Digital simulation of static converter circuits

H. Eisenack, H. Hofmeister

~IGITAL

SIMULATION OF STATIC CONVERTER CIRCUITS

H. Eisenack, Dipl.-Ing., Siemens AG., Erlangen, Germany H. Hofmeister, Dipl.-Ing., Siemens AG., Erlangen, Germany ZUSAMMENFASSUNG Es wird ein Rechenprogramm erlautert, mit dem beliebige Stromrichterschaltungen auf dem Digitalrechner nachgebildet werden. ·Ein Ubersichtlicher Datensatz beschreibt die Schaltung. Die Aufstellung der Differentialgleichungen fUr die einzelnen Schaltzustande und die Bestimmung der Schrittweite fUr die numerische Integration erfolgen programmintern. Als Ergebnis der Simulation lassen sich beliebige Funktionsverlaufe zahlenmaBig oder graphisch darstellen und Fourieranalysen der Verlaufe ausdrucken. Die Funktionsweise des Rechenprogramms wird an einem Beispiel erlautert. SUMMARY

A program for simulating static converter circuits of any kind on the digital computer is described. A clearly arranged set of data represents the circuit. The formulation of the differential equations for the individual sWitching states and the determination of the step length for the numerical integration takes place within the program. The wave form of any output quantity can, as a result of the simulation, be prp.sented numerically or graphically, and their Fourier coefficients can be printed out. The operation of the computer program is illustrated by an example.

INTRODUCTION The dynamic phenomena in static converter circuits can only in simple cases be visualized and calculated in approximation /1,2/. It is therefore desirable to check such circuits theoretically by simulation. Of the various possibilities of simulation /17/ of power electronic circuits (low power model, analogue, hybrid, digital), digital simulation stands out for its flexibility in the free choice of system parameters. Programs previously applied to the calculation of static converter circuits /3-15/ related to special arrangements; accordingly, a generally applicable program for circuit structures of any kind has not been available. Network analysis programs which have become known so far /18/, e.g. ECAP /19/, whilst tailored to the computation of general circuit arrangements, do not cover the discontinuous mode of operation of the various semiconductor valves in a complex circuit.

351

:,. Eisenack, H. Hofmeister The difficulty in computing static converter circuits is that the resistance of the converter valves var~es by several orders of magnitude depending on the switching state. The change of state takes place either at zero crossing of converter currents or voltages, or at the commencement of an igniting pulse. The possible number of switching states of a circuit can be rather high. For the cir,Mit in ~ with 18 converter valves there exist 2 = 262144 possible states. Thencomputer program would become very extensive if these 2 possibilities were to be covered explicitly. In the present computer program, the differential equations applicable to the circuit being simulated are generated within the program for each state actually occurring. This has the material advantage that the circuit need not be analysed in detail prior to the computation, a major task when developing a new circuit. The need to store previously formulated systems of differential equations for individual sWitching states is also eliminated by the generation of differential equations within the program. The numerical integration of the differential equations for each sWitching state employs an appropriate integration interval; in addition, the interval is modified to cover variations of state exactly. A similar method of static converter simulation has also been pursued by Vokler /16/. The wave form of any current or voltage in the circuit can, as a result of the simulation, be printed out numerically or presented in the form of an oscillogram by means of a plotter. A list of blocked converter valves is printed out at all instants of change of state. The Fourier Analysis of selected functions can be carried out for the steady-state condition of the circuit. SIMULATION OF INDIVIDUAL CONVERTER VALVES It is expedient to simulate converter valves by controlled passive electrical elements such as resistances and inductances, S0 that known methods of circuit analysis can be applied in the computation. A simple, and in many cases adequate, simulation consists in representing the converter valves in the on-state by a small forward resistance and in the off-state by a very high but finite blocking resistance in series with a high blocking inductance. The idea of assuming the converter valves to be ideally blocking, attractive in itself, is more complicated for a general calculation; the reason is that the structure of the circuit would change during individual switching states and differential equations of a different nature would apply. In contrast, with a blocking impedance chosen for the simulation, only the coefficients of one a 11d the same system of differential equations vary.

