136
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006
A Brushless Exciter Model Incorporating Multiple
Rectifier 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 that
includes an average-value rectifier representation that is valid for
all three rectification modes. Furthermore, magnetic hysteresis
is incorporated into the -axis of the excitation using Preisach’s
theory. The resulting model is very accurate and is ideal for
situations where the exciter’s response is of particular interest.
The model’s predictions are compared to experimental results.
Index Terms—Brushless rotating machines, magnetic hysteresis, modeling, simulation, synchronous generator excitation,
synchronous generators.
I. INTRODUCTION
B
Fig. 1. Schematic of a brushless synchronous generator.
Manuscript received October 28, 2003; revised September 29, 2004. This
work was supported by the “Naval Combat Survivability” effort under Grant
N00024-02-NR-60427. Paper no. TEC-00312-2003.
D. C. Aliprantis is with the Greek Armed Forces (e-mail:
aliprantis@alumni.purdue.edu).
S. D. Sudhoff is with the Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail:
sudhoff@ecn.purdue.edu).
B. T. Kuhn is with the SmartSpark Energy Systems, Inc., Champaign, IL
61820 USA (e-mail: b.kuhn@smartsparkenergy.com).
Digital Object Identifier 10.1109/TEC.2005.847968
[14]; it was originally devised for small-signal analyses and
its applicability to large-disturbance studies remains questionable [15]. An average-value machine-rectifier model that allows
linking of a -axes machine model to dc quantities was derived
in [16]. This model is based on the actual physical structure of
an electric machine and maintains its validity during large-transient simulations.
In this paper, the theory of [16] (which covered only mode I
operation) is extended to all three rectification modes [17].
This is necessary for brushless excitation systems, because the
exciter’s armature current—directly related to the generator’s
field current—is strongly linked to power system dynamics
[3]. During transients, the rectifier’s operation may vary from
mode I to the complete short-circuit occurring at the end of
mode III [6]. The exciter–rectifier configuration is analyzed on
an average-value basis in a later section.
The incorporation of ferromagnetic hysteresis is an additional
feature of the proposed model. Brushless synchronous generators may use the exciter’s remanent magnetism to facilitate
self-starting, when no other source is available to power the
voltage regulator. However, the magnetization state directly affects the level of excitation required to maintain a commanded
voltage at the generator terminals. Hence, representation of hysteresis enhances the model’s fidelity with respect to the voltage
regulator variables.
Hysteresis is modeled herein using Preisach’s theory [18],
[19]. The Preisach model guarantees that minor loops close to
the previous reversal point [20]–[22]. This property is essential for accurate representation of the exciter’s magnetizing path
behavior. Hysteresis models that do not predict closed minor
loops, such as the widely used Jiles–Atherton model [23], are
not appropriate. To see this, consider a brushless generator connected to a nonlinear load that induces terminal current ripple.
RUSHLESS excitation of synchronous generators offers
increased reliability and reduced maintenance requirements [1], [2]. In these systems, both the exciter machine and
the rectifier are mounted on the same shaft as the main alternator
(Fig. 1). Since the generator’s output voltage is regulated by
controlling the exciter’s field current, the exciter is an integral
part of a generator’s control loop and has significant impact on
a power system’s dynamic behavior.
This paper sets forth a brushless exciter model suitable for
use in time-domain simulations of power systems. The analysis
follows the common approach of decoupling the main generator
from the exciter–rectifier. Because of the large inductance of a
generator’s field winding, the field current is slow varying [3],
[4]. Therefore, the modeling problem may be reduced to that
of a synchronous machine (the exciter) connected to a rectifier
load.
For power system studies, detailed waveforms of rotating rectifier quantities are usually not important (unless, for example,
diode failures [5] or estimation of winding losses are of interest). Moreover, avoiding the simulation of the internal rectifier increases computational efficiency and reduces modeling
complexity [6], [7]. The machine-rectifier configuration may be
viewed as an ac voltage source in series with a constant commutating inductance [8]; however, this overly simplified model
does not accurately capture the system’s operational characteristics [9]–[13]. The widely used brushless exciter model proposed by the IEEE represents the exciter as a first-order system
0885-8969/$20.00 © 2005 IEEE
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES
Fig. 2. Interconnection block diagram (input–output relationships) for the
proposed 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 from
the correct operating point.
This paper begins with a notational and model overview.
