a brushless exciter model incorporating multiple rectifier modes and preisach's hysteresis...

136 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006

A Brushless Exciter Model Incorporating MultipleRectifier Modes and Preisach’s Hysteresis Theory

Dionysios C. Aliprantis, Member, IEEE, Scott D. Sudhoff, Senior Member, IEEE, and Brian T. Kuhn, Member, IEEE

Abstract—A brushless excitation system model is set forth thatincludes an average-value rectifier representation that is valid forall three rectification modes. Furthermore, magnetic hysteresisis incorporated into the -axis of the excitation using Preisach’stheory. The resulting model is very accurate and is ideal forsituations where the exciter’s response is of particular interest.The model’s predictions are compared to experimental results.

Index Terms—Brushless rotating machines, magnetic hys-teresis, modeling, simulation, synchronous generator excitation,synchronous generators.

I. INTRODUCTION

BRUSHLESS excitation of synchronous generators offersincreased reliability and reduced maintenance require-

ments [1], [2]. In these systems, both the exciter machine andthe rectifier are mounted on the same shaft as the main alternator(Fig. 1). Since the generator’s output voltage is regulated bycontrolling the exciter’s field current, the exciter is an integralpart of a generator’s control loop and has significant impact ona power system’s dynamic behavior.

This paper sets forth a brushless exciter model suitable foruse in time-domain simulations of power systems. The analysisfollows the common approach of decoupling the main generatorfrom the exciter–rectifier. Because of the large inductance of agenerator’s field winding, the field current is slow varying [3],[4]. Therefore, the modeling problem may be reduced to thatof a synchronous machine (the exciter) connected to a rectifierload.

For power system studies, detailed waveforms of rotating rec-tifier quantities are usually not important (unless, for example,diode failures [5] or estimation of winding losses are of in-terest). Moreover, avoiding the simulation of the internal rec-tifier increases computational efficiency and reduces modelingcomplexity [6], [7]. The machine-rectifier configuration may beviewed as an ac voltage source in series with a constant com-mutating inductance [8]; however, this overly simplified modeldoes not accurately capture the system’s operational character-istics [9]–[13]. The widely used brushless exciter model pro-posed by the IEEE represents the exciter as a first-order system

Manuscript received October 28, 2003; revised September 29, 2004. Thiswork was supported by the “Naval Combat Survivability” effort under GrantN00024-02-NR-60427. Paper no. TEC-00312-2003.

D. C. Aliprantis is with the Greek Armed Forces (e-mail:[email protected]).

S. D. Sudhoff is with the Department of Electrical and Computer Engi-neering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail:[email protected]).

B. T. Kuhn is with the SmartSpark Energy Systems, Inc., Champaign, IL61820 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TEC.2005.847968

Fig. 1. Schematic of a brushless synchronous generator.

[14]; it was originally devised for small-signal analyses andits applicability to large-disturbance studies remains question-able [15]. An average-value machine-rectifier model that allowslinking of a -axes machine model to dc quantities was derivedin [16]. This model is based on the actual physical structure ofan electric machine and maintains its validity during large-tran-sient simulations.

In this paper, the theory of [16] (which covered only mode Ioperation) is extended to all three rectification modes [17].This is necessary for brushless excitation systems, because theexciter’s armature current—directly related to the generator’sfield current—is strongly linked to power system dynamics[3]. During transients, the rectifier’s operation may vary frommode I to the complete short-circuit occurring at the end ofmode III [6]. The exciter–rectifier configuration is analyzed onan average-value basis in a later section.

The incorporation of ferromagnetic hysteresis is an additionalfeature of the proposed model. Brushless synchronous gener-ators may use the exciter’s remanent magnetism to facilitateself-starting, when no other source is available to power thevoltage regulator. However, the magnetization state directly af-fects the level of excitation required to maintain a commandedvoltage at the generator terminals. Hence, representation of hys-teresis enhances the model’s fidelity with respect to the voltageregulator variables.

