A Brushless Exciter Model Incorporating Multiple Rectifier Modes 137

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 RUSHLESS excitation of synchronous generators offers increased reliability and reduced maintenance require- B Fig. 1. Schematic of a brushless synchronous generator. ments [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 [14]; it was originally devised for small-signal analyses and controlling the exciter’s field current, the exciter is an integral its applicability to large-disturbance studies remains question- part of a generator’s control loop and has significant impact on able [15]. An average-value machine-rectifier model that allows a power system’s dynamic behavior. linking of a -axes machine model to dc quantities was derived This paper sets forth a brushless exciter model suitable for in [16]. This model is based on the actual physical structure of use in time-domain simulations of power systems. The analysis an electric machine and maintains its validity during large-tran- follows the common approach of decoupling the main generator sient simulations. from the exciter–rectifier. Because of the large inductance of a In this paper, the theory of [16] (which covered only mode I generator’s field winding, the field current is slow varying [3], operation) is extended to all three rectification modes [17]. [4]. Therefore, the modeling problem may be reduced to that This is necessary for brushless excitation systems, because the of a synchronous machine (the exciter) connected to a rectifier exciter’s armature current—directly related to the generator’s load. field current—is strongly linked to power system dynamics For power system studies, detailed waveforms of rotating rec- [3]. During transients, the rectifier’s operation may vary from tifier quantities are usually not important (unless, for example, mode I to the complete short-circuit occurring at the end of diode failures [5] or estimation of winding losses are of in- mode III [6]. The exciter–rectifier configuration is analyzed on terest). Moreover, avoiding the simulation of the internal rec- an average-value basis in a later section. tifier increases computational efficiency and reduces modeling The incorporation of ferromagnetic hysteresis is an additional complexity [6], [7]. The machine-rectifier configuration may be feature of the proposed model. Brushless synchronous gener- viewed as an ac voltage source in series with a constant com- ators may use the exciter’s remanent magnetism to facilitate mutating inductance [8]; however, this overly simplified model self-starting, when no other source is available to power the does not accurately capture the system’s operational character- voltage regulator. However, the magnetization state directly af- istics [9]–[13]. The widely used brushless exciter model pro- fects the level of excitation required to maintain a commanded posed by the IEEE represents the exciter as a first-order system voltage at the generator terminals. Hence, representation of hysteresis enhances the model’s fidelity with respect to the voltage regulator variables. Manuscript received October 28, 2003; revised September 29, 2004. This work was supported by the “Naval Combat Survivability” effort under Grant Hysteresis is modeled herein using Preisach’s theory [18], N00024-02-NR-60427. Paper no. TEC-00312-2003. [19]. The Preisach model guarantees that minor loops close to D. C. Aliprantis is with the Greek Armed Forces (e-mail: the previous reversal point [20]–[22]. This property is essen- [email protected]). S. D. Sudhoff is with the Department of Electrical and Computer Engi- tial for accurate representation of the exciter’s magnetizing path neering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail: behavior. Hysteresis models that do not predict closed minor [email protected]). loops, such as the widely used Jiles–Atherton model [23], are B. T. Kuhn is with the SmartSpark Energy Systems, Inc., Champaign, IL 61820 USA (e-mail: [email protected]). not appropriate. To see this, consider a brushless generator con- Digital Object Identifier 10.1109/TEC.2005.847968 nected to a nonlinear load that induces terminal current ripple.

