Damping torsional oscillations in a drive system

Abstract

A drive system includes at least one electrical machine and a plurality of rotating components, which are interconnected via shafts. A method for damping torsional oscillations in the drive system includes: determining angular speeds for at least one of the shafts based on measurements in the drive system; determining a damping torque from the angular speeds with a function that models at least some of the electrical machine, the rotating components and the shafts; adapting a reference torque for the at least one electrical machine by adding the damping torque; and controlling the at least one electrical machine with the adapted reference torques.

Claims

1. A method for damping torsional oscillations in a drive system, wherein the drive system comprises at least one electrical machine and a plurality of rotating components, which are interconnected via shafts, the method comprising: determining angular speeds (θ.sub.i) for at least some of the shafts based on measurements in the drive system; determining angular speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) from the angular speeds (θ.sub.i) with a drive system model, which models the drive system as a number of inertia elements (J.sub.i) interconnected via coupling elements and every angular speed difference (Δ{circumflex over ({dot over (Θ)})}.sub.i) refers to a difference of angular speeds at ends of the coupling elements; determining a damping torque (T.sub.damp) from the angular speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) with a function that models at least some of the electrical machine, the rotating components and the shafts; adapting a reference torque (T.sub.ref) for the at least one electrical machine by adding the damping torque to determine adapted reference torques (T); controlling the at least one electrical machine with the adapted reference torques (T).

2. The method of claim 1, wherein the angular speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) are composed into an angular speed difference vector (Δ{circumflex over ({dot over (Θ)})}), damping torques for more than one electrical machine are composed into a damping torque vector and the damping torque vector is the angular speed difference vector (Δ{circumflex over ({dot over (Θ)})}) multiplied with a damping matrix (R).

3. The method of claim 2, wherein entries of the damping matrix (R) are chosen, such that an angular speed difference of a coupling element is mapped to a damping torque vector, which adjusts the angular speed of the at least one electrical machine, which are connected according to the model to the coupling element.

4. The method of claim 3, wherein the damping matrix (R) comprises the transpose of an input matrix (E) as multiplicative factor; and wherein the input matrix (E) models, which electrical machine applies torque to which inertia element (J.sub.i).

5. The method of claim 3, wherein the damping matrix (R) comprises a pseudo-inverse (L.sup.+) of a difference matrix (L) as multiplicative factor; and wherein the difference matrix (L) models, which angular speeds (θ.sub.i) have to be subtracted from each other to determine the speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) for the coupling elements.

6. The method of claim 3, wherein the damping matrix (R) comprises a scaling matrix (ρ) as multiplicative factor, which determines an amount of damping for at least two electrical machines.

7. The method of claim 2, wherein the damping matrix (R) comprises the transpose of an input matrix (E) as multiplicative factor; wherein the input matrix (E) models, which electrical machine applies torque to which inertia element (J.sub.i).

8. The method of claim 7, wherein the damping matrix (R) comprises a scaling matrix (ρ) as multiplicative factor, which determines an amount of damping for at least two electrical machines.

9. The method of claim 7, wherein the damping matrix (R) comprises a pseudo-inverse (L.sup.+) of a difference matrix (L) as multiplicative factor; and wherein the difference matrix (L) models, which angular speeds (θ.sub.i) have to be subtracted from each other to determine the speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) for the coupling elements.

10. The method of claim 2, wherein the damping matrix (R) comprises a pseudo-inverse (L.sup.+) of a difference matrix (L) as multiplicative factor; wherein the difference matrix (L) models, which angular speeds (θ.sub.i) have to be subtracted from each other to determine the speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) for the coupling elements.

11. The method of claim 2, wherein the damping matrix (R) comprises a scaling matrix (ρ) as multiplicative factor, which determines an amount of damping for at least two electrical machines.

12. The method of claim 11, wherein the scaling matrix (ρ) is a multiple of the unit matrix.

13. The method of claim 1, wherein only a subset of the angular speeds (θ.sub.i) of the shafts is determined; wherein angular speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) are determined from the drive system model into which the subset of angular speeds (θ.sub.i) is input.

