Advancements in Generalized Immittance Based
Stability Analysis of DC Power Electronics Based
Distribution Systems
Scott D. Sudhoff, Jonathan M. Crider
School of Electrical and Computer Engineering
Purdue University
West Lafayette, IN, United States
Abstract—Stability is an important design consideration in
power electronics based power systems such as those on electric
ships. Generalized immittance based stability analysis is one
approach to addressing this design consideration. In this paper,
refinements to the generalized immittance based stability
analysis approach are proposed. This include the introduction
of Root Exponential Stability Criterion which offers improved
numerical robustness, and an admittance allocation algorithm
which can be used to formulate admittance specifications on the
individual loads on a common bus.
I.
INTRODUCTION
Power electronics based power distribution is a key
technology in electric ships. In general, it has been clearly
demonstrated that, if inappropriately designed, this class of
system can exhibit unstable operating points; it has also been
clearly demonstrated that there are a number of ways by which
the existence of unstable operating points can be avoided [15]. Thus, the stability of the power electronics based power
systems is a design consideration to be addressed in concert
with other design considerations such as mass, power quality,
fault protection, and efficiency.
Informally, an equilibrium point of a system is said to be
stable if, when perturbed from an equilibrium point, the
system returns to that equilibrium point.
Herein, the
equilibrium point of the system will be taken to be
synonymous with operating point. Note that for this definition
to make sense, an equilibrium point must exist in that the
system must be described in such a way that the states are
constant in the steady state. This generally involves the use of
average-value modeling techniques, and, in the case of
balanced ac systems, reference frame theory [6], and in the
case of unbalanced ac systems, multiple reference frame
theory [7].
Observe that it much easier to describe the stability of a
model of a system than the stability of an actual system; this is
because models have state variables whereas actual systems
do not. The situation is particularly difficult in power-
electronics based systems wherein the switching of the power
converters prevents the system from ever reaches a true
equilibrium point. At best operation is periodic, and it is often
the case that operation is aperiodic, even in what would be
considered a well-behaved system. In physical systems,
instability is manifested in sustained oscillations which are not
harmonically related to the switching of the power semiconductors, or, in more extreme cases, runaway behavior
leading to system shutdown through the action of over-orunder voltage or current protection controls.
That said, the consideration of stability aspects of system
behavior are often studied in the context of average-value
models in which state variables exist and have equilibrium
points. Even so, addressing stability as a design consideration
is complicated by the fact that most power converters exhibit
non-linear dynamics, and so the models representing systems
are typically both of high-order and nonlinear. Furthermore,
there is often parameter uncertainty.
A non-linear treatment of the system stability would be the
ideal approach. However, because of the high-order an
arbitrary form of the component models, methods employing,
for example, Lyapunov approaches have, while successful at
some level, been difficult to apply, even when considering a
single operating point [8-11].
As an alternative, generalized immittance stability analysis
has been used [5,12-15]. Although this technique is, at its
core, a linear systems approach, it addresses non-linearities
and parameter uncertainty by representing components as
generalized immittances (for example impedances or
admittances) which bound model behavior in the frequency
domain. A significant advantage of this analysis is that it
considers entire sets of equilibrium (operating) points at a
time, so that a single analysis can be used to explore the
stability properties of a system over its entire operating range.
In this work, two refinements of the generalized
immittance approach are set forth. The first refinement is the
introduction of the Root-Exponential Stability Criterion
This work was supported in part by the Office of Naval Research through Grant
N00014-08-1-0080.
978-1-4244-9273-2/11/$26.00 ©2011 IEEE
207
(RESC). This criterion is similar to the ESAC Criterion [1214], but has advantages from a numerical robustness point of
view. The second refinement is an admittance allocation
algorithm. This algorithm specifically addresses a source
multi-load system. As set forth in [5,12-15], generalized
admittance stability analysis can be used to, given a source,
determine a constraint on the total (aggregate) admittance of
the loads. The admittance allocation algorithm set forth herein
can be used to take the aggregate load admittance constraint
and use it to formulate load admittance constraints on the
individual loads.
The paper is organized as follows. First, in Section II a
review of the generalized immittance based stability analysis
is set forth. In Section III, the Root Extrema Stability
Criterion is discussed. In Section IV, the proposed admittance
allocation algorithm is described. The paper concludes with
an application of the method to a test system in Section V.
Before proceeding further, notes on notation are in order.
Scalar variables will appear in italics typeface. Bold variables
will be used to indicate both matrix quantities and sets. Nonbold non-italics quantities denote functions.
II.