Fig. 2. Interconnection block diagram (input–output relationships) for the proposed model. Fig. 3. Illustrations of the elementary magnetic dipole characteristic and the boundary on the Preisach domain. This ripple transfers to the exciter’s magnetizing branch current, and in the “steady-state,” a minor loop trajectory is traced on the The rotating-rectifier average-value model computes the plane. If the loop is not closed, the flux can drift away from average currents flowing in the exciter armature , based on the correct operating point. , the voltage-behind-reactance (VBR) -axis flux linkage This paper begins with a notational and model overview. and the (varying) VBR -axis inductance . (The -axis Next, a brief review of Preisach’s theory is set forth. Then VBR inductance is also used, but is considered constant.) These model development begins in earnest, with the development voltage-behind-reactance quantities are computed from the of the Preisach hysteresis model, a reduced-order machine reduced-order machine model. The hysteresis model performs model, and the rotating-rectifier average-value model. The the computations and bookkeeping required to use Preisach’s paper concludes with a validation of the model by comparison hysteresis theory. Its only input is the -axis magnetizing cur- to experimental results. rent ; its output is the incremental magnetizing inductance that represents the slope of the hysteresis loop at a given II. NOTATION AND MODEL OVERVIEW instant. The integrations of the state equations are performed Throughout this work, matrix and vector quantities appear in inside the reduced-order machine model block. The states are bold font. The primed stator quantities denote referral to the and the -axis magnetizing flux . The aforementioned rotor through the turns ratio, which is defined as the ratio of variables will be defined formally in the ensuing analysis. armature-to-field turns . The electrical rotor Notice that the proposed model is applicable whether hysteresis position and speed are times the mechanical rotor is represented or not; in case of a linear magnetizing path, the position , and speed where is the number of poles. hysteresis block is replaced by a constant inductance term. The analysis takes place in the stator reference frame (since the field winding in the exciter machine is located on the stator). III. HYSTERESIS MODELING USING PREISACH’S THEORY The transformation of rotating to stationary variables is Preisach’s theory of magnetic hysteresis is based on the con- defined by [24] cept of elementary magnetic dipoles (also called hysterons). These simple hysteresis operators may be defined by their “up” (1) and “down” switching values and , respectively (Fig. 3). where1 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- (2) tion which describes the density of hysterons is known as the Since a neutral connection is not present, . Preisach function. It is defined on and is denoted by The components of the proposed excitation model are shown or , depending on which set of coordinates is used. The in Fig. 2. The exciter model connects to the main alternator Preisach function is zero everywhere except on the shaded do- model through the field voltage and current ; it also main of Fig. 3. To explain the shape of this region, it is first requires . The voltage regulator model provides the voltage noted that . The other constraints originate from the ob- to the exciter’s field winding , and receives the current servation that a finite applied field will fully saturate the . The exciter model is comprised of three separate models, material. Thus, all dipoles must obey . Consid- namely, the rotating-rectifier average-value model, the Preisach eration of saturation in the opposite direction yields hysteresis model, and the reduced-order machine model. . These three inequalities lead to the triangular domain depicted in Fig. 3. 1The minus sign in the second row and the apparent interchange of the second The domain is divided into two parts: the upper part 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 corresponds to dipoles with negative magnetization; the lower the location of the ac windings on the rotor. part , corresponds to positive magnetization. A value for

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

Fig. 5. Simpliﬁed diagram of exciter’s magnetic ﬂux paths (d-axis on top, the total magnetization of the material may be obtained by q-axis at the bottom), and the corresponding magnetic equivalent circuits. taking into account the contribution of all elementary dipoles. Hence, the magnetization is

(3)