14. The method of claim 13, wherein the angular speeds (θ.sub.i) in the subset, which are input into the drive system model, are measured.

15. The method of claim 1, wherein the drive system model comprises inertia elements and/or coupling elements, which represent more than one rotating component and/or more than one shaft of the drive system.

16. A controller including a processor and a set of instructions stored on a memory device executable by the processor effective for damping torsional oscillations in a drive system, the drive system comprises at least one electrical machine and a plurality of rotating components, which are interconnected via shafts, comprising: determine angular speeds (θ.sub.i) for at least some of the shafts based on measurements in the drive system; determine angular speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) from the angular speeds (θ.sub.i) with a drive system model, which models the drive system as a number of inertia elements (J.sub.i) interconnected via coupling elements and every angular speed difference (Δ{circumflex over ({dot over (Θ)})}.sub.i) refers to a difference of angular speeds at ends of the coupling elements; determine a damping torque (T.sub.damp) from the angular speed differences (Δ{circumflex over ({dot over (Θ)})}.sub.i) with a function that models at least some of the electrical machine, the rotating components and the shafts; adapt a reference torque (T.sub.ref) for the at least one electrical machine by adding the damping torque to provide adapted reference torques (T); control the at least one electrical machine with the adapted reference torques (T).

17. The controller of claim 16, further comprising: an angular speed estimator for estimating the angular speed from measurements in the drive system.

18. A drive system, comprising: at least one electrical machine; at least one electrical converter for supplying the at least one electrical machine with electrical power; a plurality of rotating components; a plurality of shafts interconnecting the at least one electrical machine and the rotating components; the controller according to claim 16.

19. The drive system of claim 18, further comprising: at least one gearbox interconnecting the rotating components (14) and/or the at least one electrical machine via shafts.

20. The controller of claim 16, further comprising a torsional damping unit for determining the damping torque (T.sub.damp) for the at least one electrical machine.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The subject-matter of the invention will be explained in more detail in the following text with reference to exemplary embodiments which are illustrated in the attached drawings.

(2) FIG. 1 schematically shows a drive system according to an embodiment of the invention.

(3) FIG. 2 schematically shows a model of the drive system of FIG. 1.

(4) FIG. 3 schematically shows a control scheme for a drive system according to an embodiment of the invention.

(5) FIG. 4 shows a flow diagram for a method for damping torsional oscillations in a drive system.

(6) FIG. 5 shows a Bode diagram of a transfer function produced with the method of FIG. 4.

(7) The reference symbols used in the drawings, and their meanings, are listed in summary form in the list of reference symbols. In principle, identical parts are provided with the same reference symbols in the figures.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

(8) FIG. 1 shows a drive system 10, which comprises several electrical machines 12, several mechanical rotating components 14 and a gearbox 16, which are interconnected by shafts 18, 20, 22. It may be that electrical machines 12 have a common shaft 20, that rotating components 14 have a common shaft 22 and/or that components of the drive system 10, such as the gearbox 16, are connected to more than two shafts 18, 20, 22.

(9) The electrical machines 12 may be electrical motors, which are supplied by converters 24 with electrical current. The converters 24 may be controlled by one or more controllers 28, which also receive measurement signals from the drive system 10.

(10) With the measurement signals, such as currents, voltages and/or mechanical measurement signals, such as a speed, a rotation frequency, etc., the controller or more general control system 28 determines switching signals for the converters 24, which then generate a corresponding current.

(11) One may think of the connecting shafts 18, 20, 22 as infinitely stiff elements that introduce no twist between the components 12, 14, 16. However, in reality, the shafts 18, 20, 22 are elastic and have its own stiffness constant and damping factor, which introduce a difference between the angle and angular speeds that are seen by the electric machines 12, 14, 16 on one side and the gearbox 16 as well as the loads 14 on the other side.