352

H. Eisenack, H. Hofmeister Besides, an ideally blocking converter valve would also produce undefined voltage conditions in certain operating states, such as an intermittent current in a bridge circuit, as parts of the network would be separated with respect to potential. The blocking inductance introduced additionally to the blocking resistance has little influence on other circuit currents; it is, however, helpful in the numerical integration as later explained. The re3erse-to-forward resistance ratio may, for example, be 10 so that ideal conditions still practically exist. There is no difficulty in introducing appropriate impressed voltages should it be considered desirable to take account also of the threshold voltage of the converter valves. The transition of a thyristor from the conducting to the blocking state takes place at a current zero; a thyristor becomes conducting when an igniting pulse and a positive voltage are simultaneously present. Diodes can be represented by thyristors with permanently applied igniting pulse. The recovery time of thyristors is taken into account in the computation by an imaginary igniting pulse which for the recovery time remains on the thyristor after the blocking instant. All instants of changes of the converter state are determined by an iterative search for current and voltage zeroes of the converter valves, or by the instants of igniting pulses. If several converter valves should become conducting or blocking during a computing step, the earliest change of state has to be ascertained. From this instant on, the computation is continued with the differential equations then applying. It may be assumed that several changes of state occur simultaneously if the earliest change differs only by a small time difference from the next change or changes. CIRCUIT COMPUTATION Mesh current method The dynamic behaviour of all currents and voltages in an electrical network can be described by a system of differential equations. Since static converter circuits primarily contain inductances, the mesh current method /20,21/ is suitable on account of the continuity of the currents in the inductances. Fig. 1b shows the structure of the network to Fig. 1a; a few mesh currents have been indicated. m independent mesh currents are fixed by the complete tree; according to the theory of networks, the number m can be calculated as m

=

z - k

+

n

(1)

where z is the number of branches, k the number of nodes and n the number of galvanically separated sub-networks. The relation between branch and mesh currents i and iM is given by a linear system of equations with the coefficients 0, 1 and -1: 353

H. Eisenack, H. Hofmeister

=A

i

(2)

A is called branch-mesh matrix of incidence; it charac!erizes the structure of the circuit. A differential equation can be obtained for each mesh by adding up the voltages in a mesh which are produced by the mesh current. The m mesh currents thus yield a system of differential equations which can be written in matrix form: ~EM

=

Er.! i M

+ ~ ~ + §.M SlM

.9.M = i M where q is the charge of the condensors and S = 1/C the reciprocal value of the capacitance. The mesh quantities (subscript M) can be determined from the circuit data by the following relationships:

U

~~t,l

!
E 1)

u Ej (

u Ez (4)

s,

-lVJ

u

Ej R j L j M jk C.

354

impressed voltage in branch j resistance in branch j inductance in branch j M magnetic coupling between branches j and k kj capacitance in branch j

H. Eisenack, H. Hofmeister

Numerical integration For computation, the system of first-order differential equations given above is converted into a normal form in which the derivatives of the state variables i M and gM appear on one side. Equations suitable for num~rical integration, e.g. by the Runge-Kutta method, are then obtained:

Because in circuits with linear elements the coefficients are constant for each section, the resistance matrix EM and the inverse inductance matrix ~: needs to be recalculated only at each change of sWitcnfng state. However, if circuits with current dependent inductances are being computerl, then the matrix ~ is recalculated at instants when at least one inductancg varies. Interval of integration The numerical solution of the system of differential equations is effected in discrete time intervals. The interval h must be chosen sufficiently small so as to obtain stable results from the integration algorithm. However, if h is too small, unnecessarily long computing time and large rounding errors will result. The correct choice of integration interval depends on the smallest system time constant occurring or on the modulus of the highest eigenvalue Ipu I of the system of differential equations. For the integ~gfion method Runge-Kutta 4th order, the condition for absolute stability is /22/: h

L.

-

2.6 [p v I max

(6)

The exact calculation of the modulus of the highest eigenvalue being rather cumbersome, the following estimation is used /23/: max i

(2::::: j

[k··I) lJ

(7)

where k .. is an element of the coefficient matrix K of the comn~ned system of differential equations (5).The integration interval is recalculated for each switching state depending on the changed properties of the matrix ~.

