Simplified Modelling and Control of a Synchronous Machine with Variable Speed Six Step Drive

Transcription

1 Simplifie Moelling an Control of a Synchronous Machine with Variable Spee Six Step Drive Matthew K. Senesky, Perry Tsao,Seth.Saners Dept. of Electrical Engineering an Computer Science, University of California, Berkeley, CA 9472 Unite Defense,.P., Santa Clara, CA 955 Abstract We report on simple an intuitive techniques for the moelling of synchronous machines an their associate control systems. A scheme for control of electrical power flow is propose for machines with variable spee six step rive. The scheme is evelope with an eye towar efficient system operation, simple implementation, low latency in the control path an minimize cost of power electronics. Experimental results are presente for application of the control scheme to a homopolar inuctor motor, as use in a flywheel energy storage system. I. INTODUCTION Two techniques are presente here for simplification of the moelling an control esign process for synchronous machines. What the two techniques have in common is a reliance on the unerlying physical properties of the machine an rive system to suggest accurate yet tractable formulations of the control esign problem. By reucing the orer of the machine moel an grouping control inputs an outputs accoring to their coupling strength, engineering intuition can be improve, an classical control results can be applie with confience. The first technique uses singular perturbation theory to perform a partitioning of the state space moel into slow an fast subsystems, which correspon to the mechanical an electrical variables, respectively. Since the electrical variables are often the focus of the control effort (e.g. in implementing torque or power control, the mechanical ynamics of the moel can be suppresse. The secon technique gives a methoology for etermining whether a 2 2 multiple input multiple output (MIMO system can be approximate as iagonal, in the sense that each input is primarily couple to only one output an vice versa. If this is foun to be the case, then an opportunity exists to inepenently synthesize a stabilizing scalar feeback loop for each input output pair appearing on the iagonal, outsie of the context of the overall system. Sufficient conitions exist to show that these control loops will then stabilize the overall system in the face of cross coupling between the loops (i.e. nonzero off iagonal terms. Further, the performance of the system will remain close to that of the iagonal approximation. The motivation for this approach is the control of a flywheel energy storage system [1]. The system consists of a homopolar inuctor motor/generator (HIM whose rotor is also the flywheel energy storage element, an a six step voltage source inverter fe from a DC bus. Application of the moelling an control esign proceure escribe above to the HIM with six step rive yiels a simple an effective control of electrical power flow in the machine. Cost is a riving factor in the esign of such a system, an the power electronics associate with a variable spee machine rive are a significant component of this cost. A six step rive scheme can reuce the cost of power electronics by enabling a closer matching of the DC bus voltage an the funamental voltage waveform at the machine terminals than woul be possible with a pulse with moulation (PWM scheme. Six step rive implies higher efficiency an higher maximum synchronous frequency than PWM as well. II. VAIABE SPEED EECTIC MACHINE DIVES While the choice of rive for an electric machine may seem to be merely a etail of the harware implementation, in fact it is critical to select a rive scheme before embarking on the system moelling an subsequent control esign. Each rive type imposes unique constraints on the system inputs, an consequently the rive will etermine which moel formulations are convenient, an which are unsuitable. By far the most common approach for variable spee rives is PWM. However, six step rive carries several avantages over PWM, particularly for high spee applications: Six step rive generates a fixe amplitue excitation using the full bus voltage, resulting in semiconuctor VA requirements that are closely matche to the inverter output. PWM schemes typically o not utilize the full bus voltage, but rather regulate the bus voltage own to avoi saturation of the uty cycle. The inverter semiconuctor evices must still