3 / Generalized Immittance Based Stability Analysis

Transcription

1 3 / eneralized Immittance Based Stability Analysis In ur wrk t this pint, we have been cncentrating n design techniques that are meant t be applied t knwn system at an perating pint. In ther wrds, using the vcabulary f Chapter, we have been fcused n the prblem f lcal dynamic stability. Hwever, pwer electrnics based systems are nrmally perated thrugh a hst f perating pints, and it is desirable t, in a single analysis, shw that all perating pints are stable. In ther wrds, we wish t address the prblem f reginal dynamic stability. As it turns ut, we can apprach the prblem f reginal dynamic stability in much the same way as the prblem f lcal dynamic stability, using the cncept f generalized immittance. 3. ENERALIZED IMMITTANCE Pwer electrnics based systems exhibit nnlinear dynamics and therefre the linearized cmpnent mdel changes as lading cnditins vary. Furthermre, parameter variatins result in further variatins f the linearized plant mdel. As an example, cnsider the labratry dc pwer system depicted in Fig This system is typical f a turbine electric prpulsin system. In this system, a turbine prime mver (implemented in the labratry as a dynammeter emulating a turbine) drives a synchrnus machine rectifier system whse utput vltage is regulated at 400 V. The rectifier utput is cnnected via a shrt transmissin line t an inductin mtr drive turning a mechanical lad (emulated by a secnd dynammeter in the labratry). In this case, the inductin mtr is cntrlled using an indirect field riented cntrl s that trque cntrl is nearly instantaneus. The parameters f this system and a detailed descriptin are set frth in []. That wrk described tw cntrl systems, a nminal cntrl and a dc link stabilizing cntrl (later referred t as a Nnlinear Stabilizing Cntrl in [2]). The nminal cntrl is used herein. * vdcs Vl. Reg./ Exciter T e,des IM Cntrls ω rm,im Turbine SM v r v dcs v dci i abcs IM Mechanical Lad 3 - φ LC Tie Line Capacitive 3 - φ Uncntrlled Filter Filter Fully Cntrlled Rectifier Inverter Figure 3.-. Example system. Fig depicts the linearized surce impedance f the synchrnus machine rectifier system as pwer varies frm 0 t p.u. in 0.2 p.u. steps. Nt unexpectedly, the impedance is a functin f utput pwer due t dynamic nnlinearities and will als vary as ther aspects f the perating pint (speed, vltage 4/4/2005 Cpyright 2004 S.D. Sudhff Page 27

2 reference) and parameters vary. The techniques f the previus sectin may be extended t this practical system by representing the surce impedance and lad admittance as generalized sets (dented Z s (s) and Y l (s), respectively), which encmpass the full range f pssible plants. This is dne by characterizing the surce/lad ver the full range f perating cnditins and parameter values and then bunding the result. The generalized impedance/admittance is defined as this bunded regin. As an example, Fig depicts the generalized surce impedance f the generatr rectifier. The generalized impedance fr this example was cnstructed by defining a set f nminal speeds frm rad/s in 5 increments, defining a set f utput vltage references frm V in 5 increments, and defining a set f utput pwers frm 0 t 4.07 kw in 5 increments. The plant was linearized abut every cmbinatin f speed, vltage reference, and utput pwer btainable using these sets (therefre yielding 25 plants). The individual surce impedances f the resulting 25 plants were then bunded and the generalized surce impedance was taken t be the interir regin f this bund, which prvides a gd representatin f all pssible values f surce impedance. magnitude, db phase, degrees lg f frequency, Hz Figure Linearized surce impedance f the synchrnus machine rectifier system. 4/4/2005 Cpyright 2004 S.D. Sudhff Page 28