Hysteresis is modeled herein using Preisach’s theory [18],[19]. The Preisach model guarantees that minor loops close tothe previous reversal point [20]–[22]. This property is essen-tial for accurate representation of the exciter’s magnetizing pathbehavior. Hysteresis models that do not predict closed minorloops, such as the widely used Jiles–Atherton model [23], arenot appropriate. To see this, consider a brushless generator con-nected to a nonlinear load that induces terminal current ripple.

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ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES 137

Fig. 2. Interconnection block diagram (input–output relationships) for theproposed model.

This ripple transfers to the exciter’s magnetizing branch current,and in the “steady-state,” a minor loop trajectory is traced on the

plane. If the loop is not closed, the flux can drift away fromthe correct operating point.

This paper begins with a notational and model overview.Next, a brief review of Preisach’s theory is set forth. Thenmodel development begins in earnest, with the developmentof the Preisach hysteresis model, a reduced-order machinemodel, and the rotating-rectifier average-value model. Thepaper concludes with a validation of the model by comparisonto experimental results.

II. NOTATION AND MODEL OVERVIEW

Throughout this work, matrix and vector quantities appear inbold font. The primed stator quantities denote referral to therotor through the turns ratio, which is defined as the ratio ofarmature-to-field turns . The electrical rotorposition and speed are times the mechanical rotorposition , and speed where is the number of poles.The analysis takes place in the stator reference frame (since thefield winding in the exciter machine is located on the stator).The transformation of rotating to stationary variables isdefined by [24]

(1)

where1

(2)Since a neutral connection is not present, .

The components of the proposed excitation model are shownin Fig. 2. The exciter model connects to the main alternatormodel through the field voltage and current ; it alsorequires . The voltage regulator model provides the voltageto the exciter’s field winding , and receives the current

. The exciter model is comprised of three separate models,namely, the rotating-rectifier average-value model, the Preisachhysteresis model, and the reduced-order machine model.

1The minus sign in the second row and the apparent interchange of the secondand third columns from Park’s transformation (as defined in [24]) arises fromusing a counter-clockwise positive direction for the rotor position coupled withthe location of the ac windings on the rotor.

Fig. 3. Illustrations of the elementary magnetic dipole characteristic and theboundary on the Preisach domain.

The rotating-rectifier average-value model computes theaverage currents flowing in the exciter armature , based on

, the voltage-behind-reactance (VBR) -axis flux linkageand the (varying) VBR -axis inductance . (The -axis

VBR inductance is also used, but is considered constant.) Thesevoltage-behind-reactance quantities are computed from thereduced-order machine model. The hysteresis model performsthe computations and bookkeeping required to use Preisach’shysteresis theory. Its only input is the -axis magnetizing cur-rent ; its output is the incremental magnetizing inductance

that represents the slope of the hysteresis loop at a giveninstant. The integrations of the state equations are performedinside the reduced-order machine model block. The states are

and the -axis magnetizing flux . The aforementionedvariables will be defined formally in the ensuing analysis.Notice that the proposed model is applicable whether hysteresisis represented or not; in case of a linear magnetizing path, thehysteresis block is replaced by a constant inductance term.

III. HYSTERESIS MODELING USING PREISACH’S THEORY

Preisach’s theory of magnetic hysteresis is based on the con-cept of elementary magnetic dipoles (also called hysterons).These simple hysteresis operators may be defined by their “up”and “down” switching values and , respectively (Fig. 3).Equivalently, they may be defined by a mean value

and a loop width .The behavior of a ferromagnetic material may be thought

to arise from a statistical distribution of hysterons. The func-tion which describes the density of hysterons is known as thePreisach function. It is defined on and is denoted byor , depending on which set of coordinates is used. ThePreisach function is zero everywhere except on the shaded do-main of Fig. 3. To explain the shape of this region, it is firstnoted that . The other constraints originate from the ob-servation that a finite applied field will fully saturate thematerial. Thus, all dipoles must obey . Consid-eration of saturation in the opposite direction yields

. These three inequalities lead to the triangular domaindepicted in Fig. 3.