(12) FIG. 2 shows a model 30 for the drive system 10 of FIG. 1, which takes these considerations into account. The electrical machines 12 and the loads 14, as well as the gearbox 16 are modelled as inertia elements with inertias J.sub.i. The parts of the shafts 18, 20, 22 between the components 12, 14, 16 are modelled as coupling elements with stiffness constants k.sub.i,j.

(13) In general, there are inertia elements in the drive system 10, such as the components 12, 14, 16, i.e., which are modelled as N inertia elements with inertias
[J.sub.1 . . . J.sub.N]
and there are flexible elements, such as the shafts 18, 20, 22 with stiffness coefficients k.sub.i,j. The following formulas are for series-connected inertia elements, but can be generalized to more complex structures. In the case of series-connection, there are N−1 coupling elements with stiffness coefficients
[k.sub.12 . . . k.sub.N−1N]

(14) The differential equations governing the dynamics of inertia elements series-connected by these coupling elements are
J.sub.i{umlaut over (θ)}.sub.i=k.sub.i−1,i(θ.sub.i−1−θ.sub.i)+d.sub.i−1,i({dot over (θ)}.sub.i−1−{dot over (θ)}.sub.i)−k.sub.i,i+1(θ.sub.i−θ.sub.i+1)−d.sub.i,i+1({dot over (θ)}.sub.i−{dot over (θ)}.sub.i+1)+τ.sub.i

(15) Here, d.sub.i,j are the damping coefficients and τ.sub.i is the forcing torque due to some external mover, which in the present case is an electrical machine 12, which acts directly on the i.sup.th inertia element.

(16) The above formula can be generalized to more complex systems, which are not solely series-connected. However, as a matter of lucidity, only the more simple equation is given.

(17) In general, the equations above and the more general equations for a more complex system can be written in state-space form as a matrix equation
J{umlaut over (Θ)}=−BKB.sup.TΘ−BDB.sup.T{dot over (Θ)}+ET

(18) In this equation, Θ=[θ.sub.1 . . . θ.sub.N] are the angular positions of the inertia elements with inertia J.sub.i composed into a vector, {dot over (Θ)} are the angular speeds and {umlaut over (Θ)} the angular accelerations.

(19) T=[τ.sub.1 . . . τ.sub.M] is a vector composed of the torques produced by M electrical machines 12 or more general torque generator.

(20) The input matrix E models, which electrical machine 12 is connected to which inertia element with inertia J.sub.i. For example, in the case of three electrical machines 12 and 5 inertia elements with inertia J.sub.i, the input matrix E may be

(21) $E=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$

(22) The matrix B is called the incidence matrix and models, how the coupling elements with coefficients k.sub.i,j interconnected the inertia elements with inertia J.sub.i. For example, in the case of series-connected inertia elements J.sub.i, the incidence matrix B may be

(23) $B=\left[\begin{array}{ccc}1& 0& 0\\ -1& 1& 0\\ 0& -1& 0\\ 0& 0& \ddots \end{array}\right]$

(24) The matrix J is the inertia element matrix and encodes the inertias J.sub.i on the diagonal.

(25) $J=\left[\begin{array}{ccc}{J}_1& \text{...}& 0\\ \mathrm{.Math.}& \ddots & \mathrm{.Math.}\\ 0& \text{...}& J_N\end{array}\right]$

(26) In the case of series-connected inertia elements with inertias J.sub.i, the further matrices are defined as

(27) $K=\left[\begin{array}{ccc}{k}_{1_{\prime }2}& \text{...}& 0\\ \mathrm{.Math.}& \ddots & \mathrm{.Math.}\\ 0& \text{...}& k_{N-1,N}\end{array}\right],D=\left[\begin{array}{ccc}{d}_{1_{\prime }2}& \text{...}& 0\\ \mathrm{.Math.}& \ddots & \mathrm{.Math.}\\ 0& \text{...}& d_{N-1,N}\end{array}\right]$

(28) In general, K encodes the stiffness constants k.sub.i,j and D encodes the damping coefficients d.sub.i,j.

