Controlling a three-phase electrical converter

Abstract

A method for controlling a three-phase electrical converter comprises: selecting a three-phase optimized pulse pattern from a table of pre-computed optimized pulse patterns based on a reference flux; determining a two-component optimal flux from the optimized pulse pattern and determine a one-component optimal third variable; determining a two-component flux error from a difference of the optimal flux and an estimated flux estimated based on measurements in the electrical converter; determining a one-component third variable error from a difference of the optimal third variable and an estimated third variable; modifying the optimized pulse pattern by time-shifting switching instants of the optimized pulse pattern such that a cost function depending on the time-shifts is minimized, wherein the cost function comprises a flux error term and a third variable error term, wherein the flux error term is based on a difference of the flux error and a flux correction function providing a flux correction based on the time-shifts and the third variable error term is based on a difference of the third variable error and a third variable correction function providing a third variable correction based on the time-shifts; and applying the modified optimized pulse pattern to the electrical converter.

Claims

1. A method for controlling a three-phase electrical converter, the method comprising: selecting a three-phase optimized pulse pattern from a table of precomputed optimized pulse patterns based on a reference flux (.sub.,ref); determining a two-component optimal flux (*.sub.) from the optimized pulse pattern and determine a one-component optimal third variable (*); determining a two-component flux error from a difference of the optimal flux (*.sub.) and an estimated flux (.sub.) estimated based on measurements in the electrical converter; determining a one-component third variable error from a difference of the optimal third variable (*) and an estimated third variable (); modifying the optimized pulse pattern by time-shifting switching instants of the optimized pulse pattern such that a cost function depending on the time-shifts is minimized, wherein the cost function comprises a flux error term and a third variable error term, wherein the flux error term is based on a difference of the flux error and a flux correction function providing a flux correction based on the time-shifts and the third variable error term is based on a difference of the third variable error and a third variable correction function providing a third variable correction based on the time-shifts; applying the modified optimized pulse pattern to the electrical converter.

2. The method of claim 1, wherein the cost function comprises a third term quadratic in the time shifts; and/or wherein the optimal flux, the estimated flux, the flux error, the optimal third variable, the estimated third variable and/or the third variable error are computed over a predefined prediction horizon; and/or wherein the predefined prediction horizon is longer than two or more future time steps and only switching instants of the modified optimized pulse pattern during the next time step are applied to the electrical converter; and/or wherein the switching instants of the optimized pulse pattern are time-shifted within the prediction horizon; and/or wherein only the next future switching instants in every phase are time-shifted.

3. The method of claim 2, wherein the flux correction function and/or the third variable correction function is linear in the time-shifts and/or wherein the cost function is minimized by solving online a quadratic program.

4. The method of claim 2, wherein the third variable is based on a circulating current of a modular multi-level converter; and/or wherein the third variable is a flux determined from the circulating current.

5. The method of claim 4, wherein the optimized pulse patterns have been precalculated such that they generate no common mode voltage and an optimal third variable is directly determined from a circulating current reference.

6. The method of claim 1, wherein the flux correction function and/or the third variable correction function is linear in the time-shifts and/or wherein the cost function is minimized by solving online a quadratic program.

7. The method of claim 6, wherein the third variable is based on a circulating current of a modular multi-level converter; and/or wherein the third variable is a flux determined from the circulating current.

8. The method of claim 7, wherein the optimized pulse patterns have been precalculated such that they generate no common mode voltage and an optimal third variable is directly determined from a circulating current reference.

9. The method of claim 1, wherein the third variable is based on a neutral point potential of a DC link of a neutral point clamped converter.

10. The method of claim 9, wherein the optimal third variable is the optimal neutral point potential determined from the optimized pulse pattern and the estimated third variable is the estimated neutral point potential determined form measurements in the converter.

11. The method of claim 10, wherein a low pass filter is applied to the estimated neutral point potential and/or the estimated neutral point potential is averaged to control a drift of the neutral point potential; wherein higher order contributions to the neutral point potential are offline precomputed for each optimized pulse pattern in the table and added to the estimated neutral point potential to control a drift of the neutral point potential.

