Figure 4-1. Naval Combat Survivability Testbed. 4.1 CLASSIFICATION OF SINGLE PORT POWER CONVERTERS

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1 4 / System Analysis Thus far, our consideration of immittance based stability analysis has been limited to simple one source, one load, problems. In this chapter, we consider immittance based stability analysis of systems of power converters, such as the laboratory power system shown in Figure 4-1 [1,2]. This system, referred to as the Naval Combat Survivability Testbed, was constructed at Purdue University and the Univeristy of Missouri-Rolla in the time frame, and is representative of some notional ship power systems. Therein, PS-1 and PS-2 denote power supplies for the port and starboard bus, CM-P1, CM-P2, and CM-P3 are port bus converter modules, CM-S1, CM-S2, and CM-S3 are starboard bus converter modules. These converter modules reduce the voltage from the bus to each load and provide current limiting functions. Net, the converter marked IM is an inverter which supplies an AC bus and a load bank (LB), the MC is a motor controller, and CPL is a generic constant power load. In this chapter, the steps necessary to conduct the stability analysis of the entire system using the generalized immittance space based approach is set forth. Of particular interest in this process is the classification of power converters for stability analysis, and the derivation of mapping functions to facilitate network reductions. Figure 4-1. Naval Combat Survivability Testbed. 4.1 CLASSIFICATION OF SINGLE PORT POWER CONVERTERS The first step in the stability analysis of the systems is to categorize the components which make up the system. These components generally fall into two classes single port converters in which there is a single dc terminal, and two-port 4/4/2005 Copyright 2003 S.D. Sudhoff Page 40

2 converters in which there are two sets of terminals. Within each of the main groups, there are sub-classifications which will be important in the analysis of the system. In this section, we address single port converters which posses a single pair of dc terminals. In our work thus far, we have already addressed the two types of single port converters a source, which we will abbreviate herein as an S converter, and a load, which we refer to as a L converter. Recall that the definition of a source, or S converter, and load, or L converter, do not necessarily correspond to a component that is supplying or consuming power. This is the reason that in this chapter the S and L converter notation is used. For the purpose of defining the input output relationships for single port converters, define v as the small signal voltage across component and i as the small signal current into the positive terminal (passive sign convention) of component, where is a subscript that defines the component. Using these definitions, the two types of single port converters are reviewed below. S Converters S Converters often, but not always, represent sources. In the sample system of Fig. 4-1, the two power supplies PS-1 and PS-2 will be categorized as S Converters. These components obey the small-signal relationship v = S (4.1-1) i where S is the impedance, and which may be epressed S N S, DS, = (4.1-2) and where N S, and D S, represent the numerator and denominator of S S,. The defining property of S converters is that they are stable when the current i is fied the implication of this being that all the roots of D s, are in the open left-half plane. Comparing this work to that of the previous chapters, we see that S is the source impedance looking to the S converter, and so is nothing more or less than Z s. However, the new notation will be useful in that it will help us to separate what physically represents a source from our stability definition. L Converters L Converters often, but not always, represented loads. In our sample system of Fig , the inverter module and load band (IM and LB), motor 4/4/2005 Copyright 2003 S.D. Sudhoff Page 41

3 controller MC, and constant power load CPL will all fall into the class of L Converters. These components obey the small-signal relationship i = L (4.1-3) v where L is the admittance, which may be epressed N L, L = (4.1-4) DL, and where N L, and D L, represent the numerator and denominator of L. The defining property of L converters is that they are stable when the voltage v is fied the implication of this being that all the roots of D L, are in the open left-half plane. Again, casual inspection of L will reveal that it is simply the load admittance formally denoted Y l. However, this new notation will be useful in separating what physically constitutes a load from the mathematical definition we will use for stability analysis. 4.2 CLASSIFICATION OF MULTI-PORT POWER CONVERTERS Two port converters may be categorized by those converters that have a two pairs of dc terminals. For the purpose of defining the input output relationships for two port converters, define v, 1 and v, 2 as the small signal voltage across terminals 1 and 2 of component and i, 1 and i, 2 as the current into the positive nodes of terminals 1 and 2 (passive sign convention) of component, where is a subscript that defines the component. Using these definitions, the various types of two port converters are defined below. C Converter A C Converter, which usually represents a cable, is a passive element that obeys the relationship i1, 2, ( 1, 2, = i = C v v ) (4.2-1) where C NC, DC, = (4.2-2) 4/4/2005 Copyright 2003 S.D. Sudhoff Page 42

