝑧1 , 𝑧2 is the averaged duty cycle
estimated for the two sides of PWM switching function, and
d2 is the averaged duty cycle corresponding to the clock pulse
𝑓2 , as shown in Fig.5. In a similar manner, the zero and firstorder Fourier representation of other signals depicted in Fig. 4
are given as follows;
Fig. 6 shows the schematic diagram of the proposed GSSAbased switching function generation and inverter control. The
hardware implementation of the control signal can be done
using the combination of modulated signals shown in Fig. 4.
The required clock pulses for the voltage and frequency
regulation at inverter output are microprocessor generated.
This is one of the advantages of the proposed model;
controlling the operating frequency of the switching inverter,
as well as the required degree of accuracy associated with the
duty cycles of the modulating signals.
In Fig. 6 the SPWM control signal is first constructed using
the clock signals introduced in Fig. 5. Then, the appropriate
phase shift is applied to the generated signal in order to
provide a 12-pulse switching scheme. Each control signal then
is applied to the gate of the switching MOSFETs in both upper
and lower bridges. The predicted signals are used for 3-phase
voltage approximation at certain voltage amplitude and range
of operating frequencies. The advantages of the proposed
model include simple implementation and operational speed.
The design complexity is substantially reduced by decreasing
the circuits used for a hardware-based signal generation.
Moreover, the same clock pulses can serve other purposes by
applying the appropriate time delay to that particular control
signal.
IInv
IY
Iinv-Y
CLK_1
0
X1
S11
0
S11
S11
120
S12
S11
240
S13
S11
S12
S13
Υ
Δ
S11
X2
30
S31
S11
150
S32
S11
270
S33
Zla Zlb
Vb
Zlc
Vc
S31
PPF
S32
S33
CLK_2
CLK_1
Va
Υ
180
Iinv-Δ
Figures 8 (a) and (b) present the inverter’s input DC current.
Fig. 8(b) shows that, although the averaged GSSA-based
model does not include the higher order harmonics (for
simplicity it was not taken into account in synthesizing the
average of the input current), it provides the same pattern of
waveform as that of the computationally-intensive, exact
model real-time analysis, in terms of accuracy of the average
DC value and the fundamental oscillating component.
IΔ
100
Vdc/3
Amps
2 -1 -1
-1 2 -1
-1 -1 2
Fig. 6: Schematic diagram of the PWM switching function generator.
50
IV. RESULTS AND DISCUSSION
In this section the GSSA-developed models have been used
to obtain the 12-pulse inverter’s input/output voltage and
current profiles. Figure 7 depicts two study models of the
switching multi-pulse inverter in the aircraft system. The first
column in (a) presents an “exact” switching model, including
3-phase voltage profile at the low-pass harmonics filter’s
terminals, as well as the AC currents at the output of 12-pulse
bridge inverter before the filter and as well as the load
currents. The figures in column (b) correspond to the proposed
GSSA-based model for switching function generation and
inverter control. Moreover, the effectiveness of applying the
GSSA to switching function generation and control of high
power voltage source inverter is demonstrated in column (b)
of Fig. 7. As shown in the figures, only the fundamental
components of the original signals are considered and higher
order terms are omitted. The GSSA model results closely
agree with that of the exact time-domain implementation.
200
100
0.3
0.3005
0.301
0.3015
0.302
0.3025
Time, s
(a)
(b)
Fig. 8: DC-side current profiles (a) real time exact model (b) GSSA model.
As illustrated in Fig. 8, the dominant oscillating component
that appears on the input current of the power inverter has a
frequency of 2.4 kHz which results from the modulation of
PWM control signals fundamental component with the 5 th
order harmonic of the 6-pulse inverter bridge’s AC-side
currents, which was developed before in this article.
Fig. 9 shows the Fourier representation of the proposed
model’s generated SPWM switching function with two duty
cycles and the desired degree of accuracy. In Fig. 9, the
control signal before and after filtering is shown. The required
degree of accuracy is achieved by appropriately selecting the
duty cycles of the modulating signals as explained in the
previous sections.
0
-100
1
v ca
v cb
-300
0.3
0.302
v cc
ave
0.304
Time, s
Amplitude
1
-200
Amplitude
Volts
Iinv
Iinv, ave
0
0.5
0.5
0
0
Amps
200
0
-200
i ia
i ib
0.302
i ic
ave
0.304
Time, s
100
Amps
4
Time, ms
6
0
2
4
Time, ms
6
0
-400
0.3
0
-100
i la
i lb
-200
0.3
0.302
i lc
ave
0.304
Time, s
(a)
2
(b)
Fig. 7: Inverter output profiles (a) real time exact model (b) GSSA model.
Fig. 9: The GSSA-generated switching function with all 9 harmonics
components on the left, and with only 2 dominant components on the right.
Figures 10 (a) and (b) show the inverter’s output voltage
and current profiles. The developed Fourier representation of
the SPWM modulated signals was used for both voltage and
frequency regulation at the inverter’s output terminals. A
reduced system order of the proposed GSSA-based model for
switching function generation and inverter control has been
implemented in MATLAB® environment and it was
compared to the computationally-intensive exact real-time
switching model implemented in the POWERSIM®
environment. Investigating the graphs, one could see a high
degree of agreement.