12. The method of claim 9, wherein the optimal third variable is a neutral point flux determined from an optimal neutral point potential integrated over time and the estimated third variable is the corresponding estimated neutral point flux determined from measurements in the converter.

13. The method of claim 1, wherein the third variable is based on a circulating current of a modular multi-level converter; and/or wherein the third variable is a flux determined from the circulating current.

14. The method of claim 13, wherein the optimized pulse patterns have been precalculated such that they generate no common mode voltage and an optimal third variable is directly determined from a circulating current reference.

15. The method of claim 1, wherein a Clarke transformed three-component optimal flux is determined from the optimized pulse pattern and split into the two-component optimal flux and into a one-component flux as optimal third variable.

16. The method of claim 1, wherein the flux error term and the third variable term have different weight factors.

17. A controller for an electrical converter structured to execute instructions to control the electrical converter, comprising: select a three-phase optimized pulse pattern from a table of precomputed optimized pulse patterns based on a reference flux (.sub.,ref); determine a two-component optimal flux (*.sub.) from the optimized pulse pattern and determine a one-component optimal third variable (*); determine a two-component flux error from a difference of the optimal flux (*.sub.) and an estimated flux (.sub.) estimated based on measurements in the electrical converter; determine a one-component third variable error from a difference of the optimal third variable (*) and an estimated third variable (); modify the optimized pulse pattern by time-shifting switching instants of the optimized pulse pattern such that a cost function depending on the time-shifts is minimized, wherein the cost function comprises a flux error term and a third variable error term, wherein the flux error term is based on a difference of the flux error and a flux correction function providing a flux correction based on the time-shifts and the third variable error term is based on a difference of the third variable error and a third variable correction function providing a third variable correction based on the time-shifts; apply the modified optimized pulse pattern to the electrical converter.

18. A converter system, comprising: an electrical converter interconnected with an electrical grid; and a controller according to claim 17.

19. The converter system of claim 18, wherein the electrical converter is a neutral point clamped converter adapted for converting a three-phase AC current into a DC current in a split DC link.

20. The converter system of claim 18, wherein the electrical converter is a modular multi-level converter comprising at least two converter branches and wherein each converter branch comprises a plurality of converter cells with own cell capacity.

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 converter system according to an embodiment of the invention.

(3) FIG. 2 shows a flow diagram illustrating a method for controlling an electrical converter according to an embodiment of the invention.

(4) FIG. 3 shows a diagram with an optimized pulse pattern used in the method of FIG. 2.

(5) FIG. 4 shows a diagram illustrating shifted time instants as used in the method of FIG. 2.

(6) FIG. 5 schematically shows a neutral point clamped converter for a system according to an embodiment of the invention.

(7) FIG. 6 schematically shows a modular multi-level converter for a system according to an embodiment of the invention.

(8) FIG. 7 schematically shows a converter cell for the converter of FIG. 6.

(9) FIG. 8 shows a diagram with an optimized pulse pattern that may be used with the converter of FIG. 6.

(10) FIG. 9 schematically shows a controller according to an embodiment of the invention.

(11) FIG. 10 schematically shows a controller according to a further embodiment of the invention.

(12) 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

(13) FIG. 1 shows a converter system 10 comprising an electrical converter 12, which is connected to a three-phase electrical grid 14. For example, the electrical converter 12 may be adapted for converting an AC voltage from the grid 14 into a DC current or the electrical converter may supply the electrical grid with electrical energy from another grid.

(14) The converter system 10 and in particular the electrical converter 12 is controlled by a controller 16, which receives measurement values obtained in the system 10 and which generates switching commands for switching the converter 12. In particular, the controller 16, in pattern selector block 18 receives a reference flux .sub.,ref and selects an optimized pulse pattern 20 based on the reference flux .sub.,ref from a table 22. Furthermore, the pattern selector 18 determines an optimal flux *.sub. from the optimized pulse pattern and calculates a flux error from by subtracting an estimated flux .sub., which has been determined from measurements in the system 10, from the optimal flux *.sub..