The formation of the domain’s boundary may be visualized using the Preisach diagram, as shown in Fig. 4. First, assume that the magnetic field has the value and is increasing, forcing all dipoles with upper switching point to switch to the plus state. The switching action is graphically equivalent to the cre- ation of a sweeping front, represented by a line perpendicular to the -axis, that moves toward increasing . The shaded area that the front sweeps past becomes part of . When the field is decreasing, dipoles with a lower switching point are forced to switch to the negative state. A new front is created, this Fig. 6. Exciter’s equivalent circuit and interface mechanism to the voltage time perpendicular to the -axis and moving toward decreasing regulator and main alternator models. , claiming the area from and adding it to . The resulting boundary is formed by orthogonal line segments and is however, for or , is practi- often termed a “staircase” boundary. The shape of the boundary cally zero. The magnetization at saturation may be obtained by depends on the history of the magnetic field. integrating (4) over the right-half of Preisach plane 2 The Preisach model possesses the deletion and the congruency properties. According to the deletion property, magnetic (6) history is completely erased when the front sweeps past previous reversal points. This property is responsible for the cre- ation of closed minor loops. The congruency property states that IV. PROPOSED MODEL the shape of the minor loops depends only on the reversal points, and is independent of the material’s magnetization history. Both The exciter’s magnetic equivalent circuit is depicted in properties may be proven using geometric arguments [19]. Fig. 5. The -axis main flux path reluctance is comprised of The statistical distribution of hysterons may be approximated the stator back-iron reluctance , the pole iron reluctance by the normal distribution [19] , the air-gap reluctance , and the rotor body reluctance . In the proposed model, it is assumed that all hysteretic (4) magnetic effects are concentrated in the region of the poles; hence, magnetic nonlinearities are incorporated into . All or, in terms of , other reluctances are considered to be linear, including the reluctances of the leakage flux paths and . The -axis magnetic paths are also considered to be linear. The magnetic equivalent circuit of Fig. 5 is translated to the (5) electrical T-equivalent circuit of Fig. 6. The exciter machine does not have damper windings. As in [16], a reduced-order ma- is a magnetization constant, and are standard de- chine model is utilized, wherein the (average) armature currents viations, and is a mean value. Since for all p , the triangular Preisach domain extends to infinity; 2The error function is defined by erf@xAa@Pa %A e d$.

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 139 are injected by the rectifier model. The state variables are se- denote upwards and downwards moving (increasing and de- lected to be and . There are no states associated with the creasing) magnetic fields; an additional “ ” superscript denotes -axis, because its equation is purely algebraic. The hysteresis that the material is initially demagnetized, so that the initial model determines the incremental magnetizing inductance. In curve is being traversed. The appropriate equation is selected the following sections, the submodels are presented in detail. based on the direction of change of the magnetizing current. Since the exciter’s complete magnetic history is unknown, it A. Hysteresis Model is assumed that it is initially demagnetized. From (10)–(13), For the purposes of machine modeling, it is convenient to at the reversal points and at the origin of the initial work with electrical rather than field quantities. Hence, by curve, and is everywhere else. thus depends only analogy to , the machine’s -axis magnetizing on , , and the direction of change of (in accordance flux linkage is written as the sum of a linear and a hysteretic with the congruency property). component The Preisach model constantly monitors the direction of change of , and adds the reversal points to a last-in first-out (7) stack. The crossing of a previous reversal point signifies a minor loop closure. In this case, the two points that define The Preisach model is now expressed in terms of the magneti- this minor loop are deleted from the stack (as dictated by the zation component of flux linkage , and the magnetizing deletion property). current (instead of the magnetization , and the magnetic field ). The inductance corresponds to the slope of the B. Reduced-Order Machine Model magnetizing characteristic at saturation. This model is termed “reduced-order” because the (fast) tran- The hysteresis model’s input is the magnetizing current sients associated with the rotor windings are neglected. Its in- and its output is the incremental inductance puts are the -axes rotor currents (which will be approximated by their average value), ,3 the exciter’s field winding voltage (8) , and the incremental inductance . In this block, the integrations for the two states and are performed. Out- It will be useful to note that by combining (7) and (8) puts are the magnetizing current , the VBR -axis flux linkage , and the VBR -axis inductance . (9) In this model, an overbar is used to emphasize the approximation of a quantity by its fast-average value (its average over the The assumed normal distribution of hysterons given in (5) previous 60 ). Often, in such cases, it is appropriate to average leads, after the manipulations detailed in [19], to 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. (10) The description of the reduced-order machine model begins with the field winding flux linkage

(14)

