Current source converter with dynamical firing angle determination

Abstract

A thyristor bridge of an electrical converter is connected to at least one DC link and including at least one phase leg for each output phase and each phase leg being composed of two series-connected thyristor arms. The thyristor arms of a thyristor bridge are cyclically switched by: determining an upper bound for a firing angle of a thyristor arm, wherein the upper bound is determined from voltage and current measurements; and determining a firing angle for the thyristor bridge, which firing angle determines a switching time of the thyristor arm, wherein the firing angle is determined, such that it is less or equal to the upper bound.

Claims

1. A method for switching a thyristor bridge of an electrical converter, the thyristor bridge being connected to at least one DC link and comprising at least one phase leg for each output phase of the electrical converter and each phase leg being composed of two series-connected thyristor arms, wherein the thyristor arms of the thyristor bridge are cyclically switched by the method comprising: determining an upper bound for a firing angle of the thyristor bridge, wherein the upper bound is determined from voltage and current measurements; determining the firing angle for the thyristor bridge, which firing angle determines a switching time of the thyristor arms, wherein the firing angle is determined, such that it is less or equal to the upper bound; wherein the upper bound for the firing angle is determined, such that the upper bound together with a time window, which is at least a sum of a commutation time window and a recovery time window of a thyristor arm of the thyristor arms, is less or equal to 180° and that the time window is less or equal 360° divided by a number of cyclically switched thyristor arms; wherein the firing angle is determined based on model predictive control, by: receiving a reference for a DC link current and/or a drive torque; predicting future states of the electrical converter as a function of future inputs with a mathematical model of the electrical converter, wherein the future inputs comprise future firing angles of the thyristor arm; determining the firing angle by minimizing an objective function, which is a function of the reference, the future states and/or the future inputs.

2. The method of claim 1, wherein the commutation time window and/or the recovery time window are dependent on the firing angle; wherein the upper bound is maximized, such that the upper bound together with the time window is less or equal to 180° and that the time window is less or equal 360° divided by the number of cyclically switched thyristor arms.

3. The method of claim 2, wherein the commutation time window depends from at least one of a phase-to-phase voltage between output phases of the electrical converter, a DC link current and the firing angle; and/or wherein the recovery time window depends on at least one of a change in a DC link current, a change of a voltage applied to the thyristor arm during switching, and/or a thyristor junction temperature.

4. The method of claim 1, wherein the firing angle for a thyristor arm is determined by setting the firing angle to the determined upper bound for the firing angle.

5. The method of claim 1, which further comprises determining an unbounded firing angle based on measurements in the electrical converter, wherein the unbounded firing angle is determined independently from the upper bound; setting the firing angle to a minimum of the unbounded firing angle and the upper bound.

6. The method of claim 1, which further comprises determining the firing angle based on measurements in the electrical converter and the upper bound; wherein the upper bound is a constraint for determining the firing angle.

7. The method of claim 1, wherein the objective function is minimized with a quadratic programming solver; wherein the upper bound for the firing angle is used as a constraint for the quadratic programming solver.

8. The method of claim 1, wherein after determination of the upper bound, the upper bound is reduced by a safety margin.

9. The method of claim 1, wherein phase voltages of the electrical converter are measured and a duration of a commutation time window for a thyristor arm is determined from changes in the phase voltages; wherein the upper bound for the firing angle is determined based on the determined commutation time window.

10. The method of claim 1, wherein phase voltages of the electrical converter are measured and a commutation inductance is determined from the measured phase voltages; wherein the upper bound for the firing angle is determined based on the determined commutation inductance.

11. The method of claim 2, wherein the firing angle for a thyristor arm is determined by setting the firing angle to the determined upper bound for the firing angle.

12. The method of claim 2, which further comprises determining an unbounded firing angle based on measurements in the electrical converter, wherein the unbounded firing angle is determined independently from the upper bound; setting the firing angle to a minimum of the unbounded firing angle and the upper bound.