block the entire bus voltage however, an hence evices must be rate for a voltage higher than the intene output. Thus for a given amplitue of rive waveform, six step allows the use of evices with lower VA ratings, which are less expensive. For six step rive, the switching frequency of a given evice is the same as the rive frequency. In contrast, PWM rive requires a large ratio of switching frequency to rive frequency, resulting in lower maximum rive frequencies for a given choice of semiconuctor evice. This also implies that for a given rive frequency, six step will exhibit lower switching epenent losses. For separately excite synchronous machines, running at unity power factor with six step rive allows for zero current switching, further reucing switching epenent /4/$17. (C 24 IEEE 183

2 losses. PWM requires har switching, leaing to associate losses an evice stresses. Finally, six step rive prouces greatly reuce high frequency harmonics as compare to PWM, resulting in lower rotor an stator core losses [2]. III. SYNCHONOUS MACHINE MODE In a stationary reference frame, the stanar two axis moel for the electrical ynamics of a synchronous machine can be written as t λ abf = (θ 1 λ abf + V abf (1 where λ abf an V abf inicate the vectors of flux linkages an terminal voltages, respectively, for armature winings a an b an fiel wining f. Theterm is a iagonal matrix of wining resistances an (θ is a symmetric inuctance matrix epenent on the rotor angle θ. Electrical torque τ e is given by τ e = θ ( 1 2 λt abf (θ 1 λ abf (2 an the mechanical spee of the rotor, ω m, is governe by t ω m = 1 J (τ e B v ω m (3 where J represents rotor inertia an B v is a viscous rag term. Equation 1 can be transforme into a synchronous reference frame enote (,q, resulting in a time invariant moel. The armature voltage vector is chosen to efine the q axis; as a consequence, for sinusoial rives V q is constant an V is ientically zero. The only choice of input for a conventional fixe voltage six step rive is the rive frequency, hence ω e is one system input. If a high banwith current control loop is implemente to set i f, λ f rops out of the moel, an i f appears as a secon input. Thus the electrical ynamics simplify to t λ = λ + ω e λ q + m i f cos θ (4 t λ q = ω e λ λ q m i f sin θ + V q (5 t θ = ω e N 2 ω m (6 where Eq. 6 escribes the evolution of the angle θ between the an f axes for an N pole machine. Finally, outputs of interest are i an : i = 1 λ m i f cos θ (7 = 1 λ q + m i f sin θ. (8 Equations 4-8 can be linearize, yieling an TI system of the form tx = Ax + Bu (9 y = Cx + Du (1 with state vector x =[λ λ q θω m ] T, input u =[i f ω e ] T an output y =[i ] T. The intereste reaer will fin the exact form of the Jacobian matrices A, B, C an D in Appenix I. Mag. (B Mag. (B Mag. (B Mag. (B H H H H Frequency (Hz Fig. 1. Boe plots of synchronous machine electrical ynamics corresponing to Eqs Soli lines inicate magnitue, otte lines inicate phase. IV. TIME SCAE SEPAATION In electric machines, one expects intuitively that the time constants of the electrical variables will be much faster than those of the mechanical variables. Hence in ealing with electrical ynamics the spee of the rotor appears to be constant, while relative to the mechanical time constants electrical transients appear to settle instantaneously. The moel given by Eqs. 9-1 can thus be approximate by a partitioning into fast an slow subsystems. Such a partitioning achieves a great simplification of the moel at the expense of a slight error in the calculation of system ynamics. For example, the approximate two time scale moel of the experimental system iscusse in Sec. VII estimates the system eigenvalues to within 1% of the values given by the original moel. Singular perturbation theory provies a formal framework for such system partitioning. elevant results are presente in Appenix II; for a etaile treatment of the theory, see [3]. 9 Phase (egrees Phase (egrees Phase (egrees Phase (egrees 184