Substitution of (9) and the currents’ relationship (11) , into (14) and consideration of the ﬁeld voltage equation

(15)

yields (12)

(16)

The inductance term of the left-hand side is positive since (13) . Hence, the sign of the right-hand side determines the magnetizing current’s direction of change and which expression where is a constant with dimensions of ﬂux linkage, 3The q-axis current is not utilized by the reduced-order machine model, since is the previous reversal point, , , its dynamic behavior only involves the d-axis. However, i is computed for , and . The “ ,”“ ” superscripts 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 C. Rotating-Rectifier Average-Value Model may be obtained from (9), (15), and (16) This section contains the derivation of the rotating-rectifier average-value model, which computes the average currents flowing in the exciter armature from , the VBR -axis flux linkage , and the VBR -axis inductance . (The (17) computation also uses the VBR -axis inductance , which is assumed constant herein.) The analysis is based on the clas- and sical separation of a rectifier’s operation in three distinct modes [17]. This type of rectifier modeling is valid for a constant (or (18) 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 The derivative is estimated from the variation of . voltage set: It is convenient to approximate it by the following relationship, written in the frequency domain: (27) (19) (28)

If is relatively small (so that ), a good low-fre- (29) quency estimate is obtained. This approximation is justified by the slow-varying nature of and consequently of . Equation where .4 It is useful to define a voltage angle so (19) is readily translated into a time-domain differential equa- that the -phase voltage attains its maximum value when tion, and the problematic numerical differentiation of is thus (i.e., ). The voltage and rotor angles are thus avoided. related by The exciter’s electromagnetic torque may be computed from the well-known expression . for (30) However, since the exciter is a small machine relative to the for . main alternator, its torque is assumed negligible herein. Because of symmetry, it is only necessary to consider a 60 The armature voltage equations must be expressed in voltage- interval (for a six-pulse bridge). Consider the interval which be- behind-reactance form to be compatible with the rotating-recti- gins when diode 6 (Fig. 1) starts conducting (at , where fier average-value model. In the VBR model, the rotor flux link- is a phase delay5), and ends at . During this ages are expressed interval, current is commutated from diode 2 to diode 6 (phase (20) to phase ); if the diode resistance is negligible, a line-to-line short-circuit between phases and is in effect, so . (21) ( denotes the line-to-neutral voltage of winding ). If the where 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. and The next observation is that the average rectiﬁer output voltage may be expressed (24)

(32) These equations hold for fast current transients; hence, the overbar notation is not appropriate. In VBR form is essentially constant for fast transients. In which may be approximated as particular, if for fast transients (such as commutation processes) we assume that the ﬁeld ﬂux linkage is constant, then it can be shown that is constant as well. Upon neglecting the rotor resistance, the VBR voltage equations may be expressed (33) (25) 4The standard numbering of the diodes (Fig. 1) corresponds to the order of conduction in the case of an phase sequence. However, in this case, a reverse (26) 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 with . model section to denote the hysterons’ upper switching point.

Neglecting armature resistance makes the analysis far more tractable. As it turns out, the inaccuracy involved in this as- sumption can be largely mitigated using a correction term which will be defined in a later section. The flux linkages may be related to the phase currents 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

(34) (39) 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 sep- (41) arated into the commutation and conduction subintervals. The Knowledge of and [from (39)] allows the computation commutation lasts for less than 60 electrical degrees of the average -axes rotor currents. Equation (38) is solved , where denotes the commutation angle. During the com- for and substituted into (36), which is transformed using (1). mutation interval , three diodes are conducting The currents of the first subinterval (denoted by the superscript (1, 2, and 6); during the conduction interval , “(i)”) are thus only two diodes are conducting (1 and 6). The currents are for (36) for where is the current flowing out of the recti- (42) fier and into the generator field, and is the (positive, anode-to-cathode) current flowing through diode 6; Their average value is increases from to . The average dc voltage may be computed from (33), after (43) substituting (36) into (34); this sequence of operations yields

(37) This integral is difﬁcult to evaluate analytically, so it is evaluated numerically (e.g., using Simpson’s rule [16], [25]). On the other The term represents the effective commutating re- hand, the average value of the conduction subinterval currents sistance for mode I operation. [denoted by the superscript “(ii)”] may be computed analytically Substitution of (36) into (35) yields (44)

The total -axes currents average value is

(38) (45)

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Fig. 8. Mode II operation. Fig. 9. Mode III operation.