4 All roots of D, must be in the open left-half plane for the component to be c categorized as a C converter. Z Converter Z Converters often represent two-port components which provide power from both sets of terminals. The input-output relationship definition is that v1, Z11, = v2, Z21, Z12, i1, Z 22, i2, (4.2-3) where Z Z 11, 21, Z Z 12, 22, N N = Z11, Z 21, D N N Z, Z12, Z 22, (4.2-4) and where N Zab, and D Z, represent the ab th element of the numerator and denominator of the impedance matri. For a converter to be a Z converter, it is necessary that all the roots of D Z, be in the open left-half plane. Physically, this means that to be classified as a Z converter, the two port converter must be stable when its two terminal currents are held constant. Y Converter Y Converters often represent two-port components which are fed power from both sets of terminals. In the sample system of Fig. 4-1, CM-P2, CM-P1 and MC can together be viewed as a Z Converter. The input-output relationship definition is that i i 1, 2, Y = Y 11, 21, Y Y 12, 22, v v 1, 2, (4.2-5) where Y Y 11, 21, Y Y 12, 22, N N = Y11, Y 21, D N N Y, Y12, Y 22, (4.2-6) and where N Yab, and Y D, represent the ab th element of the numerator and denominator of the impedance matri. For a converter to be a Y converter, it is 4/4/2005 Copyright 2003 S.D. Sudhoff Page 43

5 necessary that all the roots of Y Z, be in the open left-half plane. Physically, this means the to be classified as a Y converter, the two-port converter must be stable if supplied from two ideal voltage sources. H Converter A two-port converter which accepts power from one set of terminals and transfers it to the second set of terminals is often categorized as an H converter. An eample of an H Converter will by the converter modules in Fig The inputoutput relationship for the H converter is: where H H i v 11, 21, 1, 2, H = H H H 12, 22, 11, 21, N N = H H 12, 22, H11, H 21, D v i 1, 2, N N H, H12, H 22, (4.2-7) (4.2-8) and where N Hab, and D H, represent the ab th element of the numerator and denominator of the H matri. For a converter to be an H converter, it is necessary that all the roots of D H, be in the open left-half plane. Physically, this means the to be classified as an H converter, the two-port converter must be stable if the port 1 voltage and port 2 current are constant. H Converter The H converter is a special case of the H converter in which it is possible to parallel the port 2 terminals with another H converter. Mathematically, it is a H converter in which the roots of N H 12, are in the open left-half plane (in addition to the roots of D, being in the open left-half plane). H 4/4/2005 Copyright 2003 S.D. Sudhoff Page 44

6 4.3 NETWORK REDUCTIONS Thus far, all of our work has been based on the analysis of a simple source load system. Our objective in this chapter is, however, to analyze system such as the one in Fig The application of the generalized immittance based method to interconnected systems is basically one of reducing the system, by a series of mapping functions, to a single source / single load equivalent. At each application of a mapping function, two or more components are grouped together into a single component. Often, a stability test will be required to determine whether or not the subsystem being grouped is stable as part of the mapping process. If it is not, then system stability cannot be guaranteed. A description of the most commonly used mapping functions follows. The use of the mapping functions to transform a complicated interconnected system into a simple source load system will then by illustrated by eample in the net section. SC to S Mapping Figure depicts a situation in which S converter connected to C converter y may be represented as a equivalent S converter e. Figure SC to S Mapping. Clearly, the impedance of the effective source may be readily calculated as S e S + C y = (4.3-1) 4/4/2005 Copyright 2003 S.D. Sudhoff Page 45

7 It should be kept in mind that the addition in (4.3-1) is generalized so that the generalized impedance of S E includes any impedance in S plus any impedance in C y. This operation is always valid and defined and thus no stability test is required. LC to L Mapping Figure depicts a situation in which L converter connected to C converter y may be represented as an equivalent L converter e. Figure LC to S Mapping. The admittance of the effective L converter may be epressed Le L + LC y = 1 (4.3-2) where all mathematical operations are generalized. Note that for this reduction to be valid, then the system comprised of C y as a source (the other end of the C converter is treated as an ideal source) and L as a load must be stable. This guarantees that the aggregation of the two components will satisfy the conditions required to be categorized as an L converter. Thus a subsystem stability analysis must be performed prior to making this reduction. If the subsystem is stable the analysis may proceed (note it does not have to satisfy any particular stability criterion, it merely has to be stable by the smallest degree). If the subsystem is not stable, then the reduction cannot be accomplished and the system as a whole cannot be guaranteed to be stable. 4/4/2005 Copyright 2003 S.D. Sudhoff Page 46