The domain is divided into two parts: the upper partcorresponds to dipoles with negative magnetization; the lowerpart , corresponds to positive magnetization. A value for



Fig. 4. Visualization of Preisach diagrams. (a) Increasing magnetic field.(b) Decreasing magnetic field.

the total magnetization of the material may be obtained bytaking into account the contribution of all elementary dipoles.Hence, the magnetization is

(3)

The formation of the domain’s boundary may be visualizedusing the Preisach diagram, as shown in Fig. 4. First, assume thatthe magnetic field has the value and is increasing, forcing alldipoles with upper switching point to switch to the plusstate. The switching action is graphically equivalent to the cre-ation of a sweeping front, represented by a line perpendicularto the -axis, that moves toward increasing . The shaded areathat the front sweeps past becomes part of . When the fieldis decreasing, dipoles with a lower switching point areforced to switch to the negative state. A new front is created, thistime perpendicular to the -axis and moving toward decreasing

, claiming the area from and adding it to . The re-sulting boundary is formed by orthogonal line segments and isoften termed a “staircase” boundary. The shape of the boundarydepends on the history of the magnetic field.

The Preisach model possesses the deletion and the congru-ency properties. According to the deletion property, magnetichistory is completely erased when the front sweeps past pre-vious reversal points. This property is responsible for the cre-ation of closed minor loops. The congruency property states thatthe shape of the minor loops depends only on the reversal points,and is independent of the material’s magnetization history. Bothproperties may be proven using geometric arguments [19].

The statistical distribution of hysterons may be approximatedby the normal distribution [19]

(4)

or, in terms of ,

(5)

is a magnetization constant, and are standard de-viations, and is a mean value. Since for all

, the triangular Preisach domain extends to infinity;

Fig. 5. Simplified diagram of exciter’s magnetic flux paths (d-axis on top,q-axis at the bottom), and the corresponding magnetic equivalent circuits.

Fig. 6. Exciter’s equivalent circuit and interface mechanism to the voltageregulator and main alternator models.

however, for or , is practi-cally zero. The magnetization at saturation may be obtained byintegrating (4) over the right-half of Preisach plane 2

(6)

IV. PROPOSED MODEL

The exciter’s magnetic equivalent circuit is depicted inFig. 5. The -axis main flux path reluctance is comprised ofthe stator back-iron reluctance , the pole iron reluctance

, the air-gap reluctance , and the rotor body reluctance. In the proposed model, it is assumed that all hysteretic

magnetic effects are concentrated in the region of the poles;hence, magnetic nonlinearities are incorporated into . Allother reluctances are considered to be linear, including thereluctances of the leakage flux paths and . The -axismagnetic paths are also considered to be linear.

The magnetic equivalent circuit of Fig. 5 is translated to theelectrical T-equivalent circuit of Fig. 6. The exciter machinedoes not have damper windings. As in [16], a reduced-order ma-chine model is utilized, wherein the (average) armature currents

2The error function is defined by erf(x) = (2=p�) e d�.



are injected by the rectifier model. The state variables are se-lected to be and . There are no states associated with the-axis, because its equation is purely algebraic. The hysteresis

model determines the incremental magnetizing inductance. Inthe following sections, the submodels are presented in detail.

A. Hysteresis Model

For the purposes of machine modeling, it is convenient towork with electrical rather than field quantities. Hence, byanalogy to , the machine’s -axis magnetizingflux linkage is written as the sum of a linear and a hystereticcomponent

(7)

The Preisach model is now expressed in terms of the magneti-zation component of flux linkage , and the magnetizingcurrent (instead of the magnetization , and the magneticfield ). The inductance corresponds to the slope of themagnetizing characteristic at saturation.

