IEEE Electrification Magazine - September 2017 - 59
Vbus
Vbus
Vbus
converters and m load converters. Due to the complex
levels of system damping. Figures 5(c) and (d) display
system topology, performing frequency-domain stability
two stable situations, but the system whose transient
analysis in the system of Figure 4(a) may be cumbersome
response is shown in Figure 5(d) (which is quite oscillaand tedious. Therefore, it is advantageous to derive simtory) has less damping than the one whose transient
pler equivalent models so that stability analysis typical of
response is shown in Figure 5(c). Figure 5(e) displays an
classical control theory can be applied.
unstable transient response. The stability of these three
The most common model in the literature for fredifferently damped systems can be analyzed in the frequency-domain stability analysis is the equivalent casquency domain, as shown in Figure 5(a). For the unstacade system depicted in Figure 4(b). It consists of an
ble system of Figure 5(e), the Nyquist diagram of
equivalent source subsystem (s) and an equivalent load
T MLG=Z S/Z L encircles the (−1, 0) point. For the stable
subsystem (l) defined at the MVdc-bus. The source subcases of Figures 5(c) and (d), the correspondent Nyquist
system is the parallel combination of all n source convertdiagrams of TMLG=ZS/ZL do not encircle the (−1, 0) point.
ers, and the load subsystem is the parallel combination of
Moreover, the Nyquist diagram of the highly damped
all m load converters. At the source/load interface, it is
system is farther away from the (−1, 0) point than the
possible to define the so-called minor loop gain TMLG=ZS/
case with light damping. Therefore, Figure 5(a) shows
ZL, i.e., a transfer function defined as the ratio of the
that larger GM and PM correspond to the highly
equivalent source subsystem impedance and the equivadamped system. Notice that GM and PM are positive
lent load subsystem impedance. This impedance ratio
quantities for a stable system and become negative for
contains the small-signal stability information of the casan unstable system.
cade system of Figure 4(b) and equivalently of the system
It is clear from the example described previously that
of Figure 4(a). It is called minor loop gain because it can be
applying the Nyquist stability criterion to TMLG=ZS/ZL
shown that the output impedance of the source subsysdefined at the source/load interface of the equivalent castem ZS and input impedance of the load subsystem ZL
cade model shown in Figure 4(b) provides the engineers
form a type of negative feedback system. Therefore, from
a way to design for system stability with certain stability
the control theory of linear systems, a necessary and sufmargins linked to the desired time-domain performance.
ficient condition for stability of
the system is obtained by applyFrequency-Domain Stability Analysis
Time-Domain Stability Analysis
ing the Nyquist stability criterion
Nyquist Diagram of ZS /ZL
to TMLG=ZS/ZL. The cascaded sysIm
tem is stable if and only if the
Nyquist contour of TMLG does not
encircle the (−1, 0) point in the
t
complex plane. To quantify the
(-1, j 0)
(c)
PM 0
degree of stability of the system,
1/GM
Re
two quantities called gain margin
(GM) and phase margin (PM) can
Decreasing
be defined. These quantities deDamping
fine how far the Nyquist contour
Unit Circle
of TMLG is from the critical (−1, 0)
(a)
t
point and, therefore, how far the
(d)
system is from being unstable.
Nyquist Diagram of Zbus
The quantities GM and PM are
Im
also related to the damping of the
Decreasing
system: the higher the damping,
Decreasing
Damping
Damping
the larger GM and PM are (and
vice versa; the lighter the dampt
ing, the smaller GM and PM are).
0
(e)
To better understand how the
Re
frequency-domain stability analysis is related to the transient
response performance of the system under different damping conditions, an example is reported.
(b)
Figures 5(c)-(e) depict three repreFigure 5. (a) The Nyquist stability criterion applied to the minor loop gain. (b) The PBSC applied
sentative transients (e.g., due to a
to the bus impedance in relation to three differently damped transient responses of the bus voltload step) with three different
age: (c) highly damped, (d) lightly damped, and (e) much less lightly damped (an unstable case).