13. The method of claim 2, which further comprises determining the firing angle based on measurements in the electrical converter and the upper bound; wherein the upper bound is a constraint for determining the firing angle.

14. The method of claim 2, wherein the objective function is minimized with a quadratic programming solver; wherein the upper bound for the firing angle is used as a constraint for the quadratic programming solver.

15. A controller for an electrical converter, wherein the controller is adapted for switching a thyristor bridge of the electrical converter, the thyristor bridge being connected to at least one DC link and comprising at least one phase leg for each output phase of the electrical converter and each phase leg being composed of two series-connected thyristor arms, wherein the thyristor arms of the thyristor bridge are cyclically switched by the controller operable to: determine an upper bound for a firing angle of the thyristor bridge, wherein the upper bound is determined from voltage and current measurements; determine the firing angle for the thyristor bridge, which firing angle determines a switching time of the thyristor arms, wherein the firing angle is determined, such that it is less or equal to the upper bound: wherein the upper bound for the firing angle is determined, such that the upper bound together with a time window, which is at least a sum of a commutation time window and a recovery time window of a thyristor arm of the thyristor arms, is less or equal to 180° and that the time window is less or equal 360° divided by a number of cyclically switched thyristor arms; wherein the firing angle is determined based on model predictive control, by: receiving a reference for a DC link current and/or a drive torque; predicting future states of the electrical converter as a function of future inputs with a mathematical model of the electrical converter, wherein the future inputs comprise future firing angles of the thyristor arm; determining the firing angle by minimizing an objective function, which is a function of the reference, the future states and/or the future inputs.

16. The controller of claim 15, comprising: an upper bound determination stage adapted for determining the upper bound based on measurements in the electrical converter; a firing angle determination stage adapted for determining the firing angle based on measurements in the electrical converter.

17. An electrical drive system, comprising: a line side bridge for rectifying an input multi-phase current; a machine side bridge for generating an output multi-phase current; at least one DC link interconnecting the line side bridge and the machine side bridge; a controller according to claim 15; wherein the line side bridge and/or the machine side bridge are a thyristor bridge; wherein the controller is adapted for controlling the line side bridge and/or the machine side bridge.

18. The electrical drive system of claim 17, wherein the line side bridge and/or the machine side bridge comprise two or more three-phase bridges.

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 an electrical drive system according to an embodiment of the invention.

(3) FIGS. 2A and 2B show diagrams with phase-to-phase voltages of the electrical drive system of FIG. 1.

(4) FIG. 3 schematically shows a controller according to an embodiment of the invention.

(5) FIG. 4 shows an equivalence circuit for the commutation from one phase to another phase of the electrical drive system of FIG. 1.

(6) 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

(7) System Overview

(8) FIG. 1 shows an electrical drive system 10 comprising an electrical converter 12 for converting a multi-phase AC current from an electrical grid 16 into a multi-phase current to be supplied to an electrical machine 14. It may be that the power flow through the converter 12 is from the grid to the electrical machine 14 or vice versa from the electrical machine 14 to the grid 16.

(9) The load-commutated converter 12 is connected to the grid 16 via a transformer 18, which transforms a three-phase current from the grid 16 into a current with two pairs of three phases. On the machine side, the converter 12 also generates a current with two pairs of phases, which are supplied to the double-winding synchronous machine 14.

(10) The rotor of the electrical machine 14 is excited by means of an excitation system 20 and/or is attached to a drive shaft 24 and some machinery 26, such as a pump or turbine.

(11) The converter 12 comprises a line side bridge 28, a DC link 30 with inductances 32 and a machine side bridge 34. Both the line side bridge 28 and the machine side bridge 34 are in a so-called 12-pulse configuration, meaning that they each consist of two three-phase bridges 36 with six thyristor arms 38. For the line side bridge 28 and the machine side bridge 34, the first and second three-phase bridges are indicated as 36a and 36b.