3 Figure eneralized surce impedance f the generatr rectifier. Fig depicts the generalized lad admittance f the tie line and the inductin mtr drive. In this case, the generalized lad admittance was based n 26 plants btained by every cmbinatin f linearized plants which culd be arrived frm the set f speed values (ranging frm rad/s in 6 steps), trque cmmands (ranging frm Nm in 6 steps), and input vltage (ranging frm V in 6 steps). Figure eneralized lad admittance f the tie line and inductin mtr drive. 4/4/2005 Cpyright 2004 S.D. Sudhff Page 29

4 The use f generalized impedance sets and generalized admittance sets can be used t accunt fr nnlinearities. By taking int accunt every pssible plant mdel, small signal stability f the plant can be guaranteed fr all perating pints cnsidered. Likewise, the use f generalized impedances and admittances can be used t take int accunt uncertainties, and variatins in parameters and perating cnditins. Fr example, in the case the synchrnus machine / rectifier, the surce impedance set was frmulated by linearizing the plant ver the entire range f perating cnditins and therefre represents a gd apprximatin t the full range f impedance values that culd be encuntered. In the example shwn, this invlved varying the set pints and lading cnditins, but culd have just as easily included parameter variatin as well. There are many methds t create such a bundary frm the set f linearized plants; herein the bundary is taken t be an irregular N-sided plygn. This plygn as a functin f frequency frms the generalized surce impedance r generalized lad admittance, as apprpriate. Frm a cmputatin pint f view, the autmatic calculatin f the impedances / admittances is readily achievable by linearizing time-dmain Nnlinear Average Value Mdels (NLAMs) [3-5] f the cmpnents. In particular, NLAMs are autmatically linearizable in a variety f state variable based simulatin languages such as ACSL [6] and Simulink [7]. This facilitates rapid and easy calculatin f the generalized surce impedance and the generalized lad admittance. Althugh NLAM mdels are prbably the easiest methd t use, the impedance / admittance sets required can als be btained experimentally (by measuring impedance / admittance ver a large range f perating pints) r by using a detailed cmputer simulatin t measure the input impedance. 3.2 ENERALIZED NYQUIST EVALUATION The Nyquist Immittance Criterin can be used nt nly t test fr lcal dynamic stability but fr reginal dynamic stability as well, by using generalized impedance and admittance cncepts. Whereas the impedance Z (s) (r admittance Y (s) ) is a unique cmplex number at a given frequency; the generalized impedance Z (s) (r admittance Y (s) ) is a bunded set f numbers. In ther wrds, whereas a standard impedance (r admittance) is a single pint in the s-plane, the generalized impedance (r admittance) is a bunded area in the s-plane. The prduct f tw generalized quantities X (s) and Y (s) (which culd either be impedances r admittances fr the purpses at hand) is defined as the bunded set f values that can be achieved by multiplicatin f any value x X and y Y. As a result, the Nyquist cntur Z s is nt a line in the s-plane, but rather a bunded set (representing, in essence, a thick line ). If the Nyquist cntur f Z s des nt encircle the pint than 4/4/2005 Cpyright 2004 S.D. Sudhff Page 30

5 reginal dynamic stability in ensured because it can readily be shwn that Z s culd nt have encircled the pint fr all Z s Z s and all. Hence reginal dynamic stability can be guaranteed. Let us cnsider a brief example. Cnsider the circuit shwn in Fig Figure illustrates the standard Nyquist evaluatin f Zs with Rlad = Ω and als with R lad = Ω. As can be seen, it the first case there are n encirclements f the pint and s the system is stable; in the secnd case tw clckwise encirclement are present s the system is unstable. 0.3 Ω 5 mh v s +_ 500 µ F - R lad Surce Lad Figure Example system fr Nyquist evaluatin. Unstable Case -2 - Stable Case 0 - Figure Nyquist evaluatin fr example system. Nw let us cnsider the generalized case. The generalized Nyquist evaluatin f Z s is shwn fr the circuit in Fig with R lad = Ω, with the exceptin that all parameter have been allwed t vary by ± %. The cntur itself is made up a cllectin f regins; where each regin bunds Z s fr a particular value f s. Fr example, the set f values cmprising Z s s= jω where ω = π rad/s are highlighted. This amalgamatin f all such regins as s is swept ver a range f values which encircles the right half plane makes up the generalized Nyquist evaluatin. It is 4/4/2005 Cpyright 2004 S.D. Sudhff Page 3

