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 [1-5]. 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 semi-conductors, or, in more extreme cases, runaway behavior leading to system shutdown through the action of over-or-under 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 ESA14], but has advantages from a numerical roview. The second refinement is an admialgorithm. This algorithm specifically addmulti-load system. As set forth in [5,12admittance stability analysis can be used todetermine a constraint on the total (aggregathe loads. The admittance allocation algorithcan be used to take the aggregate load admand use it to formulate load admittance cindividual loads.

The paper is organized as follows. Firsreview of the generalized immittance basedis set forth. In Section III, the Root ECriterion is discussed. In Section IV, the proallocation algorithm is described. The papean application of the method to a test system

Before proceeding further, notes on notaScalar variables will appear in italics typefacwill be used to indicate both matrix quantitiebold non-italics quantities denote functions.

II. GENERALIZED IMMITTANCE BASANALYSIS OF DC SYST

It is appropriate to begin this paper withthe generalized immittance based stabilitypower systems. This technique was set fopublications [12-14]. Although it is readcomplex multi-bus systems [5,15], it is perunderstood in terms of a simple source-loadin Fig. 1.

Fig. 1. Source-load system.

Therein, there is a source with input impedana load with admittance ),( sY ll θ where sθ aof variables which define a steady-state opersource and load respectively and wherefrequency. Formally, the source impeadmittance are linearized transfer functions in voltage to a change in current at thparticular,

ss

s

sss i

vsZθx

θ=Δ

Δ=),(

ll

l

lll v

isYθx

θ=Δ

Δ=),(

where sx and lx denote the state vectors oload.

AC Criterion [12-obustness point of ittance allocation dresses a source

2-15], generalized o, given a source, ate) admittance of hm set forth herein mittance constraint constraints on the

st, in Section II a d stability analysis Extrema Stability oposed admittance er concludes with in Section V.

ation are in order. ce. Bold variables es and sets. Non-

SED STABILITY TEMS h a brief review of y analysis of dc

orth in a series of dily extendible to rhaps most easily

d system as shown

nce ),( sZ ss θ and and lθ are vectors rating point of the e s is complex edance and load

relating a change he terminals. In

(1)

(2)

of the source and

It is known that given a sourcepoint zθ when supplying a constacompatible with lθ ) and that the loapoint lθ when fed from a constantcompatible with zθ ) then the sou

stable at operating point Tz[θθ =

Nyquist evaluation of (),( YsZ lzs θθ-1 point in the s-plane.

While the Nyquist evaluation inspection for encirclements of -1 ctest, another important application inumber of stability criteria which cdesign specifications. As an eximpedance and a stability criterionadmittance can be determined aspecification. In this context, thesediscussed in some detail in the next s

Unfortunately, as it stands the),(),( sYsZ llzs θθ only tests the stab

point θ . The concept behind generais to, in a single analysis, show thinterest are stable. To this end, let Ωof all possible operating points. impedance and load admittance are d

{ sZs zss = θ ),()(Z

{ sYs lll = θ ),()(Y

In essence, rather than being a cofrequency, a generalized immittance numbers. This set arises from the faof most power electronics based comso the linearized model is a funcFurther, the use of generalized immrepresent the effects of parameter unc

Fig. 2 depicts the source impedansystem referred to GS1 and rated output voltage of 750 V. As can bethe source impedance varies cfrequencies, passive elements dominthe impedance does not vary signpoint.

If the Nyquist evaluation of thedoes not encircle -1, then all operaload system are stable. The conversif )()( ss ls YZ partially encircles -1,system is stable. This is becau

)()( ss ls YZ is somewhat conservcorrelated with lθ in the computproduce of source impedance and loa

is stable at an operating nt current load (which is

ad is stable at an operating t voltage source (which is urce-load system will be

TTl ]θ provided that the

), sl does not encircle the

of ),(),( sYsZ llzs θθ and can be used as a stability s that it is the basis for a can be used to formulate ample, given a source

n, a constraint on a load and used as a design e stability criteria will be section.

e Nyquist evaluation of bility of a single operating alized immittance analysis at all operating points of

zΩ and lΩ denote the set The generalized source

defined as

}zz Ω∈θ) (3)

}ll Ω∈θ (4)

omplex number at a given is a closed set of complex ct that the dynamic model

mponents is non-linear and ction of operating point. mittances can be used to certainties.

nce of a generator rectifier at 60 kW with a nominal

e seen, at low-frequencies, onsiderably. At high

nate the impedance and so nificantly with operating

e product of )()( ss ls YZ ating points of the source e is not true. In particular it is still possible that the use in the computation

vative because zθ is not tation of the generalized ad admittance.

