Vector control system of a self-excited induction generator including iron losses and magnetic saturation

Control Engineering Practice 21 (2013) 395–406

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Vector control system of a self-excited induction generator including iron losses and magnetic saturation Mateo Baˇsic´ n, Dinko Vukadinovic´ University of Split, Faculty of Electrical Engineering Mechanical Engineering and Naval Architecture, R. Boˇskovic´a 32, 21000 Split, Croatia

a r t i c l e i n f o

abstract

Article history: Received 6 June 2012 Accepted 5 November 2012 Available online 18 January 2013

We propose a novel indirect rotor-flux-oriented control system of a self-excited induction generator with online calculated iron losses and magnetic saturation. Within the control system, the iron loss resistance is represented as a function of both the synchronous frequency and magnetizing flux, and the magnetizing inductance is represented as a function of the magnetizing current. Both iron losses and saturation characteristics were determined from no-load tests. The effectiveness of the proposed approach is verified through both simulations and experiments. Through a comparison with the conventional approach, insight is provided into the efficiency estimation error and detuning due to iron losses. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Self-excited induction generator Magnetic saturation Iron losses Indirect flux-oriented control Efficiency Detuning

1. Introduction In recent years, induction generators have significantly gained in importance due to the several advantages they have over other types of electric generators, such as the conventional synchronous generator (Singh, 2004). Their capability to self-excite, i.e., to excite without an external reactive power source, led to their increased application in stand-alone power generating systems where reactive power from the grid is not available. In such applications, induction generators are usually excited by three AC capacitors (capacitor bank) connected across its stator terminals and are known as self-excited induction generators (SEIGs). Although the principle of self-excitation has been known since the 1930s (Wagner, 1939; Basset & Potter 1935), until recently, it was not possible to effectively utilize it. Currently, SEIGs are particularly preferred in power generating systems that employ wind or hydro-energy of power up to 15 kW (Simoes, Chakraborty, & Wood, 2006) and are gaining in popularity. This increase in usage is partly because renewable energy systems are receiving much attention worldwide due to the rapid exploitation of fossil fuels and the associated environmental pollution (Kamalapur & Udaykumar, 2011; Degeilh & Singh, 2011; Ghedamsi & Aouzellag, 2010; Margeta & Glasnovic, 2010). The main disadvantage of a nonregulated SEIG is that its output frequency and voltage are highly dependent on the speed of the prime mover, the value of the excitation capacitor, the machine’s parameters and both the value

n

Corresponding author. Tel.: þ385 21305615; fax: þ 385 21305776. E-mail addresses: [email protected], [email protected] (M. Baˇsic´).

0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.11.004

and power factor of the load impedance. Hence, if constant generated voltage is to be maintained, a control system has to be implemented. In SEIG models and vector control systems reported in the literature, the iron losses are usually completely omitted for the sake of convenience (Idjdarene, Rekioua, Rekioua, & Tounzi, 2008; Liao & Levi, 1998; Margato, Faria, Resende, & Palma, 2011; Lin, Huang, Wang, & Teng, 2007). However, even in induction machines with minimal iron losses, their impact is not negligibly small, so neglecting them can lead to serious errors in the assessment of an actual SEIG (Baˇsic´, Vukadinovic´, & Lukacˇ, 2010; Baˇsic´, Vukadinovic´, & Lukacˇ, 2011; Baˇsic´, Vukadinovic´, & Petrovic´, 2012). Neglecting the iron losses of an induction machine is also reported to cause detuning in induction motor vector control systems (Levi, Sokola, Boglietti, & Pastorelli, 1996; Levi, 1995; Aissa & Eddine, 2009; Wee, Shin, & Hyun, 2001), so a similar effect can be expected in SEIG vector control systems as well. In Leidhold, Garcia, and Valla (2002)), the authors propose a SEIG vector control system in which the iron losses are represented by means of a constant parameter, and in Seyoum, Grantham, and Rahman (2002) and Seyoum, Grantham, and Rahman (2003), the SEIG iron losses are represented as being linearly dependent on the air-gap voltage. However, in Baˇsic´ et al. (2012), such representations are shown to be unreliable due to the dependence of iron losses on both the magnetizing flux and synchronous frequency (Boldea & Nasar, 2002). As for the magnetic saturation, when modeling a non-regulated SEIG, making an approximation of neglecting the magnetic saturation is unacceptable because it is the key factor for the voltage buildup and stabilization (Seyoum, 2003; Seyoum et al., 2003). On the other hand, there are cases of vector controlled SEIGs reported in the