(12) Each three-phase bridge 36 comprises three phase legs 40, each of which comprises an upper arm 42a and a lower arm 42b in the form of a thyristor arm 38. The phase legs 40 are connected in parallel at the side of the DC link 30 and provide a midpoint between the two arms 42a, 42b to which the corresponding phase of an AC current is connected.

(13) It has to be noted that here and in the following one thyristor arm 38 may comprise one, two or more thyristors that are connected in series and/or in parallel and that are switched simultaneously.

(14) The positive sides of the two three-phase bridges 36a, 36b of the line side bridge 28 are connected via inductors 32 of the DC link 30 with the positive sides of the two three-phase bridges 36a, 36b of the bridge 34. The negative sides of the two three-phase bridges 36a, 36b of the line side bridge 28 are connected crosswise with the negative sides of the two multi-phase bridges 36a, 36b of the bridge 34, such that a negative side is connected with the other one of the three-phase bridges as the positive side.

(15) The double-winding synchronous machine 14 has two sets of three-phase windings, mounted at a phase difference of 30° towards each other. Without loss of generality, it is assumed in the following, that the stator voltage applied to the multi-phase bridge 36b lags the stator voltage at the multi-phase bridge 36a by 30°.

(16) While the system 10 shown in FIG. 1 comprises a converter 12 with 12-pulse configuration, other configurations are also possible, such as a 6-pulse, 18-pulse or 24-pulse configuration. For example, the converter 12 of FIG. 1 may comprise one, two, three or more multi-phase bridges 36 on the line side and/or the machine side. The pulse number may be calculated by the number of multi-phase bridges 36 multiplied with 6.

(17) In the configuration shown in FIG. 1, the stator voltage from the three-phase bridge 36b may lag the stator voltage from the three-phase bridge 36a by 30 degrees. By operating two three-phase thyristor bridges 36a, 36b phase-shifted by 30 degrees, the harmonics in the drive torque may be reduced. The name 12-pulse configuration stems from the fact that for each AC voltage period on an AC side of the converter 12, each thyristor arm 38 is fired once, resulting in twelve pulses per period. For the three-phase bridges 36 of the machine side bridge 34, the thyristor arms 38 are additionally indicated as Rp, Sp, Tp, Rn, Sn, Tn connecting the phases R, S and T to the positive and negative potential of the DC link 30. The thyristor arms 38 of the three-phase bridges 36 of the line side bridge 28 may be named analogously.

(18) FIG. 2A shows a machine side phase-to-phase voltage U. FIG. 2B shows a section of FIG. 2A in more detail. The voltage U is provided in V. In FIG. 2A, the voltage U is depicted via the phase angle of one period of 360°, i.e. over one rotation of the rotor. FIG. 2 shows the section between 100° and 200°.

(19) FIGS. 2A and 2B show a commutation time window 42, due to a commutation between the phases for which the phase-to-phase voltage is shown. The commutation is due to a firing of a thyristor arm 38 at a firing angle 44. In this example, the firing angle for the thyristor arm 38 is 145° degrees, when both phases are short-circuited until the current conduction is transferred from one phase to the next. This process is called commutation, and the commutation time window 42 may be the time window, when the phase-to-phase voltage U is substantially zero. The length of the commutation window 42 may also be denoted as commutation length, commutation time or commutation angle.

(20) The other indentions in the voltage U are due to commutations between other phases. However, during one period, one thyristor arm may be fired only once and therefore, there may be only one commutation time window 42 per period.

(21) Directly after the commutation time window 42, the turn-off or recovery time window 46 starts: The turned-off thyristor arm 38 requires some recovery time with a negative thyristor voltage (which corresponds to a positive phase-to-phase voltage), in order to regain its blocking state. In other words, both the commutation time window 42 and the recovery time window 44 must be finished before the phase of the stator voltage becomes negative. The length of the recovery time window 46 may also be denoted as turn-off time, turn-off angle recovery time or recovery angle.