3 Fig. 3. Control system implementing feeback only for iagonal input output pairs. C ii inicates a compensator, H ij inicates a plant block. Fig. 2. Phasor iagram for a synchronous machine operating at unity power factor. In implementing control of electrical rather than mechanical variables, the orer of the moel can be reuce such that the slow ynamic effects of the mechanical subsystem are suppresse. This approximation of the fast subsystem neglects the effect of the evolution of ω m. The reuce orer transfer matrix H for the synchronous machine moel is thus [ ] [ ][ ] î H11 H = 12 îf (11 îq H 21 H 22 ˆω e where a hat (ˆ inicates a variable in the s omain, an the transfer functions are given by H 11 = ( m cos θ (s 2 + s + ω2 e (12 + ω e tan θ/d H H 12 = {ī q s 2 + ( īq 1 ω e λ s + ω 2 e (īq 1 λ q (13 (ī 1 λ }/sdh + ω e H 21 = ( m sin θ ( s 2 + s + ω2 e ω e cot θ /D H (14 H 22 = { ī s 2 ( ī + 1 ω e λ q s ω 2 e (ī 1 λ (15 (īq 1 λ q }/sdh + ω e ( D H = s 2 +2 s ω e. 2 The bar ( notation inicates the operating point value of a variable as use in linearizing the moel. Boe plots of the transfer functions in Eqs are shown in Fig. 1. V. INDEPENDENT SCAA FEEDBACK As a preliminary exercise to the iscussion of control esign, it is useful to examine the phasor iagram of electrical machine variables near unity power factor operation in Fig. 2. Classical analysis (as in [4] of the figure suggests strong coupling for the iagonal (in the sense of Eq. 11 input output pairs (i f,i an (ω e,, an weak coupling for the off iagonal pairs. To see this, assume that ω e, an note that a change in the magnitue of i f results in a proportional change in the magnitue of E. This causes a change in the angle between V an i with little change in the magnitue of i. If the angle between V an i remains small (near unity power factor, almost all the change in i occurs along the axis. Similarly, varying ω e will result in a slight ifference between ω e an N 2 ω m, proucing a change in the angle θ. A change in θ primarily affects the magnitue of i (rather than the angle of i, an hence most of the change in i occurs along the q axis. This qualitative analysis is supporte by the boe plots in Fig. 1. A quantitative analysis procees from Fig. 3, which shows a block iagram of the system in the s omain. Because the evelopment here is intene to be general, variables e i, u i, v i, an y i, i=1, 2, are efine as in the figure. Note that feeback, with compensators enote C 11 an C 22, is only implemente for the iagonal input output pairs (v i,y i. The following assumptions are mae concerning the transfer functions in Fig. 3[5]: 1 Each block represents a proper scalar transfer function of the form N(s/D(s. 2 No right half plane (HP pole zero cancellations between (C 11 C 22 an (H 11 H 22 occur. 3 Any HP poles that occur in (H 12 H 21 also occur (incluing multiplicity in (H 11 H 22. Consier inepenently the two single input single output (SISO feeback systems forme by the blocks on the iagonal (C 11, H 11, C 22, an H 22. Equivalently, assume H 12 =H 21 =. Classical results from SISO control theory apply to these two systems, an given assumptions 1 an 2, internally stable close loop systems can be forme by proper esign of the compensators C 11 an C 22. Disregaring inputs an outputs, the two SISO systems can be consoliate into equivalent close loop blocks C 11 S 1 an C 22 S 2, where 1 S 1 = (16 1+C 11 H 11 1 S 2 =. (17 1+C 22 H 22 Now consier the system as a whole (i.e. lift the restriction that H 12 =H 21 =. The stability of the overall system can be examine by breaking the loop between any two of the remaining four blocks, an fining the open loop transfer function G = H 12 H 21 C 11 S 1 C 22 S 2. (18 The open loop poles of G are given by the poles of S 1 an S 2, an any stable poles of H 12 an H 21 not cancelle by 185

4 H 11 an H 22. By assumption 3, any unstable poles of H 12 an H 21 are exactly cancelle by zeros of S 1 an S 2. Further, the poles of S 1 an S 2 have been explicitly place for stability an performance via pruent esign of C 11 an C 22. Thus the structure of G an the assumptions above guarantee that G has no unstable open loop poles. Note that any uncancelle poles of H 12 an H 21 are stable, but unaffecte by either SISO loop closure. While this inability to place every pole represents a isavantage of the iagonal control approach, a large subset of systems exists for which the poles of H 12 an H 21 are either acceptable or will cancel entirely. Given that G is guarantee to be open loop stable, the close loop stability of G can be emonstrate by application of the small gain theorem [6]. Specifically, if the open loop poles of G are stable, an G < 1 (19 then the close loop poles of G are stable. A slight manipulation of G can improve intuition with respect to the conition given by Eq. 19. Substituting in Eqs , an multiplying in both the numerator an enominator