(15) The pattern optimizer block 24 receives the optimized pulse pattern 20 and the flux error and furthermore receives a third variable error based on a difference of an optimal third variable * and an estimate for the third variable, also based on measurements in the system 10. From these inputs, the pattern optimizer 26 modifies time instants of the optimized pulse pattern 20 with respect to the optimization goal to minimize/compensate the errors and generates a modified optimized pulse pattern 26.

(16) General Overview of the Method

(17) FIG. 2 shows a flow diagram illustrating a method for controlling the converter 12, which may be performed by the controller 16.

(18) In step S10, based on the converter reference flux .sub.,ref, a reference flux angle *, modulation index m* and pulse number d* are determined and based on the modulation index and pulse number an appropriate optimized pulse pattern 20 is selected. Based on the flux angle *, the optimized pulse pattern 20 is read out such that it corresponds to the current time instant of a fundamental cycle/period of the voltage phases of the converter 12. The table 22 may provide the optimal flux *.sub. and the optimal switch positions in all three phases u.sub.abc. Based on the optimal switch positions, the optimal third variable, *, may also be computed.

(19) FIG. 3 shows an example of an optimized pulse pattern 20. The optimized pulse patterns 20 in the table 22 corresponding to different modulation indices have been computed offline. The optimized pulse patterns 20 may be described by the switching instances [.sub.sw,1 . . . .sub.sw,d].sup.T (reference numeral 28 indicates example) and change of switch positions at switching instances [u.sub.i . . . u.sub.d]. Here, d* denotes a pulse number, which is equal to a number of switching events that occur in a quarter-wave, and u.sub.i is defined as u.sub.i=u(t.sub.i)u(t.sub.idt), where dt denotes an infinitesimal time step.

(20) The optimized pulse patterns 20 may be defined on one quarter of the fundamental period, while the rest of the optimized pulse patterns 20 may be obtained by applying the quarter-wave symmetry rules.

(21) The modulation patterns for the three-phase system 10, characterized by switching instances t*.sub.ai,t*.sub.bi,t*.sub.ci,i=1, . . . d, are obtained by shifting the optimized pulse patterns 20 relatively to each other by one third of the fundamental period.

(22) By integrating the optimized pulse pattern 20 and transforming the obtained flux into apt coordinates, the optimal flux *.sub. is obtained. This map may be prestored (for example together with the optimized pulse pattern 20 in the table 22) or computed online.

(23) Any three-phase vector quantity .sub.abc=[.sub.a.sub.b.sub.c].sup.T may be transformed to .sub.=[.sub..sub.].sup.T in the stationary orthogonal coordinates with a two-dimensional Clarke transformation, as follows:
.sub.=P.sub.abc(1)

(24) where the matrix

(25) $P=\frac{2}{3}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\end{array}\right].$
In particular, the converter flux (at the machine/inverter side or the grid/rectifier side) is defined as integral of a three-phase voltage at the AC connection of the converter 12, v.sub.abc:

(26) $\begin{array}{cc}{}_{\mathrm{abc}}\left(t\right)={}_0^t{v}_{\mathrm{abc}}\left(\right)d.& \left(2\right)\end{array}$

(27) It follows that the converter flux in the coordinates is defined as:
.sub.=P.sub.abc.(3)

(28) In addition, the flux of a converter, denoted .sub., may be defined as integral of the common mode converter voltage:

(29) $\begin{array}{cc}{}_{}\left(t\right)={}_0^t{v}_{}\left(\right)d=\frac{1}{3}{}_0^t\left(v_a\left(\right)+v_b\left(\right)+v_c\left(\right)\right)d& \left(4\right)\end{array}$

(30) The optimal third variable is a function of pulse pattern switching times and, possibly, of other converter and modulation variables.

(31) Returning to FIG. 2, in step S12 the flux error {tilde over ()}.sub. in the system is determined by
{tilde over ()}.sub.=*.sub..sub.,(5)

(32) where *.sub. is the optimal flux and .sub. is the estimated flux.

(33) In step S14, the third variable error {tilde over ()} is determined by
{tilde over ()}=*,(6)

(34) where denotes the third variable measurement/estimate and * denotes the optimal third variable.

