Stability and transient-behavioural assessment of power-electronics based dc-distribution systems.

Transcription

1 Stabiity and transient-behavioura assessment of power-eectronics based dc-distribution systems. J O Fower DSc (Eng) CEng, FIMarEST 1 C G Hodge BSc MSc CEng FIMarEST FREng 2 1 Warwick Manufacturing Group, University of Warwick Coventry, CV4 7AL, UK Phone: Fax: Ros-Royce Nava Marine & Honorary Professor, University of Warwick PO Box 31, Fishponds, Bristo, BS16 1XY, UK Phone: Fax: Abstract: - Power eectronic systems are becoming increasingy important in marine engineering appications and their capabiity is stimuating new concepts and deveopments in eectrica machines and distribution systems. A consequence is the increasing use of DC fed inverters. One probem with dc-based systems is that they are susceptibe to instabiity, particuary when a constant-power regime is operating. This paper demonstrates that an approach to this probem, based on the root-ocus and frequency domain techniques, is we worth consideration. Key Words: DC Distribution, Stabiity, Root Locus, Nyquist Pot, Bode Diagram, Time & Frequency Domain 1 Introduction 2 The negative impedance Deveopments in power-eectronic devices in the past coupe of decades have created a renaissance of interest in researching, designing and testing nove eectrica machines and power systems. Much of the new thinking, generated by these activities, is of considerabe current and potentia interest to the marine engineering community. The form of power distribution for propusion and ships services is a key eement of the overa architecture for any marine eectrica power system and it is being expored at present whether this shoud be ac-ac, ac-dc, dc-ac or dc-dc. Much of the evoving thinking, so far as the marine industry is concerned, is chroniced in a series of papers, spreading over a decade now, on the Eectric Warship [4,5,6,7,8,9]. Many conventiona methods of anaysis and design are not readiy appicabe to these proposed systems. One of the probems with dc systems is that of stabiity. They have a propensity to exhibit negative-impedance instabiity, particuary when seeking to suppy constant power oads [10]. There has been some significant work over the ast 15 years, or so, in this area by, for exampe, Middebrook [10],and Sudhoff et a [11] which is based, essentiay, on frequency-domain techniques. This paper presents a method for examining stabiity using root ocus and frequency domain methods. It is easiy appreciated why negative-impedance instabiity, in a dc system, might be of concern by considering a system deivering constant-power to a oad. In this case we have, using the usua notation, P = IV = constant, (1) Geometricay this can be represented by the branch, in the first quadrant, of a rectanguar hyperboa in the V-I pane as in Figure 1. Taking differentias of equation 1, we have. P = V I + I V = 0, or, V I V I = = R, say, (2) That is, the incrementa resistance is negative and further varies in magnitude with the point of operation of the system. 3 Basic circuit arrangement The basic circuit arrangement considered is that shown in Fig. 2, where, in effect, the votages and currents shown are the incrementa vaues. It is firsty necessary to estabish an expression showing how the vaue of V, the votage existing across the terminas where the source and oad meet, behaves, using perturbationa quantities about some given operating point. This is easiy done, for exampe, 1

2 using Miman's circuit theorem [14] whereby there resuts, Hurwitz anaysis [12] to determine the imits of stabiity as specified by reationships between the Zs, Ys Z, Y Vs V V V Figure 1: Constant Power Operating Characteristic = ( Vs. Ys + V. Y ) ( Y + Y ) s (3) where, Y s = D s /N s and Z s = N s /D s Y = D /N and Z = N /D The N i 's and D i 's are poynomias in s, the Lapace variabe. Substitution of these in equation 3, after some minor agebraic manipuation, eads to N DsVs + N sdv V = N D (1 + Z Y ) s s Since neither N nor D s have any roots in the righthaf phase, stabiity is determined by the roots of, 1 + Zs Y = 0 (4) For a stabe system a these roots must be in the eft-haf of the compex pane. Determining the roots of such an equation, as one of the parameters is varied, is known as determining the root-ocus [12].It is convenient to do this graphicay (nowadays utiising computer graphics). This technique has been expoited in the main-stream of contro engineering work for decades. It is, of course, known as the root-ocus technique (13). However, appications are not restricted to contro activities. 4 The Circuit The circuit discussed by Sudhoff et a is shown in Fig. 3 where the votage across the oad is easiy estabished to be governed by the equation RVs V= 2 CRLs + (CRr L)s + (R r) (5) Note that, R,C,L, r are a positive quantities. Thus the stabiity can be examined by finding the position of the roots of the denominator of equation 5 in the compex pane. However, before this is done it is worthwhie doing a Routh- Figure 2: The basic circuit arrangement parameter vaues. Consider the Routh-Hurwitz tabe for this denominator. s 2 CRL (R-r) s CRr-L 0 s 0 (R-r) According to the Routh-Hurwitz criterion the condition, for stabiity is that there shoud be no changes in sign of the items in the first coumn of the tabe. Since CRL must be positive then (CRr-L) and (R-r) must aso be positive for stabiity That is: L C > and R > r. Rr Critica conditions occur when these inequaities change to equaities. This resut coud be obtained from quadratic equation theory ; however, for higher-order systems the Routh-Hurwitz criterion can be a very effective and usefu technique, hence the demonstration here. The above two inequaities are, of course, exacty as those specified by Sudhoff, [11]. Unfortunatey the Routh-Hurwitz criterion L, r Vs V C Figure 3: The Circuit indicates ony if a system is stabe or not, it does not give any other notion of the system's behaviour. And this is exacty where the root-ocus demonstrates not ony under what conditions the system is stabe, but gives information on the margin of stabiity and of the characteristic response to be expected from the system. Both of these, of course, depend on the root ocations in the compex pane. -R 2