Next, a brief review of Preisach’s theory is set forth. Then
model development begins in earnest, with the development
of the Preisach hysteresis model, a reduced-order machine
model, and the rotating-rectifier average-value model. The
paper concludes with a validation of the model by comparison
to experimental results.
II. NOTATION AND MODEL OVERVIEW
Throughout this work, matrix and vector quantities appear in
bold font. The primed stator quantities denote referral to the
rotor through the turns ratio, which is defined as the ratio of
. The electrical rotor
armature-to-field turns
position
and speed
are
times the mechanical rotor
, and speed
where is the number of poles.
position
The analysis takes place in the stator reference frame (since the
field winding in the exciter machine is located on the stator).
to stationary
variables is
The transformation of rotating
defined by [24]
(1)
where1
(2)
.
Since a neutral connection is not present,
The components of the proposed excitation model are shown
in Fig. 2. The exciter model connects to the main alternator
and current
; it also
model through the field voltage
. The voltage regulator model provides the voltage
requires
to 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 Preisach
hysteresis model, and the reduced-order machine model.
1The minus sign in the second row and the apparent interchange of the second
and third columns from Park’s transformation (as defined in [24]) arises from
using a counter-clockwise positive direction for the rotor position coupled with
the location of the ac windings on the rotor.
137
Fig. 3. Illustrations of the elementary magnetic dipole characteristic and the
boundary on the Preisach domain.
The rotating-rectifier average-value model computes the
, based on
average currents flowing in the exciter armature
, the voltage-behind-reactance (VBR) -axis flux linkage
and the (varying) VBR -axis inductance
. (The -axis
VBR inductance is also used, but is considered constant.) These
voltage-behind-reactance quantities are computed from the
reduced-order machine model. The hysteresis model performs
the computations and bookkeeping required to use Preisach’s
hysteresis theory. Its only input is the -axis magnetizing cur; its output is the incremental magnetizing inductance
rent
that represents the slope of the hysteresis loop at a given
instant. The integrations of the state equations are performed
inside the reduced-order machine model block. The states are
and the -axis magnetizing flux
. The aforementioned
variables will be defined formally in the ensuing analysis.
Notice that the proposed model is applicable whether hysteresis
is represented or not; in case of a linear magnetizing path, the
hysteresis 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 concept 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 function which describes the density of hysterons is known as the
and is denoted by
Preisach function. It is defined on
or
, depending on which set of coordinates is used. The
Preisach function is zero everywhere except on the shaded domain of Fig. 3. To explain the shape of this region, it is first
. The other constraints originate from the obnoted that
servation that a finite applied field
will fully saturate the
. Considmaterial. Thus, all dipoles must obey
eration of saturation in the opposite direction yields
. These three inequalities lead to the triangular domain
depicted in Fig. 3.
The domain is divided into two parts: the upper part
corresponds to dipoles with negative magnetization; the lower
, corresponds to positive magnetization. A value for
part
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
138
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006
Fig. 4. Visualization of Preisach diagrams. (a) Increasing magnetic field.
(b) Decreasing magnetic field.
the total magnetization of the material may be obtained by
taking into account the contribution of all elementary dipoles.
Hence, the magnetization is
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.
(3)
The formation of the domain’s boundary may be visualized
using the Preisach diagram, as shown in Fig. 4. First, assume that
and is increasing, forcing all
the magnetic field has the value
to switch to the plus
dipoles with upper switching point
state. The switching action is graphically equivalent to the creation of a sweeping front, represented by a line perpendicular
to the -axis, that moves toward increasing . The shaded area
. When the field
that the front sweeps past becomes part of
is decreasing, dipoles with a lower switching point
are
forced to switch to the negative state. A new front is created, this
time perpendicular to the -axis and moving toward decreasing
, claiming the area from
and adding it to
. The resulting boundary is formed by orthogonal line segments and is
often termed a “staircase” boundary. The shape of the boundary
depends on the history of the magnetic field.
The Preisach model possesses the deletion and the congruency properties. According to the deletion property, magnetic
history is completely erased when the front sweeps past previous reversal points. This property is responsible for the creation of closed minor loops. The congruency property states that
the shape of the minor loops depends only on the reversal points,
and is independent of the material’s magnetization history. Both
properties may be proven using geometric arguments [19].