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Nomenclature C excitation capacitance fe synchronous frequency ica and icb a-axis and b-axis capacitor current components, respectively idc and udc DC current and DC voltage, respectively iLsa and iLsb a-axis and b-axis stator inductance current components, respectively iLa and iLb a-axis and b-axis load current components, respectively ima and imb a-axis and b-axis magnetizing current components, respectively Im and IRm magnetizing and iron loss current magnitude, respectively iRma and iRmb a-axis and b-axis iron loss current components, respectively ira and irb a-axis and b-axis rotor current components, respectively isa, isb, isc stator phase currents isd and isq d-axis and q-axis stator current components, respectively isa and isb a-axis and b-axis stator current components, respectively Kra and Krb a-axis and b-axis components of the initially induced voltage due to residual rotor flux linkage, respectively kc flux factor Lm, Lr, Ls magnetizing, rotor and stator inductance, respectively Lrs rotor leakage inductance Lss stator leakage inductance

literature where the magnetizing inductance is considered constant (Idjdarene et al., 2008; Leidhold et al., 2002). Nevertheless, this approximation should not be taken for the sake of both the model’s and control system’s accuracy and reliability. It is particularly important to take magnetic saturation into account if some type of flux-adjustment-based loss minimization strategy is considered. Because inclusion of the iron losses and magnetic saturation inevitably complicates the SEIG model and, consequently, the corresponding vector control system, it needs to be implemented in a manner that is both accurate and simple. In Sokola and Levi (2000), a dynamic d–q model of an induction machine is proposed in which both the magnetic saturation and iron losses are taken into account, and, based on this model, a new RFO vector control scheme of an induction motor is developed in which both the rotor flux estimator and rotor resistance identifier are included. In the model proposed there, the iron loss resistance is represented as solely dependent on the synchronous frequency, while the dependence on the magnetizing flux is neglected. This is a valid approach for the case of an induction motor operating in the base speed region, in which the value of the magnetizing flux is held constant and equal to the nominal value, but not in the field weakening region, in which the value of the magnetizing flux is lower than the nominal value. However, within SEIGs, the value of the magnetizing flux is subject to change regardless of the rotor speed, due to the magnetic saturation, so the approach described above cannot be considered valid even in the base speed region. In this paper, a novel indirect rotor-flux-oriented (IRFO) control system of a SEIG is proposed in which both iron losses and magnetic saturation are taken into account and calculated online. The iron losses are expressed as a function of both the magnetizing

n mechanical speed in rpm p number of pole pairs Rdc and RL DC load resistance and AC load resistance, respectively Rm, Rr and Rs iron loss, rotor and stator resistance, respectively s Laplace operator Sa, Sb, Sc switching functions of the three inverter legs subscript for the Thevenin equivalent T Te induced electromagnetic torque Tr rotor time constant uca and ucb a-axis and b-axis capacitor voltage components, respectively udc DC voltage uLa and uLb a-axis and b-axis load voltage components, respectively usa, usb, usc stator phase voltages usa and usb a-axis and b-axis stator voltage components, respectively Z efficiency factor s total leakage factor crd and crq d-axis and q-axis rotor flux linkage components, respectively cra and crb a-axis and b-axis rotor flux linkage components, respectively csa and csb a-axis and b-axis stator flux linkage components, respectively ye angular position of the d–q reference frame oe, or and os synchronous, rotor and slip angular speed in electrical rad/s, respectively n superscript for the reference variable subscript for the initial condition 0

flux and synchronous frequency, whereas the level of magnetic saturation is expressed as a function of the magnetizing current. The efficacy of the proposed IRFO control system is verified both through simulations and experimentally.

2. Basic configuration of the control system Fig. 1 shows the basic configuration of the control system under consideration. The main components of the system are the induction generator, the prime mover connected to the generator’s shaft (not shown in Fig. 1), the current controlled VSI (CCVSI) with IGBTs, the DC link with the exciting capacitor and resistive load, and the IRFO controller. The objective of the control algorithm is to ensure

Fig. 1. Basic configuration of the IRFO control system.

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the DC voltage remains constant and equal to the reference value regardless of the changes in both the load and rotor speed. In Fig. 1, or, isa, isb, isc and udc denote the measured rotor speed, three phase currents and DC voltage, respectively. Hence, for realization of the control system, a total of five sensors are required. In addition, if the stator neutral is not grounded, one of the phase currents can be obtained from the other two, thus allowing the number of current sensors to be reduced to two. The battery on the DC side provides the initial voltage across the capacitor during the excitation process and as soon as the load voltage rises to a value higher than the battery voltage, the battery is automatically switched off by a diode. In this paper, the improvements within the vector control algorithm are considered (the IRFO controller in Fig. 1). These improvements are based on the modified SEIG model in which both the iron losses and magnetic saturation are taken into account, as described in the next section.

397

Thevenin equivalents for the stator resistance and the stator voltages/currents are calculated as follows: Rs Rm Rs þRm

RsT ¼ Rs 99Rm ¼

usT a ¼ usa

isT a ¼ isa

Rm , Rs þ Rm

ð1Þ

usT b ¼ usb

Rs þRm usa þ , Rm Rm

Rm Rs þRm

isT b ¼ isb

ð2Þ

Rs þRm usb þ Rm Rm

ð3Þ

The mathematical equations of the conventional SEIG model are derived from the equations of the conventional induction machine model, as described in Baˇsic´ et al. (2012). The conventional SEIG model equations expressed in the stationary reference frame are given below: usa ¼ Rs isa þ

3. Self-excited induction generator modeling In this paper, the SEIG model proposed by the authors in Baˇsic´ et al. (2012) is considered for use in the control system design because of its reported high accuracy, high numerical stability and low computational demands. These features, from the aspect of the control system design, imply low hardware and software requirements and thus low cost.

0 ¼ Rr ira þ

dcsa , dt

usb ¼ Rs isb þ

dcra þ or crb , dt

dcsb dt

0 ¼ Rr irb þ

ð4Þ

dcrb or cra dt

ð5Þ

csa ¼ Lss isa þ Lm ima , csb ¼ Lss isb þ Lm imb

ð6Þ

cra ¼ Lrs ira þ Lm ima þ cra0 , crb ¼ Lrs irb þ Lm imb þ crb0 ,

ð7Þ

3.1. Equivalent circuit and mathematical equations

ima ¼ isa þ ira ,

ð8Þ

Within the considered SEIG model, both magnetic saturation and iron losses are taken into account. The equivalent iron loss resistance is represented as a function of both the synchronous frequency and magnetizing flux. The magnetizing inductance is represented as a function of the magnetizing current, thereby taking the magnetic saturation into account. The equivalent circuit of the SEIG model in the stationary reference frame is shown in Fig. 2. Because all the physical phenomena along the b-axis are analogous to those along the a-axis, but with a phase shift of 901 el., only the equivalent circuit for the a-axis is presented. By substituting the dashed area in Fig. 2 with the Thevenin equivalents, the equivalent circuit shown in Fig. 3 is obtained. Again, only the equivalent circuit for the a-axis is presented. In Fig. 3, the iron loss resistance is included within the variable stator resistance, RsT. Such representation of the iron losses has already been proposed and considered for a non-regulated SEIG in Baˇsic´ et al. (2012), but it has not yet been considered for the control system design.

Te ¼

imb ¼ isb þ irb

 3 Lm  p cra isb crb isa 2 Lr 1 C

uca ¼ usa ¼

Z

t

0

ica dt þ usa0 ,

uLa ¼ usa ¼ RL iLa , isa ¼ iLa þica ,

ð9Þ

ucb ¼ usb ¼

1 C

Z 0

t

icb dt þ usb0

uLb ¼ usb ¼ RL iLb

isb ¼ iLb þicb

ð10Þ ð11Þ ð12Þ

Because the configuration of the equivalent circuit in Fig. 3 is the same as that of the conventional SEIG model, in which the iron losses are completely omitted, the equations of the conventional SEIG model are also valid for the model in Fig. 3, provided that the following substitutions are made: Rs -RsT usa -usT a , usb -usT b

ð13Þ

isa -isT a , isb -isT b Hence, the SEIG model shown in Fig. 3 can be fully described by the following first order differential equations, with the stator and rotor currents chosen as the state variables: 1

sisT a ¼ Fig. 2. SEIG equivalent circuit with iron loss resistance—a-axis.

sLs Lr

sisT b ¼

ðL2m or isT b Lr RsT isT a þ Lm or Lr irb þ Lm Rr ira Lr usT a Lm K ra Þ

1

sLs Lr

ð14Þ

ðLr RsT isT b L2m or isT a þ Lm Rr irb Lm or Lr ira Lr usT b Lm K rb Þ

ð15Þ sira ¼

1

sLs Lr

ðLs or Lm isT b þ Lm RsT isT a Ls or Lr irb Ls Rr ira þ Lm usT a Ls K ra Þ

ð16Þ sirb ¼ Fig. 3. SEIG equivalent circuit with the Thevenin equivalents—a-axis.

1

sLs Lr

  Lm RsT isT b þ Ls or Lm isT a Ls Rr irb þ Ls or Lr ira þ Lm usT b Ls K rb

ð17Þ

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3.2. Iron loss resistance and magnetizing inductance characteristics When an induction machine is subject to significant changes in load, magnetizing flux and rotor speed during operation, as is generally the case with SEIGs, the iron loss resistance should be, for the sake of accuracy, represented as a function of both the synchronous frequency and magnetizing flux. The iron loss resistance can be determined experimentally by performing a set of standard no-load tests over a wide range of frequencies, as described in Baˇsic´ et al. (2010), (2011), (2012). During tests, a synchronous generator driven by a DC motor was used to provide sinusoidal supply at the induction machine terminals. Measured data were acquired by means of both Fluke 435 power quality analyzer and conventional analog instruments. The equivalent iron loss resistance characteristics are shown in Fig. 4. Note that in this paper, the influence of the magnetizing flux on the iron losses is expressed by means of the corresponding iron loss current, iRm. SEIGs are known to operate at high levels of magnetic saturation (Baˇsic´ et al., 2012). Hence, for the sake of the accuracy and reliability of the model, the magnetic saturation should also be taken into account. The variation of the magnetizing inductance with respect to the magnetizing current was determined experimentally from the no-load tests. Fig. 5 shows the measured magnetizing inductance characteristics obtained at 50 Hz. The magnetizing inductance, when expressed as a function of the magnetizing current, is frequency independent (Baˇsic´ et al., 2011), so the characteristic in Fig. 5 can be used for all operating frequencies.

The magnetizing inductance characteristic that is suitable for use in simulations is obtained by linearly interpolating the measured values (look-up table) and by approximating the nonsaturated region of the characteristic (Imo r 1.437 A) with the constant value of 0.4058 H (dashed line in Fig. 5). This approximation is implemented to avoid possible numerical instabilities during the self-excitation process and is well justified by the fact that the steady-state operating point of the SEIG is always located somewhere in the saturated region of the characteristic (Imo 41.437 A).

4. IRFO control system modeling 4.1. Proposed IRFO control system In general, IRFO vector control systems are based on the following assumption: the d-axis of the rotating d–q reference frame is always kept aligned with the rotor flux space vector whose magnitude is always equal to the corresponding reference value. This assumption makes the IRFO control systems fairly easy to implement, but at the same time, it must be taken with caution due to detuning problems, i.e., errors in estimation of the rotor flux space vector. It is a well-known fact that any error in estimation of the rotor time constant, Tr, inevitably induces an error in estimation of the rotor flux and, thus, causes detuning within the control system. However, detuning problems can also be induced by neglecting the iron losses, as it is investigated in Section 5.3. Fig. 6 shows the proposed IRFO control system, with the dashed rectangle representing the IRFO controller. Note that within the proposed control algorithm, both the magnetic saturation and iron losses are taken into account and calculated online, as shown in Fig. 6b and c, respectively. The equations of the corresponding control algorithm are derived from the SEIG model shown in Fig. 3, the conditions of the indirect rotor flux orientation and the assumption of an ideal CCVSI (i.e., the actual phase currents in the machine equal their reference values). By considering a resistive load, the DC link can be represented by the following equation: udc ¼ 

Fig. 4. Measured iron loss resistance characteristics.

1 C

Z 0

t

  u idc þ dc dt þ udc0 Rdc

ð18Þ

where C is the DC-link capacitance and Rdc is the resistive load. The initial voltage on the capacitor is denoted by udc0. Stator phase voltages created by the converter are expressed in terms of the switching functions as usa ¼

1 u ð2Sa Sb Sc Þ 3 dc

ð19Þ

usb ¼

1 u ð2Sb Sc Sa Þ 3 dc

ð20Þ

usc ¼

1 u ð2Sc Sa Sb Þ 3 dc

ð21Þ

The total DC current, idc, can be expressed using the converter switching function as follows: idc ¼ Sa isa þ Sb isb þSc isc

Fig. 5. Measured magnetizing inductance characteristic.

ð22Þ

The value of the switching function for any of the three converter legs is determined by the hysteretic current controller in the given phase.

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Fig. 6. Proposed SEIG control system: (a) IRFO system with CCVSI, (b) online calculation of the magnetizing inductance and (c) online calculation of the iron loss resistance.

The rotor flux reference value is obtained as a function of the rotating speed and the DC voltage reference value, as follows:

cnr ¼

kc undc

or

ð23Þ

Determination of the rotor flux reference is explained in detail in Section 4.2. The reference frame position, ye, is calculated by integrating the synchronous angular speed, oe, which is obtained as the sum of the measured rotor angular speed, or, and the slip angular

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400

speed, os. The expression for calculating the slip angular speed is given by

os ¼

insTq

ð24Þ

T r insTd

For induction generators, the slip angular speed is negative (i.e., the rotor angular speed is greater than the synchronous angular speed) so the Thevenin equivalent q-axis components of the stator current in Eq. (24) has a negative value. The Thevenin equivalent d-axis and q-axis components of the stator current that appear in Fig. 6 are inherent to the proposed control algorithm and do not appear within the conventional control algorithm, in which the iron losses are completely omitted. Alternatively, it can be said that within the conventional algorithm, these currents are equal to the actual d-axis and q-axis components of the stator current, i.e., insd ¼ insTd

ð25Þ

insq ¼ insTq

ð26Þ

Fig. 7. Phase-to-phase no-load voltage versus magnetizing current.

In the proposed algorithm, the Thevenin equivalent d-axis component of the stator current, i*sTd, is obtained directly from the rotor flux reference and the variable magnetizing inductance (Fig. 6a), while the actual d-axis component of the stator current, i*sd, is calculated as insd ¼ insTd insTq oe

sLs Rm

ð27Þ

The term containing the iron loss resistance, Rm, on the right hand side of Eq. (27) is represented in Fig. 6a by the left gray shaded block. By setting this term equal to zero, Eq. (25) is obtained. Furthermore, in the proposed algorithm, the Thevenin equivalent q-axis component of the stator current, insTq, is calculated as insTq ¼ insq insTd oe

Ls Rm

Fig. 8. Rotor flux reference versus DC voltage.

ð28Þ

while the actual q-axis component of the stator current, insTq, is obtained directly at the output of the voltage PI controller. The term containing the iron loss resistance, Rm, on the right hand side of Eq. (28) is represented in Fig. 6a by the right gray shaded block. By setting this term equal to zero, Eq. (26) is obtained. The above considerations imply that by eliminating the terms containing the iron loss resistance, Rm, on the right hand side of Eqs. (27) and (28), and thus eliminating the gray shaded blocks in Fig. 6a, the conventional control algorithm is obtained. Of course, in this case, the current symbols in Fig. 6 should be changed as well by substituting the Thevenin equivalent d-axis and q-axis components of the stator current with the actual d-axis and q-axis components. Therefore, the transition between the proposed and the conventional control algorithm can be executed in a straightforward manner by setting the iron loss resistance value either to infinity (or, for practical purposes, to a very large number) – the conventional algorithm, or to the online calculated value (Fig. 6c) – the proposed algorithm. This transition can even be executed online, as will be explained later. 4.2. Determination of the rotor flux reference From the set of no-load tests performed at various frequencies, the stator phase-to-phase voltage versus the magnetizing current characteristics were determined, as shown in Fig. 7. The relationship between the maximum achievable magnitude of the fundamental AC line-voltage and the DC-link voltage for the case of a three-phase VSI (square-wave operation) can be

expressed as (Espinoza, 2001) pffiffiffi 2 3 U ab r U dc

p

ð29Þ

In this paper, a 10% lower value is used for the stator phase-tophase voltage to ensure the minimum stability. By using the set of no-load characteristics in Fig. 7 and setting the DC-link voltage to a desired value, the approximate required d-axis stator current component for each rotor speed (i.e., isd EImo, or E oe ¼2pfe) can be determined (Margato et al., 2011). After determining the d-axis stator current, the rotor flux value can then be determined from the measured magnetizing inductance characteristic (Fig. 5) as

cr ¼ Lm isd

ð30Þ

Repeating the same procedure for various voltages and frequencies allows the relationship between the rotor flux, DC voltage and synchronous frequency (i.e., rotor speed, or E oe) to be determined, as shown in Fig. 8. The dashed lines in Fig. 8 represent the imposed rotor flux limits. The minimum rotor flux value was determined from the knee of the magnetizing inductance characteristic (cr_min ¼0.48 Wb). This choice of the minimum value is justified by the fact that the stable steadystate operating point of the SEIG is always located in the saturated region of the characteristic. The maximum rotor flux value was defined to be 10% above the nominal value (cr_max ¼0.93 Wb). Because the characteristics in Fig. 8 are approximately linear, especially the regions of interest (i.e., the regions between the

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401

manual switch in the lower position practically eliminates the iron losses both from the induction generator model and control algorithm. If the iron losses are to be eliminated only from the control algorithm and not from the generator model, the iron loss resistance has to be separated into two parameters, one that is inherent to the induction generator model and the other that is inherent to the control algorithm. Such modification of the simulation model is applied in Section 5.3.

5. Simulation and experimental results 5.1. Performance assessment of the proposed control system The following simulations were performed using the simulation model of the proposed control system: Fig. 9. DC voltage limits for a given rotor speed.

rotor flux limits), with the slopes being approximately inversely proportional to the rotor speed, the relationship between the rotor flux reference, DC voltage and rotor speed can be expressed as Eq. (23). The mentioned inverse proportionality is expressed in Eq. (23) by multiplying the flux factor, kc, with the inverse rotor speed. In this paper, the flux factor is approximated by a constant value of 0.28. The previously imposed rotor flux limits ensure system stability in cases when the rotor flux reference value calculated from Eq. (23) exceeds the rotor flux limits. After defining the rotor flux limits and the flux factor value, it is possible to determine from Eq. (23) the minimum/maximum DC voltage for a given speed and vice versa. This relationship is presented in Fig. 9 by the gray shaded stable operating area. Hence, for any given speed of the prime mover, which is obtained by either measurement or estimation, the DC voltage limits can be defined and implemented within the control system. In this way, the system stability is ensured in steady state as well as during both system start-up and shut-down.

1. The rotor speed is fixed at n¼900 rpm, the reference voltage is fixed at undc ¼250 V, and the rotor flux reference value is fixed at cnr ¼0.743 Wb (due to the fixed speed and voltage). 2. The rotor speed is fixed at n¼ 1200 rpm, the reference voltage is fixed at undc ¼300 V, and the rotor flux reference value is fixed at cnr ¼0.669 Wb (due to the fixed speed and voltage). 3. The rotor speed is fixed at n¼ 1500 rpm, the reference voltage is fixed at undc ¼350 V, and the rotor flux reference value is fixed at cnr ¼0.624 Wb (due to the fixed speed and voltage). In all the simulations, the load is varied in a step manner as follows: at t ¼1 s, the load resistance of 220 O is connected, at t¼ 4 s, the load resistance value is changed to 500 O and at t¼ 7 s,

4.3. Simulation model of the proposed control system A simulation model of the proposed control system was built in the MATLAB/Simulink environment (Appendix A). Within the simulation model, two different sample times were used, namely Ts1 ¼1/28,000 s was used for solving the equations of the induction generator, the hysteretic controllers and the IGBT converter, while Ts2 ¼1/4000 s was used for solving the equations of the control algorithm. Using a single sample time equal to Ts1 would result in considerably slower execution of the simulation program, while the sampling accuracy of the control signals would not be notably increased. On the other hand, using a single sample time equal to Ts2 would result in an inaccurate induction generator model, and it is not able to simulate the maximum required switching frequency of the IGBTs. In this case, the maximum switching frequency of the IGBT converter is equal to 2/Ts1 ¼14,000 Hz. The hysteretic controllers are modeled with twice the hysteresis band as equal to approximately 4% of the induction machine’s rated peak current (i.e., 2 H¼0.2 A). In this way, current error, Di, in any of the three phases is, ideally, restricted to at most 70.1 A. In reality, the current error can exceed 0.1 A and reach the value of twice the hysteresis band (Lorenz, Lipo, & Nowotny, 1997). The measured iron loss resistance and magnetizing inductance characteristics (Figs. 4 and 5) are implemented by means of the look-up tables. The manual switch at the output of the block ‘‘Fig. 6c’’ enables the transition between the proposed and the conventional control system to be executed online. In the case when a single iron loss resistance is used in the entire simulation model, setting the

Fig. 10. Rotor flux reference—experiments.

Fig. 11. DC voltage response to variations in load: (a) simulations and (b) experiments.

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Fig. 12. Reference d-axis stator current: (a) simulations and (b) experiments.

Fig. 15. Online calculated iron loss resistance: (a) simulations and (b) experiments.

Fig. 13. Reference q-axis stator current: (a) simulations and (b) experiments. Fig. 16. Steady-state stator current—no load: (a) simulations and (b) experiments.

Fig. 14. Online calculated magnetizing inductance: (a) simulations and (b) experiments.

the load is disconnected. The results obtained from running the simulations are shown in Figs. 11-17. In addition, to experimentally validate the proposed control algorithm, several tests were performed over wide ranges of speed, load and DC voltage. The main components of the laboratory setup are as follows: induction generator (with parameters provided in Appendix C.1); DC motor used as a prime mover; SIMOREG DC-MASTER converter (type 6RA70), manufactured by Siemens, used for DC motor speed control; DS1104 R&D controller board, manufactured by dSpace, used for the control algorithm implementation; TMB 308 torque transducer, manufactured by Magtrol, used for the mechanical power measurements; and a 3-phase IGBT power converter. In addition, an incremental rotary encoder, type ROD 426 B, manufactured by Heidenhain, was used for acquisition of the rotor speed, two current transducers, type LA 50-P, manufactured by LEM, were used for acquisition of the phase currents and a voltage transducer, type LV 200-AW/2/800, also manufactured by LEM, was used for acquisition of the DC voltage. Both in simulations and experiments, an excitation capacitor with a capacitance of 470 mF was used. Figs. 10–17 show the

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Fig. 17. Steady-state stator current—Rdc ¼220 O: (a) simulations and (b) experiments.

experimental results obtained for the same operating regimes as in the simulations. The observation starts from the no-load steady state. In the experiments, the load step changes were applied at slightly different times compared to the simulations. Additionally, the speed transients due to the changes in load are inevitable. Because the rotor flux reference is determined online from the DC voltage reference and the measured speed signal (Eq. 23), similar transients but with the opposite sign are also present in the rotor flux reference signal, as shown in Fig. 10. Fig. 11 shows that the DC voltage closely tracks the reference value with a fast transient response to variations in load, both in the simulations and experiments. The highest transient voltage drop noted in the simulations is equal to 12.7%, whereas in the experiments, it is slightly higher (due to speed transients and additional losses) and it is equal to 15.2%. Because the steady-state value of the rotor flux reference value is held constant for each operating regime, the steady-state value of the d-axis current component is also constant, as shown in Fig. 12. On the other hand, the q-axis current component varies significantly with the connected load (decoupled control). If no detuning is present in the control system and the rotor flux value is held constant, the value of the q-axis current component is proportional to the connected load plus losses. Hence, in Fig. 13, the lowest steady-state value of the q-axis component occurs at no load, whereas the highest steady-state value occurs with the applied load resistance of 220 O. Additionally, due to additional losses that are not considered within the simulation model, such as the converter losses, stray load losses and friction losses, there is a difference between the q-axis current values obtained from the experiments and simulations (i.e., the experimental values are higher). The difference is greater with the applied load of 220 O due to the higher additional losses, whereas both at no load and with the applied load of 500 O, the agreement is remarkable. Fig. 14 shows the different values of the magnetizing inductance obtained for the considered operating regimes. The magnetizing inductance is calculated online, both in the simulations and experiments. Due to the different magnetic saturation levels, different values of the magnetizing inductance are obtained for each operating regime and can be linked to the values of the d-axis stator current component (Fig. 12) through the magnetizing inductance characteristic shown in Fig. 5 (insTd EIm).

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The iron loss resistance, shown in Fig. 15, is also calculated online from the measured iron loss characteristics and the estimated synchronous frequency and iron loss current (i.e., the magnetizing flux). Because the values of both the synchronous frequency and magnetizing flux differ significantly from one operating regime to another, the obtained values of the iron loss resistance also differ significantly. The highest value of the iron loss resistance is obtained for the highest rotor speed (i.e., the highest synchronous frequency). In addition, because the magnetizing flux is held nearly constant within each operating regime, the slight variations in the iron loss resistance steady-state values are only due to slip variations, which are reflected in the synchronous frequency variations (i.e., higher slip-lower synchronous frequency-lower iron loss resistance). In Figs. 16 and 17, the stator phase current is shown to be successfully maintained within the imposed hysteresis band limits for different load values. Note that only the current waveforms obtained for the 3rd operating regime are shown because the waveforms obtained for the other two regimes are similar. At no load, there is no significant difference between the simulated and the measured current magnitude (Fig. 16), whereas when the load resistance of 220 O is connected, the measured current magnitude is slightly higher compared to the simulated one (Fig. 17), which is due to the difference in the q-axis components. 5.2. Effect of iron losses on the efficiency estimation In this section, the comparison is performed between the estimated efficiencies obtained for the proposed and the conventional simulation model of the control system. Recall that the conventional

Fig. 18. SEIG efficiency as a function of: (a) load resistance (Udc ¼ 300 V and N ¼1200 rpm), (b) DC voltage (N¼ 1200 rpm and Rdc ¼ 220 O) and (c) rotor speed (Rdc ¼220 O and Udc ¼ 300 V).

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Table 1 SEIG efficiency versus load resistance (Udc ¼ 300 V and N ¼1200 rpm).

Measured Proposed model Conventional model

Rdc ¼ 155 O (%)

Rdc ¼ 220 O (%)

Rdc ¼500 O (%)

69.2 77.0 84.1

74.4 77.4 87.0

67.8 68.7 86.8

Table 2 SEIG efficiency versus DC voltage (N ¼1200 rpm and Rdc ¼220 O).

Measured Proposed model Conventional model

Udc ¼ 250 V (%)

Udc ¼ 300 V (%)

Udc ¼350 V (%)

73.5 77.6 87.7

74.4 77.4 87.0

74.7 76.4 85.8

operating regimes, an improvement in accuracy of the efficiency estimation by approximately 10% is obtained by using the proposed model instead of the conventional model. In Table 3, for a rotor speed of 900 rpm, the measured efficiency seems to be slightly higher (1.8%) than the one obtained for the proposed model, which clearly cannot be the case, and can be attributed to the measurement error and/or to the changes in the temperature dependent machine parameters during operation (the temperature dependency is not included within the simulation models). However, because the error is less than 2%, it can be considered negligible.

5.3. Effect of iron losses on the rotor flux estimation As stated above, in the IRFO control systems, it is assumed that the d-axis of the rotating d-q reference frame is always aligned with the rotor flux space vector. Hence, if this assumption is valid, d-axis component of the rotor flux space vector should be equal to the rotor

Table 3 SEIG efficiency versus rotor speed (Rdc ¼ 220 O and Udc ¼ 300 V).

Measured Proposed model Conventional model

N ¼ 900 rpm (%)

N ¼ 1200 rpm (%)

N ¼1500 rpm (%)

67.8 66.0 76.2

74.4 77.4 87.0

75.3 80.4 89.2

simulation model implies neglecting the iron losses within the equations of both the induction generator and control algorithm. Some of the simulation results are also compared with the experimental results. The following ranges of rotor speed, DC voltage and load resistance were considered: 900–1500 rpm, 200–400 V and 110–500 O, respectively, taking into account that both the voltage and speed are consistent with the characteristics shown in Fig. 9. Fig. 18 shows the SEIG efficiency variation with respect to load resistance, DC voltage and rotor speed, respectively, obtained for both the proposed and conventional models. The efficiency estimated by the proposed model is somewhat lower compared to the conventional model for all the considered speeds, voltages and load values. Because the electrical output power is independent of the model and is solely defined by the connected load and DC voltage, the difference in the obtained efficiencies can only be attributed to the difference in the SEIG input power, i.e., to the difference in the SEIG overall losses. This attribution is obviously a consequence of including the iron losses within the proposed model. In Fig. 18a, the difference in efficiencies between the two models significantly increases with the load resistance value. This behavior is due to the fact that when the SEIG is lightly loaded, the iron losses contribute to a larger fraction of the SEIG overall losses (Baˇsic´ et al., 2012). On the other hand, in Fig. 18b and c, for variations in the DC voltage and the rotor speed, respectively, the difference in efficiencies does not change significantly. In addition, for both the proposed and conventional models, the efficiency slightly decreases with the DC voltage for a constant load resistance and rotor speed, whereas the efficiency significantly increases with the rotor speed for a constant load resistance and DC voltage. Tables 1–3 show the comparison between the efficiencies obtained from the simulations and experiments for various rotor speed, load resistance and DC voltage values. In general, the proposed model provides significantly improved estimation of the actual efficiency compared with the conventional model. For the considered operating regimes, the highest efficiency estimation error obtained for the conventional model is equal to 19%, the lowest error is equal to 8.4%, and the average error is equal to 13.4%. For the proposed model, the highest error is equal to 7.8%, the lowest error is equal to 0.9%, and the average error is equal to 3.5%. Hence, for the considered

Fig. 19. Rotor flux linkage components: (a) d-axis and (b) q-axis.

Fig. 20. Steady-state stator current—detuning effect: (a) proposed control system and (b) conventional control system.

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Fig. A.1. Proposed control system model in Simulink (black lines-Ts1 and gray lines-Ts2).

flux space vector magnitude, whereas q-axis component should be equal to zero. However, this condition is fulfilled only if there is no detuning present within the control system. The detuning is usually caused by an error in the estimation of the rotor resistance, which is a temperature dependent parameter, but it can also be caused by neglecting the iron losses within the control system. For the purpose of the control system detuning analysis, the simulation model in Appendix A.1 was modified as follows: the iron loss resistance was separated into two parameters, one that is inherent to the induction generator model (Rm1 ¼f(oe, IRm)) and the other that is inherent to the control algorithm (Rm2 ¼1012 O). This approach allowed the iron losses to be neglected only within the equations of the control algorithm and emulates the situation where the iron losses are present within the machine (which is generally the case), but are neglected within the control algorithm. Fig. 19 illustrates the error in the estimation of the rotor flux linkage space vector in the case when the iron losses are neglected only within the control system. The results for both the d-axis and q-axis components of the rotor flux linkage space vector are shown and compared with the results obtained for the proposed control system. Only the simulation results obtained for the 1st operating regime, described in Section 5.1, are shown because the results obtained for the other two operating regimes are similar. It is evident that neglecting the iron losses within the control system can result in detuned operation and, consequently, in loss of control of the machine stator currents. This effect is clearly shown in Fig. 20, in which two experimentally obtained steadystate stator current waveforms are presented: one for the proposed control system and the other for the conventional control system. The current waveforms are obtained for the 1st operating regime with the applied load resistance of 220 O. The same rotor flux reference value of 0.836 Wb is used in both control systems.

Fig. B.1. IRFO control system experimental setup.

Finally, note that in the experiments, the transition between the proposed and the conventional control system can be executed online in a similar manner as in the simulations, i.e., by means of a switch that enables the iron loss resistance to be either calculated online or set to be equal to 1012 O and thereby be practically eliminated from the control system.

6. Conclusions This paper presents a novel approach for the design of the IRFO control system of a SEIG. The proposed control system was derived from the SEIG mathematical model in which both iron

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losses and magnetic saturation effects are taken into account. Within the proposed control system, the calculation of the variable iron loss resistance and magnetizing inductance is executed online. The same approach in the control system design could be easily extended to other types of induction machine vector control systems. However, this option is not considered in the paper. From the extensive simulation and experimental analysis performed, the following conclusions have been drawn: 1. The simulation model of the proposed control system provides a significantly more accurate assessment of the actual control system compared with the simulation model of the conventional control system (the efficiency estimation error is reduced by approximately 10%). 2. Neglecting the iron losses in the SEIG vector control system can lead to detuned steady-state operation. 3. The proposed control system provides successful compensation of the iron losses with only slightly increased complexity. 4. The proposed control system provides excellent DC voltage control with fast transient response and no steadystate error. Appendix A.1 Fig. A.1 Appendix B.1 Fig. B.1 Appendix C.1 Pn ¼1.5 kW, Un ¼380 V, p¼2, Y, In ¼3.81 A, nn ¼1391 r/min, Lnm ¼0.4058 H, Lss ¼0.01823 H, Lrs ¼0.02185 H, Rs ¼4.293 O, Rr ¼3.866 O (at 20 1C), Tn ¼10.5 Nm, J¼ 0.0071 kgm2, crn ¼0.845 Wb. References Aissa, K., & Eddine, K. D. (2009). Vector control using series iron loss model of induction, motors and power loss minimization. World Academy of Science, Engineering and Technology, 52, 142–148. Basset, E. D., & Potter, F. M. (1935). Capacitive excitation for induction generators. AIEE Transactions on Electrical Engineering, 54, 540–545. Baˇsic´, M., Vukadinovic´, D., & Lukacˇ, D. (2010). Analysis of an enhanced SEIG model including iron losses. In: Proceedings of sixth WSEAS international conference on energy, environment, ecosystems and sustainable development (pp. 37–43). Timisoara, Romania. Baˇsic´, M., Vukadinovic´, D., & Lukacˇ, D. (2011). Novel dynamic model of self-excited induction generator with iron losses. International Journal of Mathematical Models and Methods in Applied Sciences, 5, 221–229.

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