(22) The lengths of the commutation time window 42 and of the recovery time window 46 depend on the current state of the electrical drive system 10, in particular on the phase voltages in the phases R, S, T, the DC link current, a junction temperature of one or more thyristors of the thyristor arm 38 and/or the firing angle 44.

(23) Control System

(24) FIG. 3 shows an electrical drive system 10 with an electrical converter 12, with a line side bridge 28 and a machine side bridge 34 interconnected by a DC link 30. In FIG. 3, only two three-phase bridges 36 are shown. However, the electrical converter 12 may be designed like the one shown in FIG. 1.

(25) FIG. 3 furthermore shows a controller 48 with several control modules and/or control stages for controlling the electrical drive system 10. The modules and/or stages may be implemented in one control unit or in several control units, i.e. in the same or different hardware devices.

(26) A speed controller stage 50 receives the actual speed 52 of the electrical machine 14 and a reference speed 54 and determines a reference 56 for a firing angle determination stage 58. For example, the reference 56 may be a reference torque and/or a reference DC link current. The actual speed 52 is also supplied to an excitation controller stage 60 that controls the excitation system 20 of the electrical machine 14. In addition, measurement values of phase voltages 62 measured in the output phases of the converter 12 are supplied to the excitation controller stage 60.

(27) The reference 56 of the speed controller 50 is supplied to a firing angle determination stage 64, which determines firing angles 44 for the thyristor bridges 28, 34. The firing angles 44 are supplied to firing logic stages 68, which determine firing signals 70 from the firing angles. In addition, voltage measurements of phase voltages 62 in the output phases or input phases of the thyristor bridges 28, 34 are supplied to the firing logic stages 68, which voltage measurements are also used for determining the firing signals 70. The firing signals 70 may comprise firing instances, i.e. times, when the respective thyristor arms 38 are to be fired.

(28) An upper bound determination stage 72 receives current measurements of a DC link current 74, which are also supplied to the firing angle determination stage 64. The upper bound determination stage 72 determines an upper bound 76 of the firing angle 44. For example, the upper bound 76 is determined, such that it permits the commutation and the recovery of the thyristor arms 38 without misfiring. Based on the current operating point of the converter 12, a maximal possible line side and/or machine side firing angle, i.e. an upper bound 76 for the respective firing angle 44, is determined. It has to be noted that the upper bound 76, determined for the line side bridge 28, usually is different from an upper bound determined for the machine side bridge 34. In this case, the upper bound determination stage 72 determines two different upper bounds.

(29) The upper bound 76 is supplied to the firing angle determination stage 64 to support the determination of the firing angle 44. For example, the upper bound 76 may be used as a constraint, when the firing angle determination stage 64 is implemented based on model predictive control.

(30) It has to be noted that the voltage measurements 62 of the phase voltage in the input and/or output phases also may be supplied to the firing angle determination stage 64 and/or the upper bound determination stage 72.

(31) Alternatively, the firing angle determination stage 64 may be omitted and the upper bound(s) 76 (for the line side and/or the machine side) may be supplied directly to the firing logic stage(s) 68. As a further alternative, the firing angle(s) 44 may be determined independently from the upper bound(s) 76 and a minimum of the firing angle(s) 44 and the respective upper bound 76 may be supplied to the firing logic stage(s) 68.

(32) Control Method

(33) In the following, a control method for the electrical drive system shown in FIG. 1 will be described. The method may be performed by the controller shown in FIG. 3.

(34) In general, the thyristor arms 38 of a thyristor bridge 28, 34 are cyclically switched by the respective firing logic stage 68 of the line side bridge 28 or the machine side bridge 34. Both firing logic stages 68 are provided with a firing angle 44, which determines the switching signals or switching time instants 70 of the thyristor arms. The firing angle 44 may be provided relative to a phase angle of the respective output phase, i.e. the AC phases of the line side bridge 28 or the AC phases of the machine side bridge 34.

(35) It may be that a new value for the firing angle 44 is determined every time instant a thyristor arm 38 is switched, or for longer time intervals. For example, the control stages 64 and 68 may operate asynchronously.