(35) In step S16, the optimized pulse pattern 20 is modified by time-shifting switching instants 28. FIG. 4 shows an example of an optimized pulse pattern 22 that has been determined over a specific prediction horizon 30. The diagram of FIG. 4 depicts the switching instants 28 and voltage levels u.sub.a, u.sub.b, u.sub.c for three phases a,b,c over time t.

(36) The switching instants 28 are moved (shifted) in time t, such that an objective function J (explained below) is minimized.

(37) The objective function J may penalize both the uncorrected error {tilde over ()}.sub., {tilde over ()} of the controlled variables and the changes of the switching instants 28 (the manipulated variable), using the diagonal weight matrix R, whose components may be very small. Specifically, the optimization problem may be formulated as

(38) $\begin{array}{cc}\underset{t}{\min }J\left(t\right)={.Math.{\stackrel{~}{}}_{}-{}_{,\mathrm{corr}}\left(t\right).Math.}_2^2+.Math.\stackrel{~}{}-{{}_{\mathrm{corr}}(t.Math.}_2^2+t^{T}Rts.t.& \left(7a\right)\\ {\mathrm{kT}}_s{t}_{a1}{t}_{a2}\mathrm{.Math.}{t}_{{\mathrm{an}}_a}{t}_{a\left(n_a+1\right)}^{*}& \left(7b\right)\\ {\mathrm{kT}}_s{t}_{b1}{t}_{b2}\mathrm{.Math.}{t}_{{\mathrm{bn}}_b}{t}_{b\left(n_b+1\right)}^{*}& \left(7c\right)\\ {\mathrm{kT}}_s{t}_{c1}{t}_{c2}\mathrm{.Math.}{t}_{{\mathrm{cn}}_c}{t}_{c\left(n_c+1\right)}^{*}.& \left(7d\right)\end{array}$

(39) Here, , 0, and R.sup.(n.sup.a.sup.+n.sup.b.sup.+n.sup.c.sup.)x(n.sup.a.sup.+n.sup.b.sup.+n.sup.c.sup.), R0, are weighting factors. The corrections of switching instants 28 are aggregated in the vector
t=[t.sub.a1t.sub.a2 . . . t.sub.an.sub.at.sub.b1 . . . t.sub.bn.sub.bt.sub.c1 . . . t.sub.cn.sub.c].sup.T.(8)

(40) For phase a, for example, the correction of the i-th transition time is given by
t.sub.ai=t.sub.ait*.sub.ai,(9)

(41) where t*.sub.ai denotes the nominal switching instant of the i-th transition u.sub.ai. Again, the latter is defined as u.sub.a=u.sub.a(t*.sub.ai)u.sub.a(t*.sub.aidt) with dt being an infinitesimally small time step. Moreover, n.sub.a denotes the number of switching instants 28 in phase a that area within the prediction horizon 30, and t*.sub.a(n.sub.a.sub.+1) refers to the first nominal switching instants 28 beyond the horizon 30. The quantities for phases b and c are defined accordingly.

(42) The switching instants 28 may not be modified arbitrarily. For the three phases, the set of constraints (7b)-(7d) may be imposed, which constraints the switching instants 28 in two ways. Firstly, the current time-instant kT.sub.s, i.e. transitions cannot be moved into the past. Secondly, the neighboring switching instants in the same phase may not be moved beyond each other, ensuring that the correct sequence of switching instants 28 is kept.

(43) For example, in FIG. 4, the first switching instant 28 in phase b is constrained to lie between kT.sub.s and the nominal switching instant of the second transition in phase b,t*.sub.b2. The second switching instant in phase b can only be delayed up to the nominal switching instant of the third transition in the same phase, t*.sub.b3. In this example, the number of transitions that fall within the prediction horizon 30 is n.sub.a=2,n.sub.b=3andn.sub.c=1. Note that the switching instants 28 in a given phase may be modified independently from those in the other phases.

(44) The horizon length T.sub.p is a design parameter. If required, T.sub.p is increased so as to ensure that switching instants in at least two phases fall within the horizon 30. For example, in FIG. 4, in case T.sub.p is smaller than t*.sub.a1kT.sub.s, it is increased to this value.