(29) For rather complex drive systems 10, for example with a large number of inertia elements J.sub.i, it may be attractive to reduce the size of the model equations by aggregating inertia elements and coupling elements into fewer dominant inertia elements.

(30) Such an aggregation can be defined with the aid of a partition matrix P, which encodes, how specific inertia elements are aggregated into one replacing inertia element. Each column of the matrix P may define a partition, where 1 indicates the inertia elements with inertias J.sub.i that are included into a specific partition.

(31) For example,

(32) $P=\left[\begin{array}{cccc}1& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

(33) encodes that the first and second inertia element as well as the third and fourth inertia element are aggregated into one inertia element, respectively.

(34) Based on this, a reduced-order model can be defined
Ĵ{circumflex over ({umlaut over (Θ)})}=−{circumflex over (B)}{circumflex over (K)}{circumflex over (B)}.sup.T{circumflex over (Θ)}−{circumflex over (B)}{circumflex over (D)}{circumflex over (B)}.sup.T{circumflex over ({dot over (Θ)})}+ÊT

(35) where Ĵ=P.sup.TJP, Ê=P.sup.TE, {circumflex over (B)}=non_zero_columns(P.sup.TB).

(36) The stiffness coefficients matrix {circumflex over (K)} and the damping coefficients matrix {circumflex over (D)} may be defined based on the partition, for example as series equivalents of the original coefficients. For example in the case of series-connected inertia elements by applying the formulas

(37) ${\hat{k}}_{i,j}=\frac{1}{\underset{\mathrm{partion}}{.Math.}\frac{1}{k_{i,j}}}\mathrm{and}{\hat{d}}_{i,j}=\frac{1}{\underset{\mathrm{partion}}{.Math.}\frac{1}{d_{i,j}}}$

(38) It has to be noted that the reduced model may be used, when the matrix size should be reduced. The determination of the damping torques as described herein may be done with the full model or the reduced model. Both models have the same structure.

(39) For defining a torsional damping part for the model (either the full or the reduced model), in the following, a differential model is constructed. This can be achieved via defining a transformation with a difference matrix L, which defines, which speeds θ.sub.i have to be subtracted from each other, such that the corresponding difference Δ{dot over (Θ)}.sub.a for the corresponding coupling element (with index a) is output. For example, in the case of series-connected inertia elements, the difference matrix may be

(40) $L=\left[\begin{array}{cccc}1& -1& 0& \\ 0& 1& -1& \mathrm{.Math.}\\ & \mathrm{.Math.}& & \ddots \end{array}\right]$

(41) With the difference matrix L, the dynamics of the model (here the reduced model) can be written as
Δ{circumflex over ({umlaut over (Θ)})}=−LĴ.sup.−1{circumflex over (B)}{circumflex over (K)}{circumflex over (B)}.sup.TL.sup.+Δ{circumflex over (Θ)}−LĴ.sup.−1{circumflex over (B)}{circumflex over (D)}{circumflex over (B)}.sup.TL.sup.+Δ{circumflex over ({dot over (Θ)})}+LĴ.sup.−1ÊT

(42) L.sup.+=L.sup.T(LL.sup.T).sup.−1 is the pseudo-inverse.

(43) The dynamics are now in a form, such that a possible damping term can be included. A damping term can be included via the input matrix Ê and may have a similar structure as the term comprising the factor Δ{circumflex over ({dot over (Θ)})}.

(44) The overall feedback T may take the form
T=T.sub.ref+T.sub.damp

(45) T, T.sub.ref and T.sub.damp are vectors with the same number M of components as electrical machines 12 are present.

(46) T.sub.ref is the reference torque that is to be given to the electric machines 12 that we can control and that may come from a different control loop (such as a speed and/or torque control loop).