The statistical distribution of hysterons may be approximated
by the normal distribution [19]
(4)
or, in terms of ,
(5)
is a magnetization constant,
and
are standard deviations, and is a mean value. Since
for all
, the triangular Preisach domain extends to infinity;
Fig. 6. Exciter’s equivalent circuit and interface mechanism to the voltage
regulator and main alternator models.
however, for
or
,
is practically zero. The magnetization at saturation may be obtained by
2
integrating (4) over the right-half of Preisach plane
(6)
IV. PROPOSED MODEL
The exciter’s magnetic equivalent circuit is depicted in
Fig. 5. The -axis main flux path reluctance is comprised of
, the pole iron reluctance
the stator back-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;
. All
hence, magnetic nonlinearities are incorporated into
other reluctances are considered to be linear, including the
and
. The -axis
reluctances of the leakage flux paths
magnetic paths are also considered to be linear.
The magnetic equivalent circuit of Fig. 5 is translated to the
electrical T-equivalent circuit of Fig. 6. The exciter machine
does not have damper windings. As in [16], a reduced-order machine model is utilized, wherein the (average) armature currents
2The
p
error function is defined by erf (x) = (2= )
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
e
d .
ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES
are injected by the rectifier model. The state variables are seand
. There are no states associated with the
lected to be
-axis, because its equation is purely algebraic. The hysteresis
model determines the incremental magnetizing inductance. In
the following sections, the submodels are presented in detail.
A. Hysteresis Model
For the purposes of machine modeling, it is convenient to
work with electrical rather than field quantities. Hence, by
analogy to
, the machine’s -axis magnetizing
flux linkage is written as the sum of a linear and a hysteretic
component
(7)
The Preisach model is now expressed in terms of the magneti, and the magnetizing
zation component of flux linkage
(instead of the magnetization , and the magnetic
current
field ). The inductance
corresponds to the slope of the
magnetizing characteristic at saturation.
The hysteresis model’s input is the magnetizing current
and 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)
139
denote upwards and downwards moving (increasing and decreasing) magnetic fields; an additional “ ” superscript denotes
that the material is initially demagnetized, so that the initial
curve is being traversed. The appropriate equation is selected
based on the direction of change of the magnetizing current.
Since the exciter’s complete magnetic history is unknown, it
is assumed that it is initially demagnetized. From (10)–(13),
at the reversal points and at the origin of the initial
is everywhere else.
thus depends only
curve, and
,
, and the direction of change of
(in accordance
on
with the congruency property).
The Preisach model constantly monitors the direction of
, and adds the reversal points to a last-in first-out
change of
stack. The crossing of a previous reversal point signifies a
minor loop closure. In this case, the two points that define
this minor loop are deleted from the stack (as dictated by the
deletion property).
B. Reduced-Order Machine Model
This model is termed “reduced-order” because the (fast) transients associated with the rotor windings are neglected. Its inputs are the -axes rotor currents (which will be approximated
,3 the exciter’s field winding voltage
by their average value),
, and the incremental inductance
. In this block, the integrations for the two states
and
are performed. Out, the VBR
puts are the magnetizing current
-axis flux linkage
, and the VBR -axis inductance
.
In this model, an overbar is used to emphasize the approximation of a quantity by its fast-average value (its average over the
previous 60 ). Often, in such cases, it is appropriate to average
the entire model, thereby yielding a formalized average-value
model. However, because of the nonlinearities involved with the
hysteresis model, formal averaging of the model would prove
awkward. Therefore, the interpretation applicable herein is that
quantities indicated as instantaneous (without overbars) are also
being approximated by their fast-average value.
The description of the reduced-order machine model begins
with the field winding flux linkage
(14)
(11)
Substitution of (9) and the currents’ relationship
, into (14) and consideration of the field voltage equation
(15)
yields
(12)
(16)
(13)
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
is a constant with dimensions of flux linkage,
where
is the previous reversal point,
,
,
, and
. The “ ,” “ ” superscripts
3The q -axis current is not utilized by the reduced-order machine model, since
its dynamic behavior only involves the d-axis. However, i is computed for
completeness.
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
140
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006
for
is to be selected from (10)–(13). The state equations
may 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:
C. Rotating-Rectifier Average-Value Model
This section contains the derivation of the rotating-rectifier
average-value model, which computes the average currents
from
, the VBR -axis
flowing in the exciter armature
flux linkage
, and the VBR -axis inductance
. (The
, which
computation also uses the VBR -axis inductance
is assumed constant herein.) The analysis is based on the classical separation of a rectifier’s operation in three distinct modes
[17]. This type of rectifier modeling is valid for a constant (or
slow-varying) dc current.