3 From the circuit arrangement in Fig. 3, the component vaues chosen were r = 300mOhms, L = 10mH and R = 24.3 Ω, where the system is deivering, under constant power conditions, 3.7 kw at 300V, fig. 1. The probem is to determine suitabe vaues of the capacitor, C, to produce acceptabe behaviour. We can see that, from the above inequaities, the R>r condition is satisfied, and that the critica vaue of C is given by, ( Ls + R r) 1 + = 0, CRs(Ls + r) (7) which has the form used in contro engineering, shown previousy, as, N(s) 1 + k = 0 D(s), with the 1/C term paying the roe of the k term. This is aso the form required in expoiting MATLAB. At Figure 4a: Root Locus for varying C L C = = 1.37mF rr For this exampe, in Sudhoff s paper [11], using a deveopment of the Nyquist criterion, it is concuded that C = 40mF is a stabe situation, and, C = 0.5mF, represents an unstabe situation. This is exacty what woud be expected from the above resut. Obviousy, for stabiity of some acceptabe degree, C, must be greater than 1.37mF. The question remains as to what vaues of, C, give desired or, at east, acceptabe behaviour. It woud be convenient if there was a graphica way of dispaying how the roots of the governing equation vary as C is varied. This is precisey what the root-ocus pot achieves. These roots being, of course, the roots of the characteristic equation for the system. Now the denominator of equation 5, when equated to zero, is the Lapace transformation of the characteristic equation, viz, RCLs 2 + (CRr - L)s + (R - r) = 0 (6) This equation is not presented in a form suitabe for the appication of the root-ocus technique, but simpe manipuation renders it into such a form;.i.e C[RLs 2 + Rrs]- sl - r + R = 0 or, Figure 4b: Detai of Figure 4a this stage the root-ocus commands for the computer programme MATLAB (version 5.3) were utiised in dispaying this root-oci on a VDU. The pot resuting can be interrogated for specific vaue of k (i.e.1/c) and the corresponding roots. This whoe process can be done with ess than ten ines of computer code. So entering the specific vaues of the coefficients of the numerator and denominator poynomias of equation (7), for the probem in hand, in to the rocus (n,p) command of MATLAB, eads to Fig. 4a. Because of the vaues of the coefficients, this pot is interesting but not particuary usefu for the purpose required. The vaues of the two poes of the equation 7 are actuay situated on the rea axis at - 30 and 0 but they ook to be coincident in Fig. 4a. It can be seen, however, that ony a sma portion of the root-ocus, that to the eft of the imaginary axis, indicates a stabe region. By using the zoom faciity provided by MATLAB the area of interest, as shown in Fig. 4b, can be examined in detai. Since roots of the equations, if compex, occur in compex pairs the part of the root-ocus beow the rea axis is shown notionay. The rocfind command provided by MATLAB prints out the vaue of any roots specified by the cursor on the computer screen, as we as the vaue of k (or 1/C in our case). Fig. 4b shows seven specific points on the graph - the figure egend specifies the vaues of C found for these particuar points and the corresponding damping factor [12]. 3