by the quantity (H 11 H 22 gives G = H 12H 21 H 11 H 22 Defining the terms C 11 H 11 1+C 11 H 11 C 22 H 22 1+C 22 H 22. (2 = H 12H 21 H 11 H 22 (21 T 1 = C 11H 11 1+C 11 H 11 (22 T 2 = C 22H 22 (23 1+C 22 H 22 the small gain conition for stability becomes or equivalently T 1 T 2 < 1 (24 max <jω< { T 1 T 2 } < 1. (25 The conition given by Eq. 25 implies an elegant esign methoology for stabilizing 2 2 MIMO systems. (We note the similarity to iniviual channel esign escribe extensively in [7], [8], [9], [1], [11]. The magnitue of the term gives a measure of the iagonal ness of the system that is, a measure of the relative coupling strengths of iagonal versus off iagonal input output pairs. If the magnitue of is much less than unity over the esire control banwith, then chances are goo that the system can be stabilize by two inepenently esigne feeback loops. (As a corollary, a magnitue much larger than unity suggests that exchanging the input output pairs will result in favorable conitions for inepenent control. It is then a straightforwar task to esign compensators C 11 an C 22 to meet given specifications for Mag. (B Frequency (Hz Fig. 4. Plot of (ashe, T 1 (ash otte, T 2 (otte, an T 1 T 2 (soli. the two inepenent SISO systems. If each SISO system can be stabilize an mae to exhibit a suitably well ampe response, then the MIMO system is guarantee to be stable as well. Consier Fig. 4, which uses values from the experimental system of Sec. VII. The figure makes explicit the banwith limitations impose by the structure of the plant. Examining the plot of, it is clear that well ampe close loop transfer functions T 1 an T 2, esigne to have approximately the same corner frequency as, will satisfy Eq. 25. Note that the small gain theorem gives a sufficient, but not necessary, conition for stability of the close loop system. A necessary an sufficient conition is given by the Nyquist theorem [12]. Compare to the Nyquist theorem, the small gain theorem is overly restrictive; Nyquist esign constraints permit open loop gain to excee unity for phase angles that are far from 18. However, the restriction impose by the small gain theorem proves to be beneficial, in that it also keeps close loop performance close to the performance preicte by the inepenent esign of C 11 an C 22. Equations give the close loop transfer functions for Fig. 3. The transfer functions have been arrange such that in each case the first term represents what might be calle the ecouple result, while the term in parentheses represents a multiplicative perturbation resulting from loop interactions via off iagonal terms. y 1 = C ( 11H 11 1 T2 (26 v 1 1+C 11 H 11 1 T 1 T 2 ( y 1 S1 S 2 = C 22 H 12 (27 v 2 1 T 1 T 2 ( y 2 S1 S 2 = C 11 H 21 (28 v 1 1 T 1 T 2 y 2 = C ( 22H 22 1 T1 (29 v 2 1+C 22 H 22 1 T 1 T 2 Boe plots of Eqs using values from the experimental system of Sec. VII are presente in Fig. 5. For each equation, the figure shows the ecouple term (which is the 186

5 5 (a Mag. (B (b Fig. 6. Control system block iagram. Mag. (B Mag. (B Mag. (B (c ( Implementation of this scheme is shown in Fig. 6. An attractive feature of the control scheme is its simplicity in both architecture an implementation. The reference frame is efine by the inverter voltage, hence the reference frame angle (φ in Fig. 6 is explicitly known six times per perio of the electrical frequency every time that the inverter switches. By sampling armature currents at these instants, sample current values can be transforme into the rotating reference frame an compare to commans. Unlike flux oriente vector base control schemes, no observer is require to resolve the reference frame, an no sampling of terminal voltages is require. Scalar control laws an integrator for C 11 an proportional integral (PI form for C 22 yiel the esire response characteristics. Because only a small number of control calculations nee to be performe, latency in the control path is small. Assuming a six step rive operating from a fixe bus voltage, this current control amounts to instantaneous control of active an reactive power in the machine Frequency (Hz Fig. 5. Boe plots of (a Eq. 26, (b Eq. 27, (c Eq. 28, ( Eq. 29. Dashe lines inicate ecouple terms, ash otte lines inicate perturbation terms, soli lines inicate combine results. Note that the soli an ashe lines in (a an ( coincie up to the resolution of the figure. result obtaine by consiering each transfer function inepenently, the