(36) Upper Bound Determination

(37) In general, the upper bound determination stage 72 determines upper bounds 76 for firing angles 66 of the thyristor arms 38 of the line side and/or machine side bridges 28, 34. The upper bound 76 may be determined from voltage and current measurements 74, 62 in the electrical converter 12.

(38) An objective of the upper bound determination stage 72 may be to calculate, for a current operating point of the electrical converter 12, an upper bound 76 for the firing angle 44, such that a misfiring of the thyristor arms is avoided. The upper bound 76 of the firing angle 44 may be determined by solving the following optimization problem:

(39) $\underset{\stackrel{\_}{\vartheta }}{\max }\stackrel{\_}{\vartheta }$$\begin{array}{cc}s.t.& \stackrel{\_}{\vartheta }+t_c+t_r\le \pi \\ & t_c+t_r\le 2\pi /n\end{array}$

(40) Here, ϑ denotes the upper bound 76 on the firing angle 44, t.sub.c the duration of the commutation time window 42 and t.sub.r the duration of the recovery time window 46 (all in radians, i.e. 180°=π). Below it will be described that both t.sub.c and t.sub.r are functions depending on the firing angle 44. n is the number of cyclically switched thyristor arms, for example n=12 for one of the thyristor bridges 28, 34 shown in FIG. 1. In general, every three-phase bridge 36 accounts for 6 thyristor arms, such that n=6m, when in is the number of three-phase bridges 36.

(41) The constraints ensure that both commutation time window 52 and recovery time window 46 take place before the concerned phase-to-phase voltage (see FIGS. 2A and 2B) becomes negative, either because of reaching 180° degrees or because of the subsequent firing of the next thyristor arm 38.

(42) In summary, the upper bound ϑ, 76 for the firing angle 44 is maximized, such that the upper bound 76 together with a time window 42, 46, i.e. t.sub.c+t.sub.r, depending on the firing angle 44 is less or equal to 180° and that the time window 42, 56 is less or equal 360° divided by the number of cyclically switched thyristor arms 38.

(43) The time window t.sub.c+t.sub.r may be the sum of the commutation time window t.sub.c, 42 and the recovery time window t.sub.r, 46 for a thyristor arm 38.

(44) In order to calculate the upper bound ϑ, 76, the length of the commutation time window 42 and the recovery time window 46 of the thyristor arms 38 may need to be calculated and/or may be provided as formulas.

(45) For example, the calculation may be based on an equivalence model for the two commutated phases, such as shown in FIG. 4. Here, the synchronous electrical machine 14 is modeled as an AC voltage source and a commutation inductance 78 for each phase. As example, two thyristor arms 38 are denoted as Rp and Sp. During the commutation, both thyristor arms Rp, Sp are conducting and the length t.sub.c of the commutation time window 42 is the time the voltage source 14 requires to reduce the current in one phase to zero, while ramping up the current in the other phase to the DC current.

(46) From this equivalence model, an equation for length t.sub.c of the commutation time window 42 can be derived:

(47) $t_c=\arccos \left(-\frac{4\pi f\mathrm{LI}}{\sqrt{2}\stackrel{.Math.}{U}}+\cos \stackrel{\_}{\vartheta }\right)-\stackrel{\_}{\vartheta }$

(48) In this equation, f denotes the rotor frequency (or grid frequency), L the commutation inductance 78 of each phase, I the DC link current, and Ū the root mean square value of the phase-to-phase voltage.

(49) Additionally or alternatively, it may be that phase voltages 62 of the electrical converter 12 are measured and the duration t.sub.c of the commutation time window 42 for a thyristor arm 38 is determined from changes in the phase voltages 62.

(50) Furthermore, the commutation inductance L, 78 may be determined from the measured phase voltages 62.