4 What is seen immediatey from this pot is that increasing the capacitor vaue increases the damping effect and decreases the frequency of the osciation. As wi become apparent, roots aong the rea axes are achievabe ony with very arge, and unreaistic, vaues of capacitance. Athough one can gean from the position of the roots the sort of response expected from different vaues of C, MATLAB has a faciity for dispaying the response to a unit-step and unitimpuse disturbing a system. In the present appication we wish to disturb the system in some way to observe what happens. As an iustration in 0.45 s.. Fig 5.c is drawn for a vaue of C equa to 2mF and the frequency of the osciation has increased markedy from that of 8.1mF case, Fig 5.b and it is seen to have a much more osciatory character. The overshoot is some 95%. Fig. 5d shows the resut obtained using a capacitor of vaue C=4.1 mf. The overshoot is about 82%, but again the transient is effectivey over in 0.5s. The vaue of the capacitor at which sustained osciation is predicted is C = 1.37 mf. Fig. 5d indicates the response for a vaue of capacitor of1.4mf, sighty above this critica vaue, but continuous osciation is seen to be very cose indeed. Figure 5a: Time response. C = 0.44 F Figure 5b: Time response. C = 8.1 mf Figure 5c: Time response. C = 2 mf this contribution shows what happens when a step-disturbance is introduced in the suppy votage. Because of the MATLAB faciities this can be simuated with very itte effort on the designer's part. Fig. 5a predicts this response when C = 0.44F. The response for this vaue is criticay-damped, ie. it has two equa rea roots. However, this capacitor vaue is very arge. Athough the rootocus has branches aong the rea axis, the roots here can ony be obtained with even arger capacitor vaues, and so are of very imited interest in practice. Fig 5b predicts the response when empoying a capacitor of 8.1mFvaue. The response is now osciatory, over-shooting by about 60%but the transient has virtuay died out 4 Figure 5d: Time response. C = 1.4 mf This series of graphs showing the behavioura changes with decreasing capacitance is what one coud roughy predict from a preiminary viewing of the root-ocus curve. The root-ocus not ony indicates critica vaues of components to ensure stabiity, but indicates at a gance, the main characteristic features of the responses to be expected. Two features are particuary interesting. The first is that a change of capacitance from 0.44F to 20mF changes the rea-part of the characteristic roots from 15s -1 to 14s -1. However, the imaginary parts change markedy. The second feature is that in the stabe region where the ocus is moving rapidy towards unstabe behaviour, for exampe, from 20mF to1.4mf, the decay decreases very markedy,

5 but the osciatory nature of the responses do not change much in frequency. 5 Frequency Domain Techniques The root ocus is essentiay a time domain anaysis technique and indeed one of its main advantages is in being abe to deveop some understanding of the ikey time domain behaviour. The root ocus assumes that an approximate knowedge of the system s governing equations are known, which may not be the case, and, even if these are known, then for higher order systems the technique becomes unwiedy and therefore ess attractive. However, these disadvantages can be overcome by using other contro engineering techniques such as Bode diagrams or Nyquist Pots. These frequency domain techniques can again be used in the noncontro context of eectrica power systems. As an iustration, this section wi use the same circuit given in Fig 3. The ceebrated Nyquist stabiity criterion deas with determining whether, or not, a poynomia equation written in the form, ( ) 1+ G s = 0, has roots situated in the right-haf pane of the compex pane [12]. The criterion was originay deveoped in the 1930 s as a means of predicting the stabiity of ampifiers empoying feedback. It subsequenty became estabished as a major topic in feedback contro studies, where G(s) generay represents the open-oop transfer function of the system. Nyquist theory deveopment invoves the appication of compex-variabe theory in particuar contour integration. Mathematicay it is a topic in poynomia theory, and feedback systems anaysis may be viewed as an appication of this theory. The practica significance of the Nyquist theory is that by potting on the compex pane, the frequency-response function ocus G( j ω ), as ω goes from 0 to, and noting this ocus s disposition, reative to the ( 1, j0) point, enabes the stabiity to be assessed. Simpe rues have been estabished in the contro iterature for actuay doing this [13]. These rues differ depending on whether, G(s), represent a minimum phase or non-minimum phase system. In addition, if the system is stabe, then the coseness of approach of the G( jω ) contour to the ( 1, j0) point gives a notiona indication of the system s response, to a disturbance, that is to be expected, e.g. osciatory, suggish, etc. Whist the Nyquist diagram is exceent for quaitative discussion of system behaviour, it is not particuary convenient for quantitative studies. For these, Bode diagrams are superior normay (12). Note that G( jω ) can be written in poar form as, ( ω) =Η exp( jθ ) G j., where, Η and θ are both functions of ω, the frequency. Aso note that ( ω) n G j = n Η+ jθ. Bode diagrams are no more than a pot of Η (in dbs) against the og ω, and a second pot of θ (usuay in degrees) against og ω. Normay these two pots are potted one above the other, Fig. 6 shows the Bode Pot for a case of C = 1.4mF which matches the root ocus pot shown in Fig 5.d. The advantages of going to this seemingy compex potting procedure are discussed, in detai, in amost every eementary contro text book [12]. Suffice it to say that it is just an equivaent way of drawing a Nyquist pot. If consideration is given ony to the condition where the denominator of, G(s), has no positive rea-part roots, a common condition, then there are two points on the Bode diagrams of particuar significance. The first is the vaue of the gain at the frequency where the phase has reached Figure 6: Bode Pots for C = 1.4 mf For stabiity, this gain must be ess than 0dBs, at this frequency. This vaue is known as the gainmargin, i.e. it is the increase in gain required to make the system go unstabe. In a simiar fashion the phase-margin is measured at the frequency where the gain crosses the 0dB point, and it is the increase in negative phase that woud be required to make the phase 180 0, at this cross-over frequency. Thus for stabe systems both the gain- and phase-margins must be positive. As can be seen from Fig 6, the vaue of C = 1.4mF is cose to instabiity, both phase and gain margins are cose to zero. Whie, generay, it is difficut to provide a quantitative correation 5