The hysteresis model’s input is the magnetizing currentand its output is the incremental inductance

(8)

It will be useful to note that by combining (7) and (8)

(9)

The assumed normal distribution of hysterons given in (5)leads, after the manipulations detailed in [19], to

(10)

(11)

(12)

(13)

where is a constant with dimensions of flux linkage,is the previous reversal point, , ,

, and . The “ ,” “ ” superscripts

denote upwards and downwards moving (increasing and de-creasing) magnetic fields; an additional “ ” superscript denotesthat the material is initially demagnetized, so that the initialcurve is being traversed. The appropriate equation is selectedbased on the direction of change of the magnetizing current.Since the exciter’s complete magnetic history is unknown, itis assumed that it is initially demagnetized. From (10)–(13),

at the reversal points and at the origin of the initialcurve, and is everywhere else. thus depends onlyon , , and the direction of change of (in accordancewith the congruency property).

The Preisach model constantly monitors the direction ofchange of , and adds the reversal points to a last-in first-outstack. The crossing of a previous reversal point signifies aminor loop closure. In this case, the two points that definethis minor loop are deleted from the stack (as dictated by thedeletion property).

B. Reduced-Order Machine Model

This model is termed “reduced-order” because the (fast) tran-sients associated with the rotor windings are neglected. Its in-puts are the -axes rotor currents (which will be approximatedby their average value), ,3 the exciter’s field winding voltage

, and the incremental inductance . In this block, the in-tegrations for the two states and are performed. Out-puts are the magnetizing current , the VBR

-axis flux linkage , and the VBR -axis inductance .In this model, an overbar is used to emphasize the approxima-

tion of a quantity by its fast-average value (its average over theprevious 60 ). Often, in such cases, it is appropriate to averagethe entire model, thereby yielding a formalized average-valuemodel. However, because of the nonlinearities involved with thehysteresis model, formal averaging of the model would proveawkward. Therefore, the interpretation applicable herein is thatquantities indicated as instantaneous (without overbars) are alsobeing approximated by their fast-average value.

The description of the reduced-order machine model beginswith the field winding flux linkage

(14)

Substitution of (9) and the currents’ relationship, into (14) and consideration of the field voltage equation

(15)

yields

(16)

The inductance term of the left-hand side is positive since. Hence, the sign of the right-hand side determines the

magnetizing current’s direction of change and which expression

3The q-axis current is not utilized by the reduced-order machine model, sinceits dynamic behavior only involves the d-axis. However, i is computed forcompleteness.



for is to be selected from (10)–(13). The state equationsmay be obtained from (9), (15), and (16)

(17)

and

(18)

The derivative is estimated from the variation of .It is convenient to approximate it by the following relationship,written in the frequency domain:

(19)

If is relatively small (so that ), a good low-fre-quency estimate is obtained. This approximation is justified bythe slow-varying nature of and consequently of . Equation(19) is readily translated into a time-domain differential equa-tion, and the problematic numerical differentiation of is thusavoided.

The exciter’s electromagnetic torque may be computed fromthe well-known expression .However, since the exciter is a small machine relative to themain alternator, its torque is assumed negligible herein.

The armature voltage equations must be expressed in voltage-behind-reactance form to be compatible with the rotating-recti-fier average-value model. In the VBR model, the rotor flux link-ages are expressed

(20)

(21)

where

(22)

(23)

and

(24)

These equations hold for fast current transients; hence, theoverbar notation is not appropriate.

In VBR form is essentially constant for fast transients. Inparticular, if for fast transients (such as commutation processes)we assume that the field flux linkage is constant, then it can beshown that is constant as well. Upon neglecting the rotorresistance, the VBR voltage equations may be expressed

(25)

(26)

with .

C. Rotating-Rectifier Average-Value Model

This section contains the derivation of the rotating-rectifieraverage-value model, which computes the average currentsflowing in the exciter armature from , the VBR -axisflux linkage , and the VBR -axis inductance . (Thecomputation also uses the VBR -axis inductance , whichis assumed constant herein.) The analysis is based on the clas-sical separation of a rectifier’s operation in three distinct modes[17]. This type of rectifier modeling is valid for a constant (orslow-varying) dc current.

The transformation of the no-load versions of (25) and (26)to the rotor reference frame yields the following three-phasevoltage set:

(27)

(28)

(29)

where .4 It is useful to define a voltage angle sothat the -phase voltage attains its maximum value when

(i.e., ). The voltage and rotor angles are thusrelated by

forfor .