208

Fig. 2. GS-1 source impedance.

III. ROOT EXPONENTIAL STABILITY CRITERION While the inspection of Nyquist evaluation of

( )sYsZ llzs ,),( θθ or )()( ss ls YZ 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).

Fig. 3. Stability criteria.

Mathematically, the RESC may be defined by first introducing a function

( ) ( )Re( )( ) Im( )nn snf s s eγα β= + (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 ( ) 1f s = (6) with

1

2 sinnPM

αφ

= (7)

ln 2

cos 1

ngm

gm pm

kk

γφ

=−

(8)

and

cos12

pm

neγ φβ = (9)

and where pmφ is the desired phase margin, and gmk is the desired gain margin. It should be noted that in order for the RESC to be defined, it is necessary that cos 1gm pmK φ > (10)

It can be shown that the points defining the gain and phase margin, ( ) 1f s = , so that these points are on the stability criterion. In particular, by direct substitution,

1 2 3( ) ( ) ( ) 1f s f s f s= = = (11) where 1,2 cos sinpm pms jφ φ= − ± (12)

209

and 3 1 / gms k= − , respectively.

Additional properties of the RESC may expressing (5) as

( ) ( )nnnf yx βα= + where Re( )sx eγ= Im( )y s= Clearly, in the x y− plane, the RESC is a hynote that from the definition of x , only positgenerated, which infers that in the s-plane not close. Rather as Re( )s → −∞ , 0x → anapproaches the horizontal lines Im( ) 1/s β→ ±

Fig. 4 depicts the level sets of ( )f s . Nof its construction from a hyperellipse, a simpoint tps falls within the allowed region is to

Using the methods set forth in [12-1source impedance (load admittance) ancriterion, a design specification on the (source impedance) may be found. The specification based on the source impedanceand the RESC is shown in Fig. 5. Therein,phase margin, and exponent are set at 6 respectively.

Fig. 4. Level sets of ( )f s .

IV. ADMITTANCE ALLOCATION AIn this section, the proposed admit

algorithm is introduced. To illustrate the algorithm, consider Fig. 5. Therein, the specification is shown based on a given sand stability criterion. However, that specification is the aggregate load admittanadmittance of all the loads on a bus. If tloads, it is desirable to derive a specifindividual load.

(13)

be observed by

(14)

(15) (16)

yperellipse. Also tive values can be the criterion will

nd so the criterion

(17) Note that because

mple test to see if a o see if ( ) 1tpf s ≥ . 4], based on the nd the stability load admittance load admittance

e shown in Fig. 2 , the gain margin,

dB, 45o, and 4,

ALGORITHM ttance allocation need for such an load admittance

source impedance load admittance

nce – i.e. the total there are multiple fication for each

Fig. 5. Aggregate load admi

The approach to allocating the aspecification into individual loads iFig. 7. Fig. 6 depicts a complex plagiven frequency. At the given frmust always fall in the allowed regmethod of [12-14]. The boundarforbidden admittance is labeled bndY

Fig. 6. Admittanc

Fig. 7. Mapped admit

ittance constraint.

aggregate load admittance s illustrated in Fig. 6 and ane lY of admittance at a requency, the admittance gion determined using the ry between allowed and d .

ce plane.

ttance plane.