The transformation of the no-load versions of (25) and (26)
to the rotor reference frame yields the following three-phase
voltage set:
(27)
(19)
(28)
If is relatively small (so that
), a good low-frequency estimate is obtained. This approximation is justified by
and consequently of . Equation
the slow-varying nature of
(19) is readily translated into a time-domain differential equais thus
tion, and the problematic numerical differentiation of
avoided.
The exciter’s electromagnetic torque may be computed from
.
the well-known expression
However, since the exciter is a small machine relative to the
main alternator, its torque is assumed negligible herein.
The armature voltage equations must be expressed in voltagebehind-reactance form to be compatible with the rotating-rectifier average-value model. In the VBR model, the rotor flux linkages are expressed
(29)
(20)
(21)
where
where
.4 It is useful to define a voltage angle
so
that the -phase voltage attains its maximum value when
(i.e.,
). The voltage and rotor angles are thus
related by
for
for
.
(30)
Because of symmetry, it is only necessary to consider a 60
interval (for a six-pulse bridge). Consider the interval which begins when diode 6 (Fig. 1) starts conducting (at
, where
is a phase delay5), and ends at
. During this
interval, current is commutated from diode 2 to diode 6 (phase
to phase ); if the diode resistance is negligible, a line-to-line
short-circuit between phases and is in effect, so
.
denotes the line-to-neutral voltage of winding ). If the
(
rotor’s resistance is also neglected, Faraday’s law implies
(22)
(31)
(23)
where is a constant. This relationship will prove useful in the
analysis that follows.
The next observation is that the average rectifier output
voltage may be expressed
and
(24)
(32)
These equations hold for fast current transients; hence, the
overbar notation is not appropriate.
is essentially constant for fast transients. In
In VBR form
particular, if for fast transients (such as commutation processes)
we assume that the field flux linkage is constant, then it can be
is constant as well. Upon neglecting the rotor
shown that
resistance, the VBR voltage equations may be expressed
which may be approximated as
(33)
(25)
(26)
with
.
4The standard numbering of the diodes (Fig. 1) corresponds to the order of
conduction in the case of an abc phase sequence. However, in this case, a reverse
acb 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 Preisach
model section to denote the hysterons’ upper switching point.
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES
141
Neglecting armature resistance makes the analysis far more
tractable. As it turns out, the inaccuracy involved in this assumption can be largely mitigated using a correction term
which will be defined in a later section.
flux linkages may be related to the phase currents
The
and the VBR flux linkage by transforming (20) and (21) using
(1) and (30). After manipulation
Fig. 7. Mode I operation.
Evaluating this expression at
and
, we obtain
(39)
(34)
and
(40)
respectively. By equating (39) and (40), the following nonlinear
equation is obtained, which may be solved numerically for the
commutation angle :
(35)
To proceed further, the rectification mode must be considered.
1) Mode I Operation: Mode I operation (Fig. 7) may be separated into the commutation and conduction subintervals. The
commutation lasts for less than 60 electrical degrees
, where denotes the commutation angle. During the com, three diodes are conducting
mutation interval
(1, 2, and 6); during the conduction interval
,
currents are
only two diodes are conducting (1 and 6). The
for
for
(41)
Knowledge of and [from (39)] allows the computation
of the average -axes rotor currents. Equation (38) is solved
for and substituted into (36), which is transformed using (1).
The currents of the first subinterval (denoted by the superscript
“(i)”) are thus
(36)
where
is the current flowing out of the rectiis the (positive,
fier and into the generator field, and
anode-to-cathode) current flowing through diode 6;
increases from
to
.
The average dc voltage may be computed from (33), after
substituting (36) into (34); this sequence of operations yields
(37)
represents the effective commutating reThe term
sistance for mode I operation.
Substitution of (36) into (35) yields
(42)
Their average value is
(43)
This integral is difficult to evaluate analytically, so it is evaluated
numerically (e.g., using Simpson’s rule [16], [25]). On the other
hand, 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
(38)
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
(45)
142
Fig. 8.
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006
Mode II operation.
Fig. 9. Mode III operation.
2) Mode II Operation: In mode II operation (Fig. 8), the
commutation angle is 60 , but commutation is auto-delayed by
the angle
. There are always three diodes concurrents are
ducting, and the
, there are three diodes conducting (1, 2,
and 6), and a line-to-line short circuit is imposed on the exciter.
. The
curDue to symmetry considerations,
rents are
(46)
to
.
The current increases from
The average dc voltage is computed similarly to mode I by
substituting (46) into (34) and (33)
,
.