(47) T.sub.damp is the damping torque and is given by
T.sub.damp=−ρÊ.sup.TL.sup.+Δ{circumflex over ({dot over (Θ)})}

(48) which results in the following closed-loop equation system
Δ{circumflex over ({umlaut over (Θ)})}=−LĴ.sup.−1{circumflex over (B)}{circumflex over (K)}{circumflex over (B)}.sup.TL.sup.+Δ{circumflex over (Θ)}−LĴ.sup.−({circumflex over (B)}{circumflex over (D)}{circumflex over (B)}.sup.T+ρÊÊ.sup.T)L.sup.+Δ{circumflex over ({dot over (Θ)})}+LĴ.sup.−1ÊT.sub.ref

(49) The matrix ({circumflex over (B)}{circumflex over (D)}{circumflex over (B)}.sup.T+ρÊÊ.sup.T determines the amount of damping in the closed-loop equation system.

(50) It is important to note that the value of the torsional natural frequencies after the feedback may be the same as before the feedback.

(51) The scaling matrix ρ, which may also may be a single coefficient, can be seen as a tuning factor. FIG. 3 shows a possible controller 28, which controls the electromechanical parts 30 of the drive system 10 with the damping torque T.sub.damp as defined above.

(52) A speed and/or torque controller 32, in which, for example, a reference speed 34 and measurements 36 from the electromechanical parts 30 of the drive system 10 may be input provides the reference torque vector T.sub.ref. To the reference torque vector T.sub.ref, a damping torque vector T.sub.damp is added and the resulting torque vector T is input into a converter controller 37, which produces the switching commands 38 for the converters 24.

(53) The damping torque vector T.sub.damp is provided by a torsional damping unit 40, which receives a difference speed vector Δ{circumflex over ({dot over (Θ)})} and calculates the damping torque vector T.sub.damp according to T.sub.damp=−ρÊ.sup.TL.sup.+Δ{circumflex over ({dot over (Θ)})}.

(54) It has to be noted that the matrices J, K, D and B from above do not need to be known for calculating the damping torques from the difference speeds.

(55) The difference speed vector is Δ{circumflex over ({dot over (Θ)})} provided by an angular speed estimator 42. The angular speed estimator 42 determines the difference speed vector Δ{circumflex over ({dot over (Θ)})} based on measurements 36 in the drive system 10. For example, the speeds θ.sub.i may be directly measured by angular speed sensors that are connected to the respective electrical machines 12 and/or the rotating components 14.

(56) If the required (differential) speed measurements θ.sub.i are not available, it is possible that the angular speed estimator 42 estimates the speed differences Δ{circumflex over ({dot over (Θ)})}.sub.i from a subset of the speeds θ.sub.i and optional further measurements 36 in the drive system 10,

(57) For example, an estimation scheme may be used that utilizes the system model 30, measured angular displacements and (differential) angular speeds to estimate the unmeasured quantities.

(58) FIG. 4 shows a flow diagram for a method for damping torsional oscillations in the drive system 10.

(59) In step S10, the speed controller 32 determines the reference torques T.sub.ref for the electrical machines 12 from one or more reference speeds 34 of the drive system 10.

(60) In step S12, some or all of the angular speeds θ.sub.i for at least some of the shafts 18 are determined based on measurements in the drive system 10.

(61) The angular speed estimator 42 determines the angular speed differences Δ{circumflex over ({dot over (Θ)})}.sub.i from the angular speeds θ.sub.i with the drive system model 30. Some or all of the angular speeds θ.sub.i in the subset may be measured.

(62) It may be that only a subset of the angular speeds θ.sub.i of the shafts 18, 20, 22 is determined and that the angular speed estimator 42 determines the angular speed differences Δ{circumflex over ({dot over (Θ)})}.sub.i from the drive system model 30 into which the subset of angular speeds θ.sub.i is input.