GENERALIZED IMMITTANCE BASED STABILITY
ANALYSIS OF DC SYSTEMS
It is appropriate to begin this paper withh a brief review of
the generalized immittance based stability analysis of dc
power systems. This technique was set forth in a series of
publications [12-14]. Although it is readily extendible to
complex multi-bus systems [5,15], it is perhaps most easily
understood in terms of a simple source-load system as shown
in Fig. 1.
Fig. 1. Source-load system.
Therein, there is a source with input impedance Z s (θ s , s ) and
a load with admittance Yl (θl , s) where θ s and θl are vectors
of variables which define a steady-state operating point of the
source and load respectively and where s is complex
frequency.
Formally, the source impedance and load
admittance are linearized transfer functions relating a change
in voltage to a change in current at the terminals. In
particular,
Z s (θ s , s ) =
Yl (θl , s ) =
Δvs
Δis
x s =θ s
Δil
Δvl
xl =θl
It is known that given a source is stable at an operating
point θ z when supplying a constant current load (which is
compatible with θl ) and that the load is stable at an operating
point θl when fed from a constant voltage source (which is
compatible with θ z ) then the source-load system will be
stable at operating point θ = [θTz θTl ]T provided that the
Nyquist evaluation of Z s (θ z , s)Yl (θl , s) does not encircle the
-1 point in the s-plane.
While the Nyquist evaluation of Z s (θ z , s)Yl (θl , s) and
inspection for encirclements of -1 can be used as a stability
test, another important application is that it is the basis for a
number of stability criteria which can be used to formulate
design specifications. As an example, given a source
impedance and a stability criterion, a constraint on a load
admittance can be determined and used as a design
specification. In this context, these stability criteria will be
discussed in some detail in the next section.
Unfortunately, as it stands the Nyquist evaluation of
Z s (θ z , s)Yl (θl , s) only tests the stability of a single operating
point θ . The concept behind generalized immittance analysis
is to, in a single analysis, show that all operating points of
interest are stable. To this end, let Ω z and Ωl denote the set
The generalized source
of all possible operating points.
impedance and load admittance are defined as
{
Y ( s ) = {Y (θ , s )
Z s ( s ) = Z s (θ z , s ) θ
l
l
l
}
}
z ∈Ω z
θ l ∈Ω l
(3)
(4)
In essence, rather than being a complex number at a given
frequency, a generalized immittance is a closed set of complex
numbers. This set arises from the fact that the dynamic model
of most power electronics based components is non-linear and
so the linearized model is a function of operating point.
Further, the use of generalized immittances can be used to
represent the effects of parameter uncertainties.
Fig. 2 depicts the source impedance of a generator rectifier
system referred to GS1 and rated at 60 kW with a nominal
output voltage of 750 V. As can be seen, at low-frequencies,
At high
the source impedance varies considerably.
frequencies, passive elements dominate the impedance and so
the impedance does not vary significantly with operating
point.
If the Nyquist evaluation of the product of Z s ( s)Yl ( s)
does not encircle -1, then all operating points of the source
load system are stable. The converse is not true. In particular
if Z s ( s)Yl ( s) partially encircles -1, it is still possible that the
(1) system is stable. This is because in the computation
Z s ( s)Yl ( s) is somewhat conservative because θ z is not
correlated with θl in the computation of the generalized
(2) produce of source impedance and load admittance.
where x s and xl denote the state vectors of the source and
load.
208
Fig. 2. GS-1 source impedance.
III.
Fig. 3. Stability criteria.
ROOT EXPONENTIAL STABILITY CRITERION
While the inspection of Nyquist evaluation of
Z s (θ z , s )Yl (θ l , s ) or Z s ( s )Yl ( s) can be useful as a stability
test, it is often the case that the principal of the test is used in
conjunction with a stability criterion design to ensure a
certain measure of robustness and/or to translate the stability
test into a design specification.
Numerous Nyquist based stability criterion have been
suggested in the literature [12-14,16-18]. In each case, the
function of the stability criterion is to prevent the Nyquist
evaluation of a source-impedance load-admittance product
from circling the -1 point. Thus systems that meet these
criteria have models which have stable equilibrium points.
Fig. 3 illustrates a number of these criteria in the s-plane.
Perhaps the first proposed criterion was that of Middlebrook
[16]. Later criterion included that proposed by Carrol [17],
the Gain and Phase Margin Criterion [18], and later the ESAC
Criterion [12-14]. Historically, each new criterion opened up
more of the s-plane for the source-impedance load-admittance
product. This reduced ‘artificially conservativeness’ as
described in [12-14]. In essence, restricting the behavior in
the s-plane unnecessarily can increase the cost of the design
without improving system performance.