(45) Returning to FIG. 2, in step S18, switching instants 28 that will occur within the same sampling interval are removed from the modified pulse pattern 26. This may be accomplished by updating a pointer to the table 22 that stores the switching angles of the optimized pulse pattern 22 and the respective three-phase potential values.

(46) In step S20, the switching commands for the semiconductor switches of the converter 12 are derived from the modified pulse pattern 26 over the sampling interval, i.e. the switching instants 28 and the associated switch positions. The switching commands may be applied to the converter 12 by sending them to gate units of the semiconductor switches in the converter 12.

(47) Correction of the Converter Flux Vector

(48) The flux error term
J.sub.1(t)={tilde over ()}.sub..sub.,corr(t).sub.2.sup.2(10)
penalizes the difference between the converter flux error at time t, which is given by (5), and a flux correction function applied to the stator flux over the time interval [tt+T.sub.p]. The difference between these two quantities is the uncorrected stator flux error at t+T.sub.p. This quantity is penalized using a quadratic penalty in (10), where for any vector x:x.sub.2.sup.2=x.sup.Tx.

(49) The converter flux correction function in as a function of the modifications to the switching time instants t may be written as

(50) $\begin{array}{cc}{}_{,\mathrm{corr}}\left(t\right)=-v_{\mathrm{dc}}P\left[\begin{array}{c}\underset{i=1}{\overset{n_a}{.Math.}}{u}_{\mathrm{ai}}{t}_{\mathrm{ai}}\\ \underset{i=1}{\overset{n_b}{.Math.}}{u}_{\mathrm{bi}}{t}_{\mathrm{bi}}\\ \underset{i=1}{\overset{n_c}{.Math.}}{u}_{\mathrm{ci}}{t}_{\mathrm{ci}}\end{array}\right],& \left(11\right)\end{array}$
where v.sub.dc corresponds to the voltage level of one step of the converter. This can be expanded to
.sub.,corr(t)=V.sub.1t,(12)
where

(51) $V_1=-{\frac{v_{\mathrm{dc}}}{3}\left[\begin{array}{cc}2u_{a1}& 0\\ \mathrm{.Math.}& \mathrm{.Math.}\\ 2u_{{\mathrm{an}}_a}& 0\\ -u_{b1}& \sqrt{3}{u}_{b1}\\ \mathrm{.Math.}& \mathrm{.Math.}\\ -u_{{\mathrm{bn}}_b}& \sqrt{3}{u}_{{\mathrm{bn}}_b}\\ -u_{c1}& -\sqrt{3}{u}_{c1}\\ \mathrm{.Math.}& \mathrm{.Math.}\\ -u_{{\mathrm{cn}}_c}& -\sqrt{3}{u}_{{\mathrm{cn}}_c}\end{array}\right]}^T.$

(52) Inequality Constraints

(53) The inequality constraints (7b)-(7d) may be written in matrix form. With the definition (9), it is straightforward to rewrite the constraints (7b) on the switching instants for phase a as

(54) $\begin{array}{cc}{G}_{a}t{g}_a,\mathrm{with}& \left(13\right)\\ G_a=\left[\begin{array}{cccccc}-1& 0& \mathrm{.Math.}& & & \\ 1& -1& 0& \mathrm{.Math.}& & \\ 0& 1& -1& 0& \mathrm{.Math.}& \\ & & & & & \\ & \mathrm{.Math.}& 0& 1& -1& 0\\ & & \mathrm{.Math.}& 0& 1& -1\\ & & & \mathrm{.Math.}& 0& 1\end{array}\right],g_a=\left[\begin{array}{c}{t}_{a1}^{*}\\ t_{a2}^{*}-t_{a1}^{*}\\ t_{a3}^{*}-t_{a2}^{*}\\ \mathrm{.Math.}\\ t_{a\left(n_a-2\right)}^{*}-t_{a\left(n_a-3\right)}^{*}\\ t_{a\left(n_a-1\right)}^{*}-t_{a\left(n_a-2\right)}^{*}\\ t_{{\mathrm{an}}_a}^{*}-t_{a\left(n_a-1\right)}^{*}\end{array}\right].& \left(14\right)\end{array}$

(55) Similarly, the constraints associated with phases b and c, (7c) and (7d), can be expressed by
G.sub.btg.sub.b(15a)
G.sub.ctg.sub.c(15b)

(56) The matrices G.sub.b, G.sub.c and vectors g.sub.b, g.sub.c are defined according to (14).