(63) In step S14, the torsional damping unit 40 determines a damping torque T.sub.damp for each electrical machine 12 from the angular speed differences, wherein the angular speed differences Δ{circumflex over ({dot over (Θ)})}.sub.i are composed into an angular speed difference vector Δ{circumflex over ({dot over (Θ)})}, the damping torques are composed into a damping torque vector T.sub.damp and the damping torque vector T.sub.damp is the angular speed difference vector Δ{circumflex over ({dot over (Θ)})} multiplied with a constant damping matrix R. As derived above, the damping matrix may have the form
R=−ρÊ.sup.TL.sup.+

(64) In general, entries of the damping matrix R may be chosen, such that a angular speed difference Δ{circumflex over ({dot over (Θ)})}.sub.i of the coupling element corresponding to k.sub.ij is mapped to a damping torque vector T.sub.damp, which adjusts the angular speeds of the electrical machines 12, which are connected according to the model 30 to the coupling element corresponding to k.sub.ij.

(65) The damping matrix R may comprise the transpose E.sup.T of an input matrix E as multiplicative factor. As described above, the input matrix E models, which electrical machine 12 applies torque to which inertia element with inertia J.sub.i.

(66) It may be that also the transpose of the input matrix of a reduced model Ê.sup.T=Ê=Ê.sup.TP is used. In this case, the drive system model 30 may comprise inertia elements and/or coupling elements, which represent more than one rotating component 14 and/or more than one shaft 18, 20, 22 of the drive system 10.

(67) The damping matrix R may comprise a pseudo-inverse L.sup.+ of a difference matrix L as multiplicative factor. As described above, the difference matrix L models, which angular speeds θ.sub.i have to be subtracted from each other to determine the speed differences Δ{circumflex over ({dot over (Θ)})}.sub.i for the coupling elements.

(68) The damping matrix R may comprise a scaling matrix ρ as multiplicative factor, which determines an amount of damping for each electrical machine 12. The scaling matrix ρ may be a diagonal matrix and/or may be a multiple of the unit matrix.

(69) In the end, the reference torque vector T.sub.ref for the electrical machines 12 is adapted by adding the damping torque vector T.sub.damp.

(70) In step S16, the electrical machines 12 are controlled with the adapted reference torque vector T. The one or more converter controllers 37 determine switching states 38 for the electrical converters 24 of the drive system 10 from the adapted reference torque T.

(71) FIG. 5 shows a Bode diagram of the transfer function from one input torque to one speed output, without torsional damping (dashed) and with torsional damping (continuous). The upper part of the diagram shows the attenuation of the magnitude and the lower part the phase shift.

(72) The Bode diagram was generated for a 15 mass-spring-damper drive system model with 3 torque inputs. A sinusoidal disturbance was injected to one of the torques at the first torsional natural frequency. One sees that applying the torsional damping to the drive system attenuates the peak of the first 3 torsional natural frequencies.

(73) While the invention has been illustrated and described in detail in the drawings and foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive; the invention is not limited to the disclosed embodiments. Other variations to the disclosed embodiments can be understood and effected by those skilled in the art and practising the claimed invention, from a study of the drawings, the disclosure, and the appended claims. In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. A single processor or controller or other unit may fulfil the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims should not be construed as limiting the scope.

LIST OF REFERENCE SYMBOLS

(74) 10 drive system

(75) 12 electrical machine

(76) 14 rotating component

(77) 16 gearbox

(78) 18 shaft

(79) 20 shaft

(80) 22 shaft

(81) 24 converter

(82) 28 controller/control system

(83) 30 drive system model

(84) 32 speed controller

(85) 34 speed reference

(86) 36 measurements

(87) 37 converter controller

(88) 38 switching states

(89) 40 torsional damping unit

(90) 42 angular speed estimator

(91) J.sub.i inertia

(92) k.sub.ij stiffness constant of coupling element

(93) θ.sub.i angular speed

(94) Δ{circumflex over ({dot over (Θ)})} angular speed difference

(95) T.sub.ref reference torque

(96) T.sub.damp damping torque

(97) T adapted reference torque