In this work, a new stability criterion is defined, the Root
Exponential Stability Criterion (RESC). The RESC is similar
to the ESAC in terms of the region of the s-plane which is
forbidden. However, it is an improvement in that it may be
represented by a continuous function. From a design impact,
the difference between the ESAC and RESC criterion is
minimal. However the RESC has an advantage over the
ESAC criterion when computing a load admittance (source
impedance) constraint based on the source impedance (load
admittance). In particular, it is more numerically robust
because the path of the RESC in the s-plane has a normal
direction which is always well-defined; the same cannot be
said of the ESAC criterion (or the Gain and Phase Margin
Criterion).
Mathematically, the RESC may be defined by first
introducing a function
f ( s) =
n
(α Im(s) )
n
+ ( β eγ Re( s ) )
n
(5)
where n is an even integer equal to or greater than 2, Re( s )
and Im( s) denote the real and imaginary parts of a complex
variable s , and α , β , and γ are positive real constants.
The RESC is defined by the set of points in the s - plane
which satisfy
f ( s) = 1
(6)
with
1
α=n
(7)
2 sin φPM
γ=
k gm ln n 2
k gm cos φ pm − 1
(8)
and
β=
1
n
2
e
γ cos φ pm
(9)
and where φ pm is the desired phase margin, and k gm is the
desired gain margin. It should be noted that in order for the
RESC to be defined, it is necessary that
K gm cos φ pm > 1
(10)
It can be shown that the points defining the gain and phase
margin, f ( s ) = 1 , so that these points are on the stability
criterion. In particular, by direct substitution,
f ( s1 ) = f ( s2 ) = f ( s3 ) = 1
(11)
s1,2 = − cos φ pm ± j sin φ pm
(12)
where
209
and
s3 = −1 / k gm ,
(13)
respectively.
Additional properties of the RESC may be observed by
expressing (5) as
f = n (α x ) n + ( β y )
n
(14)
where
x = eγ Re( s )
(15)
(16)
y = Im( s )
Clearly, in the x − y plane, the RESC is a hyperellipse. Also
note that from the definition of x , only positive values can be
generated, which infers that in the s-plane the criterion will
not close. Rather as Re( s) → −∞ , x → 0 and so the criterion
approaches the horizontal lines
Im( s) → ±1/ β
(17)
Fig. 4 depicts the level sets of f ( s ) . Note that because
of its construction from a hyperellipse, a simple test to see if a
point stp falls within the allowed region is to see if f ( stp ) ≥ 1 .
Using the methods set forth in [12-14], based on the
source impedance (load admittance) and the stability
criterion, a design specification on the load admittance
(source impedance) may be found. The load admittance
specification based on the source impedance shown in Fig. 2
and the RESC is shown in Fig. 5. Therein, the gain margin,
phase margin, and exponent are set at 6 dB, 45o, and 4,
respectively.
Fig. 5. Aggregate load admittance constraint.
The approach to allocating the aggregate load admittance
specification into individual loads is illustrated in Fig. 6 and
Fig. 7. Fig. 6 depicts a complex plane Yl of admittance at a
given frequency. At the given frequency, the admittance
must always fall in the allowed region determined using the
method of [12-14]. The boundary between allowed and
forbidden admittance is labeled Ybnd .
Fig. 6. Admittance plane.
Fig. 4. Level sets of f ( s ) .
IV.
ADMITTANCE ALLOCATION ALGORITHM
In this section, the proposed admittance allocation
algorithm is introduced. To illustrate the need for such an
algorithm, consider Fig. 5. Therein, the load admittance
specification is shown based on a given source impedance
and stability criterion. However, that load admittance
specification is the aggregate load admittance – i.e. the total
admittance of all the loads on a bus. If there are multiple
loads, it is desirable to derive a specification for each
individual load.
210
Fig. 7. Mapped admittance plane.
The first step in deriving the proposed admittance
allocation algorithm is to consider a mappinng in the form of a
rotation. In particular, the mapped admittance is defined as
ˆ = Y e jθ
Y
(18)
l
l
where the angle of rotation, θ , is defined such that maximum
ˆ , denoted Yˆ
real part of the mapped boundary Y
, is
bnd
minimized. Mathematically,
θ = arg min(Yˆbnd , mx (θ ))
bnd , mx
(MVDCT) facility located at Purdue University [19] as shown
in Fig. 8. This portion of the system consists of a generation
system (GS-1), and two loads, a ship propulsion system (SPS)
and a pulsed power load (PPL). Descriptions, parameter
values, and control schemes for the generation and propulsion
systems can be found in [19]. The parameter values and
description for the pulsed power load can be found in [20].