355

.,

Eisenack, H. Hofmeister

Integration interval for various converter states Very high eigenvalues, or small blocking time constants (Tb ~ L/R ), would result if blocked converter valves in ~tatic 8b~verter circuits were re p resented 0nly b¥ a high ohmic reverse resistance (e.g. R f = 10 5 • R ). This in turn would lead to unacceptab~f long compu££ng time and large rounding errors. As already mentioned, a large blocking inductance L is therefore additionally introduced so that the integr~ffon interval is appropriate to the phenomenon of principal interest. The computing time thus gained is substantial; the error introduced by the blocking inductance is negligible, the reverse current ~eing very small and not substantially influencing the rest of the circuit. The necessary integration intervals may differ very much from each other, since the system of differential equations has different coefficients depending on the state of the converter. For exact determination of the instants of changes of converter state, the last integration interval during the old converter state must be suitably chosen. This interval is given either by the commencement of the next igniting pulse or by the current or voltage zeroes of the individual converter valves. Newton's iteration method is used for the computation of the zeroes. Calculation of the desired results After each integration interval the values of all mesh currents and capacitor charges, and of their derivatives, are available. The currents and charges and their derivatives, and thus also the voltages on resistances, inductances and capacitances, can be computed by the relationships: (8) A different method is used for the computation of the reverse voltages of the converter valves as the reverse current may possibly have been computed by taking the difference of large currents and may thus possess large relative errors. It is advisable for this reason to calculate the reverse voltage by taking the sum of those branch voltages which constitute the reverse voltage. A tree is automatically determined which starts from one of the nodes of the blocked branch and reaches the second node of the blocked branch by a tree branch. The Fourier integrals of the functions to be analysed are calculated along with the functions themselves. Following on the simulation, this enables the magnitudes and phase angles of harmonics to be determined as well as the r.m.s. value and the fundamental and harmonic content of the waveform.

356

H. Eisenack, H. Hofmeister COMPUTING PROGRAM The computing method described in the preceding sections is compiled in the problem-oriented programming language ALGOL. Memory capacity of about 12 000 words is required for the program, not included additional space for the data specific to the circuit. The number of storage words first increases about linearly and then with the square of the number of branches. The computing time of the currents and voltages shown in Fig. 2 amounted to 15 minutes on the Siemens 306 Computer (core store 16 k, cycle time 0.6 JUs). With regard to program organization, no limits are set to the size of the circuit; however, a limit is set by the storage capacity of the computer used. It is not absolutely necessary that the computed circuit be a static converter circuit. An electric circuit without converters is a special case included in the general formulation of the program. The input of a given circuit takes the form of a list. The branch-mesh matrix of incidence is entered to characterize the structure of the circuit. The values of the components used, such as resistors, reactors, capacitors, diodes, thyristors and voltage sources, and the initial values of currents, voltages on capacitors and converter states are assigned to each branch number. The data record also contains information on the recovery time, magnetic coupling, non-linear inductances, sequence of igniting pulses, control parameters, number of Fourier analyses to be carried out and a specification of the output quantities. EXAMPLE OF COMPUTATION Fig. 1a shows a dc link inverter circuit supplying a rotating field machine. The synchronous machine used here is represented by a simplified electric network containing reactances and voltage sources. For the purpose of simulating this circuit, the network had to be converted into an equivalent scheme to suit the program. Fig. 1b shows the equivalent scheme with the complete tree chosen and the consecutively numbered branches and nodes. The result of the simulation is shown in ~ . The inverter circuit is in this case operated at a frequency of 10 Hz. The computed oscillogram shows the transition from self-commutated to externally commutated operation which at this speed of the machine i~ achieved by advancing the igniting pulses by 30 .

357

H. Eisenack, H. Hofmeister

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n41

n61

nil

n21

n31

C,

---==---.. . . . UN

a)

14

18

19

10

11

r

1'"-------.... 11 ,15 I

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17

,

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-----, .

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: !~-.....I/ t: 9<>---=---+-+-..",2:O:'==.....

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I~-----.----,

5

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I ______ ./

4

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b)

0

c::::::> node

•

,re" bniLch

frEe tranch nesn Fig. 1 :

dc link inverter a) circuit diagram b) equivalent scheme

358

H. Eisenack, H. Hofmeister sell-

externally commutated

c omruu ta ted

p

i:i"~ ···1,: ..i'~'.r::=x",

................._

'!~~.1..

.! [~

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-. .... __ "__ .,-n11 ..,- ..,- ..11._.,_. __ . ,,,._ .... _...