300
[2]
d'Aparo, R.D.; Crostella, G.; Nicoletti, D.; Orcioni, S.; Conti, M.; , "DCAC power converter using sigma-delta modulation," 8th Workshop on
Intelligent Solutions in Embedded Systems, 8-9 July 2010
[3]
Rezwan Khan, M.; Al-Sammak, A.I.; , "A 12-step voltage source
inverter control circuit for induction motor drives," Industry
Applications Conference, IAS’96, 6-10, Oct 1996
[4]
Bose, Bimal K.; Sutherland, Hunt A.; , "A High-Performance
Pulsewidth Modulator for an Inverter-Fed Drive System Using a
Microcomputer," IEEE Transactions on Industry Applications, vol. IA19,no.2, pp.235-243.
[5]
Lee, B.K.; Ehsani, M.; , "A simplified functional model for 3-phase
voltage source inverter using switching function concept," Industrial
Electronics Society, IECON '99 Proceedings, 1999.
[6]
Jianping Xu; Lee, C.Q.; , "A unified averaging technique for the
modeling of quasi-resonant converters," Power Electronics, IEEE
Transactions on , vol.13, no.3, pp.556-563, May 1998
[7]
S. Sanders, J. Noworolski, X. Liu, and G. Verghese, “Generalized
averaging method for power conversion systems,” IEEE Transactions
Power Electroics, vol. 6, no. 4, pp. 251–259, Apr. 1991.
[8]
Lee, B.K.; Ehsani, M.; , "A simplified functional model for 3-phase
voltage source inverter using switching function concept," IEEE
Industrial Electronics Society, IECON '99 Proceedings , 1999.
[9]
K.W.E. Cheng, “Comparative study of AC/DC converters for more
electric aircraft,” IEE Power Electronics and Variable Speed drives
Conference, pp. 299-304, Sep. 1998.
Voltage, V
200
100
0
-100
-200
-300
0.3
Modeled Simulation
0.305
0.31
Time, s
0.315
0.32
100
Current, A
50
0
-50
-100
Modeled Simulation
-150
0.3
0.305
0.31
Time, s
0.315
0.32
150
Voltage, V
100
50
0
-50
-100
Modeled Simulation
-150
0.3
(a)
0.305
0.31
Time, s
0.315
(b)
Fig. 10. The approximated PWM switching function with all 9 harmonics
components, shown at left, and with two dominant components shown at
right.
IV. CONCLUSIONS
A novel method for generating the switching function and
control of high power inverters for advanced aircraft electric
power systems is presented. The GSSA model is applied for
the modeling, control, and characterization of SPWM twelve
pulse inverter in AAEPS. The proposed GSSA model has been
applied to derive the corresponding averaged model of the
reduced-order system for the voltage source inverter, while
taking into consideration the interaction with the entire aircraft
electric power system. An accurate model of the switching
function approximations has been derived by developing
average duty cycle intervals over several regions of the PWM
switching pulses. The first order approximations of the
modulated signals are subsequently generated and the
inverter’s input and output profiles have been captured
according to the GSSA model. The GSSA model has been
utilized to derive the key characteristics of the reduced-order
system. The advantages of the proposed model include simple
implementation and operational speed. The design complexity
is substantially reduced by reducing the circuitry used for a
hardware-based signal generation. The proposed model
eliminates the need for time consuming, computationallyintensive real time simulation.
REFERENCES
[1]
Wu, T.S.; Bellar, M.D.; Tchamdjou, A.; Mahdavi, J.; Ehsani, M.; , "A
review of soft-switched DC-AC converters," Industry Applications
Conference, IAS’96, 6-10, Oct 1996
[10] Eid, A.; El-Kishky, H.; Abdel-Salam, M.; El-Mohandes, M.T.; , "On
Power Quality of Variable-Speed Constant-Frequency Aircraft Electric
Power Systems," IEEE Transactions on Power Delivery, vol.25, no.1,
pp.55-65, Jan. 2010.
[11] Liqiu Han; Jiabin Wang; Howe, D.; , "State-space average modeling of
6- and 12-pulse diode rectifiers," European Conference on Power
Electronics and Applications, vol., no., pp.1-10, 2-5 Sept. 2007
[12] L. Andrade and C. Tenning, " Design of the Boeing 777 electric
system", IEEE Aerospace and Electronic Systems Magazine, Vol. 7, No.
7, pp. 4 – 11, July 1992.
[13] A. Eid, H. El-Kishky, M. Abdel-Salam, T. El-Mohandes Constant
frequency aircraft electric power systems with harmonic reduction, 34th
Annual Conference of Industrial Electronics, September, 2008.
[14] A. Eid, M. Abdel-Salam, H. El-Kishky, and T. El-Mohandes Simulation
and transient analysis of conventional and advanced aircraft electric
power systems with harmonics mitigation, Elsevier Electric Power
Systems Research, vol. 01, 2009
[15] A. Eid, M. Abdel-Salam, H. El-Kishky, and T. El-Mohandes Active
power filters for harmonic cancellation in conventional and advanced
aircraft electric power systems, Elsevier Electric Power Systems
Research vol. 01, 2009
[16] H. El-Kishky, H. Ibrahimi, On modeling and control of advanced
aircraft electric power systems: System stability and bifurcation
analysis, International Journal of Electrical Power & Energy Systems
vol 63, pp. 246-259, December 2014.
[17] H. Ibrahimi, J. Gatabi, and H. El-Kishky, "An Auxiliary Power Unit for
Advanced Aircraft Electric Power Systems." Elsevier, Electric Power
Systems Research, vol. 119 pp. 393-406, 2015.
[18] H. Ibrahimi and H. El-Kishky, "A Novel Generalized State-Space
Averaging (GSSA) model for Advanced Aircraft Electric Power
Systems." Elsevier, Energy Conversion and Management, vol. 89, pp.
507-524, 2015.