(50)
During the first subinterval,
and
. Inserting the corresponding part of
(50) into (34) and (35)
(47)
The commutating resistance now depends on , as well as the
VBR -axes inductances.
and
, and equating
Evaluating (35) at
the two results yields the following nonlinear equation, which
is solved numerically for :
(51)
(48)
The expression (42) for
mutation, and the average
is valid throughout com-axes currents are
(49)
(52)
respectively. Substitution of the values of
separating angles
,
into (38), (51), and (52), yields
3) Mode III Operation: In mode III operation (Fig. 9), commutation is delayed by
and
.
This mode may be split into two subintervals. During
, two commutations are taking place simultaneously; four diodes are conducting (3, 1, 2, and 6), and a
.
three-phase short-circuit is applied to the exciter, so
At
, the commutation of diode 1 is at a further commutation stage than the commutation of diode 6, which is just
,
). At
, the
starting (
;
commutation of diode 3 to diode 1 finishes
the current of diode 6 has increased to
. During
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
and at the three
, and
,
(53)
(54)
(55)
ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES
143
The dc current flows through the (ideally) zero-resistance path
formed by the conducting diodes that belong to the same leg
(diodes 3 and 6 in this case). The average -axes currents may
be found by substituting
and
in (60), which
yields
(56)
Equation (56) is solved numerically for . Using (53)–(55) in
conjunction with (33), it can be shown that
(57)
Analytic formulas for the commutating currents during the
first subinterval may be obtained by solving the linear system
formed by (51) and (52)
(62)
5) Determining the Mode of Operation: Determination of
the mode of operation is the first step of the averaging subroutine
and it guides the algorithm to the correct set of formulas. In
to a set of
particular, the mode is determined by comparing
increasing current values that define the mode boundaries.
At the boundary between modes I and II, both nonlinear reand
yield
lations
(63)
(58)
At the boundary between modes II and III, the evaluation of
and
yields
(64)
At the point of complete short-circuit occurring at the edge of
becomes
mode III,
(65)
(59)
The average first subinterval -axes currents may thus be evaluated analytically. After manipulation
This mode separation is valid if the boundaries are well oris always true; on the other
dered. Note that
hand,
is satisfied only for the following range of
VBR inductance parameters:
(66)
(60)
are given by
The second subinterval -axes currents
(42) and may be evaluated by numerical integration
(61)
4) Mode IV Operation: Traditionally, a rectifier’s operation
is divided into three distinct modes; these modes naturally occur
when the rectifier is feeding a passive resistive load. Herein,
however, an additional fourth mode (mode IV) needs to be considered. This mode is an extension to mode III, and occurs when
the rectifier’s dc current exceeds the maximum possible current
that the ac source alone (i.e., the exciter) may supply. This sitis decreased rapidly
uation may arise, for example, when
enough, while
decays at a much slower pace, constrained by
the main alternator field inductance.
During mode IV, a constant three-phase short circuit is im, and at any given instant there
posed on the exciter
are four diodes conducting (diodes 3, 1, 2, and 6 during the time
frame considered in this analysis). The auto-delay and commutation 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.
At first glance, (66) imposes a significant constraint on the
model parameters. However, in the proposed model,
assumes values closer to a leakage inductance, while
is
dominated by a magnetizing inductance term. Hence, it is generally expected that (66) will be satisfied for all “reasonable”
inductance values.
6) Solving the Nonlinear Equations: According to the operation mode, a numerical solution to one of the nonlinear equations (41), (48), or (56) needs to be obtained. Recall that a conhas a root
if
tinuous function
. In this case, it suffices to show i)
, ii)
, and iii)
.
, it may be
For mode I operation, where
shown that
(67)
(68)
For mode II,
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
and
(69)
(70)
144
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006
For mode III,
and
(71)
(72)
Hence, a solution to all three equations will always exist. Further
algebraic manipulations—not shown herein—reveal that the solution is unique. It may thus be obtained with arbitrary precision
in a finite number of steps using the bisection algorithm [25].
7) Incorporating Resistive Losses: The model’s accuracy
may be improved by taking into account the resistive losses
of the armature and the voltage drop of the rotating rectifier
diodes. Their incorporation affects the magnitude of the brushless exciter steady-state field current, as well as the transient
behavior of the synchronous generator.