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Fig. 2:

_

Currents and voltages of a dc link converter (result of simulation)

359

H. Eisenaek, H. Hofmeister

.~EFERENCES

/1/

Moltgen, G.: NetzgefUhrte Stromrichter. Siemens AG., Berlin-MUnchen: Siemens AG., 1970

/2/

Heumann, K., Stumpe, A.C.: Thyristoren. Stuttgart: B.G. Teubner 1969

/3/

Holtz, J.: Naehbildung einer Hochspannungsgleichstrom-Ubertragung auf dem Digitalrechner. ETZ-A, vol 89, p. 438-439, 1968

/4/

Holtz, J.: Digitalreehenprogramm zur Naehbildung einer Hoehspannungs-Gleichstrom-Ubertragung. ETZ-A, vol. 90,p. 195-199, 1969

/5/

Holtz, J.: Uber den Einsatz des Digitalreehners bei der Nachbildung statiseher und dynamiseher Vorgange in Verbundnetzen unter besonderer BerUeksiehtigung der Hoehspannungs-GleichstromUbertragung. Thesis TU-Braunsehweig, 1969

/6/

Leonhard, W.: Digitalreehner-Untersuehung dynamischer Vorgange bei der Gleiehstromkopplung von Drehstromnetzen. ETZ-A, vol. 91, p 83-87, 1970

/7/

Sadek, K., Schmidt, H. H.: Neuere Ergebnisse bei der Nachbildung von HGU-Systemen. ETZ-A, vol. 91, p. 642-643, 1970

/8/

Hay, J. L., Hingorani, N.G.: Digital simulation of a multiconverter hvdc system by digital computer. I. Mathematical model. 11. Computer program. IEEE/Trans. on Power Appar. & Syst., vol. 89, ~18-228, 1970

/9/

Michelakakis, P.: Neues Verfahren zur digitalen Bereehnung von stationaren und nichtstationaren Vorgangen in 12-pulsigen Stromrichteranlagen. ETZ-A, vol. 92, no. 6, p. 348-353, 1971

/10/

AnschUtz, W.: Die Naehbildung der HoehspannungsGleiehstrom-Ubertragung mit mehr als zwei Stationen und ihre Regelung mit dem Digitalreehner. Thesis TH Darmstadt, 1971

/11/

MUller, R.: Erzeugung des Differentialgleiehungssystems fUr die Simulation einer HGU-Station mit mehreren DrehstrombrUckensehaltungen. ETZ-A, vol. 95, p. 144-147, 1972

/12/

Leonhard, W., Schmidt, H. H.: Digital simulation of multi terminal HVDe links. Proe. IV Power Systems Computation Conference, Grenoble, 1972

360

H. Eisenack, H. Hofmeister-

/13/

Sadek, K.: Digitale Simulation einer HochspannungsGleichstrom-Ubertragung. Thesis TU-Braunschweig, 1972

/14/

Sadek, K.: Digitale Nachbildung eines zweipoligen erdsymmetrischen HGU-Mehrpunktnetzes mit vier Stationen und Gleichstromleitungen. ETZ-A, vol. 93, p. 139-143, 1972

/15/

Schmidt, H. H.: Digitale Nachbildung umfangreicher dynamischer Systeme am Beispiel eines Netzmodells mit Drehstrom- und Gleichstrom-Obertragung. Thesis TU-Braunschweig, 1973

/16/

Vokler, K. D.: Digitale Simulation von StromrichterStellgliedern. Elektrie, vol. 27, no. 1, p. 12-14, 1973

/17/

Vogt, F.: Die Simulation von Stromrichtern. ETZ-A, vol. 94, no. 3, p. 479-482, 1973

/18/

Renk, K. D., Steinkopf, U.: Programme zur Analyse elektrischer Schaltungen - eine vergleichende Ubersicht. NTZ-Kurier 3/73, p. K37-K43

/19/

1620 Electronic Circuit Analysis Program (ECAP) (1620-EE-02X) User's Manual, IBM, 1965

/20/

Edelmann, H.: Berechnung elektrischer Verbundnetze. Berlin-Gottingen-Heidelberg: Springer 1963

/21/

Weh, H.: Elektrische Netzwerke und Maschinen in Matrizendarstellung. Mannheim: BI Hochschultaschenbuch 108, 1968

/22/

Jentsch, W.: Digitale Simulation kontinuierlicher Systeme. Mlinchen-Wien: Oldenburg 1969

/23/

Zurmlihl, R.: Matrizen. Berlin-Gottingen-Heidelberg: Springer 1961

/24/

Eisenack, H., Cordes, D.: Digitale Nachbildung der Vorgange in Stromrichterschaltungen. ETZ-A, vol 92, no. 2, p. 120-122, 1971

/25/

Eisenack, H., Hofmeister, H.: Digitale Nachbildung von elektrischen Netzwerken mit Dioden und Thyristoren, Arch. Elektrotech., vol. 55, p. 32-43, 1972

361