8 HL to L Mapping Figure illustrates the mapping of H converter feeding L converter y into an effective load converter L e. Figure HL to L Mapping. The effective L converter admittance may be epressed Le H12, H21, Ly H11, 1+ H22, Ly = (4.3-3) In order for this operation to be valid, then a subsystem with H 22, as a source, and Ly as a load must be stable. If this is not the case, system stability cannot be guaranteed and further analysis along this path is not valid. SH to S Mapping The mapping of S converter connected to H converter y into an equivalent S converter S e is illustrated in Fig The effective source impedance may be epressed S H 21, y H12, y S e H 22, y 1+ S H11, y = (4.3-4) 4/4/2005 Copyright 2003 S.D. Sudhoff Page 47

9 Figure SH to S Mapping. The subsystem formed by S as a source and H 11, y as a load must be shown to be stable (thought not necessarily to satisfy a given stability criterion) for this operation to be valid. Parallel L to L Mapping Paralleling an arbitrary number of L converters is depicted in Fig Therein L converter α, L-converter β through L converter L Ω are combined to form an effective L converter admittance L e given by L e = Lα + Lβ + + LΩ (4.3-5) 4/4/2005 Copyright 2003 S.D. Sudhoff Page 48

10 Figure Parallel L to L Mapping. No stability test is required this operation is always valid. As in the case with all mathematical operations in this discussion, it should be kept in mind that the basic algebraic operations of addition, subtraction, multiplication, and division are modified to operate on generalized sets. Parallel Y to Y Mapping Figure illustrates the mapping of Y converter α through Y converter Ω into an effective Y converter e whose Y parameters are given by Y e = Y + Yβ + + YΩ α (4.3-6) 4/4/2005 Copyright 2003 S.D. Sudhoff Page 49

11 Figure Parallel Y to Y Mapping. As is the case of the Parallel L to L Mapping, this operation is always valid no stability test is required. HLH to Y Mapping In zonal power systems, there is often a need to collapse an entire zone or part of a zone into a single equivalent Y converter. The HLH to Y mapping depicted in Fig illustrates the procedure by which H converter α, H - converter β, and L converter are combined to form an effective Y converter Y e. In terms of the effective converter, i1, α Y11,e Y12,e v1,a = (4.3-7) i1, β Y21,e Y22,e v1, β 4/4/2005 Copyright 2003 S.D. Sudhoff Page 50

12 where ( 1+ L H ) H12, α H 21, α 22, β Y11, e = H11, α H12, β + H 22, α + L H 22, α H 22, β (4.3-8) H12, α H 21, β Y12, e H12, β + H 22, α + L H 22, α H 22, β (4.3-9) H 21, α H12, β Y21, e H12, β + H 22, α + L H 22, α H 22, β H12, β H 21, β ( 1+ L H 22, α ) Y22, e = H11, β (4.3-11) H12, β + H 22, α + L H 22, α H 22, β Figure HLH to Y Mapping. 4/4/2005 Copyright 2003 S.D. Sudhoff Page 51

13 For this operation to be valid two stability tests must be met. First, the subsystem 1 comprised of H 22, α as a source and H 22, β as a load must be stable. Second, the subsystem comprised of the parallel combination of H 22, α and H 22, β as a source and L as a load must be stable. YS to L Mapping Figure depicts the mapping of Y converter connected to S converter y into an effective L converter e. The load admittance of the effective L converter is given by S yy12, Y21, y Le = Y11, (4.3-12) 1+ Y S 22, y Figure YS to L Mapping. The validity of this mapping is contingent upon the subsystem consisting of source and Y 22, as a load being stable. S y as a Y to L Mapping The final mapping function depicted herein is that depicted in Fig In this case, the two ports of the Y converter are tied together. The load admittance of the effective L converter L is given by This operation is always valid. e L e Y11, + Y12, + Y21, + Y22, = (4.3-13) 4/4/2005 Copyright 2003 S.D. Sudhoff Page 52