In the previous sections, the armature resistance and the
diodes were ignored. The rigorous incorporation of these terms
in the model would entail considerable modifications and possibly would make the algebra intractable. Hence, to simplify
the analysis, the computation of the losses is decoupled from
the computation of the average dc voltage. Thus, the average
voltage applied across the main generator field is
(73)
is computed by averaging the drop
The average voltage loss
across diodes 1 and 6, and the ohmic drop of the armature’s
resistance, that is
(74)
A diode’s voltage–current characteristic is represented herein
by the following function:
(75)
The parameters , , and
procedure.
are obtained with a curve-fitting
D. Model Summary
In summary, the algorithm proceeds as follows.
1) Initialize model, assume the material is demagnetized.
from (19).
2) Compute
using (16), and
3) Determine the direction of change of
check for the reversal of direction. In case of direction
reversal, add a point to the magnetic history stack.
4) Detect the crossing of a previous reversal point (minor
loop closure). In this case, delete two points from the
history stack.
using one of (10)–(13).
5) Compute
from (24).
6) Compute
7) Determine from (30).
8) Determine the mode of operation, using (63)–(65).
9) If mode I:
from (37).
a) Compute
b) Solve (41) for .
c) Compute average currents from (42)–(45).
Fig. 10. Schematic of experimental setup; the brushless synchronous
generator 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:
from (57).
a) Compute
b) Solve (56) for .
c) Compute average currents from (60) and (61).
If mode IV:
.
a) Set
b) Compute average currents from (62).
from (73) and (74).
10) Compute
11) Compute
from (17).
from (18).
12) Compute
13) Go to step (2).
Steps (3)–(5) are specific to the Preisach model. If a linear magnetizing 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, model
LSA 432L7. The exciter is an eight-pole machine, whose field
is rated for 12 V, 2.5 A. The generator’s prime mover is a Dyne
Systems 110-kW, vector-controlled, induction-motor-based
dynamometer, programmed to maintain constant rated speed
(1800 r/min). The voltage regulator uses a proportional-integral
control strategy to maintain the commanded voltage [560 V,
line-to-line, fundamental, root mean square (rms)] at the generator terminals; the brushless exciter’s field current is controlled
with a hysteresis modulator. The generator is loaded with an
uncontrolled rectifier that feeds a resistive load through an
filter.
The exciter’s parameters (listed in Table I) were identified
from rotating tests, as described in [26]. The time constant of
. The load parameters are
,
(19) is
, and
. The remaining components are
documented in [27]–[29]. (In particular, the voltage regulator
model and control diagram is described in detail in Appendix D
of [29].) The quantities of the internal rotating parts (Fig. 1) are
not measurable because slip rings were not installed. Hence, the
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES
Fig. 12.
angle.
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 characterized
by 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 correspond
to root mean square (rms) quantities].
TABLE I
LIST OF EXCITER MODEL PARAMETERS
model is judged based on terminal quantities only, namely the
synchronous generator voltage and the exciter’s field current.
The simulations were conducted using Advanced Continuous
Simulation Language (ACSL) [30].
In this case study, the generator’s voltage reference is modified according to the profile shown in Fig. 11. This series of
commanded voltage steps creates an extended period of significant disturbances and tests the model’s validity for large-transients simulations. The terminal voltage exhibits an overshoot,
which is more pronounced for the faster slew-rate steps. Moreover, due to the exciter’s magnetically hysteretic behavior, it
does not fall to zero. The varying levels of remanence in the
exciter machine reflect on the magnitude of the voltage and are
captured fairly accurately. The standard IEEE model [14] does
145
Variation of rectification mode, commutation angle, and auto-delay
not predict hysteretic effects. The higher ripple in the experimental voltage waveform is attributed to slot effects, not incorporated in the synchronous machine model [27].
The corresponding variation of rectification mode is depicted
in Fig. 12. Under steady-state conditions, the exciter operates
in mode II; however, the auto-delay angle varies with the operating point. During transients, operation in all modes takes
place. Therefore, a simple mode I model would have been insufficient to predict this behavior. The observed rapid mode alternations and ripple in the waveforms of and result from
the ripple in the main alternator field current which, in turn, is
caused by the rectifier load on the main alternator.
Simulated versus experimental waveforms of the exciter’s
field current command are shown in Fig. 13. The first plot depicts a situation where the controller’s current limit (3 A) is
reached. Such nonlinear control strategies may not be studied
using the IEEE model, which does not calculate the exciter’s
field current. The proposed model is able to predict both steadystate values and transient behavior.