(51) The length t.sub.r of the recovery time window 42 may be determined by a number of contributing factors, such as the rate of change dI/dt of the DC current during the commutation, the rate of change dv/dt of the voltage applied to the thyristor arm 38 during the switching process, or the junction temperature Tj of the thyristor. More formally, the relation can be stated as

(52) $t_r=F\left(\frac{\mathrm{dI}}{\mathrm{dt}},\frac{\mathrm{dv}}{\mathrm{dt}},T_j\right)$

(53) The precise relation F may depend on the used type of thyristors. The rate of change dI/dt of the DC current during the commutation may depend on the operating point of the converter 12, and may be stated as

(54) $\frac{\mathrm{dI}}{\mathrm{dt}}=\frac{\sqrt{2}\stackrel{\_}{U}\sin \vartheta }{2L}$
where ϑ denotes the firing angle 44.

(55) A snubber circuit may be connected in parallel to a thyristor arm 38, limiting the rate of change of the thyristor voltage dv/dt.

(56) Summarized, the commutation time window 42 and/or the recovery time window 46 may depend on the firing angle 44. The commutation time window 42 may depend from at least one of a phase-to-phase voltage between output phases of the electrical converter 12, a DC link current and the firing angle 44. The recovery time window 46 may depend on at least one of a change in a DC link current, a change of a voltage applied to the thyristor arm 38 during switching, and/or a thyristor junction temperature.

(57) The temperature at the junction Tj of the thyristor arm 38 is typically not measured, however, if temperature measurements at the vicinity of the thyristor arm 38 are available, the junction temperature Tj may be estimated depending on the operating point of the converter 12, i.e. the DC current, the firing angle, the stator voltage and the rotor speed.

(58) In the end, after determination, the upper bound ϑ, 76 may be reduced by a safety margin. This may be a constant value that is subtracted from the upper bound ϑ, 76.

(59) Firing Angle Determination

(60) In general, the firing angle 44 for the thyristor arm 38 is determined, such that it is less or equal to the upper bound 76. There are several possibilities, how the firing angle 44 is determined with the aid of the upper bound 76.

(61) In a first embodiment, the firing angle 44 for a thyristor arm 38 is determined by setting the firing angle 44 to the determined upper bound 76 for the firing angle 44. In this embodiment, the stage 64 of the controller 48 is omitted.

(62) In a second embodiment, an unbounded firing angle is based on the measurements 62, 74 in the electrical converter 12, wherein the unbounded firing angle is determined independently from the upper bound 76. In this embodiment, the upper bound 76 is not input to the controller stage 64, but the firing angle 44, which is applied to the firing logic stage 68, may be set to the minimum of the unbounded firing angle and the upper bound 76.

(63) In a third embodiment, the firing angle 44 is determined based on measurements 62, 74 in the electrical converter 12 and the upper bound 76: In this case, the upper bound 76 may be input to the controller stage 64 and/or the upper bound 76 may be a constraint for determining the firing angle 44.

(64) In the following, a firing angle determination stage 76 according to the third embodiment will be described.

(65) Model Predictive Control Firing Angle Determination

(66) The firing angle determination stage 64 may comprise a current controller to determine the firing angle 44 less or equal to the upper bound 76 provided by the upper bound determination stage 72. In model predictive control, a mathematical model of the converter 12 is employed to predict the evolution of its states, i.e. its future states, such as the DC link current over a finite time horizon as a function of the firing angles 44.

(67) This prediction is formulated as an optimization problem with the objectives to minimize the deviation of the actual DC current from its reference 56, and/or the deviation of the actual drive torque from its reference 56. Other objectives may be the maximization of the power factor of the electrical machine 14 and/or the smooth change of firing angles 44. The objectives may be encoded into an objective function, which has to be minimized to achieve the goals. The objectives also may be weighted, while the weights of these objectives may determine a prioritization between those goals.

(68) In the following, a possible embodiment of a possible model predictive control method is described, which may be performed by the firing angle determination stage 64. However, other implementations also may be possible.