6 interesting (and imprtant) t bserve that even thugh the range f parameter variatin was small, the Nyquist cntur thickened cnsiderably. Z s s= jω -2-0 Figure eneralized Nyquist evaluatin fr example system. 3.3 ENERALIZED MIDDLEBROOK The Middlebrk design cnstraint is based n cnfining the Nyquist cntur f Z s t the interir f a circle r radius / M in the s-plane, where the gain margin ( M ) is greater than ne. Recall that in Sectin 2, wherein we discussed lcal dynamics stability, the Middlbrk Criteria culd be stated - Zs < (3.3-) M When examining reginal dynamic stability using generalized admittances, it is cnvenient t define a generalized impedance (admittance) nrm n X as X = max x (3.3-2) x X whereupn the Middlebrk design cnstraint t ensure reginal dynamic stability becmes Zs < / M (3.3-3) Prvided that (3.3-3) is satisfied then it fllws that (3.3-) will be satisfied fr all Z s Z s and all Y, which in turn means that the Nyquist l 4/4/2005 Cpyright 2004 S.D. Sudhff Page 32

7 criteria will be satisfied fr all Z s Z s and all, thereby insuring reginal dynamic stability. Frm (3.3-3), we can readily frm design specificatins. In particular, suppse the generalized surce impedance is knwn. Frm (3.3-3) we require that the generalized lad admittance satisfy Y l < (3.3-4) M Z Alternately, if the lad admittance is knwn, we can frmulate the fllwing specificatin n the surce impedance Z l s s < (3.3-5) M Y 3.4 ENERALIZED AIN AND PHASE MARIN CRITERIA The ain Margin, Phase Margin (MPM) design methd is based n cnfining the Nyquist cntur f Z s t the allwable regin indicated in Figure Frm Sectin 2.5, the MPM design cnstraint is satisfied in a lcal sense prvided that (3.4-) r (3.4-2) is satisfied Y Z s < (3.4-) l M + Z s + Z s ( 80 PM ) ( 80 + PM ) and (3.4-2) Similarly, the MPM criterin can be used t ensure reginal dynamic stability prvided that either (3.4-3) r (3.4-4) is satisfied. Z s < (3.4-3) M max + max Z s ( 80 PM ) and (3.4-4) min + minz s ( 80 + PM ) 4/4/2005 Cpyright 2004 S.D. Sudhff Page 33

8 where the definitin f max X and min X is illustrated in Fig In particular, therein shws the bundary f a generalized set X in the Nyquist plane. The angles φ max and φmin are measured in the clckwise and cunterclckwise directins, respectively, and are bth restricted t the interval [ 0 2π ). They are selected s that the indicated lines are tangent t the generalized set, whereupn max X is defined as φ max as mapped t the interval [ π π ) and min X is defined as φmin as mapped t the interval [ π π ). X φmin φ max Figure Definitin f max X and min X Frm (3.4-3)-(3.4-4), if the surce impedance is knwn, the design specificatins n the lad are such that either (3.5-5) r (3.4-6) is satisfied. max min < (3.4-5) Zl M ( 80 PM ) max Z s ( 80 PM ) Z min s and (3.4-6) Cnversely, if the lad is characterized then the design cnstraints n the lad are that either (3.4-7) r (3.4-8) are satisfied. Z s < (3.4-7) M 4/4/2005 Cpyright 2004 S.D. Sudhff Page 34