IEEE Elec trific ation Magazine / S EP T EM BE R 2 0 1 7
59
Table of Contents for the Digital Edition of IEEE Electrification Magazine - September 2017
IEEE Electrification Magazine - September 2017 - Cover1
IEEE Electrification Magazine - September 2017 - Cover2
IEEE Electrification Magazine - September 2017 - 1
IEEE Electrification Magazine - September 2017 - 2
IEEE Electrification Magazine - September 2017 - 3
IEEE Electrification Magazine - September 2017 - 4
IEEE Electrification Magazine - September 2017 - 5
IEEE Electrification Magazine - September 2017 - 6
IEEE Electrification Magazine - September 2017 - 7
IEEE Electrification Magazine - September 2017 - 8
IEEE Electrification Magazine - September 2017 - 9
IEEE Electrification Magazine - September 2017 - 10
IEEE Electrification Magazine - September 2017 - 11
IEEE Electrification Magazine - September 2017 - 12
IEEE Electrification Magazine - September 2017 - 13
IEEE Electrification Magazine - September 2017 - 14
IEEE Electrification Magazine - September 2017 - 15
IEEE Electrification Magazine - September 2017 - 16
IEEE Electrification Magazine - September 2017 - 17
IEEE Electrification Magazine - September 2017 - 18
IEEE Electrification Magazine - September 2017 - 19
IEEE Electrification Magazine - September 2017 - 20
IEEE Electrification Magazine - September 2017 - 21
IEEE Electrification Magazine - September 2017 - 22
IEEE Electrification Magazine - September 2017 - 23
IEEE Electrification Magazine - September 2017 - 24
IEEE Electrification Magazine - September 2017 - 25
IEEE Electrification Magazine - September 2017 - 26
IEEE Electrification Magazine - September 2017 - 27
IEEE Electrification Magazine - September 2017 - 28
IEEE Electrification Magazine - September 2017 - 29
IEEE Electrification Magazine - September 2017 - 30
IEEE Electrification Magazine - September 2017 - 31
IEEE Electrification Magazine - September 2017 - 32
IEEE Electrification Magazine - September 2017 - 33
IEEE Electrification Magazine - September 2017 - 34
IEEE Electrification Magazine - September 2017 - 35
IEEE Electrification Magazine - September 2017 - 36
IEEE Electrification Magazine - September 2017 - 37
IEEE Electrification Magazine - September 2017 - 38
IEEE Electrification Magazine - September 2017 - 39
IEEE Electrification Magazine - September 2017 - 40
IEEE Electrification Magazine - September 2017 - 41
IEEE Electrification Magazine - September 2017 - 42
IEEE Electrification Magazine - September 2017 - 43
IEEE Electrification Magazine - September 2017 - 44
IEEE Electrification Magazine - September 2017 - 45
IEEE Electrification Magazine - September 2017 - 46
IEEE Electrification Magazine - September 2017 - 47
IEEE Electrification Magazine - September 2017 - 48
IEEE Electrification Magazine - September 2017 - 49
IEEE Electrification Magazine - September 2017 - 50
IEEE Electrification Magazine - September 2017 - 51
IEEE Electrification Magazine - September 2017 - 52
IEEE Electrification Magazine - September 2017 - 53
IEEE Electrification Magazine - September 2017 - 54
IEEE Electrification Magazine - September 2017 - 55
IEEE Electrification Magazine - September 2017 - 56
IEEE Electrification Magazine - September 2017 - 57
IEEE Electrification Magazine - September 2017 - 58
IEEE Electrification Magazine - September 2017 - 59
IEEE Electrification Magazine - September 2017 - 60
IEEE Electrification Magazine - September 2017 - 61
IEEE Electrification Magazine - September 2017 - 62
IEEE Electrification Magazine - September 2017 - 63
IEEE Electrification Magazine - September 2017 - 64
IEEE Electrification Magazine - September 2017 - 65
IEEE Electrification Magazine - September 2017 - 66
