981255

An Experimentally Validated Nonlinear Stabilizing Control forPower Electronics Based Power Systems

Steven F. GloverP.C. Krause and Associates

Scott D. SudhoffPurdue University

Copyright © 1997 Society of Automotive Engineers, Inc.

ABSTRACT

High performance high bandwidth control of powerelectronic converters, inverters, and motor drives hasbecome feasible over the past decade. These devicesbehave as constant power loads over large bandwidthswhen they are tightly regulated. However, constantpower loads have a severe side affect known asnegative impedance instability. In order to mitigate theproblem of negative impedance instability a newnonlinear system stabilizing controller has beendeveloped. The details of how this controller worksalong with its implementation is discussed anddemonstrated in hardware.

INTRODUCTION

Power electronics based power and propulsionsystems are topics of great interest to industry and themilitary. Applications include submarines, ships, hybridelectric and electric vehicles, aircraft, and spacecraft.The flexibility and potential for high bandwidth control inthese systems, which is afforded by the use of powerelectronic converters/inverters, reduces the need forhuman operators and at the same time permits a greaterdegree of optimization, reducing fuel costs. As a resultof fuel savings, these systems also offer loweremissions. However, with these benefits come onedisadvantage in that these systems tend to be inherentlydynamically unstable.

In such a system, most loads take the form of electricdrive systems, dc/ac converters, or dc/dc converters. Inthe past, many different control schemes have beeninvestigated including Sliding Mode Control [2], FuzzyLogic [3 and 4], Nonlinear Proportional Integral (PI) [5],as well as many others linear and nonlinear [6 and 7].Regardless of the strategy, these different control lawsshare the property that if they tightly regulate theconverter/inverter output, the converter/inverter presentsa constant power load to the system. Thus, the currententering the converter/inverter increases if the voltage atthe input decreases, causing the converter/inverter to

display a negative impedance characteristic as viewedfrom the rest of the system - clearly a destabilizingeffect.

This has led to an impedance based stability criteriathat predicts, on an operating point basis, when theoverall system will be unstable. Using this criteria thereare several techniques that can be used to eliminate thenegative impedance instability. The first of which isdesigning the converter regulatory control with very lowbandwidth so that the negative impedance characteristicis reduced. This can be effective but results in poorregulation of the output of the converters/inverters.Another approach is to incorporate a passive RCdamping network to stabilize the system. This is alsoeffective but the extra hardware and space required canbe quite expensive, and system efficiency is reduced.The path chosen here is to modulate the commandedinput power of the load converters at the appropriatefrequencies with a nonlinear system stabilizing controller(NSSC). One possibility to accomplish this is to monitorthe power passed down the AC transmission line [8] andfrom this measurement modulate the commandedconverter/inverter input power. This requires additionalsensors placed long distances from theconverter/inverter. Another possibility is the use of alinear feed forward pole-zero cancellation control asdiscussed in [9] to eliminate the instability. This linearfeed forward control requires a duty cycle controlledconverter/inverter with the assumption that the inputvoltage times the on time of the upper transistor is aconstant. The technique chosen here, [10 and 11], usesa new nonlinear feed forward control law that is not onlysimple to implement but also has minimal effect on thedesired performance of the system and at the same timeguarantees system stability. This nonlinear control lawworks well with current controlled converters/inverters,which allow better control in terms of safety and currentlimiting. The NSSC is augmented to the existingcontrols which are designed solely on the basis of outputregulation. This new control scheme is verified on boththe induction motor drive (IMD) of a propulsion system,

and the dc/dc buck boost converter (BBC) of a DCpower system.

BASE TEST SYSTEMS

INDUCTION MOTOR BASED ELECTRICPROPULSION SYSTEM - Figure 1 illustrates the type ofelectric propulsion system considered herein, [1]. Thepower source of the system is a diesel engine or turbine(emulated by a dynamometer), which serves as a primemover for the 3-phase synchronous machine (SM). The3-phase output of the machine is rectified using anuncontrolled rectifier. The rectifier output voltage isdenoted rv . An LC circuit serves as a filter, and the

output of this filter is denoted dcsv . A voltage regulator /exciter adjusts the field voltage of the SM in such a waythat the source bus voltage dcsv is equal to the