An illustration of hysteretic behavior is shown in Fig. 14. As
can be seen, the trajectories move through four “steady-state”
. These points do not
points, labeled , , , and
lie on a straight line. This complex behavior could not have
been 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 at
20 000 V/s (not shown in Fig. 11). The exciter’s flux is forced
to a higher-than-normal level (Fig. 14). According to the deletion property, the previous magnetic history is erased. Furthermore, on account of the congruency property, the return path decurve. Hence,
pends only on the reversal point on the
this initialization procedure is guaranteed to bring the material
back to the same state, regardless of the previous operating history. This theoretically predicted behavior was experimentally
verified.
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
146
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006
VI. CONCLUSION
The described brushless exciter model was successfully evaluated against experimental results. The modeling of all rectification modes, the prediction of the exciter’s field current, and
the representation of magnetic hysteresis, are important features
that are not included in the standard IEEE exciter model. The
proposed model is thus a high-fidelity alternative for large-disturbance simulations, where a computationally efficient exciter
representation is necessary. Hence, it is recommended for transient stability studies and voltage regulator design.
REFERENCES
Fig. 13. Plots of the commanded exciter field current i
to the seven command steps of Fig. 11.
. These correspond
Fig. 14. Illustration of hysteresis. The depicted transient corresponds to the
first trapezoid of Fig. 11. The upper right-hand plot depicts the magnetization
(i
);
component of the magnetizing flux versus magnetizing current the upper left and lower right plots depict (t) and i
(t), respectively.
The i (t) plot has been rotated 90 .
[1] R. W. Ferguson, R. Herbst, and R. W. Miller, “Analytical studies of the
brushless excitation system,” AIEE Trans., Part III-B, Power App. Syst.,
vol. 79, pp. 1815–1821, Feb. 1960.
[2] E. C. Whitney, D. B. Hoover, and P. O. Bobo, “An electric utility brushless excitation system,” AIEE Trans., Part III-B, Power App. Syst., vol.
79, pp. 1821–1828, Feb. 1960.
[3] H. W. Gayek, “Behavior of brushless aircraft generating systems,” IEEE
Trans. Aerosp.-Support Conf. Procedures, vol. 1, no. 2, pp. 594–621,
Aug. 1963.
[4]
, “Transfer characteristics of brushless aircraft generator systems,”
IEEE Trans. Aerosp., vol. 2, no. 2, pp. 913–928, Apr. 1964.
[5] T. Zouaghi and M. Poloujadoff, “Modeling of polyphase brushless exciter behavior for failing diode operation,” IEEE Trans. Energy Convers.,
vol. 13, no. 3, pp. 214–220, Sep. 1998.
[6] M. A. Abdel-Halim and C. D. Manning, “Modeling a laminated brushless exciter-alternator unit in all modes of operation,” Proc. Inst. Elect.
Eng. B, vol. 138, no. 2, pp. 87–94, Mar. 1991.
[7] A. Darabi and C. Tindall, “Brushless exciter modeling for small salient
pole alternators using finite elements,” IEEE Trans. Energy Convers.,
vol. 17, no. 3, pp. 306–312, Sep. 2002.
[8] T. H. Warner and J. G. Kassakian, “Transient characteristics and modeling of large turboalternator driven rectifier/inverter systems based on
field test data,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 7, pp.
1804–1811, Jul. 1985.
[9] P. W. Franklin, “Theory of the three phase salient pole type generator
with bridge rectified output—parts I and II,” IEEE Trans. Power App.
Syst., vol. PAS-91, no. 5, pp. 1960–1975, Sep./Oct. 1972.
[10] W. J. Bonwick and V. H. Jones, “Performance of a synchronous generator with a bridge rectifier,” Proc. Inst. Elect. Eng., vol. 119, no. 9, pp.
1338–1342, Sep. 1972.
[11]
, “Rectifier-loaded synchronous generators with damper windings,”
Proc. Inst. Elect. Eng., vol. 120, no. 6, pp. 659–666, Jun. 1973.
[12] W. J. Bonwick, “Characteristics of a diode-bridge-loaded synchronous
generator without damper windings,” Proc. Inst. Elect. Eng., vol. 122,
no. 6, pp. 637–642, Jun. 1975.
[13] I. Jadrić, D. Borojević, and M. Jadrić, “Modeling and control of a synchronous generator with an active DC load,” IEEE Trans. Energy Convers., vol. 15, no. 2, pp. 303–311, Mar. 2000.
[14] Recommended Practice for Excitation System Models for Power System
Stability Studies, IEEE Standard 421.5, Mar. 1992.