14 Figure Y to L Mapping. 4.4 CASE STUDY PART 1: SYSTEM DESCRIPTION In order to illustrate the use of these concepts and their incorporation into a system analysis, let us perform a stability analysis on the system shown in Figure 4-1. Before we can do this, however, we need to set forth a more detailed description of the system and the mathematical models used to represent it. The NCS Testbed depicted in Figure 4-1 will be the eample system for this work. In this system, there are two power supplies (PS-1 and PS-2), that can each be fed from an independent power source. One power supply feeds the port bus, and the other feeds the starboard bus. There are three zones of dc distribution. Each zone is fed by a converter module (CM) on the port bus (CM-P1,CM-P2, or CM-P3) and a converter module on the starboard bus (CM-S1, CM-S2, or CM-S3). Diodes prevent a fault from one bus from being fed by the opposite bus. The converter modules feature a droop characteristic so that they can share power. The three loads consist of an inverter module (IM) which in turn feeds an ac load bank (LB), a motor controller (MC), and a generic constant power load (CPL). Robustness in this system is achieved as follows. First, in the event that either a power supply fails, or a distribution bus is lost, then the other bus can pick up full system load without interruption in service. Faults between the converter module and diode are mitigated by imposing current limits on the converter modules; and again the bus opposite the fault can supply the component. Finally, 4/4/2005 Copyright 2003 S.D. Sudhoff Page 53

15 faults within the components are mitigated through the converter module controls. The result is a highly robust system. The discussion of the system components will begin with the power supply. The power supplies PS-1 and PS-2 are identical, but have three operating modes. For the studies to be considered herein, the uncontrolled rectifier mode will be considered. A schematic of the power supply is depicted in Figure Figure Power Supply Schematic. The nonlinear average value model (NLAM) [4,5] equivalent circuit of the power supply appears in Figure Figure Power Supply Equivalent Circuit. Variables of interest which are not directly defined by the figures include the rms primary line-to-line voltage, v pll, the radian source frequency ω e, the transformer primary to secondary turns ratio, n ps, and the transformer commutating inductance L c (the sum of the primary and secondary leakage inductances referred to the secondary winding). Parameter values for v pll, ω e, n ps, and L c are V, rad/s, 1.30, and 1.24 mh, respectively. Note that since generalized parameters are used, all combinations of input voltage and frequency are considered in the analysis. The circuit diagram for the converter modules along with circuit element values is depicted in Fig /4/2005 Copyright 2003 S.D. Sudhoff Page 54

16 Figure Converter Module Schematic. The control algorithm is illustrated in Figure Figure Converter Module Control. The principal variables not defined by Figure and Figure are the commanded output voltage v out * and the commanded inductor current i l *. This current command, in conjunction with the measured current i l is used by a hysteresis modulator so that the actual current closely tracks the measured current. The transfer function of the stabilizing feedback H sf (s) [6,7] is given by H sf ( s) = K sf ( τ sf 1 τ sf 1s s + 1)( τ sf 2 s + 1) (4.4-1) Parameter values are listed in Table The final two values listed therein, v in, mn and v in, m, represent the minimum and maimum epected input voltage this is used for determining the generalized immittance description of the converter. The numbers are based on the range of power supply parameters and input and load conditions. 4/4/2005 Copyright 2003 S.D. Sudhoff Page 55

17 Table Converter Module Control Parameters τ invc = ms τ invout = ms τ iniout = 0. 17ms d = 1A/V K = AV K = 197 AS/V pv K sf = 0.1 τ sf 1 = 20 ms τ sf 2 = 4 ms K ii = 15.7 ka/s * v outma = 20 V i limit = 20 A iintlim = 2 A v in, mn = 548 V v in, m = 450 V iv For the purposes of this stability analysis the inverter module (including its load bank), motor controller, and constant power load are treated as a capacitor with capacitance C and effective series resistance r in parallel with a ideal constant power load of p as depicted in Fig Parameters for the three loads are set forth in Table The nominal input voltage v 0, and power p are included in this table as a range; every possible value within this range in every combination will be considered. Figure Load Model. Table Load Converter Parameters Component C, µf r, mω v 0, V p, kw IM MC CPL /4/2005 Copyright 2003 S.D. Sudhoff Page 56