(57) The single-phase constraints (13) and (15) can be aggregated to

(58) $\begin{array}{cc}Gtg\mathrm{with}& \left(16\right)\\ G=\left[\begin{array}{ccc}{G}_a& 0& 0\\ 0& G_b& 0\\ 0& 0& G_c\end{array}\right]\mathrm{and}g=\left[\begin{array}{c}{g}_a\\ g_b\\ g_c\end{array}\right],& \left(17\right)\end{array}$

(59) where 0 denotes zero matrices of appropriate dimensions.

(60) Control of the Third Variable

(61) The third variable error term is
J.sub.2(t)={tilde over ()}.sub.corr(t).sub.2.sup.2(18)

(62) where the third variable correction function .sub.corr may be assumed to be linear in the time-shifts.
.sub.corr(t)=V.sub.2t(19)

(63) Examples for the third variable , as well as the matrix V.sub.2, will be given below for neutral point potential control and circulating current control.

(64) Optimization Problem

(65) Collecting all the previous derivations, the cost function J may be written

(66) $\begin{array}{cc}J\left(t\right)=J_1\left(t\right)+J_2\left(t\right)+t^{T}Rt=& \left(20a\right)\\ {.Math.\left[\begin{array}{c}{\stackrel{~}{}}_{}-{}_{,\mathrm{corr}}\left(t\right)\\ \stackrel{~}{}-{}_{\mathrm{corr}}\left(t\right)\end{array}\right].Math.}_Q^2+t^{T}Rt,& \left(20b\right)\end{array}$
where x.sub.Q.sup.2=x.sup.TQx and Q=diag([1 1 ]). By inserting (12) and (19) into (20b), the latter equation can be rewritten as
J(t)={tilde over (y)}+vt.sub.Q.sup.2+t.sup.TRt ,(21)
where

(67) 0$\tilde{y}=\left[\begin{array}{c}{\stackrel{~}{}}_{}\\ \stackrel{~}{}\end{array}\right]\mathrm{and}V=\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right].$

(68) Expanding the cost function (21) leads to
J(t)=({tilde over (y)}+Vt).sup.TQ({tilde over (y)}+Vt)+t.sup.TRt(22)
which can be further simplified to
J(t)=t(V.sup.TQV+R)t+2{tilde over (y)}.sup.TQVt+{tilde over (y)}.sup.TQ{tilde over (y)}(23)
by completing the squares. Defining
H=V.sup.TQV+R andc=V.sup.TQ.sup.T{tilde over (y)}(24)
and neglecting the constant offset term {tilde over (y)}.sup.TQ{tilde over (y)} in (23), the cost function J may be written in standard form
J(t)=t.sup.THt+2c.sup.Tt.(25)

(69) Thus, minimizing the cost function (25) subject to the inequality constraints (16) leads to the Quadratic Program (QP)

(70) $\begin{array}{cc}\underset{t}{\min }{t}^{T}Ht+2c^{T}t& \left(26a\right)\\ \mathrm{subj}.\mathrm{to}Gtg.& \left(26b\right)\end{array}$

(71) This Quadratic Program is equivalent to the formulation (7), if the third variable correction function is linear in the time-shifts.

(72) The Quadratic Program may be solved online in the controller 16 or may be solved offline and the algebraic offline solution may be evaluated in the controller 16 to get a solution for the Quadratic Program.

(73) Deadbeat Control

(74) There also is a computationally less demanding version of the optimization problem (7), which may be seen as a deadbeat control method.

(75) In the deadbeat control method, the weight matrix R is set to zero in the cost function J, and the optimization horizon is set as a minimum time interval starting at the current time instant, such that three phases exhibit switching instants.