9 max Z s minz s ( 80 PM ) max ( 80 PM ) Y min l and (3.4-8) 3.5 ENERALIZED ESAC CRITERIA Frm ur wrk in Sectin 2.7, the reader will recall that the frmulatin f immittance specificatins using the ESAC Criterin is mre invlved that fr the Middlebrk r MPM Criteria. Nt surprisingly, this is als true in the generalized case. The prcess fr cnstructing the lad admittance cnstraint frm a generalized surce impedance is illustrated in Figure Therein the ESAC Criteria stability cnstraint is shwn, a partial Nyquist evaluatin f the (nngeneralized) surce impedance f a nminal plant (the same as shwn as is shwn in Figure 2.6-(a)) alng with generalized surce impedance at a frequency f a (this is based n the nminal surce plant with a parameter variatin f ± 0 %. Fr the purpses at hand it is cnvenient t define Z s, a = Z s ( j2πf a ) ; it is als P cnvenient t define Z s, a as the path in the s-plane that bunds Z s, a. Pint S b in Fig represents a pint n the stability criteria. The immediate prblem at hand is t find the lad admittance Y l, ab which will cause Y l, abz s t be tangent t the stability criteria curve at pint Sb fr ne value Z s Z s, a, and be in the allwable regin fr all ther value Z s Z s, a. As in turns ut, this may be calculated as Sb, ab = (3.5-) Z s, ab where Z s, ab is the value f surce impedance which will define be used t calculate the lad admittance cnstraint at pint S b and at frequency f a. In rder t determine Z s, ab, it is cnvenient t define a pint t b which is an incremental distance ε frm S b, in the frbidden regin, and lcated in a directin nrmal t the stability criteria curve. emetrical cnsideratins reveal that Z s, ab is the value f P Z s Z s, a which maximizes s Z P Z s, a s t b b Zs, a( s) ds Zs, ab (3.5-2) 4/4/2005 Cpyright 2004 S.D. Sudhff Page 35

10 Nyquist Evaluatin f Nminal Surce Impedance sc s d s b ε { t b ESAC Stability Criteria Z s, ab P Z s, a Z s, a Real Axis Imaginary Axis Figure Cnstructin f lad admittance cnstraint frm a generalized surce impedance. By sweeping pint S b ver the entire stability cnstraint curve (which is truncated at pints S c t S d fr bundedness) a cnstraint is placed n the lad admittance at frequency f a. Repeating this prcess ver all frequencies f interest results in the three dimensinal lad admittance cnstraint curve. As can be seen, except fr the calculatin Z s, ab this part f the prcedure is identical t the prcedure fr lcal dynamic stability. The cnverse prblem f determining a design specificatin n the surce impedance frm the lad impedance utilizes an identical prcedure with the exceptin that the rles f surce impedance and lad admittance are interchanged and the result is pltted in impedance rather than in admittance space. 3.6 CASE STUDY In rder t illustrate sme f the cncepts presented, let us return t the sample system first discussed in Sectin 3.. Figure 3.6- depicts the resulting lad admittance cnstraint and generalized lad admittance. Therein, the ESAC Criterin was used with a 3 db gain margin and a 20 phase margin. In this case it can be seen that the generalized lad admittance intersects the lad admittance cnstraint, which culd result in instability. 4/4/2005 Cpyright 2004 S.D. Sudhff Page 36

11 Fig Lad admittance cnstraint and generalized lad admittance fr system shwn in Fig 3.-. Fig depicts the measured system perfrmance during a ramp increase in trque cmmand. Initially the system is perating in the steady state with a cmmanded bus vltage f 400 V, the generatr speed is at rad/s, and the inductin mtr speed is at 88.5 rad/s (the mechanical speed was fixed by a dynammeter). Apprximately 00 ms int the study, a ramped trque cmmand is issued. The traces are the cmmanded current, the actual current, and the dc bus vltage at the input t the cnverter. As can be seen, an instability results, which is cnsistent with Figure i * as,a i as,a v dci,v ms Fig Measured system perfrmance during a ramp increase in trque cmmand. It shuld be kept in mind that satisfying the stability criterin is a sufficient but nt necessary cnditin fr stability, and s it wuld have been pssible fr the system t be stable even thugh a vilatin f the stability criterin was bserved. If a necessary and sufficient cnditin criterin is desired, this can be apprximated by using the ESAC Criterin (because it intrduces less artificial cnservatism than the ther methds) with very lw gain and phase margins. 4/4/2005 Cpyright 2004 S.D. Sudhff Page 37