18 4.5 CASE STUDY PART 2: SYSTEM ANALYSIS With the basic tools, network reduction mapping functions, and system description in place, the stability analysis of the system shown in Figure 4-1 can now be conducted. In order to facilitate this process, first consider Figure 4.5-1, which again depicts the system architecture, but focuses on the classification of the components of the system. Therein, S ps represents the generalized impedance of the power supplies, H cm (which will be shorthand for the generalized H-parameters of the converter modules that is H 11, cm, H 12, cm, H 21, cm, and H 22, cm ), and L im, L mc, and L cpl the generalized admittance of the inverter module, the motor controller, and constant power load respectively. Figure NCS Testbed (Showing Converter Classification) As stated previously, the basic approach to system wide stability analysis is to methodically reduce the system to a single-source single-load system, checking to make sure that each reduction is valid. The mathematical manipulation of the generalized immittances is readily accomplished using a Matlab toolbo [8]. The first set of reductions to be made in this case is to reduce each zone to an equivalent Y-Converter. To this end, consider Zone 1. Our first step will be to combine the converter module P1, converter module S1, and the inverter module IM into an equivalent Y converter characterized by the generalized Y-parameters Y 11, z1, Y 12, z1, Y 21, z1, and Y 22, z1. To do this, we must first eamine the stability of a source with generalized 1 impedance H 22, cm and a load with generalized load admittance H 22, cm. This is illustrated in Fig , wherein generalized admittance H 22, cm is plotted in along 4/4/2005 Copyright 2003 S.D. Sudhoff Page 57

19 1, cm with a constraint based upon H 22 in conjunction with the ESAC stability criterion with a 30 db gain margin and a 30 degree phase margin. In this figure, hybrid coordinates are used. These coordinates constitute a mied linear / logarithmic representation. Mathematically, if and y denote the real and imaginary part of an immittance at a frequency, then the representation of this point in hybrid coordinates is given by + jy 2 2 if + y 20 B / jy 2 2 ' + jy' = 20log + y + 20 B if (4.5-1) y y 20 where B is a breakpoint in db. In this system if a comple number has a magnitude less than B db, it is in the linear region. A number with a magnitude of B db will always have a magnitude of 20 in this coordinate system. Finally, for numbers with a magnitude greater than B db will have a magnitude equal to its db value plus 20 B (the offset is for continuity). Figure Admittance Space Plot of CM to CM Interface (Testing 1 H 22, cm of CM-P1 as load admittance with admittance constraint based on H 22, cm of CM-S1 as a source). 4/4/2005 Copyright 2003 S.D. Sudhoff Page 58

20 Upon eamination of Fig , it is seen that H 22, cm does not intersect the forbidden region thus stability of this load source system is guaranteed. Recall that to make a H LH to Y conversion, a second stability test is required. In particular, given the subsystem consisting of H 22, cm of the CM-P1 in parallel with H 22, cm (from the CM-S1, which is identical to that of the CM-P1) as a source and L im as a load must be stable. This admittance constraint based on this effective load and the chosen stability criterion (the ESAC criterion, with a 3 db gain margin and 30 degree phase margin) is illustrated in Fig Figure Admittance Space Plot of Dual CM to IM Interface (Testing L as load admitance with admittance constraint based im H, cm on 22 of CM-P1 in parallel with H 22, cm of CM-S1 as a source). The reduction of the other two zones to Y converters is identical and is not shown for brevity. After this first round of system reduction, the resulting reduced system has the topology shown in Figure As indicated therein, the net step is to conduct a parallel Y to Y mapping. This step does not require a subsystem analysis. The result is the further reduced system shown in Figure /4/2005 Copyright 2003 S.D. Sudhoff Page 59

21 Figure System After HLH to Y Mapping. Figure System After Parallel Y to Y Mapping. The net step in the reduction is to reduce the effective Y converter representing the three zones (with admittance Y 3z and the starboard power supply PS-2 with impedance S ps to an effective load converter with admittance L 3 zps. This is accomplished with a YS to L mapping, an operation that requires that a subsystem with a source with impedance S ps and a load with admittance Y 22, 3 z to be stable. Figure depicts this stability test; therein a load admittance constraint based on source impedance S ps and the ESAC stability criterion is plotted along with the load admittance. As can be seen there is no intersection and hence the 4/4/2005 Copyright 2003 S.D. Sudhoff Page 60