commanded bus voltage *dcsv . The source bus is

connected via a tie line to the load bus, the voltage atwhich is denoted dciv . The load bus consists of acapacitive filter (which includes both electrolytic andpolypropylene capacitance) as well as a 3-phase fullycontrolled inverter, which in turn supplies an inductionmotor. The induction motor drives the mechanical load,which is rotating at a speed imrm,ω . Based upon the

mechanical rotor speed, and the desired electromagnetictorque deseT , (which is determined by the controller

governing the mechanical system), the induction motorcontrols specify the on/off status of each of the invertersemiconductors in such a way that the desired torque isobtained. Although this system is quite robust withregard to over currents, and simple to design from theviewpoint that the controller governing the mechanicalsystem is decoupled from the control of the electricalsystem (since the torque can be controlled nearlyinstantaneously), such systems are prone to be subjectto a limit cycle behavior in the dc bus voltage known asnegative impedance instability [12].

Before setting forth the implementation of theproposed NSSC controller, it is appropriate to firstconsider a standard field oriented control such as therotor flux indirect field oriented control illustrated in

Figure 2 (note that the control proposed in this paper, is,however, independent of whether or not the fieldoriented control is direct or indirect). Therein, aninstantaneous torque command *

eT is the input to thecontroller. This torque command is equal to the torquedesired by the controller governing the mechanicaldynamics, deseT , . As can be seen, based on the torque

command *eT and desired d-axis rotor flux level *e

drλ , the

desired q- and d- axis stator currents, *eqsi and *e

dsi , are

determined. This calculation is a function of theinduction motor rotor magnetizing inductance mL , theinduction motor rotor inductance (rotor leakage plusmagnetizing) rrL' , the rotor resistance rr ' , and thenumber of poles. Based on the q- and d- axis statorcurrents the electrical radian slip frequency, ims ,ω , is

determined, which is then added to the electrical rotorspeed imr ,ω in order to determine the electrical speed of

the synchronous reference frame ime,Θ . In addition to

the algorithm illustrated in Figure 2, especially in largedrives, the field oriented control will often include an online parameter identification algorithm to compensate forvariations of the rotor time constant [13-14].

Figure 1. System Configuration

∑

Figure 2. Rotor Flux Oriented Indirect Field OrientedControl.

Once the q- and d- axis current commands and theposition of the synchronous reference frame areestablished, these currents may be synthesized in avariety of ways. Herein, the q- and d- axis currentcommand was transformed back into an abc variablecurrent command, which is an input to a hysteresis typecurrent control.

System Behavior – The performance of thepropulsion system was tested by ramping the desiredelectromagnetic torque of the induction motor from 2 to19 Nm over a period of 100 ms. Figure 3 depicts thecommanded a-phase current asi* , the actual a-phasecurrent asi , and the dc inverter voltage dciv . Theincrease in torque can be associated with the linearincrease in asi . It can be seen that as the powercommand increases the dc bus voltage becomesunstable, stressing both the semiconductors and thecapacitors. In a typical system such behavior couldeasily result in the semiconductor and/or capacitorfailure.

DC POWER SYSTEM - In order to investigate thecontrol of dc power systems, the small butrepresentative system depicted in Figure 4 was utilized,[1]. As can be seen, this 3.7kW system consists of ageneration system, a distribution system, and loads.The loads are the dc/dc converter, which is of specialconcern herein, and a permanent magnet synchronousmotor drive.

The generation system consists of a dynamometerthat acts as a prime mover, a 3-phase synchronousmachine, an LC filter, and a solid state exciter/voltageregulator. The output of the rectifier is filtered by an LCcircuit creating a nearly ripple free dc source. Thegeneration system output is connected to a generationbus, 1dcv , which is attached to the transmission line.The transmission line transfers energy to the load bus,

2dcv , which distributes power to the remainder of thesystem.

Figure 3. Performance of standard field oriented controlduring ramp increase in desired torque.

Vdc3Vdc2

Vdc1

Figure 4. DC Power system

The dc system has two loads, a permanent magnetsynchronous machine drive (brushless dc motor) and aBBC, Figure 5. The synchronous motor drive is used torepresent a propulsion load whereas the BBC representsa distribution type of load in which a dc/dc powerconverter is used to interface between two voltagelevels, and/or provide a voltage regulated bus from anunregulated dc source. The operation and parametervalues of this motor drive are as set forth in [17] with theexception that the current control was delta modulated ata frequency of 30 kHz rather than hysteresis modulated[18]. The modeling of this type of drive is set forth in[17].

The nominal converter control scheme utilizedherein consists of two separate feedback levels. Theouter level consists of a regulating nonlinear PIcontroller, Figure 6, used to maintain a constant outputvoltage. It consists of second order low pass filters usedto eliminate aliasing in the measured inputs and toremove discritization noise in the controller outputs. Thenonlinear block following the PI control converts theoutput current command to an input current command.System components are protected by limiting the rangeof the commanded input current following the controller.The conditional block is used to limit the valid operatingrange of the converter based on the level of thedistribution bus voltage, providing additional systemprotection. The regulating PI controller was designed

based on the linearized average value model of thesystem [15], resulting in controller gains of Kp and Ki

being equal to 0 and 3.1 respectively.

The inner level consists of a hysteresis currentcontroller that regulates the input current of the converterto with in plus or minus a given hysteresis level of thecommanded input current. Advantages of usinghysteresis current control are that current ripple isindependent of operating conditions and the tightregulation of the input current provides for highlyeffective current limiting. The two main disadvantages inusing this type of control are variable switchingfrequency and an undefined duty cycle, which makes theaverage value model difficult to derive as can be seen bythe almost complete avoidance of this type of control inliterature. However, recently an appropriate averagevalue model has been set forth [15 and 16] and is theapproach used herein.

System Behavior – Figure 7 depicts the loadvariables associated with the BBC for a step change inBBC load from 129.3 to 60.1 Ohms. Depicted are theoutput voltage of the BBC 3dcv , the distribution bus

Figure 5. Buck/Boost Converter Schematic

Figure 6. Nonlinear PI controller

0 0.1 0.2 0.3 0.4 0.5200

300

400

0 0.1 0.2 0.3 0.4 0.50

10

20

0 0.1 0.2 0.3 0.4 0.5300

400

500

vdc3

vdc2

idcdc

Time (sec)

Figure 7. Step in BBC load

voltage 2dcv , and the current entering the BBC dcdci . Itis shown that the system remains stable and recovers.The current entering the BBC is shown limiting as set bythe controls but also recovers as 2dcv approachessteady state.

Figure 8 depicts the load variables for a study inwhich the filter capacitance connected to the generationbus is stepped from 1315.5 µF to 1.4 µF. Note how

2dcv and dcdci begin to oscillate violently once thecapacitance is removed, demonstrating that the originalcontrol cannot operate without significant generation buscapacitance. This is not surprising in view of the factthat the control design assumed the presence of thegeneration bus capacitance. This raises the question ofhow much bandwidth would have to be sacrificed inorder to eliminate the generation bus capacitance fromthe system with the standard PI control.

As it turns out, the instability demonstrated in Figure8 was found to be uncorrectable by adjustment of theproportional integral control gains. In order todemonstrate this, the system nonlinear average valuemodel (NLAM) was linearized and the loci of systemroots were plotted as the gains kp and ki were variedbetween zero and nine million as shown in Figures 9 and10. Once the generation bus capacitance is removedthere are three pole pairs which can contribute to systeminstability. The shaded regions in Figure 9 (pole pair 1)shows where the first pole pair can be placed whileFigure 10 (pole pairs 2 and 3) shows the areas in whichone pole of each of the remaining two interesting polepairs can be placed. Each pole pair can be made stablebut it is impossible to move all poles into the left handplane simultaneously. A similar situation can arise withother converter configurations as well. This would seemto mandate that at least some level of generation buscapacitance must be present. However, this is not thecase. The NSSC algorithm introduced in the nextsection can achieve system stability when it isaugmented to the regulating control regardless of thegeneration bus capacitance without significantlydegrading the transient performance.

NONLINEAR SYSTEM STABILIZING CONTROLLER

The effect of regulating the constant power output ofthe IMD and BBC results in the input of each deviceappearing as a constant power, negative impedanceload. Negative impedance loads in many powersystems have a destabilizing effect known as negativeimpedance instability. The physical cause of, a measurefor prediction, and a means of mitigating this type ofinstability are set forth in this section.

NEGATIVE IMPEDANCE INSTABILITY - In order togain insight into the nature of negative impedanceinstability, consider the highly simplified representationof both systems depicted in Figure 11. Therein, thesource is modeled as an ideal source followed by a lowpass filter; the constant power load could be consideredas either the IMD or the BBC. It is assumed that theIMD as well as the BBC both compensateinstantaneously to changes in the dc link voltage, inv ,allowing them to be modeled as single dependentcurrent sources which are formulated by assuming thatthe input power is equal to the commanded input power,

*P .

vdc3

vdc2

idcdc

Time (sec)

0 0.1 0.2 0.3 0.4 0.5200

300

400

0 0.1 0.2 0.3 0.4 0.50

10

20

0 0.1 0.2 0.3 0.4 0.5300

400

500

Figure 8. Loss of Generation Bus Capacitance

-1 -0.5 0 0.5 1

x 105

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 x 105

Real Axis

Imag.Axis

Feasible Region forPole Pair 1

Figure 9. System Root Loci for Varying Kp and Ki

-2000 0 2000 4000

-2000

-1000

0

1000

2000

Real Axis

Imag.Axis

Feasible Region forPole Pair 2(Conjugate Area Not Shown)

Feasible Region forPole Pair 3(Conjugate Area Not Shown)

Figure 10. System Root Loci for Varying Kp and Ki

The stability of this simplified system can bedetermined by calculating its pole locations. Thedifferential equations governing the system can beexpressed as

piv R i v

Ldceqdceq eq dceq in

eq=

− − (1)

and

pvi

Cindceq

Pv

eq

in=− *

(2)

where p denotes differentiation with respect to time.Linearizing (1) - (2), finding the eigenvalues, anddetermining the conditions for which they are in the LHPyields the following necessary and sufficient conditionsfor stability:

1) PR C v

Leq eq ino

eq* <

2

(3)

and

2) PvRino

eq* <

2 (4)

Physically speaking, (4) is normally satisfied inpractice and so (3) is the most important constraint. It isconvenient to state the stability criteria in terms of thesmall signal input impedance of the constant power load.This impedance is defined as the linearized transferfunction between the constant power load input voltageand input current. In particular, for a constant powerload

ZvPin

ino= − 2

* (5)

Note that in a small signal sense the constant powerload appears as a negative resistance, which wouldsuggest a destabilizing effect. In terms of the linearinput impedance the stability criteria (3) may beexpressed

− >ZL

R Cineq

eq eq (6)

As *P increases inZ− decreases and so eventually thesystem becomes unstable. Equation (6) immediatelysuggests several methods for manipulating systemstability. First, increasing the capacitance to anappropriate level can insure stability. However, suchmeasures can be expensive in terms of capitol, space,weight and reliability. Alternatively, reducing eqL is also

a means of satisfying (6). However this technique islimited because eqL and eqR are both tied to the

subtransient inductances of the synchronousmachine/generator and the ratio is not readilymanipulated. Another method is to manipulate the inputimpedance of the converter/motor drive. This can beaccomplished by adding passive filters at the inputs,although this can again be an expensive solution.Herein an NSSC, Figure 12, is investigated which altersthe input impedance characteristics and adds noadditional cost. For the IMD drive ini* and inci* should

be replaced by edesT and eT * , respectively.

EFFECT OF STABILIZING CONTROL ON LOADIMPEDANCE - In this section the effect of the NSSC onthe load input impedance is explored. The commandedinput current of the converter can be written in terms ofthe desired input current as shown in Figure 12 or interms of the desired input power as

iv

vPinc

inn

n* *( )

=− 1

inf (7)

Assuming that the desired power is constant linearizingequation (7) yields

inf)1(inf0

)1(0

0inf

)2(0

*)(

*)1(*

vv

Pvn

vv

Pvni

n

nin

inn

nin

inc

∆−+

∆−=∆

+

−

−

(8)

From Figure 12,

∆ ∆v H s vinf in= ( ) (9)

This filter is designed such that 1)0( =H . Therefore

Figure 11. Simplified System Model

v vin0 inf0= (10)

Incorporating (9) and (10) into (8) yields

∆ ∆ ∆in P

vv

n P

vH s vinc in*

( ) * ( ) *( )= − + −1

2 2in0 in0

in (11)

The input admittance can then be determined about theoperating point, assuming that the actual input current isalways equal to the commanded input current. Inparticular,

)(*)(*)1(

*)(*

20

2in0

in

sHv

Pnv

Pnv

isY

in

incinc

−+−=

∆∆=

(12)

Inverting the admittance yields the input impedance:

[ ]Z sv

n H s Pincinc( )

( ( )) *=

− −

2

1 1 (13)

As can be seen, the NSSC offers many possibilities forinput impedance control by adjusting ‘n ’ and )(sH .First, setting ‘n ’ equal zero yields

Z svP

nincin( )*

( )= − =02

0 (14)

whereupon it can be seen that the stabilizing control hasno effect. If ‘n ’ is set equal to one, it can be seen thatwith proper choice of )(sH the input impedance can bereadily manipulated.

Zv

H s Pninc

in= − =02

1( ) *

( ) (15)

Setting ‘n ’ equal to two yields

[ ]ZvH s P

nincin=

−=0

2

1 22

( ) *( ) (16)

In this case, and for higher powers, it can be seen that‘n ’ acts as a gain on the filter. Although only integervalues have been considered herein, ‘n ’ is in the set ofreal numbers and does not have to be an integer. It isinteresting to observe that if )(sH is set equal to

dB02.6− over the frequency range in which thestability criteria is failing, infinite input impedance wouldoccur alleviating the problem. If )(sH continued to getsmaller in magnitude the input impedance would thenbecome positive.

TEST SYSTEMS WITH NSSC

INDUCTION MOTOR BASED ELECTRICPROPULSION SYSTEM - The advantage of using thissimple though nonlinear stabilizing control algorithm isthat it is extremely straightforward to implement yethighly effective in mitigating negative impedanceinstabilities. In order to illustrate the effect of thealgorithm on the system, note that using the control law,input power into the inverter is given by

des

n

dci

dci Pvv

P

= ~ (17)

where

rmdesedes TP ω,= (18)

From (24) the input current may be expressed

desdci

n

dcin

dci Pvv

i ~1−

= (19)

Linearizing (26) about the desired operating point( dcsdci vv *= ) yields

des

dcsdci P

vn

i2*

11−

= (20)

If the low pass filter time constant, τ , (used indetermining dciv~ ) were so great as to not interact withthe dc link dynamics and n is selected to be unity thenthe input impedance presented by the inverter is infinitefor the frequency range over which negative impedanceinstabilities occur, thus avoiding this type of instability.

In order to illustrate the effects of varying n and τ ,

consider the case of a system in which Vv dceq 400* = ,

Ω= 58.4eqR , mHLeq 9.13= , and FCeq µ4.51= .

These parameters correspond to a test system that wasused for laboratory verification. Figure 13 illustrates theroot loci the characteristic equation as τ is varied from

Figure 12. Nonlinear system stabilizing control

0.1 ms to 1s for n =1,3,5, and 7. As can be seen, ineach case the root locus contains an unstable complexpole (denoted A and A*) for small values of τ whichbecomes stable as τ is increased. For all n shown inFigure 13 the real part of the eigenvalues becomes morenegative as τ is increased. In addition, initially thecomplex part also decreases. In the case of 5=n ,eventually the complex pair becomes real (point B) andthen one of these real roots meets the rootcorresponding to the filter at point C, at which this pair ofeigenvalues becomes complex. In the case of 7=nthe two complex poles eventually become real at pointD, after which the pair moves away from each other onthe real axis.

System Behavior - Incorporating the link stabilizingcontrol into the field oriented control is quitestraightforward. In particular, the only difference in thecontrol is that the instantaneous torque command isgenerated using Figure 11 (with ini* , inci* , and inv

replaced by deseT , , eT * , and dciv respectively) rather

than being set equal to the desired torque, as isillustrated in Figure 2. The study performed on thepropulsion system earlier is repeated in Figure 14 exceptthat now the nonlinear stabilizing control is included.The link stabilizing control parameters were set to 1=nand ms4=τ , based on the rootlocus generated inFigure 13. As predicted, the dc bus voltage is wellbehaved and the dc link bus voltage is stable.