(30)

Because of symmetry, it is only necessary to consider a 60interval (for a six-pulse bridge). Consider the interval which be-gins when diode 6 (Fig. 1) starts conducting (at , where

is a phase delay5), and ends at . During thisinterval, current is commutated from diode 2 to diode 6 (phase

to phase ); if the diode resistance is negligible, a line-to-lineshort-circuit between phases and is in effect, so .( denotes the line-to-neutral voltage of winding ). If therotor’s resistance is also neglected, Faraday’s law implies

(31)

where is a constant. This relationship will prove useful in theanalysis that follows.

The next observation is that the average rectifier outputvoltage may be expressed

(32)

which may be approximated as

(33)

4The standard numbering of the diodes (Fig. 1) corresponds to the order ofconduction in the case of an abc phase sequence. However, in this case, a reverseacb phase sequence is obtained, and the diodes conduct in a different order.

5This � should not be confused with the symbol that was used in the Preisachmodel section to denote the hysterons’ upper switching point.



Neglecting armature resistance makes the analysis far moretractable. As it turns out, the inaccuracy involved in this as-sumption can be largely mitigated using a correction termwhich will be defined in a later section.

The flux linkages may be related to the phase currentsand the VBR flux linkage by transforming (20) and (21) using(1) and (30). After manipulation

(34)

(35)

To proceed further, the rectification mode must be considered.1) Mode I Operation: Mode I operation (Fig. 7) may be sep-

arated into the commutation and conduction subintervals. Thecommutation lasts for less than 60 electrical degrees

, where denotes the commutation angle. During the com-mutation interval , three diodes are conducting(1, 2, and 6); during the conduction interval ,only two diodes are conducting (1 and 6). The currents are

forfor

(36)

where is the current flowing out of the recti-fier and into the generator field, and is the (positive,anode-to-cathode) current flowing through diode 6;increases from to .

The average dc voltage may be computed from (33), aftersubstituting (36) into (34); this sequence of operations yields

(37)

The term represents the effective commutating re-sistance for mode I operation.

Substitution of (36) into (35) yields

(38)

Fig. 7. Mode I operation.

Evaluating this expression at and , we obtain

(39)

and

(40)

respectively. By equating (39) and (40), the following nonlinearequation is obtained, which may be solved numerically for thecommutation angle :

(41)

Knowledge of and [from (39)] allows the computationof the average -axes rotor currents. Equation (38) is solvedfor and substituted into (36), which is transformed using (1).The currents of the first subinterval (denoted by the superscript“(i)”) are thus

(42)

Their average value is

(43)

This integral is difficult to evaluate analytically, so it is evaluatednumerically (e.g., using Simpson’s rule [16], [25]). On the otherhand, the average value of the conduction subinterval currents[denoted by the superscript “(ii)”] may be computed analytically

(44)

The total -axes currents average value is

(45)



Fig. 8. Mode II operation.

2) Mode II Operation: In mode II operation (Fig. 8), thecommutation angle is 60 , but commutation is auto-delayed bythe angle . There are always three diodes con-ducting, and the currents are

(46)

The current increases from to .The average dc voltage is computed similarly to mode I by

substituting (46) into (34) and (33)

(47)

The commutating resistance now depends on , as well as theVBR -axes inductances.

Evaluating (35) at and , and equatingthe two results yields the following nonlinear equation, whichis solved numerically for :

(48)

The expression (42) for is valid throughout com-mutation, and the average -axes currents are

(49)

3) Mode III Operation: In mode III operation (Fig. 9), com-mutation is delayed by and .This mode may be split into two subintervals. During

, two commutations are taking place simul-taneously; four diodes are conducting (3, 1, 2, and 6), and athree-phase short-circuit is applied to the exciter, so .At , the commutation of diode 1 is at a further com-mutation stage than the commutation of diode 6, which is juststarting ( , ). At , thecommutation of diode 3 to diode 1 finishes ;the current of diode 6 has increased to . During

Fig. 9. Mode III operation.