22 system is stable. Thus the YS to L Mapping can be used to reduce the system shown in Fig to that shown in Figure Figure Admittance Space Plot of Three Zones / PS-2 Interface (Testing Y 22, 3z as a load admittance with admittance constraint based on S as a source). ps Figure System After YS to L Mapping. In Fig , the entire system has been reduced to two components. The source is PS-1 has a generalized source impedance S ps. The load represents the aggregation of the remainder of the system the three zones plus the starboard power supply and has an admittance L 3 zps. In Fig 4.5-8, the generalized admittance L 3 zps is shown along with the admittance constraint calculated from the source impedance S ps in conjunction with the selected stability criterion. As can be seen, there is no intersection and hence the system is stable in a small-signal sense for all operating points. 4/4/2005 Copyright 2003 S.D. Sudhoff Page 61

23 Figure Admittance Space Plot of PS-1 / Rest of System Interface. (Testing L 3 zps as a load admittance with admittance constraint based on S ps as a source). In order to see a situation in which an instability can arise, let us consider the same system with the following modifications: (1) the output capacitance of the two power supplies is removed, (2) the input capacitance of all of the converter modules ( C in Figure 4.4-3) is reduced to 100 µf, and (3) the stabilizing gain in K sf (see Table 4.4-1) is reduced to zero. The analysis proceeds much as before; however when the analysis reaches the point where the interface of the 3 zone equivalent to the starboard power supply ( Y 22, 3 z as a load, S ps as a source), it is found that stability of this subsystem (three zones and the starboard power supply) cannot be guaranteed. This is illustrated in Fig wherein the admittance Y 22, 3 z enters the forbidden region calculated from. Thus stability of this subsystem cannot be guaranteed. Note that further S ps analysis cannot be conducted because the 3-zone equivalent / starboard power supply do not constitute a valid L converter. 4/4/2005 Copyright 2003 S.D. Sudhoff Page 62

24 Figure Admittance Space Plot of Three Zones / PS-2 Interface for Modified System (Testing L 3 zps as a load admittance with admittance constraint based on S ps as a source). 4.6 CLOSING REMARKS In this chapter, the methodology to apply immittance based stability analysis was set forth. This began with a classification scheme for power converters in terms of their stability properties. Net, a variety of network reductions were introduced which allow a comple system to be reduced to a simple one. The chapter concluded with an eample. It should be noted that the eample analysis presented here is considered and viewed in terms of measured system behavior. It was found that the techniques shown here were consistent with eperimentally observed behavior. 4.7 ACKNOWLEDGEMENTS A monograph supported by grant N , National Naval Responsibility for Naval Engineers: Education and Research for the Electric Naval Engineer 4/4/2005 Copyright 2003 S.D. Sudhoff Page 63

25 4.8 REFERENCES [1] S.D. Sudhoff, S.D. Pekarek, B.T. Kuhn, S.F. Glover, J. Sauer, D.E. Delisle, Naval Combat Survivability Testbeds for Investigation of Issues in Shipboard Power Electronics Based Power and Propulsion Systems, proceedings of the IEEE Power Engineering Society Summer Meeting, July 21-25, 2002,Chicago, Illinois, USA [2] S. Pekarek, S.D. Sudhoff, D.E. Delisle, J. Sauer, E.J. Zivi, Overview of a Naval Combat Survivability Program, Paper 233, 7-9 April 2003, Orlando, Florida, USA [3] 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. [4] S.D. Sudhoff, S.F. Glover, Modeling Techniques, Stability Analysis, and Design Criteria for DC Power Systems with Eperimental Validation, Proceedings of the 1998 SAE Aerospace Power Systems Conference, pp [5] S.D. Sudhoff and K.A. Corzine, H.J. Hegner, D.E. Delisle, Transient and Dynamic Average-Value Modeling of Synchronous Machine Fed Load- Commutated Converters, IEEE Transactions on Energy Conversion, Vol. 11, No. 3, September 1996, pp [6] S.D. Sudhoff et. al., Control of Zonal DC Distribution Systems: A Stability Perspective, Sith IASTED International Multi-Conference On Power and Energy Systems, May 12-15, 2002, Marina del Rey, California, USA [7] 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 [8] S.D. Sudhoff, DC Stability Toolbo Version 2.1, January 4, 2002, Purdue University. [9] S.P. Pekarek et. al., Development of a Testbed for Design and Evaluation of Power Electronic Based Generation and Distribution System, SAE2002 Power Systems Conference (Paper Offer #: 02PSC-28), October 29-31, 2002, Coral Springs, Florida, USA 4/4/2005 Copyright 2003 S.D. Sudhoff Page 64