2) Mode II Operation: In mode II operation (Fig. 8), the , there are three diodes conducting (1, 2, commutation angle is 60 , but commutation is auto-delayed by and 6), and a line-to-line short circuit is imposed on the exciter. the angle . There are always three diodes con- Due to symmetry considerations, . The cur- ducting, and the currents are rents are (46) , The current increases from to . (50) The average dc voltage is computed similarly to mode I by . substituting (46) into (34) and (33) 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. Evaluating (35) at and , and equating the two results yields the following nonlinear equation, which (51) is solved numerically for :

(48) (52) The expression (42) for is valid throughout commutation, and the average -axes currents are respectively. Substitution of the values of and at the three separating angles , , and , into (38), (51), and (52), yields (49)

(53) 3) Mode III Operation: In mode III operation (Fig. 9), commutation is delayed by and . (54) 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 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 starting ( , ). At , the commutation of diode 3 to diode 1 ﬁnishes ; (55) the current of diode 6 has increased to . During

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) (62) Equation (56) is solved numerically for . Using (53)–(55) in conjunction with (33), it can be shown that 5) Determining the Mode of Operation: Determination of (57) the mode of operation is the first step of the averaging subroutine and it guides the algorithm to the correct set of formulas. In Analytic formulas for the commutating currents during the particular, the mode is determined by comparing to a set of first subinterval may be obtained by solving the linear system increasing current values that define the mode boundaries. formed by (51) and (52) 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 of (58) and yields (64)

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

(65) (59) This mode separation is valid if the boundaries are well or- The average ﬁrst subinterval -axes currents may thus be eval- dered. Note that is always true; on the other uated analytically. After manipulation hand, is satisﬁed only for the following range of VBR inductance parameters:

(66) (60) The second subinterval -axes currents are given by At first glance, (66) imposes a significant constraint on the (42) and may be evaluated by numerical integration 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 gen- (61) erally expected that (66) will be satisfied for all “reasonable” inductance values. 4) Mode IV Operation: Traditionally, a rectifier’s operation 6) Solving the Nonlinear Equations: According to the oper- is divided into three distinct modes; these modes naturally occur ation mode, a numerical solution to one of the nonlinear equa- when the rectifier is feeding a passive resistive load. Herein, tions (41), (48), or (56) needs to be obtained. Recall that a con- however, an additional fourth mode (mode IV) needs to be continuous function has a root if sidered. This mode is an extension to mode III, and occurs when . In this case, it suffices to show i) , ii) the rectifier’s dc current exceeds the maximum possible current , and iii) . that the ac source alone (i.e., the exciter) may supply. This sit- For mode I operation, where , it may be uation may arise, for example, when is decreased rapidly shown that enough, while decays at a much slower pace, constrained by (67) the main alternator field inductance. During mode IV, a constant three-phase short circuit is im- (68) posed on the exciter , and at any given instant there are four diodes conducting (diodes 3, 1, 2, and 6 during the time For mode II, and frame considered in this analysis). The auto-delay and commutation angles are at their maximum possible values ( (69) and ), and the currents become purely sinusoidal, as may be readily seen by analyzing the mode III equations. (70)