210

The first step in deriving the propallocation algorithm is to consider a mappinrotation. In particular, the mapped admittanc ˆ j

l leθ=Y Y

where the angle of rotation, θ , is defined sureal part of the mapped boundary ˆ

bndY , dminimized. Mathematically, ,

ˆarg min( ( ))bnd mxYθ θ= Fig. 7 depicts the load admittance const

the rotated coordinate system, with ,bnd mxY

,bnd mxY is negative. In order to allocate thadmittance specification to the loadsspecification is replaced with the simconservative) requirement that the mappedadmittance satisfy ,

ˆ ˆreal( )l bnd mxY>Y Now consider the case in which there ar

bus. The generalized load admittance of denoted ,

ˆl kY . The aggregate load admittanc

is guaranteed to be satisfied if each individua , ,

ˆ ˆreal( )l k k bnd mxYα>Y where kα is a load allocation factor, which h 0 1kα≤ ≤ and

1

1K

kk

α=

≤∑

If restrictions (21-23) are satisfied, the minimthe mapped aggregate load admittance is Ysignal stability is guaranteed.

Although there may be several methodallocation factor, one simple and intuitive mthe allocation factors in accordance with

,

,1

b kk K

b kk

P

Pα

=

=∑

where ,b kP is rated power of the k’th loadchoice automatically insures that (22) and (2

Although the development herein has beemapped coordinate system, it may be readinto the original coordinate system. In parti , ,

ˆjbnd k k bnd mxe θα −=Y Y

where ,ˆ

bnd mxY is a set of points satisfying

, ,ˆreal( )bnd mx bnd mxY=Y

V. EXAMPLE APPLICATI

The system used to illustrate these advportion of the Electric Ship Research aConsortium’s Medium Voltage DC (repres

posed admittance ng in the form of a ce is defined as

(18) uch that maximum denoted ,bnd mxY , is

(19) traint in terms of indicated. Note

he aggregate load s, the original

mpler (and more d aggregate load

(20) re K loads on the f the k’th load is ce constraint (20) al load satisfies

(21) has the properties

(22)

(23)

mum real value of

,bnd mxY , and small-

ds to choose the method is to assign

(24)

d. Note that this 23) are satisfied. en put in terms of dily mapped back icular

(25)

(26)

ION vancements is the and Development sentative) Testbed

(MVDCT) facility located at Purduein Fig. 8. This portion of the systemsystem (GS-1), and two loads, a shipand a pulsed power load (PPL). values, and control schemes for the gsystems can be found in [19]. Tdescription for the pulsed power load

Fig. 8. System desc

The generalized admittance basconducted using the open-source MToolbox [21]. The generalized sourcin Fig. 2, and the aggregate load admconstraint on the sum of all load

6gmk dB= , 30opmφ = , and 4n =

Fig. 9 depicts the simplified modadmittances for the PPL and SPS loa

Fig. 9. Load mo

In these models, the resistor/inductorcorresponds to an equivalent repreinput filter inductor in the system. Tfor both the PPL and SPS are: dcr

47acL mH= , 340L Hμ= , 210ler =40dr k= Ω , 1300inC Fμ= , and

constant power load varies from 10the SPS constant power load varies

University [19] as shown m consists of a generation p propulsion system (SPS) Descriptions, parameter generation and propulsion he parameter values and

d can be found in [20].

cription. sed stability analysis was

Matlab-based DC Stability ce impedance is illustrated

mittance constraint (i.e. the s) using the RESC with 4 is depicted in Fig. 5.

dels used to obtain the load ads.

odels.

r network to the left of dr esentation of the physical The component parameters

20 m= Ω , 120acr m= Ω , 0 Ω , 11leL mH=

53esrr m= Ω . The PPL 0 watts to 13.5 kW and s from 100 watts to 37.7

211

kilowatts. The SPS loads also includes an additional high-frequency capacitor, 9.0invC Fμ= .

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.

Fig. 10. SPS load admittance constraint and load admittance.

Fig. 11. PPL load admittance constraint and load admittance.

REFERENCES [1] 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

[2] 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.

[3] 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

[4] 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

[5] 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

[6] P.C. Krause, Analysis of Electric Machinery, McGraw-Hill, New York, 1986.

[7] 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.

[8] 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.

[9] 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.

[10] 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

[11] 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.

[12] 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

[13] 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.

[14] 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.

[15] 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.

[16] R.D. Middlebrook, “Input Filter Considerations in Design and Application of Switching Regulators,” IEEE Proc. IASAM, 1976.

[17] 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.

[18] 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.

[19] 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.

[20] 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

[21] DC Stabilty Toolbox; source code aand manual available at https://engineering.purdue.edu/ECE/Research/Areas/PEDS

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