[15] S. M. L. Kabir and R. Shuttleworth, “Brushless exciter model,” Proc.
Inst. Elect. Eng. C, vol. 141, no. 1, pp. 61–67, Jan. 1994.
[16] S. D. Sudhoff and O. Wasynczuk, “Analysis and average-value modeling of line-commutated converter-synchronous machine systems,”
IEEE Trans. Energy Convers., vol. 8, no. 1, pp. 92–99, Mar. 1993.
[17] R. L. Witzke, J. V. Kresser, and J. K. Dillard, “Influence of A-C reactance on voltage regulation of 6-phase rectifiers,” AIEE Trans., Part I,
Commun. Electron., vol. 72, pp. 244–253, Jul. 1953.
[18] F. Preisach, “Über die magnetische Nachwirkung,” Zeitschrift für
Physik, vol. 94, pp. 277–302, Mar./May 1935.
[19] E. Della Torre, Magnetic Hysteresis, 1st ed. Piscataway, NJ: IEEE
Press, 1999.
[20] Members of the Staff of the Department of Electrical Engineering—
Massachusetts Institute of Technology, Magnetic Circuits and Transformers, 15th ed. Cambridge, MA: MIT Press, 1965.
[21] E. P. Dick and W. Watson, “Transformer models for transient studies
based on field measurements,” IEEE Trans. Power App. Syst., vol. PAS100, no. 1, pp. 409–419, Jan. 1981.
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.
ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES
147
[22] J. G. Frame, N. Mohan, and T. Liu, “Hysteresis modeling in an electromagnetic transients program,” IEEE Trans. Power App. Syst., vol. PAS101, no. 9, pp. 3404–3412, Sep. 1982.
[23] D. C. Jiles and D. L. Atherton, “Ferromagnetic hysteresis,” IEEE Trans.
Magn., vol. MAG-19, no. 5, pp. 2183–2185, Sep. 1983.
[24] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric
Machinery. New York: IEEE Press, 1995.
[25] W. Gautschi, Numerical Analysis, an Introduction, 1st ed. Boston,
MA: Birkhäuser, 1997.
[26] D. C. Aliprantis, S. D. Sudhoff, and B. T. Kuhn, “Genetic algorithmbased parameter identification of a hysteretic brushless exciter model,”
IEEE Trans. Energy Convers., to be published.
[27]
, “A synchronous machine model with saturation and arbitrary rotor
network representation,” IEEE Trans. Energy Convers., vol. 20, no. 3,
Sep. 2005.
, “Experimental characterization procedure for a synchronous ma[28]
chine model with saturation and arbitrary rotor network representation,”
IEEE Trans. Energy Convers., vol. 20, no. 3, Sep. 2005.
[29] D. C. Aliprantis, “Advances in electric machine modeling and evolutionary parameter identification,” Ph.D. dissertation, Purdue University,
West Lafayette, IN, Dec. 2003.
[30] Advanced Continuous Simulation Language (ACSL) Reference Manual,
AEgis Technologies Group, Inc., Huntsville, AL, 1999.
Scott D. Sudhoff (SM’01) received the B.S. (Hons.),
M.S., and Ph.D. degrees in electrical engineering
from Purdue University, West Lafayette, IN, in 1988,
1989, and 1991, respectively.
Currently, he is a Full Professor at Purdue University. From 1991 to 1993, he was Part-Time Visiting
Faculty with Purdue University and as a Part-Time
Consultant with P. C. Krause and Associates, West
Lafayette, IN. From 1993 to 1997, he was a Faculty
Member at the University of Missouri-Rolla. He has
authored many papers. His interests include electric
machines, power electronics, and finite-inertia power systems.
Dionysios C. Aliprantis (M’04) received the electrical and computer engineering diploma from the
National Technical University of Athens, Athens,
Greece, in 1999 and the Ph.D. degree in electrical
and computer engineering from Purdue University,
West Lafayette, IN, in 2003.
Currently, he is serving in the armed forces of
Greece. His interests include the modeling and
simulation of electric machines and power systems,
and evolutionary optimization methods.
Brian T. Kuhn (M’93) received the B.S. and M.S.
degrees in electrical engineering from the University
of Missouri-Rolla in 1996 and 1997, respectively.
He was a Research Engineer at Purdue University,
West Lafayette, IN, from 1998 to 2003. Currently, he
is a Senior Engineer with SmartSpark Energy Systems, Inc., Champaign, IL. His research interests include power electronics and electrical machinery.
Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply.