Control of inverter-assisted single-phase induction generators

This thesis investigates the problem of producing power using a single-phase induction machine. Single-phase induction motors are typically nonsymmetric two-phase motors, also called split-phase motors, with a main winding and an auxiliary winding. We consider the control of power delivered by the main winding through the auxiliary winding. A state-space model of an asymmetric split-phase generator has been developed. This model is used to characterize and identify a 1/3 hp induction machine. The parameters are used to predict suitable operating regions for the generator. It is found that there exists a viable operating region in a band just above the synchronous speed of the motor. Four control algorithms are presented. The first is a simple open-loop algorithm for a sinusoidal reference signal. The second is a controller based on the inverse-G adaptive algorithm. The third is an algorithm featuring an adaptive estimate of the system frequency response. Finally, a PI control method is considered for situations where the phase of the generated voltage does not need to be controlled. Simulated and experimental results are presented for each of the control algorithms. The closed-loop algorithms were able to track a desired reference. These algorithms were also able to reject a disturbance caused by a 20 percent increase in speed.

CONTROL OF INVERTER-ASSISTED SINGLE-PHASE INDUCTION GENERATORS by Maxwell Myers A thesis submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical and Computer Engineering The University of Utah December 2012 Copyright © Maxwell Myers 2012 All Rights Reserved The Unive rs i ty of Utah Graduat e School STATEMENT OF THESIS APPROVAL The thesis of Maxwell Myers has been approved by the following supervisory committee members: Marc Bodson , Chair 4/20/12 Date Approved Faisal Khan , Member 4/05/12 Date Approved Jake Abbott , Member 4/17/12 Date Approved and by Gianluca Lazzi , Chair of the Department of Electrical and Computer Engineering and by Charles A. Wight, Dean of The Graduate School. ABSTRACT This thesis investigates the problem of producing power using a single-phase induction machine. Single-phase induction motors are typically nonsymmetric two-phase motors, also called split-phase motors, with a main winding and an auxiliary winding. We consider the control of power delivered by the main winding through the auxiliary winding. A state-space model of an asymmetric split-phase generator has been developed. This model is used to characterize and identify a 1/3 hp induction machine. The parameters are used to predict suitable operating regions for the generator. It is found that there exists a viable operating region in a band just above the synchronous speed of the motor. Four control algorithms are presented. The first is a simple open-loop algorithm for a sinusoidal reference signal. The second is a controller based on the inverse-G adaptive algorithm. The third is an algorithm featuring an adaptive estimate of the system frequency response. Finally, a PI control method is considered for situations where the phase of the generated voltage does not need to be controlled. Simulated and experimental results are presented for each of the control algorithms. The closed-loop algorithms were able to track a desired reference. These algorithms were also able to reject a disturbance caused by a 20 percent increase in speed. TABLE OF CONTENTS ABSTRACT.......................................................................................................................iii ACKNOWLEDGMENTS..................................................................................................vi INTRODUCTION...............................................................................................................1 Problem Statement...................................................................................................1 Importance of the Problem.......................................................................................1 Review of Previous Work........................................................................................2 PARAMETER IDENTIFICATION.....................................................................................4 Asymmetric Induction Machine Model...................................................................4 Parameter Identification without Load....................................................................6 Generator Model with Load...................................................................................11 OPERATING CONDITIONS............................................................................................13 Viable Region Estimation......................................................................................13 Verification of Power Curves.................................................................................16 Selection of Operation Point..................................................................................16 CONTROL DEVELOPMENT..........................................................................................18 Open-Loop Control................................................................................................18 Adaptive Algorithm for Known Plant....................................................................20 Adaptive Algorithm with Plant Adaptation............................................................21 PI Control...............................................................................................................22 SIMULATION COMPARISON........................................................................................24 Open-Loop Control................................................................................................24 Adaptive Algorithm for Known Plant....................................................................26 Adaptive Algorithm with Plant Adaptation............................................................28 PI Control...............................................................................................................30 EXPERIMENTAL RESULTS...........................................................................................32 Experimental Setup................................................................................................32 Open-Loop Control................................................................................................34 Adaptive Algorithm for Known Plant....................................................................35 Adaptive Algorithm with Plant Adaptation............................................................37 PI Control...............................................................................................................39 CONCLUSIONS................................................................................................................43 REFERENCES..................................................................................................................45 v ACKNOWLEDGMENTS I would like to thank Dr. Bodson for his invaluable support and patience throughout the course of this project. I would also like to thank Dr. Khan and Dr. Abbott for their input to this project. This thesis is dedicated to my family who taught me the value of education. CHAPTER 1 INTRODUCTION 1.1 Problem Statement This research focuses on the development of a method to obtain usable electric power from commercially available single-phase induction machines (SPIM). There exists an extensive collection of research papers investigating the phenomenon of self-excitation in induction machines. However, it is difficult to extract power from the machines due to the inherent instability and nonlinearity of the self-excitation phenomenon. The current project considers controlled generation when a signal excitation voltage is applied to the auxiliary phase of a nonsymmetric machine and power is produced on the main winding. The goal of this research is to develop a control algorithm that produces 120 V at 60 Hz across the generating phase of the machine. It is assumed that the speed of the rotor is a measured quantity and can be adjusted close to the synchronous speed. 1.2 Importance of the Problem Single-phase induction machines (SPIM) are one of the most ubiquitous types of power converter. Although they are called single phase machines, most are actually nonsymmetric two-phase machines. They are attractive because they are inexpensive, low-maintenance, and lightweight when compared to their synchronous cousins. An important development of the research is a method to model and identify asymmetric single-phase machines. There are standardized tests for the parametrization of symmetric SPIMs, but no such tests exist for asymmetric machines. The proposed parameter identification method can be applied to any 2 induction machine and can be automated. This research in the control of residential power generation using off-the-shelf induction machines is particularly relevant given recent interest in alternative energy technology. Although residential generation solutions exist, single-phase induction generators may provide similar performance at a fraction of the cost. Furthermore, these machines are readily available as surplus or salvage and could be utilized as generators in developing countries where access to electrical power is limited. 1.3 Review of Previous Work The idea of using a voltage-source inverter to control the electrical output of an induction generator was first proposed in a paper by Novotny, Gritter, and Studtmann in 1977 [1]. This paper develops equivalent circuit models for the inverter-generator system. This paper uses an inverter to control the power drawn from a three-phase induction generator operating in the self-excited mode. The paper does not provide any experimental results. Results were obtained much later by Murthy in 1993 [2]. Murthy also extended the analysis to consider self-excited single-phase machines. The work was continued by Olorunfemi Ojo at Tennessee Technological University in a series of papers from 1995 to 2001. Ojo's most significant contribution to the work is to use the inverter to excite the auxiliary phase of the machine [4]. This inverter-assisted topology is qualitatively different than self-excited generation. In this configuration, there is a linear relationship between the voltage applied at the excited phase and the voltage produced by the generating phase. An additional feature of this topology is that control is performed on only a single phase so only one inverter is needed. Furthermore, the voltage generated by the machine will have fewer induced harmonics. In [5], Ojo explores the effect of different loading on this generator's performance. He explores the effect of more advanced modulation schemes in [6]-[8]. Finally, Ojo uses the inverter-generator topology to identify the parameters of an induction 3 machine [9]. Despite his thorough investigation of this topology, Ojo never attempted closed-loop control of the inverter-generator system. CHAPTER 2 PARAMETER IDENTIFICATION The first step in any control problem is to develop an accurate representation of the plant. This chapter presents a state-space model of an asymmetric induction machine. This model is then simplified to examine operation when one phase is left open while an excitation voltage is applied across the opposite phase. Two frequency response functions are identified as a function of these state-space parameters. One frequency response corresponds to a winding impedance and the other corresponds to the transmittance from one winding to another. Parameters are determined by fitting measured frequency response functions using least-squares algorithms. Finally, the model is expanded to include parameter identification when an RC load is attached in parallel to the generating winding. 2.1 Asymmetric Induction Machine Model Begin by considering the two-phase induction motor shown in Fig. 1. The rotor windings are assumed to be identical, but the stator windings may be different from each other. The rotor windings typically represent a squirrel-cage rotor, rather than actual windings. Based on the geometry of the motor, the total flux linkages in the four windings are assumed to be of the form 5 Figure 1: Two-phase induction motor ψS A=LA iS A+M A cos( np θ )iR X−M Asin (npθ )iRY ψS B=LB iS B+M B sin(np θ )iR X+M A cos(npθ )iRY ψR X=LRiR X+M A cos( npθ )iS A+M B sin( np θ )iS B ψRY=LRiRY−M B sin( np θ )iS A+M B cos( npθ )iS B (1) It is assumed that there is no mutual inductance between stator windings A and B or between rotor windings X and Y. The electrical equations for the stator and rotor windings are given by d ψS A d t =vS A−RA iS A , d ψS B d t =vS B−RB iS B d ψR X d t =vR X−RR iR X=−RR iR X , d ψR Y d t =vRY−RR i RY=−RR iR Y (2) where the right side of the last two equations assume that the rotor windings are short-circuited. Combining (1) and (2) gives a set of four equations which may be organized in the following form i i i i SB RY RX SA q 6 ( LA 0 M Acos( npθ) −M A sin( npθ ) 0 LB M B sin (np θ) MB cos( npθ) M A cos( npθ ) M B sin (np θ) LR 0 −M A sin (np θ ) M B cos( npθ ) 0 LR )d d t (iS A iS B iR X iRY )= (vS A−RA iS A+npωM A(i R X sin ( npθ )+iR Y cos( npθ )) vS B−RB iS B−npωMB (i R X cos( npθ )−i RY sin( npθ )) −RR iR X+npω(MA iS A sin (np θ )−M B iS B cos(np θ)) −RR iRY+npω(M Ai S A cos( npθ )+MB iS B sin ( npθ ))) (3) The rotor currents can be described in a coordinate frame attached to the stator, so that (i R A i R B)=(cos (np θ ) −sin ( npθ) sin (npθ ) cos (npθ ) )(iR X i RY ) (4) Applying this transformation, the electrical equations of the motor become (LA 0 M A 0 0 LB 0 MB M A 0 LR 0 0 MB 0 LR )d d t (iS A iS B iR A iR B)=( vS A−RA iS A vS B−RB iS B −RR iR A−npω( LR iR B+M B iS B ) −RR iR B+npω( LR iR A+MA iS A )) (5) 2.2 Parameter Identification without Load Now, consider operation on a single phase, A, with the other phase left open. Thus, iSB = 0. The speed is assumed to be constant. The system can then be studied with Laplace transforms, so that ( sLA+RA sM A 0 sM A s LR+RR npω LR −npωM A −n pω LR s LR+RR)(iS A iR A iR B)=(vS A 00 ) (6) while vS B=sM B iR B (7) Let the determinant of the matrix be D( s)=( s LA+RA )(( s LR+RR )2+( npωLR )2 )−sM A 2 ( s( s LR+RR )+( npω)2 LR) (8) Then 7 (i S A i R A i R B)=( ( s LR+RR)2+( npω LR )2 −M A( s( s LR+RR )+( npω)2 LR ) npωMA RR ) vS A D( s ) (9) and vS A= D( s ) ( s LR+RR )2+(npωLR)2 i S A ,i R B= npωM ARR ( s LR+RR )2+(npω LR)2 i S A (10) If we define the impedance ZA(s) of phase A and the transmittance GAB(s) from the current in A to the voltage in B, we have vS A=Z A( s)i S A , vS B=GA B( s)iS A (11) where ZA( s)=( s LA+RA )−sM A 2 s( s LR+RR)+(npω)2 LR ( s LR+RR )2+( npω LR )2 (12) GA B( s)= s npωMAMB RR ( s LR+RR )2+( npωLR )2 (13) Knowing the two transfer functions ZA and GAB, it is possible to determine the following five constants C1= npωM AM B RR LR 2 ,C2= RR LR ,C3=( npω)2 ,C4=LA ,C5=RA (14) Indeed, then transfer functions (18) and (19) can be expressed as ZA( s)=(C4 s+C5 )− C1 s C2√C3 s2+C2 s+C3 ( s+C2 )2+C3 (15) GA B( s)= C1 s ( s+C2 )2+C3 (16) and the constants C1 … C5 are uniquely determined from the transfer functions. For nonsymmetric motors, phases A and B can be swapped to obtain two more parameters: RB and LB (as well as validation of the two common parameters, C1 and C2). First, the method was tested with a symmetric machine so that its results could be 8 compared to those of conventional methods. The symmetrical split-phase induction motor was mechanically coupled to a dc motor. The test motor had the nameplate data given in Table 1. A sinusoidal excitation voltage was applied to phase A of the generator and, for a fixed mechanical speed, this excitation frequency was swept from 5 Hz to 100 Hz. For each excitation frequency, the current through phase A and the voltage generated across phase B were measured. The magnitude and the phase of these signals at the excitation frequency were calculated on a data acquisition and processing board. Two tests were performed in order to demonstrate the repeatability of the technique for different speeds. The first test was performed at 1200 RPM and the second at 3600 RPM. For each test, Bode plots of the measured impedance and transmittance were prepared. Transfer functions (15) and (16) were manually fitted to the data. The resulting Bode plots are displayed in Figs. 2 and 3. From the definitions in (14), four motor parameters can be identified. Note that all of the parameters of a squirrel-cage induction machine cannot be independently determined. The parameters are displayed in Tables 2 and 3. For comparison, the parameters of the motor obtained using classical standstill and synchronous tests are shown in Table 4. Agreement between the results is very good. A set of tests was also performed for an asymmetric machine at 2400 RPM. The resulting Bode plot with the excitation applied across the auxiliary winding is shown in Figure 4. The Bode plot with the excitation across the main winding is shown in Figure 5. Each fit was obtained using a least-squares algorithm. The parameters are displayed in Tables 5 and 6. TABLE 1. NAMEPLATE OF SYMMETRICAL MOTOR Volts Phases Power Speed 24 VAC 1 7.5 W 3350 RPM 9 Figure 2: Fit of magnitude and phase of transmittance (left) and impedance (right) data for 1200 RPM. The data are shown by the dotted line, the fit is shown by the solid line. Figure 3: Fit of magnitude and phase of impedance (top) and transmittance (bottom) data for 3600 RPM. The data are shown by the dotted line, the fit is shown by the solid line. 10 TABLE 2. MEASURED PARAMETERS FOR 1200 RPM RA LA RR/LR M2/LR 48.9 Ω 0.365 H 69.0 Ω/H 0.316 H TABLE 3. MEASURED PARAMETERS FOR 3600 RPM RA LA RR/LR M2/LR 48.9 Ω 0.350 H 69.0 Ω/H 0.302 H TABLE 4. PARAMETERS MEASURED USING CLASSICAL TESTS RA LA RR/LR M2/LR 49.5 Ω 0.332 H 72.3 Ω/H 0.280 H Figure 4: Fit of magnitude and phase of impedance (right) and transmittance (left) data for auxiliary-winding excitation. The rotor velocity was 2400 RPM. 11 Figure 5: Fit of magnitude and phase of impedance (right) and transmittance (left) data for main-winding excitation. The rotor velocity was 2400 RPM. TABLE 5. PARAMETERS OBTAINED USING AUXILIARY-WINDING EXCITATION RA LA RR/LR MA2/LR MAMB/LR 5.38 Ω 0.199 H 0.103 Ω / H 0.177 H 0.098 H TABLE 6. PARAMETERS OBTAINED USING MAIN-WINDING EXCITATION RB LB RR/LR MB2/LR MAMB/LR 1.34 Ω 0.122 H 0.136 Ω / H 0.113 H 0.096 H 2.3 Generator Model with Load Next, consider operation when a load is connected in parallel with the generator. Adding a capacitor C and a resistor R in parallel to phase B of the system described by (5) gives the following modified equations. E d x d t =F x+Gu (17) where 12 E=(LA 0 M A 0 0 0 LB 0 M B 0 M A 0 LR 0 0 0 M B 0 LR 0 0 0 0 0 C), x=(iS A iS B iR A iR B vS B) F=(−RA 0 0 0 0 0 −RB 0 0 1 0 −npωM B −RR −npω LR 0 npωM A 0 npω LR −RR 0 0 −1 0 0 −1/RL) G=(10000 ), u=vS A (18) In the Laplace domain, the vector transfer function H(s) from VSA(s) to X(s) is given by H( s)=( E s−F )−1G (19) CHAPTER 3 OPERATING CONDITIONS The next step of the research is the identification of viable operating regions. A region is considered viable if more electrical power is produced than consumed. The size of the viable region is determined by several factors, including load, capacitance, and mechanical frequency. The large number of variables involved makes the identification of these regions non-trivial. Once a viable region is identified, an operating point within the region must be selected. Operating points could be chosen to provide maximum power, to provide optimum efficiency, or to satisfy other design requirements. This chapter uses the model developed in the previous section to estimate the viable region of an induction generator with an RC load. The simulation is confirmed with experimental power data. The final section of the chapter argues for operating the system around the point where the real power consumed by the excited phase is zero. 3.1 Viable Region Estimation The transfer function defined in (19) can be used to estimate the power generated by the machine. Assuming a sinusoidal voltage is applied to winding A, vS A(t )=Asin (ωe t ) (20) The following power variables can be computed. PA= A2 2 R[ H1 ( j ωe )] (21) QA=−A2 2 I [H1( jωe)] (22) 14 PB= A2 2 R[ H2( jωe)H5(− j ωe )] (23) QB=−A2 2 I [H2 ( j ωe )H5 (− jωe) ] (24) These transfer functions are dependent on electrical frequency, mechanical frequency, load resistance and load capacitance. In order to restrict the design space, the electrical frequency was selected to be 60 Hz and VSB = 110 Vpk. The load resistance was chosen to be RL = 100 Ω. This was a choice of necessity as the converter used to drive the prime mover began to behave unpredictably at heavier loads. A load capacitor was placed in parallel with the load resistance, C = 200 μF. The size of the capacitor was selected based on an estimate of the reactive power requirements of the generating phase. This leaves the mechanical frequency, ω, as the only undetermined parameter. Using Matlab, values of PA, QA, PB, and QB were calculated for a range of ω. Parameters for the calculation were taken from Table 5. The total power generated by both phases of the machine is shown in Figure 6. The computation predicts a viable region just above the synchronous speed of the generator, 188.5 rad/sec. The region is approximately 30 rad/sec wide and peak power is delivered at approximately 207 rad/sec. The real and reactive power of the excited phase is shown in Figure 7. It is important to note that the computation predicts that power can be either consumed or generated by the excited phase. 3.2 Verification of Power Curves In order to confirm the computation, an experiment was performed. Phase A of the induction generator was connected to the grid via a variac. The mechanical frequency of the machine was manually adjusted using a laser tachometer and then the generated voltage was regulated to 110 Vpk. This procedure was performed for ten speeds above the synchronous speed of the motor. At each point, the power consumption of the machine was measured using a 15 Figure 6: Predicted total power generated by an asymmetrical induction machine Figure 7: Predicted real and reactive power consumed by phase A 16 Voltech PM1000 Power Analyzer. The total power generated is plotted alongside the predicted values in Figure 8. Figure 9 shows the active and reactive power consumed by the excited phase. The data are in good agreement with the predicted values. The data follow the trend of the simulation. The maximum deviation from the predicted total power generated is 32 percent. 3.3 Selection of Operation Point Once the feasible region is identified, the next step is to choose an operating point within the region. As shown in Figure 7, the point PA = 0 appears within the range of viable frequencies at approximately ω = 193.25 rad/sec. This is an attractive operating point since the excited phase does not consume any real power. A battery could supply the reactive power to the phase. Changes to the state of charge of the battery could be obtained through small speed adjustments around the operating point. This is especially significant if the generator is designed to be operated off-grid. This operating point is also attractive because the excited phase does not generate any real power. If a standard off-the-shelf amplifier is used for the controller, it will not be able to regenerate power. Power flowing back to the amplifier will cause significant distortions in the input voltage. The alternative to these amplifiers is to design an ad hoc power converter to control power flow between the battery and generator. This is unattractive since it would increase the cost and complexity of the design. Due to these properties, PA = 0 was chosen as the operating point for the control design. 17 Figure 8: Total power generated by the induction machine Figure 9: Real and reactive power generated by phase A CHAPTER 4 CONTROL DEVELOPMENT The final phase of the project was the development of methods to produce a controlled voltage across the generating phase of the machine. This problem presents several challenges. First, the system is highly resonant. This means that a pair of poles is near the jω-axis and the system is close to instability. Another challenge is that transfer function of the plant is dependent on the mechanical velocity. Changes in this velocity can cause the system to depart significantly from original conditions. Finally, the model includes equations for the rotor currents. In practice, these currents cannot be measured. The robust controller must function without knowledge of these currents and uncertainties in the parameters, speed, and load. Four controllers have been considered: an open-loop controller, two adaptive controllers and a PI controller. The first three controllers are designed to match the amplitude and phase of a reference signal. The ability to track both the amplitude and the phase is important in order to connect the generator to the grid. The PI controller is designed to generate a sinusoidal voltage at a desired amplitude. This method is applicable to systems operating as stand-alone generators where phase is not a consideration. All four methods are discussed in detail below. The analyses are based on steady-state sinusoidal responses. 4.1 Open-Loop Control The system is assumed to be described by y(t )=P( s )[ u(t )] (25) where y(t) is the plant output, u(t) is the control input, and P(s)[(·)] denotes the time-domain 19 output of the plant with transfer function P(s). In this context, u is the voltage applied to the auxiliary phase (vSA) and y is the voltage at the main phase (vSB). The plant is assumed to be stable. The objective is for the plant output to track a sinusoidal reference signal r (t )=rccos(ωe t )+r s sin(ωe t ) (26) where rc, rs are reference parameters determining the magnitude and phase of the reference signal, and ωe is the frequency of the reference signal. Specifically,and ωe=2 π 60. The control signal is chosen to be u(t )=uccos(ωe t )+us sin(ωe t ) (27) where uc, us are control parameters to be determined. For fixed control parameters, the steady-state output of the plant is ys s=PR( uc cos(ωe t )+us sin (ωe t ))+PI (−uc sin(ωe t )+uscos(ωe t )) (28) where PR, PI are the real and imaginary parts of the plant's frequency response evaluated at ωe. It follows that yss(t) = r(t) for all t if uc, us are equal to the so-called nominal control parameters (uc * us *)=G*−1(rc rs) (29) where G* is given by G*=( PR PI −PI PR) (30) In practice, equation (27) may be implemented as a control algorithm with (uc us)=G−1(rc rs) (31) where G is an estimate of the matrix G∗. This algorithm will work well if precise estimates of PR and PI are known. Estimates can be computed from the model of section 3.1, but depend on speed and load parameters. 20 4.2 Adaptive Algorithm for Known Plant The first adaptive method is the so-called inverse-G adaptive algorithm [16], which is a special case of the filtered-x LMS algorithm in signal processing. This method searches for control parameters that minimize the error between the reference signal and the plant output. For this purpose, it is convenient to define rv=(rc rs),uv=(uc us),w(t )=(cos(ωe t ) sin(ωe t )) (32) so that r (t )=wT ( t )rv ,u( t )=wT (t ) uv , ys s=wT (t )G*uv (33) in steady state. The adaptive algorithm for uv is then defined as u˙v( t )=2 g G−1w(t )( r( t )−y (t )) (34) where g > 0 is an arbitrary adaptation gain to be adjusted for optimal performance. A motivation for (34) can be found using the theory of averaging. A simplified explanation follows. Approximating y by its steady-state value for fixed control parameters r (t )−y( t )≃wT (t )( rv−G* uv ) (35) Neglecting sinusoidal terms of frequency 2ωe (to be "averaged" by the differential equation for uv), one has 2 w(t )wT (t )≃I (36) where I is the identity matrix and therefore 2 w( t )(r ( t )−y (t ))≃rv−G* uv (37) With these approximations u˙v( t )=g G−1G* (uv *−uv ) (38) In ideal conditions (G = G*), u˙v( t )=g( uv *−uv ) (39) 21 so that the control vector converges to the nominal vector with the dynamics of a first-order system with dimension 2 and poles at s = −g. The advantage of the algorithm is that convergence to the nominal vector still occurs if G = G*, as long as stability is preserved. In general, the stability of the system is determined by the roots of det ( s I−gG−1G* )=0 (40) which remain in the open left-half plane if G is sufficiently close to G*. 4.3 Adaptive Algorithm with Plant Adaptation The second adaptive algorithm is a simplified form of the algorithm by Pigg and Bodson [17]. This simplification occurs because the reference parameters rc, rs are known and do not need to be estimated. The algorithm is obtained by defining a vector of plant parameter estimates x, whose nominal value is x*=(PR PI ) (41) The control law is defined from the parameter estimates as if they were the nominal parameters, i.e., uv=G−1 rv (42) with G=( x1 x2 −x2 x1) (43) Then, (35) can be rewritten as 2 w(t )( r( t )−y(t ))≃rv−W x* (44) where W=( uc us −us uc)= 1 x1 2 +x2 2 (x1 rc−x2 rs x1 rs+x2 rc x1 rs+x2 rc −x1 rc+x2 rs) (45) 22 Next, define the error signal ea=W x−rv+2 w(t )( r (t )−y(t )) (46) Within approximations made earlier, the error signal is equal to ea=W ( x−x* ) (47) so that the error vector ea is related linearly to the parameter error, and standard gradient or least-squares algorithms can be used for parameter estimation. For example, a gradient algorithm is given by ˙x =−gWT ea (48) where g > 0 is an adaptation gain to be selected arbitrarily. The algorithm is nonlinear because W is a function of x. However, it is guaranteed to be stable because d d t ∣x−x*∣2 = −2 g( x−x* )T WT ea = −2 g(W ( x−x* )TW ( x−x* )) = −2 g∣W ( x−x* )∣2 ≤ 0 (49) Further, the parameter error and the tracking error converge to zero if W is nonsingular, which requires that rc or rs ≠ 0. There is a potential problem in the computation of u if x12 + x22 = 0, but it can be avoided by replacing the computation of uv by uv= 1 max ( ε , x1 2 +x2 2 ) (x1 rc−x2 r s x1 rs+x2 rc) (50) where ε is a small positive number such that ε < x1*2 + x2*2. 4.4 PI Control The previous three controllers considered tracking the magnitude and phase of a reference sinusoid. The controller design can be simplified when the phase is unimportant. A straightforward approach is to calculate the difference between a measured amplitude and a 23 reference amplitude and then apply the techniques of PID controller design. The problem is reduced to calculating a continuous, accurate estimate of the amplitude of the generated voltage. One solution to obtain the envelope of a signal is to use a diode detector. These types of detectors were common in early AM radios. Unfortunately these types of detectors are sensitive to noise and have an asymmetric voltage response. The diode detector was found to be unsuitable for this application due to these drawbacks. Another way of calculating the amplitude is to define an estimate of the generated voltage, ̂y(t )=yccos(ωt )+ys sin (ωt ) (51) The values of yc and ys are updated by performing the continuous averaging defined below. y˙c=g( y( t )−̂y (t ))cos(ωt ) (52) ˙y s=g ( y(t )−̂y ( t ))sin (ωt ) (53) where g > 0 is a gain that represents the "memory" of the filter. The amplitude of the estimate is simply Y=√( yc 2+ys 2). CHAPTER 5 SYSTEM SIMULATION The next step in the design process was to develop simulation models to test the validity of the control algorithms. This chapter presents Simulink simulations of each of the control methods discussed in the previous chapter. The simulations all use a solver based on the Euler method. The step size of the solver was set to 0.0001 seconds in order to correspond to the sampling frequency of the experimental equipment. The reference signal was chosen to be r (t )=rccos(ωt )+rs sin (ωt )=110cos( 2π 60 t ) (54) Note that the amplitude of the reference voltage is 110 Vpk instead of 110 VRMS. The smaller reference was used because the amplifier used for the experiments began to saturate at 150 Vpk. The voltage was slightly derated to avoid clipping leading to unwanted distortion. The first step was to develop a suitable model of the induction generator. The state-space model of the generator presented in 2.3 was coded in Simulink. This system is shown in Figure 10. Parameters for the model were taken from Table 5. The mechanical frequency was set to be 1845 rpm under load. This places the system near the operating point discussed in section 3.3. 5.1 Open-Loop Control The open-loop control method was the first system simulated. The Simulink block diagram of the system is shown in Figure 11. For the given reference signal, (30) yields uc = -51.15, us = 50.36. The results of the simulation are shown in Figure 12. 25 Figure 10: Simulink model of the induction generator Figure 11: Block diagram of the open-loop control algorithm 26 Figure 12: Simulation results of the open-loop algorithm The simulation produces 110 V across the generating winding as expected. The rise time of the system is approximately 0.1 seconds. The predicted gain of the system can be calculated by measuring the amplitudes of the steady-state voltages. GS S=(V B ) (V A)= 110 74.3=1.48 (55) 5.2 Adaptive Algorithm for Known Plant The second algorithm simulated was the adaptive algorithm for known plant introduced in 4.2. The Simulink block diagram of the system is shown in Figure 13. The results of the simulation are shown in Figure 14. The system settles to 110 V as expected. The system is only slightly slower than the open-loop method. It also exhibits more overshoot than the open-loop method. The overshoot can be eliminated by reducing the gain, but this increases the rise time of the system. Figure 13: Block diagram of the adaptive algorithm for known plant 27 28 Figure 14: Simulation results of the adaptive method for known plant 5.3 Adaptive Algorithm with Plant Adaptation Next, the algorithm for unknown plant was simulated. This system consists of three subsystems: the induction machine, the error estimator (46) and a gradient descent algorithm (48). A block diagram of the system is shown in Figure 15. The adaptation gain was set to g = 0.00005. The results are shown in Figure 16. Since the plant parameters are initialized exactly, this algorithm converges extremely rapidly. The response will be much slower in practice. 5.4 PI Control Finally, the PI control method discussed in 4.4 was implemented in Simulink. The feedback gains were set to KP = 0.5 and KI = 8.5. The averaging gain was set to g = 150. This Figure 15: Block diagram of the adaptive algorithm with plant adaptation 29 30 Figure 16: Simulation results of the adaptive method for unknown plant gain was made much larger than the feedback gains in order to ensure that the dynamics of the averaging filter did not interfere significantly with overall system performance. A block diagram of the system is shown in Figure 17. The results of the simulation are shown in Figure 18. The PI controller responds much more rapidly than the other algorithms - even though the adaptive controllers have exact knowledge of plant parameters. 31 Figure 17: Block diagram of the PI control method Figure 18: Simulation results of the PI controller CHAPTER 6 EXPERIMENTAL RESULTS 6.1 Experimental Setup An experimental testbed was built in order to test the control methods described in Chapter 4. The testbed consisted of a prime mover connected to a generator via a spider coupling. Both machines were manufactured by the Worldwide Electric Corporation. The prime mover had the nameplate data shown in Table 7. The three-phase voltage required by the prime mover was provided by a GE 300 Mini adjustable frequency drive. The 300 Mini is a V/f drive and therefore the speed of the motor is poorly regulated under varying torque. Figure 19 shows the speed response due to a 110 Vpk step in the generated voltage. The measurements were obtained using a Compact Instruments laser tachometer. The data were filtered using a six-pole Butterworth low-pass filter with a 40 Hz cutoff. The filter does not introduce any significant dynamic response. The wringing of the motor drive is evident in many of the experimental responses. This behavior can be eliminated by initializing the speed of the motor nearer its steady-state value. This would mean operating the system farther from the operating point chosen in Chapter 3. TABLE 7. NAMEPLATE DATA OF PRIME MOVER Power Speed Phases Hz Voltage 1/2 HP 3450 RPM 3 60 Hz 208-230/460 V 33 Figure 19: Response of the motor drive system to a step change in torque The generator under test had the nameplate data shown in Table 8. The start capacitor of the generator was removed in order to provide access to the auxiliary phase of the machine. Both of the machines were fastened to a heavy metal plate for stability. The setup is shown in Figure 20. The excitation signal was generated using an onboard digital-to-analog converter of a dSPACE DS1104 controller board. This signal was passed to a power amplifier. The generated voltage was measured using the analog-to-digital converter on the DS1104. In order to compare with the simulations presented in Chapter 5, the speed of the prime mover was set near 1845 rpm under load. This was difficult in practice due to the nature of the drive. As before, the reference was set to rc = 110, rs = 0. TABLE 8. NAMEPLATE DATA OF GENERATOR UNDER TEST Power Speed Phases Hz Voltage 1/3 HP 1725 1 60 115/230 V 34 Figure 20: Experimental setup 6.2 Open-Loop Control The open-loop control method was the first implemented. The results of the experiment are shown in Figure 21. An estimate of the amplitude of the generated voltage, Y, is included in the plot. The estimate is calculated using (52) and (53), but is not used in the control law. The steady-state gain of the system can be calculated as before GS S=(V B ) (V A)= 110 83.9=1.31 (56) This is approximately 11 percent less than the theoretical value. Note that the model predicts overdamped behavior when the system exhibits slightly underdamped responses. This behavior is due to oscillations in the speed of the generator-motor system. 35 Figure 21: Response of the open-loop control algorithm 6.3 Adaptive Algorithm for Known Plant Next, the adaptive method for known plant was implemented in C and loaded on to the dSPACE system. The adaptation gain was set to g = 3. The results are shown in Figure 22. The experiment was repeated at a slower speed to obtain the response shown in Figure 23. Larger gains improve the transient response of the generator, but introduce steady-state oscillations to the output. It is also informative to examine the response of the algorithm to variations in the plant parameters. The speed of the generator, ω, is the only parameter that can vary significantly during normal operating conditions. The generator was started and allowed to reach steady state at a speed of 1800 rpm. The mechanical frequency was increased to 2160 rpm and the system was allowed to return to steady state. The resulting transients are shown in Figure 24. The maximum voltage deviation is five percent. 36 Figure 22: Response of the adaptive algorithm for known plant Figure 23: Response of the adaptive algorithm for known plant at slower speed 37 Figure 24: Response of the adaptive algorithm for known plant for a step change in speed at t = 0.65 sec. 6.4 Adaptive Algorithm with Plant Adaptation The third experiment implemented on the system was the adaptive algorithm for an unknown plant. The adaptation gain was set to g = 0.0001. The initial values of the parameters x1 and x2 were estimated using the real and imaginary parts of the transmittance. The response of the controller to a step reference is shown in Figure 25. The peak overshoot of the controller is approximately 148 V. This is a significant overshoot and could damage connected loads. The adaptive action of this method is apparent in Figure 26. The controller's estimate of the system transfer function converges to x1 = -0.54, x2 = -0.77. A larger adaptive gain causes these estimates to converge quicker, but introduces more overshoot to the system. The experiments were repeated at a slower speed to reduce drive dynamics. The step response of the controller is shown in Figure 27. The overshoot is reduced to approximately five percent. Once again, the controller's response to a step change in plant parameters was tested by increasing the speed of the generator from 30 Hz to 36 Hz. The results are shown in Figure 28. 38 Figure 25: Results of the adaptive algorithm for unknown plant Figure 26: Plant estimates of the adaptive controller for unknown plant 39 Figure 27: Response of the adaptive algorithm for unknown plant at slower speed 6.5 PI Control The PI control method discussed in section 4.4 was implemented in a similar fashion. The gain used in the amplitude estimator was chosen to be g = 150. The selected feedback gains were kP = 0.5 and kI =6.5. The results of the experiment are shown in Figure 29. Larger values of g increase the response time and decrease the maximum overshoot. However, if g is too large it causes steady-state oscillations in the output voltage. Once again, the experiment was repeated at a slower speed. The results of the slower experiment are shown in Figure 30. The rise time of the response is in good agreement with the simulation presented in Figure 18. The response of the PI controller to a change in the plant parameters was also investigated. Figure 31 shows the response of the system to a 20 percent increase in speed at t = 1.5 sec. Since the action of the controller is faster than the acceleration of the prime mover, there is no significant change in the generated voltage. 40 Figure 28: Response of the adaptive algorithm for unknown plant to step change in speed A final experiment was performed to examine the behavior of the generator under different loading conditions. The 100 Ω load resistor was replaced with a universal motor. The reactive power requirements of the motor are more typical of residential loads. Figure 32 shows the response of the system to a step in voltage. The inductance of the motor coils allows the load to increase gradually, eliminating some of the overshoot. The same gains were used for each of the experiments. 41 Figure 29: Response of the PI controller Figure 30: Response of the PI controller at slower speed 42 Figure 31: Results of the PI controller to a step change in speed Figure 32: Response of the PI controller with an inductive load CHAPTER 7 CONCLUSIONS This thesis examined the problem of producing a controlled voltage from a single-phase induction machine. A state-space model of a SPIM was presented. The parameters of a symmetric induction machine were obtained by fitting data to this model. These estimates were in good agreement with measurements taken using standard tests. The method was used to identify the parameters of each phase of an asymmetric induction machine. In order to predict power generation, the state-space model was expanded to include an RC load connected across the main phase. The identified parameters were used to estimate the operating region. It was concluded that power can be produced in a limited region above the synchronous speed of the motor. Power is produced by both phases of the machine in most of the operating region. The point PA = 0 was chosen as the operating point in order to eliminate costly power electronics. Finally, four control methods were investigated. The three closed-loop algorithms were able to reliably track a reference voltage. Furthermore, each control algorithm was robust to a 20 percent increase in speed. In practice, the adaptive algorithm for unknown plant was faster than the adaptive algorithm for known plant. Variations in the speed of the system caused dangerous transients when the adaptive algorithm for known plant was used. In both cases, the adaptation gain can be increased to speed up the response, but this comes at the cost of greater transients on the line. In situations where phase is not a consideration, the PI control method is preferable. 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