, there are three diodes conducting (1, 2,and 6), and a line-to-line short circuit is imposed on the exciter.Due to symmetry considerations, . The cur-rents are

,.

(50)

During the first subinterval, and. Inserting the corresponding part of

(50) into (34) and (35)

(51)

(52)

respectively. Substitution of the values of and at the threeseparating angles , , and ,into (38), (51), and (52), yields

(53)

(54)

(55)



(56)

Equation (56) is solved numerically for . Using (53)–(55) inconjunction with (33), it can be shown that

(57)

Analytic formulas for the commutating currents during thefirst subinterval may be obtained by solving the linear systemformed by (51) and (52)

(58)

(59)

The average first subinterval -axes currents may thus be eval-uated analytically. After manipulation

(60)The second subinterval -axes currents are given by(42) and may be evaluated by numerical integration

(61)

4) Mode IV Operation: Traditionally, a rectifier’s operationis divided into three distinct modes; these modes naturally occurwhen the rectifier is feeding a passive resistive load. Herein,however, an additional fourth mode (mode IV) needs to be con-sidered. This mode is an extension to mode III, and occurs whenthe rectifier’s dc current exceeds the maximum possible currentthat the ac source alone (i.e., the exciter) may supply. This sit-uation may arise, for example, when is decreased rapidlyenough, while decays at a much slower pace, constrained bythe main alternator field inductance.

During mode IV, a constant three-phase short circuit is im-posed on the exciter , and at any given instant thereare four diodes conducting (diodes 3, 1, 2, and 6 during the timeframe considered in this analysis). The auto-delay and commu-tation angles are at their maximum possible values (and ), and the currents become purely sinusoidal,as may be readily seen by analyzing the mode III equations.

The dc current flows through the (ideally) zero-resistance pathformed by the conducting diodes that belong to the same leg(diodes 3 and 6 in this case). The average -axes currents maybe found by substituting and in (60), whichyields

(62)

5) Determining the Mode of Operation: Determination ofthe mode of operation is the first step of the averaging subroutineand it guides the algorithm to the correct set of formulas. Inparticular, the mode is determined by comparing to a set ofincreasing current values that define the mode boundaries.

At the boundary between modes I and II, both nonlinear re-lations and yield

(63)

At the boundary between modes II and III, the evaluation ofand yields

(64)

At the point of complete short-circuit occurring at the edge ofmode III, becomes

(65)

This mode separation is valid if the boundaries are well or-dered. Note that is always true; on the otherhand, is satisfied only for the following range ofVBR inductance parameters:

(66)

At first glance, (66) imposes a significant constraint on themodel parameters. However, in the proposed model,assumes values closer to a leakage inductance, while isdominated by a magnetizing inductance term. Hence, it is gen-erally expected that (66) will be satisfied for all “reasonable”inductance values.

6) Solving the Nonlinear Equations: According to the oper-ation mode, a numerical solution to one of the nonlinear equa-tions (41), (48), or (56) needs to be obtained. Recall that a con-tinuous function has a root if

. In this case, it suffices to show i) , ii), and iii) .

For mode I operation, where , it may beshown that

(67)

(68)

For mode II, and

(69)

(70)



For mode III, and

(71)

(72)

Hence, a solution to all three equations will always exist. Furtheralgebraic manipulations—not shown herein—reveal that the so-lution is unique. It may thus be obtained with arbitrary precisionin a finite number of steps using the bisection algorithm [25].

7) Incorporating Resistive Losses: The model’s accuracymay be improved by taking into account the resistive lossesof the armature and the voltage drop of the rotating rectifierdiodes. Their incorporation affects the magnitude of the brush-less exciter steady-state field current, as well as the transientbehavior of the synchronous generator.

In the previous sections, the armature resistance and thediodes were ignored. The rigorous incorporation of these termsin the model would entail considerable modifications and pos-sibly would make the algebra intractable. Hence, to simplifythe analysis, the computation of the losses is decoupled fromthe computation of the average dc voltage. Thus, the averagevoltage applied across the main generator field is

(73)

The average voltage loss is computed by averaging the dropacross diodes 1 and 6, and the ohmic drop of the armature’sresistance, that is

(74)

A diode’s voltage–current characteristic is represented hereinby the following function:

(75)

The parameters , , and are obtained with a curve-fittingprocedure.