(19)
Fig. 7 depicts the load admittance consttraint in terms of
the rotated coordinate system, with Yˆbnd , mx indicated. Note
Yˆbnd ,mx is negative. In order to allocate the aggregate load
admittance specification to the loads, the original
specification is replaced with the simpler (and more
Fig. 8. System description.
conservative) requirement that the mapped aggregate load
admittance satisfy
The generalized admittance based stability analysis was
ˆ ) > Yˆ
conducted
using the open-source Matlab-based DC Stability
real(Y
(20)
l
bnd , mx
Toolbox [21]. The generalized source impedance is illustrated
Now consider the case in which there are K loads on the in Fig. 2, and the aggregate load admittance constraint (i.e. the
bus. The generalized load admittance of the k’th load is constraint on the sum of all loads) using the RESC with
ˆ . The aggregate load admittance constraint (20)
denoted Y
l ,k
k gm = 6dB , φ pm = 30o , and n = 4 is depicted in Fig. 5.
is guaranteed to be satisfied if each individual load satisfies
Fig. 9 depicts the simplified models used to obtain the load
ˆ ) > α Yˆ
real(Y
(21)
l ,k
k bnd , mx
admittances for the PPL and SPS loads.
where α k is a load allocation factor, which has the properties
0 ≤ αk ≤ 1
(22)
and
K
∑α
k
≤1
(23)
k =1
If restrictions (21-23) are satisfied, the minimum real value of
the mapped aggregate load admittance is Yˆbnd , mx , and smallsignal stability is guaranteed.
Although there may be several methods to choose the
allocation factor, one simple and intuitive method is to assign
the allocation factors in accordance with
P
α k = K b,k
(24)
P
∑ b, k
k =1
where Pb , k is rated power of the k’th loadd. Note that this
choice automatically insures that (22) and (23) are satisfied.
Although the development herein has been put in terms of
mapped coordinate system, it may be readily mapped back
into the original coordinate system. In partiicular
Fig. 9. Load models.
ˆ
Ybnd , k = α k e − jθ Y
(25)
bnd , mx
In these models, the resistor/inductor network to the left of rd
ˆ
where Ybnd , mx is a set of points satisfying
corresponds to an equivalent representation of the physical
input filter inductor in the system. The component parameters
ˆ
real(Ybnd , mx ) = Ybnd , mx
(26)
for both the PPL and SPS are: rdc = 20 mΩ , rac = 120 mΩ ,
Lac = 47 mH , L = 340 μ H , rle = 210 Ω ,
Lle = 11 mH
V.
EXAMPLE APPLICATION
rd = 40 k Ω , Cin = 1300 μ F , and resr = 53 mΩ . The PPL
The system used to illustrate these advancements is the
constant power load varies from 100 watts to 13.5 kW and
portion of the Electric Ship Research and Development
the SPS constant power load varies from 100 watts to 37.7
Consortium’s Medium Voltage DC (representative) Testbed
211
kilowatts. The SPS loads also includes an additional highfrequency capacitor, Cinv = 9.0 μ F .
Based on a rated SPS power of 37.7 kW and a PPL of 13.5
kW, the load allocation factors of the two loads were taken to
be 0.736 and 0.264, respectively. Using the admittance
allocation factor, the load admittance allocations and
generalized load admittances for the SPS and PPL are
depicted in Fig. 10 and Fig. 11, respectively. As can be seen,
both loads satisfy their respective admittance constraints.
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
Fig. 10. SPS load admittance constraint and load admittance.
[14]
[15]
[16]
[17]
[18]
Fig. 11. PPL load admittance constraint and load admittance.
[19]
REFERENCES
[1]
[2]
[3]
S.D. Sudhoff, K.A. Corzine, S.F. Glover, H.J. Hegner, and H.N. Robey,
“DC Link Stabilized Field Oriented Control of Electric Propulsion
Systems,” IEEE Transactions on Energy Conversion, Vol. 13, No. 1,
March 1998
S.F. Glover, S.D. Sudhoff, “An Experimentally Validated Nonlinear
Stabilizing Control for Power Electronics Based Power Systems,” 1998
SAE Transactions, Journal of Aerospace, Section 1, pp. 68-77.