One concern which may arise is a possible reductionin torque bandwidth since a drop in inverter voltage willresult in a transient dip in torque. Using a detailedcomputer simulation this effect is depicted in Figures 15and 16 with and with out the stabilizing control,respectively. This study tests the performance of thefield oriented control to a step change in commandedtorque from 2 to 19Nm. As can be seen, theelectromagnetic torque, in Figure 15, reaches thecommanded value in approximately 5ms. The torqueresponse is not instantaneous due to the fact that a step

change in current cannot be achieved in practice andbecause the dip in link voltage causes a temporary lossof current tracking in the hysteresis current control.When the stabilizing controller is implemented, Figure16, the electromagnetic torque reaches the commandedvalue in the order of 8ms. Although the link stabilized

control is somewhat slower than the standard fieldoriented control, this slight reduction in bandwidth is nota significant disadvantage in view of the improved dcbus voltage. This is particularly true due to the fact thatmost propulsion systems have mechanical inertia suchthat in either case the torque response may beconsidered to be instantaneous.

2000

1500

1000

500

0

-500

-500 500

-1000

-1000

-1500

-1500-2000

0

Imag

inar

y Pa

rt

Real Part

Figure 13. Root Locus as τ and n are varied.

Figure 14. Measured performance of link stabilizedfield oriented control during ramp increase in desired

torque

Figure 15. Simulated performance of standard fieldoriented control during step change in desired torque.

Figure 16. Simulated performance of link stabilizedfield oriented control during step change in desired

torque.

DC POWER SYSTEM - The NSSC controller usedhere sets ‘n ’ equal to one leaving only the time constantin )(sH to be chosen, assuming a first order low passfilter with unity gain at dc. In order to facilitate a meansof making this choice a root locus as 'τ ' was varied wascalculated by linearizing the NLAM model. This wasdone with the bus filter capacitance effectively removedfrom the system. The dominant poles were then plottedin the complex plane creating a rootlocus in terms of thefilter time constant, Figure 17. From the root locus avalue of ‘ ms2=τ ’ was chosen that offered significantlyhigh damping but still maintained a fairly high cutofffrequency so that system stability was guaranteed andsystem performance degradation was minimized. Theresulting nonlinear PI controller with the augmentedNSSC is illustrated in Figure 18.

System Behavior – The studies presented earlier onthe DC power system are repeated in Figures 19 and 20except that now the stabilizing controller is included. InFigure 19 the almost identical transient response isobserved, compared to Figure 7. While in Figure 20 thesystem remains stable. Notice that the generation busvoltage does undergo increased variation; however thisis due to increased rectifier harmonics, since the sourcefilter capacitance has been effectively removed. The

stabilizing controller has the desired effect of maintainingsystem stability for conditions (very low generation buscapacitance) in which it was determined that aproportional integral control alone could not maintain

stability.

CONCLUSION

Power systems and electric propulsion systems withdc links are becoming more prominent in industry andthe military. This trend will continue as the need formore efficient and versatile methods of moving energyand designing drive systems progresses. A simplifiedmodel of a constant power load similar to what istypically found in many of these electric power/drivesystems was used to develop a stability criteria and anonlinear system stabilizing controller (NSSC).Verification of the control was accomplished usingtransient time domain studies on two such systems, aDC power system and electric propulsion system. It wasfound that the NSSC offered a means to guaranteesystem stability without sacrificing significant dynamicperformance or introducing extra passive components.In addition, the proposed strategy convenientlyseparates the component regulatory aspects of thecontrol from the negative impedance system stability

Figure 18. Nonlinear PI controller with attached NSSC

-500 -400 -300 -200 -100 0 100-2000

-1500

-1000

-500

0

500

1000

1500

2000

Figure 17. System Root Locus for Varying ‘τ’

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300

400

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10

20

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400

500

vdc3

vdc2

idcdc

Time (sec)

Figure 19. Step in BBC load with NSSC.

aspects.

ACKNOWLEDGMENTS

This research has been supported in part by the NavalSea Systems Command (Contracts N00024-93-C-4180and N61533-95-C-0107) and P.C. Krause andAssociates' "Electric drive and finite inertia power systemanalysis, modeling, simulation, and design Task 6". Thesupport and technical interest of Henry Hegner andHenry Robey, the technical monitors of this project, isgratefully acknowledged.

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0 0.1 0.2 0.3 0.4 0.5200

300

400

0 0.1 0.2 0.3 0.4 0.50

10

20

0 0.1 0.2 0.3 0.4 0.5300

400

500

vdc3

vdc2

idcdc

Time (sec)

Figure 20. Loss of generation bus capacitance withNSSC