D. Model Summary

In summary, the algorithm proceeds as follows.

1) Initialize model, assume the material is demagnetized.2) Compute from (19).3) Determine the direction of change of using (16), and

check for the reversal of direction. In case of directionreversal, add a point to the magnetic history stack.

4) Detect the crossing of a previous reversal point (minorloop closure). In this case, delete two points from thehistory stack.

5) Compute using one of (10)–(13).6) Compute from (24).7) Determine from (30).8) Determine the mode of operation, using (63)–(65).9) If mode I:

a) Compute from (37).b) Solve (41) for .c) Compute average currents from (42)–(45).

Fig. 10. Schematic of experimental setup; the brushless synchronousgenerator is feeding a nonlinear rectifier load.

If mode II:

a) Solve (48) for .b) Compute from (47).c) Compute average currents from (42) and (49).

If mode III:

a) Compute from (57).b) Solve (56) for .c) Compute average currents from (60) and (61).

If mode IV:

a) Set .b) Compute average currents from (62).

10) Compute from (73) and (74).11) Compute from (17).12) Compute from (18).13) Go to step (2).

Steps (3)–(5) are specific to the Preisach model. If a linear mag-netizing inductance is used instead, set and

.

V. EXPERIMENTAL VALIDATION

The experimental setup (shown in Fig. 10) contains a 59-kW,600-V, Leroy–Somer brushless synchronous generator, modelLSA 432L7. The exciter is an eight-pole machine, whose fieldis rated for 12 V, 2.5 A. The generator’s prime mover is a DyneSystems 110-kW, vector-controlled, induction-motor-baseddynamometer, programmed to maintain constant rated speed(1800 r/min). The voltage regulator uses a proportional-integralcontrol strategy to maintain the commanded voltage [560 V,line-to-line, fundamental, root mean square (rms)] at the gener-ator terminals; the brushless exciter’s field current is controlledwith a hysteresis modulator. The generator is loaded with anuncontrolled rectifier that feeds a resistive load through anfilter.

The exciter’s parameters (listed in Table I) were identifiedfrom rotating tests, as described in [26]. The time constant of(19) is . The load parameters are ,

, and . The remaining components aredocumented in [27]–[29]. (In particular, the voltage regulatormodel and control diagram is described in detail in Appendix Dof [29].) The quantities of the internal rotating parts (Fig. 1) arenot measurable because slip rings were not installed. Hence, the



Fig. 11. Plots of the commanded and actual line-to-line voltage “envelope,”computed from the synchronous reference frame voltages v =[3(v + v )] . Each of the seven trapezoid shaped blocks is characterizedby a different slope (the same for rise and fall) and peak voltage: (1) 20 000 V/s,560 V; (2)–(4) 2000 V/s, 560 V, 420 V, 280 V, respectively; (5)–(7) 400 V/s,560 V, 420 V, 280 V, respectively. [Note: the above voltage values correspondto root mean square (rms) quantities].

TABLE ILIST OF EXCITER MODEL PARAMETERS

model is judged based on terminal quantities only, namely thesynchronous generator voltage and the exciter’s field current.The simulations were conducted using Advanced ContinuousSimulation Language (ACSL) [30].

In this case study, the generator’s voltage reference is mod-ified according to the profile shown in Fig. 11. This series ofcommanded voltage steps creates an extended period of signif-icant disturbances and tests the model’s validity for large-tran-sients simulations. The terminal voltage exhibits an overshoot,which is more pronounced for the faster slew-rate steps. More-over, due to the exciter’s magnetically hysteretic behavior, itdoes not fall to zero. The varying levels of remanence in theexciter machine reflect on the magnitude of the voltage and arecaptured fairly accurately. The standard IEEE model [14] does

Fig. 12. Variation of rectification mode, commutation angle, and auto-delayangle.

not predict hysteretic effects. The higher ripple in the experi-mental voltage waveform is attributed to slot effects, not incor-porated in the synchronous machine model [27].