12 It is interesting t bserve the changes that can ccur in the stability f the example system as the cntrl is mdified. In particular, incrprating the dc link stabilizing cntrl first intrduced in [] and later referred t as a nnlinear stabilizing cntrl (NSC) in [2] yields a dramatic change in the generalized lad admittance, as seen in Fig In this case, there is n intersectin f the generalized lad admittance and s stability is guaranteed (althugh there is an apparent intersectin in Fig , rtatin f the figure will reveal that the intersectin is merely a line-f-sight effect). The experiment shwn in Fig is repeated in Fig with the Nnlinear Stabilizing Cntrl demnstrating that fr this scenari system peratin is stable. Fig Lad admittance cnstraint and generalized lad admittance fr system shwn in Fig 3.- with additin f Nnlinear Stabilizing Cntrl. 20 i * as,a i as,a v dci,v ms Fig Measured system perfrmance during a ramp increase in trque cmmand while utilizing Nnlinear Stabilizing Cntrl. 4/4/2005 Cpyright 2004 S.D. Sudhff Page 38

13 3.7 CLOSIN REMARKS In this Sectin, we have mdified ur riginal immittance base apprach s that we can treat nnlinearities, and uncertainties, and thus cnsider the prblem f reginal dynamic stability. Hwever, at this pint, we are still nly cnsidering simple surce lad systems. In the next sectin, we will extend this technique t the cnsideratin f full systems. 3.8 ACKNOWLEDEMENTS A mngraph supprted by grant N , Natinal Naval Respnsibility fr Naval Engineers: Educatin and Research fr the Electric Naval Engineer 3.9 REFERENCES [] S.D. Sudhff, K.A. Crzine, S.F. lver, H.J. Hegner, and H.N. Rbey, DC Link Stabilized Field Oriented Cntrl f Electric Prpulsin Systems, IEEE Transactins n Energy Cnversin, Vl. 3, N., March 998, pp [2] S.F. lver and S.D. Sudhff, An Experimentally Validated Nnlinear Stabilizing Cntrl fr Pwer Electrnics Based Pwer Systems, Prceedings f the 998 SAE Aerspace Pwer Systems Cnference, pp [3] S.D. Sudhff, K.A. Crzine, H.J. Hegner, D.E. Delisle, Transient and Dynamic Average-Value Mdeling f Synchrnus Machine Fed Lad-Cmmutated Cnverters, IEEE Transactins n Energy Cnversin, Vl., N. 3, September 996, pp [4] S.F. lver, S.D. Sudhff, H.J Hegner, H.N. Rbey, Average Value Mdeling f Hysteresis Cntrlled DC/DC Cnverter fr Use in Electrmechanical System Studies, Prceedings f the 997 Naval Sympsium n Electric Machines, July 28-3, 997, Newprt, RI pp [5] S.D. Sudhff, S.F. lver, Mdeling Techniques, Stability Analysis, and Design Criteria fr DC Pwer Systems with Experimental Verificatin, Prceedings f the 998 SAE Aerspace Pwer Systems Cnference, pp [6] Mitchell and authier Assciates, 99, Advanced Cntinuus Simulatin Language Reference Manual, Cncrd Massachusetts. [7] The Math Wrks, Inc., Simulink Dynamic System Simulatin Sftware Users Manual, Natick, Massachusetts, /4/2005 Cpyright 2004 S.D. Sudhff Page 39