coupling term (the term in parentheses, which represents a multiplicative perturbation resulting from off iagonal coupling, an the combine result. The perturbation of the iagonal terms is close to unity over a wie range of frequencies, preserving the performance of the SISO loop esigns. The perturbation of the off iagonal terms serves to increase the ecoupling of the two control loops over the control banwith, improving upon the original assumption of weak coupling. VI. CONTO STATEGY Given the analysis of Sec. V, it is clear that the tracking of commans for i an in the synchronous machine can be achieve with two inepenent scalar feeback loops. VII. IMPEMENTATION AND ESUTS The above control scheme was implemente on a homopolar inuctor machine (a separately excite synchronous machine with six step voltage source inverter rive. The machine is part of a flywheel energy storage system in which the motor/generator rotor also serves as the energy storage element. The system is escribe in etail in [1], [13], [14]. Figures 7-11 show system response to step commans of from 8 A to 8 A an from 8 A to 8 A, while i is commane to a constant zero. Bus voltage was 7 V, an rotor spee range from 15 krpm at low to high transitions to 3 krpm at high to low transitions. Figure 7 shows experimental ata for electrical frequency an real current. However, since the machine is synchronous an the bus voltage is fixe, the figure can be thought of as showing flywheel spee an power flow in the machine. When the current (power is positive, the machine spees up, storing energy. Similarly, when current is negative, the machine slows own, returning its store kinetic energy to the bus. Note the small transients that occur at each peak an valley of the ω e trajectory. These represent instantaneous epartures from synchronous operation to change the angle between reference frame an rotor. 187

6 ω e (Hz time (s Fig. 7. Experimental response to step commans, showing ω e (top an (bottom Fig. 9. Step response of, for an step comman from 8 A to 8 A an i comman of constant zero, showing comman (soli, experimental result (otte, nonlinear moel simulation (ashe, an linear moel simulation (ash otte. The lower figure shows the plot from the upper figure in an expane time scale i Fig. 8. Step response of,foran step comman from 8 A to 8 A an i comman of constant zero, showing comman (soli, experimental result (otte, nonlinear moel simulation (ashe, an linear moel simulation (ash otte. The lower figure shows the plot from the upper figure in an expane time scale. Figures 8-11 focus on the transients, showing the i an commans, experimental i an outputs, simulations using the nonlinear ynamics presente in Eqs. 2-8 an simulations using the linear reuce orer system Eqs Note that the comman is not a pure step input, but rather an exponential rise with very short time constant. This input was use in the experimental system to reuce sharp transient spikes, an hence simulations were performe with the same input. The nonlinear moel shows excellent agreement with the experimental results, except for sharp transients on i. These were cause by fluctuations in the bus voltage not inclue in the moel. The linear moel accurately captures the rise an fall times of the output currents, although it fails to preict some overshoot an steay state error. VIII. CONCUSIONS Analysis of a synchronous machine an variable spee rive system was performe, an insights were provie as to useful simplifications that ai the control esign process. A control scheme was presente that permits six step inverter operation, which has several avantages over PWM. The i Fig. 1. Step response of i, for an step comman from 8 A to 8 A an i comman of constant zero, showing i comman (soli, experimental result (otte, nonlinear moel simulation (ashe, an linear moel simulation (ash otte. The lower figure shows the plot from the upper figure in an expane time scale. i i Fig. 11. Step response of i, for an step comman from 8 A to 8 A an i comman of constant zero, showing i comman (soli, experimental result (otte, nonlinear moel simulation (ashe, an linear moel simulation (ash otte. The lower figure shows the plot from the upper figure in an expane time scale. 188