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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 so- Fig. 10. Schematic of experimental setup; the brushless synchronous lution is unique. It may thus be obtained with arbitrary precision generator is feeding a nonlinear rectifier load. in a finite number of steps using the bisection algorithm [25]. 7) Incorporating Resistive Losses: The model’s accuracy If mode II: may be improved by taking into account the resistive losses of the armature and the voltage drop of the rotating rectifier a) Solve (48) for . diodes. Their incorporation affects the magnitude of the brush- b) Compute from (47). less exciter steady-state field current, as well as the transient c) Compute average currents from (42) and (49). behavior of the synchronous generator. In the previous sections, the armature resistance and the If mode III: diodes were ignored. The rigorous incorporation of these terms a) Compute from (57). in the model would entail considerable modifications and pos- b) Solve (56) for . sibly would make the algebra intractable. Hence, to simplify c) Compute average currents from (60) and (61). the analysis, the computation of the losses is decoupled from the computation of the average dc voltage. Thus, the average If mode IV: voltage applied across the main generator field is a) Set . (73) b) Compute average currents from (62). The average voltage loss is computed by averaging the drop 10) Compute from (73) and (74). across diodes 1 and 6, and the ohmic drop of the armature’s 11) Compute from (17). resistance, that is 12) Compute from (18). 13) Go to step (2). (74) Steps (3)–(5) are specific to the Preisach model. If a linear mag- A diode’s voltage–current characteristic is represented herein netizing inductance is used instead, set and by the following function: . (75) V. E XPERIMENTAL VALIDATION The parameters , , and are obtained with a curve-fitting The experimental setup (shown in Fig. 10) contains a 59-kW, procedure. 600-V, Leroy–Somer brushless synchronous generator, model LSA 432L7. The exciter is an eight-pole machine, whose field D. Model Summary is rated for 12 V, 2.5 A. The generator’s prime mover is a Dyne In summary, the algorithm proceeds as follows. Systems 110-kW, vector-controlled, induction-motor-based 1) Initialize model, assume the material is demagnetized. dynamometer, programmed to maintain constant rated speed 2) Compute from (19). (1800 r/min). The voltage regulator uses a proportional-integral 3) Determine the direction of change of using (16), and control strategy to maintain the commanded voltage [560 V, check for the reversal of direction. In case of direction line-to-line, fundamental, root mean square (rms)] at the gener- reversal, add a point to the magnetic history stack. ator terminals; the brushless exciter’s field current is controlled 4) Detect the crossing of a previous reversal point (minor with a hysteresis modulator. The generator is loaded with an loop closure). In this case, delete two points from the uncontrolled rectifier that feeds a resistive load through an history stack. filter. 5) Compute using one of (10)–(13). The exciter’s parameters (listed in Table I) were identified 6) Compute from (24). from rotating tests, as described in [26]. The time constant of 7) Determine from (30). (19) is . The load parameters are , 8) Determine the mode of operation, using (63)–(65). , and . The remaining components are 9) If mode I: documented in [27]–[29]. (In particular, the voltage regulator a) Compute from (37). model and control diagram is described in detail in Appendix D b) Solve (41) for . of [29].) The quantities of the internal rotating parts (Fig. 1) are c) Compute average currents from (42)–(45). not measurable because slip rings were not installed. Hence, the