(69) The model predictive control may determine the firing angles 44 of both line side bridge 28 and machine side bridge 34. The control actions of line side bridge 28 and machine side bridge 34 may thus be determined by the same controller stage 64, allowing a systematic coordination of control actions.

(70) The mathematical prediction model may be stated in ordinary differential equations of the form
x(k+1)=ƒ(x(k),u(k))
y(k)=g(x(k),u(k))

(71) where k represents the discrete time (instants); x(k) represents the state of the converter 12 at time k; u(k) its inputs at time k such as firing angles 44; and y(k) represents the measurable outputs of the converter 12 and the synchronous electrical machine 14 at time k, such as voltages, currents or the rotational speed. While ƒ(.) is typically a non-linear function describing the dynamic behavior of the converter 12, g(.) is typically a non-linear function describing how the outputs depend on the states and inputs of the converter 12.

(72) The functions ƒ(.) and g(.) have to be known a-priori, and they may be different (structural and parameter differences are possible) for each electric drive configuration. Therefore, a modeling procedure may have to be executed before operation of the proposed controller 48.

(73) The model predictive control may be performed by a collection of software routines on a real-time computing platform. The collection of software routines may include a non-linear mathematical prediction model of the drive system 10, a quadratic programming (QP) preparation algorithm, and a quadratic programming (QP) solver.

(74) Note that in this section only one possible embodiment is described, whereas a number of alternative embodiments exist. For instance, the non-linear optimization problem may be solved directly by means of a non-linear problem solver, or the derivation of the quadratic programming from the non-linear problem may be achieved by linearizing the mathematical prediction model at each sampling instance. Instead of solving the full non-linear problem, one may alternatively also solve a simplified problem, which still contains non-linearities such as bilinear terms. Furthermore, one may also decide to not solve the optimization problem online, but to follow a so-called explicit model predictive control approach, in which the optimization problem is solved parametrically offline and the online procedure is reduced to an evaluation of the solution for the current state estimate.

(75) Before execution, the software routines may need to be prepared by means of an initialization phase, afterwards the software routines may be run in an on-line phase.

(76) Initialization Phase (Offline)

(77) In the initialization phase, the model predictive control is prepared for application to the drive system 10. This preparation may comprise two steps: 1. providing a dynamic model of the electric drive system 10, and 2. selecting the objectives and constraints for the electrical drive system 10.

(78) Solving the Constrained Optimal Control Problem (Online)

(79) After initialization, the software routines may be executed periodically on-line in the controller stage 64, such as a real-time computing platform, for example every few hundred microseconds. The software routines may be carried out in the following sequence:

(80) Step 1: On-Line Integration and Formulation of a Quadratic Programming (QP) Sub-Problem

(81) The first step linearizes the constrained finite-time optimal control problem to arrive at a sub-problem in form of a quadratic programming (QP), which is easier to solve than the original problem. For doing so, the non-linear model is integrated along the prediction starting at the current estimates of the initial state to determine future states. In addition, first-order derivatives of the predicted future states with respect to the initial state as well as the control inputs are determined.

(82) This integration of the non-linear model and the computation of the first-order derivatives delivers a linearized formulation of the original model:
ξ(k)=A.sub.kξ(k)+B.sub.ku(k)+ƒ.sub.k
η(k)=C.sub.kξ(k)+D.sub.ku(k)+g.sub.k

(83) Based on this linearized model formulation, a linearized version of the model predictive control problem can be formulated, which is equivalent to a convex quadratic programming (QP):
minimize (z−r).sup.tQ(z−r) with respect to z
subject to G.sub.inz<=b.sub.in and G.sub.eqz=b.sub.eq.

(84) Therein, the states ξ(k) and control input u(k) are gathered at all time instants along the prediction horizon (of length p) within a vector
z=[ξ(k),u(k),ξ(k+1),u(k+1), . . . ,ξ(k+p−1),u(k+p−1),ξ(k+p)]
and the reference values for the future states and control inputs within a vector
r=[ξref(k),uref(k),ξref(k+1),uref(k+1), . . . ,ξref(k+p−1),uref(k+p−1),ξref(k+p)].