6 between these two margins and the time behaviour of the system, various rues-of-thumb have been devised. For exampe, some given vaues of gainand phase-margins combinations are known to produce, usuay, acceptabe time responses. Indeed, specifications for contro system performance are often drawn up in terms of these two parameters. Further if the system response is dominated by 2 compex conjugate roots it is possibe to estimate the damping factor and ω from ζ and φ m [15]. 6 Other Contro Engineering n A further advantage of the frequency response approach is that, if measured frequency response data is avaiabe, the design may be done using these directy i.e. the mathematica form governing the behaviour may be unnecessary. References: [1] Voyce, J. and Norton, P. "The advancement for the AIM system, IMarEST INEC., [2] Hodge, C G., Mattick, D.J., and Voyce, J., "The transverse-fux machine", 3rd Nava Symposium or Eectrica Machines, Phiadephia, USA, 2000 Techniques [3] Poock, C., 'Low-cost switched-reuctance drives", Trans IMarE, Vo. 110, part 4, 1998 This paper has provided a practica demonstration of the appication of traditiona contro engineering techniques to an eectrica power system and used a simpe dc circuit with an inherent constant power instabiity as an exampe. It is to be recognized that the techniques iustrated here can be appied to other forms of eectrica power system. It shoud aso be noted that the remaining contro engineering toos can aso be appied. This work is continuing and is being more fuy reported in the Proceedings of the Institute of Marine Engineering, Science & Technoogy. Other deveopments yet to be reported incude parameter uncertainty or variation, compensation and non-inearity. 7 Concusion It has been demonstrated that the root-ocus technique can be a usefu too in deaing with design and stabiity assessment of dc distribution systems. This has been verified using a particuar exampe, but is generay appicabe to any such system whose behaviour may be assumed inear (at east around some operating point), and whose dynamica equation is known. The avaiabiity of computer programs, such as MATLAB, enabes the process to be carried out without any cacuation being done by the designer. Information concerning suitabe parameter vaues and the corresponding characteristic-root vaues are immediatey avaiabe by interrogating the dispayed root-ocus. The response of the system to step changes in, say, the suppy votage is aso immediatey avaiabe using the MATLAB software. It has aso been demonstrated that Frequency Domain techniques such as Nyquist Pots and Bode Diagrams can aso be appied to an eectrica power system and that the Bode Pot in particuar can be used to gain quantified and physicay meaningfu insights into the conditions of stabiity of the circuit in question. 6 [4] Hodge, C.G. and Mattick, D.J. "The eectric warship", Trans IMarE, Vo 108, part 2, 1996 [5] Hodge, C.G. and Mattick, D.J. "The eectric warship II", Trans IMarE, Vo1109, part 2, 1997 [6] Hodge, C.G. and Mattick, D.J. "The eectric warship III", Trans IMarE, Vo 110, part 2, 1998 [7] Newe, J, Mattick, D J & Hodge, C.G. and Mattick, D.J. "The eectric warship IV", Trans IMarE, Vo 112, part 2, [8] Hodge, C.G. and Mattick, D.J. "The eectric warship V", Trans IMarE, Vo 113, part 2, [9] Hodge, C.G. and Mattick, D.J. "The eectric warship VI Trans IMarE, Vo1 114, part 2, 2001 [10] Middebrook, RD. "Input fiter considerations in design and appication of switching reguations", IEEE Proc., IASAM, [11] Sudhoff, S.D., Gover, S.F., Lamm, P.T, Schmucker, D.H., and Deise, D.E. "Stabiity anaysis of power eectronics distribution systems using admittance space constraints", IEEE Trans. On Aero. and Eect. Sys., Vo 36, No 3, pp , 2000 [12] Dorf, R C., and Bishop, RH. "Modern Contro Systems", 7 th edition, Addison Wesey, 1995 [13] Evans, W.R "Graphica anaysis of contro systems", Trans.AIEE, Vo1 67, pp , 1948 [14] Bogart, TF., "Eectrica Circuits", Gencoe, [15] Fower J O & Hodge C G, Stabiity and transient-behavioura assessment of powereectronics based dc-distribution systems. Part2: the frequency response approach to be pubished, IMarEST Proceedings Part A, 2004.