The corresponding variation of rectification mode is depictedin Fig. 12. Under steady-state conditions, the exciter operatesin mode II; however, the auto-delay angle varies with the op-erating point. During transients, operation in all modes takesplace. Therefore, a simple mode I model would have been in-sufficient to predict this behavior. The observed rapid mode al-ternations and ripple in the waveforms of and result fromthe ripple in the main alternator field current which, in turn, iscaused by the rectifier load on the main alternator.

Simulated versus experimental waveforms of the exciter’sfield current command are shown in Fig. 13. The first plot de-picts a situation where the controller’s current limit (3 A) isreached. Such nonlinear control strategies may not be studiedusing the IEEE model, which does not calculate the exciter’sfield current. The proposed model is able to predict both steady-state values and transient behavior.

An illustration of hysteretic behavior is shown in Fig. 14. Ascan be seen, the trajectories move through four “steady-state”points, labeled , , , and . These points do notlie on a straight line. This complex behavior could not havebeen captured by a linear magnetization model (where

).In order to “initialize” the magnetic state, the commanded

voltage is stepped from 0 to 560 V and then back to 0 V at20 000 V/s (not shown in Fig. 11). The exciter’s flux is forcedto a higher-than-normal level (Fig. 14). According to the dele-tion property, the previous magnetic history is erased. Further-more, on account of the congruency property, the return path de-pends only on the reversal point on the curve. Hence,this initialization procedure is guaranteed to bring the materialback to the same state, regardless of the previous operating his-tory. This theoretically predicted behavior was experimentallyverified.



Fig. 13. Plots of the commanded exciter field current i . These correspondto the seven command steps of Fig. 11.

Fig. 14. Illustration of hysteresis. The depicted transient corresponds to thefirst trapezoid of Fig. 11. The upper right-hand plot depicts the magnetizationcomponent of the magnetizing flux versus magnetizing current � (i );the upper left and lower right plots depict � (t) and i (t), respectively.The i (t) plot has been rotated 90 .

VI. CONCLUSION

The described brushless exciter model was successfully eval-uated against experimental results. The modeling of all recti-fication modes, the prediction of the exciter’s field current, andthe representation of magnetic hysteresis, are important featuresthat are not included in the standard IEEE exciter model. Theproposed model is thus a high-fidelity alternative for large-dis-turbance simulations, where a computationally efficient exciterrepresentation is necessary. Hence, it is recommended for tran-sient stability studies and voltage regulator design.

REFERENCES

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Dionysios C. Aliprantis (M’04) received the elec-trical and computer engineering diploma from theNational Technical University of Athens, Athens,Greece, in 1999 and the Ph.D. degree in electricaland computer engineering from Purdue University,West Lafayette, IN, in 2003.

Currently, he is serving in the armed forces ofGreece. His interests include the modeling andsimulation of electric machines and power systems,and evolutionary optimization methods.

Scott D. Sudhoff (SM’01) received the B.S. (Hons.),M.S., and Ph.D. degrees in electrical engineeringfrom Purdue University, West Lafayette, IN, in 1988,1989, and 1991, respectively.

Currently, he is a Full Professor at Purdue Univer-sity. From 1991 to 1993, he was Part-Time VisitingFaculty with Purdue University and as a Part-TimeConsultant with P. C. Krause and Associates, WestLafayette, IN. From 1993 to 1997, he was a FacultyMember at the University of Missouri-Rolla. He hasauthored many papers. His interests include electric

machines, power electronics, and finite-inertia power systems.

Brian T. Kuhn (M’93) received the B.S. and M.S.degrees in electrical engineering from the Universityof Missouri-Rolla in 1996 and 1997, respectively.

He was a Research Engineer at Purdue University,West Lafayette, IN, from 1998 to 2003. Currently, heis a Senior Engineer with SmartSpark Energy Sys-tems, Inc., Champaign, IL. His research interests in-clude power electronics and electrical machinery.


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