(76) If the optimization problem is formulated as a Quadratic Program (such as (26a) and (26b) above), the solution of the deadbeat control method may reduce to a simple projection operation.

(77) In the following, two applications of the control method are described and explicit formulas for the third variable are provided.

(78) Control of Neutral Point Potential

(79) FIG. 5 schematically shows a neutral point clamped converter 12a, which may be part of the system 10. The neutral point clamped converter 12a comprises a DC link 32 that is common to the three converter phase circuits 34. The split DC link 32 comprises a positive point 36, a negative point 38 and a neutral point 40, between a positive and a negative DC voltage, v.sub.dc.sup.+ and v.sub.dc.sup. is provided. The voltages v.sub.dc.sup.+ and v.sub.dc.sup. are switched in the circuits 34 via power electronic semiconductor switches.

(80) Each circuit 34 comprises four series-connected semiconductor switches 40, which are connected in parallel to the positive point 36 and the negative point 38 and which provide one output phase in their middle. Between the two upper and the two lower series-connected semiconductor switches 40, a series-connected pair of diodes 42 is connected, which are connected in their middle to the neutral point 40.

(81) A problem that may arise in this topology is that any imbalance in the operation may generate a difference between the currents charging the capacitors C.sub.dc, i.sub.dc.sup.+ and i.sub.dc.sup., which may lead to uneven charging/discharging of the two capacitors. The imbalance may be caused by imperfect switching, differences in the DC capacitors characteristics due to manufacturing tolerances, ageing, inconsistencies in switching device characteristics or imbalances in operation between the three phases. The consequence is a difference between the upper and lower DC link voltages, resulting in a non-zero neutral point potential. The neutral point potential is defined as
v.sub.np=v.sub.dc.sup.+v.sub.dc.sup.,(27)

(82) i.e. as the difference between the voltages of the upper and the lower DC link capacitors C.sub.dc.

(83) Fluctuations and drifts in the neutral point potential may cause overvoltages in sensitive components of a converter and may introduce undesired harmonics. Therefore, a control of the neutral point potential may be an important part of high performance drives.

(84) The dynamic of the neutral point potential is given by

(85) $\begin{array}{cc}\frac{d}{\mathrm{dt}}{v}_{\mathrm{np}}=-\frac{1}{C_{\mathrm{dc}}}{i}_{\mathrm{np}},& \left(28\right)\end{array}$

(86) where i.sub.np denotes the neutral point current. A phase contributes to the neutral point current, when the switch position in this phase is zero, allows to write
i.sub.np=(1|u.sub.a|)i.sub.a+(1|u.sub.b|)i.sub.b+(1|u.sub.c|)i.sub.c.(29)

(87) In each phase x{a,b,c}, we define the new variable

(88) $\begin{array}{cc}{s}_x=1-.Math.u_x.Math.=\{\begin{array}{cc}0& \mathrm{if}{u}_x\left\{-1,1\right\}\\ 1& \mathrm{if}{u}_x=0\end{array}.& \left(30\right)\end{array}$

(89) With this, the neutral current can be written in the compact form
i.sub.np=s.sub.ai.sub.a+s.sub.bi.sub.b+s.sub.ci.sub.c.(31)

(90) The control objective is to control a drift or the DC component in the neutral point potential, rather than a harmonic or an AC component. In the following, two versions of neutral point control are described.

(91) Control of Instantaneous Neutral Point Potential

(92) In this embodiment, the third controlled variable is the instantaneous neutral point potential v.sub.np.

(93) The neutral point potential at a future time instant t.sub.1>t can be controlled by manipulating the time instants of switching instants within the time window [t t.sub.1]. More specifically, in phase x, consider the i-th switching instant u.sub.xi(t.sub.xi). Its transition time can be modified by t.sub.xi=t.sub.xit*.sub.xi, where t*.sub.xi denotes the nominal switching instant of this switching instant and t.sub.xi refers to the modified switching instant.

(94) The neutral point potential at t.sub.1 can be computed by inserting (31) into (28) and integrating both sides from t to t.sub.1.