7 feasibility of the control scheme, as well as the usefulness of the esign methoology, was emonstrate through application to experimental harware. APPENDIX I The Jacobians A, B, C an D escribe in Sec. III are given by Eqs The bar ( notation inicates the operating point value of a variable as use in linearizing the moel. A = B = C = D = ω e m īf sin θ ω e m īf cos θ N 2 a 41 a 42 a 43 Bv J (3 a 41 = N 1 m 2 J ī f sin θ a 42 = N 1 m 2 J ī f cos θ a 43 = N 1 m 2 J ī f ( λ cos θ λ q sin θ m cos θ λq m sin θ λ 1 (31 N 1 m 2 J ( λ sin θ + λ q cos θ [ 1 m ī f sin θ ] 1 m ī f cos θ (32 [ m cos θ ] m sin θ (33 APPENDIX II Here results from singular perturbation analysis are evelope. Consiering Eqs. 9 an 1, the system is partitione accoring to the ouble lines in Eqs. 3-33, where x f =[λ λ q θ] T an x s=ω m. A small parameter ε> explicitly inicates that x f changes quickly with respect to x s, such that ε x t f =A 11x f + A 12x s + B 1u (34 xs=a21x t f + A 22x s + B 2u. (35 An invertible change of variables can be efine such that w f,s = Mx f,s,wherem is given by [ ] If M M = 12. (36 I s Applying M to Eqs gives where ε t w f =(A 11 + εm 12A 21w f + f(m 12,εw s (37 +(B 1 + εm 12B 2u t ws=a21w f +(A 22 A 21M 12w s + B 2u (38 f(m 12,ε=A 12 A 11M 12 + εm 12A 22 (39 εm 12A 21M 12. From here, a block triangular form can be obtaine by fining a value of M 12 such that f(m 12,ε =. This is possible only if A 1 11 exists. Assuming this is the case, M12 can be foun via a Taylor series expansion, giving M 12(ε =A 1 11 A 12 + εa 2 11 A 12A + O(ε 2 (4 where A = A 22 A 21A 1 11 A12. Hence the slow subsystem is governe by ws(t=a21w t f (t+(a 22 A 21M 12(εw s(t (41 +B 2(u an the fast subsystem is governe by w τ f (τ=(a 11 + εm 12A 21w f (τ (42 +(B 1 + εm 12B 2u(τ where the time scale for the fast ynamics has been stretche by efining τ= t to. Notice that as ε, Eq. 42 approaches ε w τ f (τ =A 11w f (τ+b 1u(τ (43 which is the fast subsystem that one intuitively expects. Further, Eq. 41 approaches ws(t=a21w t f (t+(a 22 A 21A 1 11 A 12w s(t (44 +B 2(u giving an approximation of the slow subsystem. It can be shown for Eq. 43 that if all the eigenvalues of A 11 (nonzero by previous assumption have negative real parts, then w f (τ converges exponentially to a finite solution epenent on u(τ. Hence a necessary requirement for a stable reuce orer system is e(λ i(a 11 < c < i (45 where λ i(a 11 inicates the ith eigenvalue of A 11, anc>. This conition is obviously not met for the A 11 matrix given in Eq. 3; in fact this matrix is singular. However, it is possible to satisfy the conition given by Eq. 45 by applying feeback control to the fast subsystem. Hence the moelling effort may still procee with the reuce orer system given by A 11, B 1, C 1,anD. EFEENCES [1] P. Tsao, M. Senesky, an S. Saners, An integrate flywheel energy storage system with homopolar inuctor motor/generator an high frequency rive, IEEE Trans. In. Applicat., vol. 39, pp , Nov./Dec. 23. [2] A. Boglietti, P. Ferraris, M. azzari, an F. Profumo, Energetic behavior of soft magnetic materials in the case of inverter supply, IEEE Trans. In. Applicat., vol. 3, pp , Nov [3] P. Kokotovic, H. Khalil, an J. O eilly, Singular Perturbation Methos in Control: Analysis an Design. Acaemic Press, [4] A. Bergen an V. Vittal, Power Systems Analysis. Prentice-Hall, 2. [5] J. Doyle, B. Francis, an A. Tannenbaum, Feeback Control Theory. MacMillan, [6] C. Desoer an M. Viyasagar, Feeback Systems: Input Output Properties. Acaemic Press, [7] J. O eilly an W. E. eithea, Multivariable control by iniviual channel esign, International Journal of Control, vol. 54, pp. 1 46, July [8] W. E. eithea an J. O eilly, Performance issues in the iniviual channel esign of 2 input 2 output systems - i. structural issues, International Journal of Control, vol. 54, pp , July [9], Performance issues in the iniviual channel esign of 2 input 2 output systems - ii. robustness issues, International Journal of Control, vol. 55, pp. 3 47, Jan [1], Performance issues in the iniviual channel esign of 2 input 2 output systems - iii. non iagonal control an relate issues, International Journal of Control, vol. 55, pp , Feb [11], m input m output feeback control by iniviual channel esign, International Journal of Control, vol. 56, pp , Dec [12] H. Nyquist, egeneration theory, Bell System Tech. J., vol. 11, pp , [13] M. Senesky, Control of a synchronous homopolar machine for flywheel applications, Master s thesis, University of California, Berkeley, 23. [14] P. Tsao, A homopolar inuctor motor/generator an six-step rive flywheel energy storage system, Ph.D. issertation, University of California, Berkeley,