IEEE Electrification Magazine - September 2017 - 67
IEEE Electrification Magazine - September 2017 - 68
IEEE Electrification Magazine - September 2017 - 69
IEEE Electrification Magazine - September 2017 - 70
IEEE Electrification Magazine - September 2017 - 71
IEEE Electrification Magazine - September 2017 - 72
IEEE Electrification Magazine - September 2017 - 73
IEEE Electrification Magazine - September 2017 - 74
IEEE Electrification Magazine - September 2017 - 75
IEEE Electrification Magazine - September 2017 - 76
IEEE Electrification Magazine - September 2017 - 77
IEEE Electrification Magazine - September 2017 - 78
IEEE Electrification Magazine - September 2017 - 79
IEEE Electrification Magazine - September 2017 - 80
IEEE Electrification Magazine - September 2017 - 81
IEEE Electrification Magazine - September 2017 - 82
IEEE Electrification Magazine - September 2017 - 83
IEEE Electrification Magazine - September 2017 - 84
IEEE Electrification Magazine - September 2017 - 85
IEEE Electrification Magazine - September 2017 - 86
IEEE Electrification Magazine - September 2017 - 87
IEEE Electrification Magazine - September 2017 - 88
IEEE Electrification Magazine - September 2017 - Cover3
IEEE Electrification Magazine - September 2017 - Cover4
https://www.nxtbook.com/nxtbooks/pes/electrification_december2022
https://www.nxtbook.com/nxtbooks/pes/electrification_september2022
https://www.nxtbook.com/nxtbooks/pes/electrification_june2022
https://www.nxtbook.com/nxtbooks/pes/electrification_march2022
https://www.nxtbook.com/nxtbooks/pes/electrification_december2021
https://www.nxtbook.com/nxtbooks/pes/electrification_september2021
https://www.nxtbook.com/nxtbooks/pes/electrification_june2021
https://www.nxtbook.com/nxtbooks/pes/electrification_march2021
https://www.nxtbook.com/nxtbooks/pes/electrification_december2020
https://www.nxtbook.com/nxtbooks/pes/electrification_september2020
https://www.nxtbook.com/nxtbooks/pes/electrification_june2020
https://www.nxtbook.com/nxtbooks/pes/electrification_march2020
https://www.nxtbook.com/nxtbooks/pes/electrification_december2019
https://www.nxtbook.com/nxtbooks/pes/electrification_september2019
https://www.nxtbook.com/nxtbooks/pes/electrification_june2019
https://www.nxtbook.com/nxtbooks/pes/electrification_march2019
https://www.nxtbook.com/nxtbooks/pes/electrification_december2018
https://www.nxtbook.com/nxtbooks/pes/electrification_september2018
https://www.nxtbook.com/nxtbooks/pes/electrification_june2018
https://www.nxtbook.com/nxtbooks/pes/electrification_december2017
https://www.nxtbook.com/nxtbooks/pes/electrification_september2017
https://www.nxtbook.com/nxtbooks/pes/electrification_march2018
https://www.nxtbook.com/nxtbooks/pes/electrification_june2017
https://www.nxtbook.com/nxtbooks/pes/electrification_march2017
https://www.nxtbook.com/nxtbooks/pes/electrification_june2016
https://www.nxtbook.com/nxtbooks/pes/electrification_december2016
https://www.nxtbook.com/nxtbooks/pes/electrification_september2016
https://www.nxtbook.com/nxtbooks/pes/electrification_december2015
https://www.nxtbook.com/nxtbooks/pes/electrification_march2016
https://www.nxtbook.com/nxtbooks/pes/electrification_march2015
https://www.nxtbook.com/nxtbooks/pes/electrification_june2015
https://www.nxtbook.com/nxtbooks/pes/electrification_september2015
https://www.nxtbook.com/nxtbooks/pes/electrification_march2014
https://www.nxtbook.com/nxtbooks/pes/electrification_june2014
https://www.nxtbook.com/nxtbooks/pes/electrification_september2014
https://www.nxtbook.com/nxtbooks/pes/electrification_december2014
https://www.nxtbook.com/nxtbooks/pes/electrification_december2013
https://www.nxtbook.com/nxtbooks/pes/electrification_september2013
https://www.nxtbookmedia.com