(85) The quadratic objective function (z−r).sup.tQ(z−r) penalizes the deviation of the predicted states and inputs z from their reference r, where Q is a positive semidefinite quadratic weight matrix used to tune the model predictive control.

(86) The linearized model equations are incorporated into the quadratic programming (QP) problem by means of the equality constraints G.sub.in z<=b.sub.in.Math.G.sub.eq and b.sub.eq may represent the linearized state-update equations over the whole prediction horizon in a compact form. Physical and/or desired limitations on the states and control inputs may be incorporated by means of the inequality constraints given by G.sub.eg and b.sub.eg.

(87) Note that the above quadratic programming (QP) formulation may be adapted to incorporate the predicted measurable outputs η(k) instead of the predicted states ξ(k).

(88) Note further that many variations and extensions of the above-mentioned quadratic programming (QP) formulation may be provided. For example, soft constraints to avoid non-feasibilities of the optimization problem, move blocking and multiplex model predictive control may be used to reduce the size of the optimization problem.

(89) Moreover, the objective function may comprise a linear objective term of the form gT(z−r), where the gradient g is an additional tuning parameter to provide more flexibility in penalizing set-point deviations. Furthermore, instead of providing set points for the control inputs, one can also use a so-called δu-formulation, and minimize the difference between successive control inputs instead of their deviation from a given steady-state value.

(90) Finally note that the quadratic programming (QP) problem formulation may exhibit a special sparsity structure, as it comprises optimization variables for both states and control inputs. One way to exploit this structure in terms of computational efficiency may be to employ a sparse quadratic programming (QP) solver. Another approach is to use the linearized state-update equations to remove all but the initial state ξ(k) from the vector z and thus from the quadratic programming (QP) problem formulation. This may lead to a smaller-scale, but dense quadratic programming (QP) problem. This second approach may be more efficient on short prediction horizons.

(91) Step 2: Solving the Quadratic Programming (QP) Sub-Problem and Implementation of Control Action

(92) Solving the convex quadratic programming (QP) as described in step 1 (in either form) may be done in a fast and reliable way using existing quadratic programming (QP) solvers. A quadratic programming (QP) solver may be able to solve small-scale quadratic programming (QP) problems in the range of milliseconds or less. By doing so, a possibly approximately optimal solution z.sub.opt(k) is obtained at time instant k, which comprises the optimized control input action u.sub.opt(k) at time instant k.

(93) Only this first piece u.sub.opt(k) of the optimized control input profile may be implemented in a moving horizon fashion. The optimized firing angle 44 may be distributed as reference values to the firing logic stages 66.

(94) At the next sampling instant, the whole procedure may be repeated starting with step 1.

(95) 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

(96) 10 electrical drive system 12 electrical converter 14 electrical machine 16 electrical grid 18 transformer 20 excitation system 24 drive shaft 26 machinery 28 line side bridge 30 DC link 32 inductances 34 machine side bridge 36 three-phase bridge, twelve-pulse bridge 36a first three-phase bridge, six-pulse bridge 36b second three-phase bridge, six pulse bridge 38 thyristor arm 10 phase leg R phase S phase T phase Rp thyristor of positive arm of phase leg Sp thyristor of positive arm of phase leg Tp thyristor of positive arm of phase leg Rn thyristor of negative arm of phase leg Sn thyristor of negative arm of phase leg Tn thyristor of negative arm of phase leg 42 commutation time window 44 firing angle 46 recovery time window 48 controller 50 speed controller stage 52 actual speed 54 reference speed 56 reference for a firing angle determination stage 58 firing angle determination stage 60 excitation controller stage 62 phase voltage measurements 64 firing angle determination stage 68 firing logic stage 70 firing signals 72 upper bound determination stage 74 DC link current measurements 76 upper bound 78 commutation inductance