S.D. Sudhoff, S.F. Glover, “Modeling Techniques, Stability Analysis,
and Design Criteria for DC Power Systems with Experimental
Validation,” 1998 SAE Transactions, Journal of Aerospace, Section 1,
pp. 52-67
[20]
[21]
212
S.D. Sudhoff, S.F. Glover, S.H. Żak, S.D. Pekarek, E.J Zivi, D.Clayton,
D.E. Delisle, “Analysis Methodologies for DC Power Distribution
Systems,” Thirteenth International Ship Control Systems Symposium,
Paper 235, 7-9 April 2003, Orlando, Florida, USA
R.R. Chan, Y. Lee, S.D. Sudhoff, E.L. Zivi, “Evolutionary
Optimization of Power Electronics Based Systems,” IEEE Transactions
on Power Electronics, Vol. 23, No. 4, pp. 1907-1917, July 2008
P.C. Krause, Analysis of Electric Machinery, McGraw-Hill, New
York, 1986.
P.C. Krause, “Method of Multiple Reference Frames Applied to the
Analysis of Symmetrical Induction Machinery,” IEEE Transactions on
Power Apparatus and Systems, Vol. 87, pp. 227-234, January 1968.
Benjamin Loop, S.D Sudhoff, S.H. Zak, E. Zivi, “An Optimization
Approach to Estimating Stability Regions Using Genetic Algorithms,”
2005 American Control Conference, 8-10 June 2005, Portland, Oregon.
B.P. Loop, S.D. Sudhoff, S. H. Zak, E.L. Zivi, “Estimating Regions of
Asymptotic Stability of Power Electronics Systems Using Genetic
Algorithms,” IEEE Transactions on Control Systems Technology. vol.
18, no. 5, pg(s) 1011-1021, September 2010.
C. Sullivan, S. Sudhoff, E. Zivi, S. Zak, “Methods of Optimal
Lyapunov Function Generation With Application to Power Electronic
Converters and Systems,”
IEEE Electric Ship Technologies
Symposium, Arlington, VA, May 21-23, 2007
C. Sullivan, S.D. Sudhoff, S.H. Zak, “Region of Asymptotic Stability
Estimation with Lyapunov Function Optimization and the Average
Derivative Method,” SAE 2010 Power Systems Conference.
S.D. Sudhoff, D.H. Schmucker, R.A. Youngs, H. J. Hegner, “Stability
Analysis of DC Distribution Systems Using Admittance Space
Constraints,” Proceedings of The Institute of Marine Engineers All
Electric Ship 98, London, September 29-30, 1998
S.D. Sudhoff, S.F. Glover, “Three Dimensional Stability Analysis of
DC Power Electronics Based Systems,” Proceedings of the Power
Electronics Specialist Conference, Galway, Ireland, June 19-22, 2000,
pp. 101-106.
S.D. Sudhoff, S.F. Glover, P.T. Lamm, D.H. Schmucker, D.E. Delisle,
“Admittance Space Stability Analysis of Power Electronic Systems,”
IEEE Transactions on Aerospace and Electronics Systems, Vol. 36. No.
3. July 2000, pp. 965-973.
S.D. Sudhoff, S.D. Pekarek, S.F. Glover, S.H. Zak, E. Zivi, J.D. Sauer,
D.E Delisle, “Stability Analysis of a DC Power Electronics Based
Distribution System,” SAE2002 Power Systems Conference (Paper
Offer #: 02PSC-17) , October 29-31, 2002, Coral Springs, Florida,
USA.
R.D. Middlebrook, “Input Filter Considerations in Design and
Application of Switching Regulators,” IEEE Proc. IASAM, 1976.
J. Carrol, “An Input Impedance Stability Criterion Allowing More
Flexibility For Multiple Loads Which Are Independently Designed,”
Naval Air Warefare Center, Aircraft Division, Indianapolis, B/812,
January 22, 1992.
C.M. Wildrick, F.C.Lee, B.H. Cho, and B. Choi, “A Method of
Defining the Load Impedance Specification for a Stable Distributed
Power System,” IEEE Transactions on Power Electronics, pp. 280-285,
May 1995.
M. Bash, R.R. Chan, J. Crider, C. Harianto, J. Lian, J.
Neely, S.D. Pekarek, S.D. Sudhoff, N. Vaks,
Medium
Voltage
DC
Testbed
for
Ship
Power
System
Research, IEEE Electric Ship Technologies Symposium,
Baltimore, MD, April 20-22, 2009.
Crider, J.M.; Sudhoff, S.D.; , "Reducing Impact of Pulsed Power Loads
on Microgrid Power Systems," Smart Grid, IEEE Transactions on ,
vol.1, no.3, pp.270-277, Dec. 2010
DC Stabilty Toolbox; source code aand manual available at
https://engineering.purdue.edu/ECE/Research/Areas/PEDS