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

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 Fig. 11. Plots of the commanded and actual line-to-line voltage “envelope,” in Fig. 12. Under steady-state conditions, the exciter operates computed from the synchronous reference frame voltages v a in mode II; however, the auto-delay angle varies with the op- Q@v C v A . Each of the seven trapezoid shaped blocks is characterized erating point. During transients, operation in all modes takes 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, place. Therefore, a simple mode I model would have been in- 560 V, 420 V, 280 V, respectively. [Note: the above voltage values correspond sufficient to predict this behavior. The observed rapid mode al- to root mean square (rms) quantities]. ternations and ripple in the waveforms of and result from the ripple in the main alternator field current which, in turn, is TABLE I caused by the rectifier load on the main alternator. LIST OF EXCITER MODEL PARAMETERS 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 steady- state 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” points, labeled , , , and . These points do not model is judged based on terminal quantities only, namely the lie on a straight line. This complex behavior could not have synchronous generator voltage and the exciter’s field current. been captured by a linear magnetization model (where The simulations were conducted using Advanced Continuous ). Simulation Language (ACSL) [30]. In order to “initialize” the magnetic state, the commanded In this case study, the generator’s voltage reference is mod- voltage is stepped from 0 to 560 V and then back to 0 V at ified according to the profile shown in Fig. 11. This series of 20 000 V/s (not shown in Fig. 11). The exciter’s flux is forced commanded voltage steps creates an extended period of signif- to a higher-than-normal level (Fig. 14). According to the dele- icant disturbances and tests the model’s validity for large-tran- tion property, the previous magnetic history is erased. Further- sients simulations. The terminal voltage exhibits an overshoot, more, on account of the congruency property, the return path de- which is more pronounced for the faster slew-rate steps. More- pends only on the reversal point on the curve. Hence, over, due to the exciter’s magnetically hysteretic behavior, it this initialization procedure is guaranteed to bring the material does not fall to zero. The varying levels of remanence in the back to the same state, regardless of the previous operating his- exciter machine reflect on the magnitude of the voltage and are tory. This theoretically predicted behavior was experimentally captured fairly accurately. The standard IEEE model [14] does verified.

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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

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[22] J. G. Frame, N. Mohan, and T. Liu, “Hysteresis modeling in an electro- Scott D. Sudhoff (SM’01) received the B.S. (Hons.), magnetic transients program,” IEEE Trans. Power App. Syst., vol. PAS- M.S., and Ph.D. degrees in electrical engineering 101, no. 9, pp. 3404–3412, Sep. 1982. from Purdue University, West Lafayette, IN, in 1988, [23] D. C. Jiles and D. L. Atherton, “Ferromagnetic hysteresis,” IEEE Trans. 1989, and 1991, respectively. Magn., vol. MAG-19, no. 5, pp. 2183–2185, Sep. 1983. Currently, he is a Full Professor at Purdue Univer- [24] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric sity. From 1991 to 1993, he was Part-Time Visiting Machinery. New York: IEEE Press, 1995. Faculty with Purdue University and as a Part-Time [25] W. Gautschi, Numerical Analysis, an Introduction, 1st ed. Boston, Consultant with P. C. Krause and Associates, West MA: Birkhäuser, 1997. Lafayette, IN. From 1993 to 1997, he was a Faculty [26] D. C. Aliprantis, S. D. Sudhoff, and B. T. Kuhn, “Genetic algorithm- Member at the University of Missouri-Rolla. He has based parameter identification of a hysteretic brushless exciter model,” authored many papers. His interests include electric IEEE Trans. Energy Convers., to be published. machines, power electronics, and finite-inertia power systems. [27] , “A synchronous machine model with saturation and arbitrary rotor network representation,” IEEE Trans. Energy Convers., vol. 20, no. 3, Sep. 2005. [28] , “Experimental characterization procedure for a synchronous machine 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.

Dionysios C. Aliprantis (M’04) received the elec- Brian T. Kuhn (M’93) received the B.S. and M.S. trical and computer engineering diploma from the degrees in electrical engineering from the University National Technical University of Athens, Athens, of Missouri-Rolla in 1996 and 1997, respectively. Greece, in 1999 and the Ph.D. degree in electrical He was a Research Engineer at Purdue University, and computer engineering from Purdue University, West Lafayette, IN, from 1998 to 2003. Currently, he West Lafayette, IN, in 2003. is a Senior Engineer with SmartSpark Energy Sys- Currently, he is serving in the armed forces of tems, Inc., Champaign, IL. His research interests in- Greece. His interests include the modeling and clude power electronics and electrical machinery. simulation of electric machines and power systems, and evolutionary optimization methods.

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