(95) $\begin{array}{cc}{v}_{\mathrm{np}}\left(t_1\right)=v_{\mathrm{np}}\left(t\right)-\frac{1}{C_{\mathrm{dc}}}{}_t^{t_1}\left(i_a\left(\right)s_a\left(\right)+i_b\left(\right)s_b\left(\right)+i_c\left(\right)s_c\left(\right)\right)d.& \left(32\right)\end{array}$

(96) Consider the contribution from phase a over the time interval [t t.sub.1]

(97) $\begin{array}{cc}{v}_{\mathrm{np},a}=-\frac{1}{C_{\mathrm{dc}}}{}_t^{t_1}{i}_a\left(\right)s_a\left(\right)d.& \left(33\right)\end{array}$

(98) Assume that one switching instant occurs in phase a in this time interval. Let u.sub.a=u.sub.a1u.sub.a0 denote the switching instant, where u.sub.a is a nonzero integer.

(99) Correspondingly,
s.sub.a=s.sub.a1s.sub.a0.(34)

(100) The nominal switching time is t*.sub.a, while the actual or modified switching time is
t.sub.a=t*.sub.a+t.sub.a.(35)

(101) In light of the assumption of one switching instant occurring in phase a within [t t.sub.1], (33) can be rewritten as

(102) $\begin{array}{cc}{v}_{\mathrm{np},a}=-\frac{1}{C_{\mathrm{dc}}}{i}_a\left(t\right)\left({}_t^{t_a}{s}_{a0}d+{}_{t_a}^{t_1}{s}_{a1}d\right),& \left(36\right)\end{array}$
where it is assumed that the phase a current i.sub.a is constant over the time interval [t t.sub.1]. With the help of (34) and (35), (36) can be reformulated to

(103) $\begin{array}{cc}{v}_{\mathrm{np},a}=-\frac{1}{C_{\mathrm{dc}}}{i}_a\left(t\right)\left(s_{a0}\left(t_1-t\right)+s_a{t}_1-s_a{t}_a^{*}-s_a{t}_a\right).& \left(37\right)\end{array}$

(104) As can be seen, the phase a contribution to the neutral point potential at time t.sub.1 can be manipulated through the last term in (37), using the switching time modifications t.sub.a. We interpret this last term in (37) as a correction to the neutral point potential, and the neutral point potential correction function for one time-shift is

(105) $\begin{array}{cc}{v}_{\mathrm{np},\mathrm{corr},a}\left(t_a\right)=\frac{1}{C_{dc}}{i}_a\left(t\right)s_a{t}_a.& \left(38\right)\end{array}$

(106) More specifically, modifying the switching instant by t.sub.a changes the phase a contribution to the neutral point potential by v.sub.np,corr,a(t.sub.a).

(107) In a next step, the correction is generalized to the neutral point potential of three phases and to an arbitrary number of switching instants. Consider the neutral point potential correction over the time interval [tt+T.sub.p]. Assume that n.sub.x switching instants are located within the time interval [tt +T.sub.p] in phase x. As before, assume that the phase currents are constant within [tt+T.sub.p]. This leads to the correction to the neutral point potential

(108) $\begin{array}{cc}{v}_{\mathrm{np},\mathrm{corr}}\left(t\right)=\frac{1}{C_{dc}}\left(i_a\left(t\right)\underset{i=1}{\overset{n_a}{.Math.}}{s}_{\mathrm{ai}}{t}_{\mathrm{ai}}+i_b\left(t\right)\underset{i=1}{\overset{n_b}{.Math.}}{s}_{\mathrm{bi}}{t}_{\mathrm{bi}}+i_c\left(t\right)\underset{i=1}{\overset{n_c}{.Math.}}{s}_{\mathrm{ci}}{t}_{\mathrm{ci}}\right)& \left(39\right)\end{array}$

(109) as a function of the phase currents, switching instants and modifications to the switching time instants. Equivalently, (39) can be stated as the scalar product

(110) 0$\begin{array}{cc}{v}_{\mathrm{np},\mathrm{corr}}\left(t\right)=\frac{1}{C_{dc}}\left[i_a{s}_{a1}i{s}_{a2}\mathrm{.Math.}{i}_a{s}_{{\mathrm{an}}_a}{i}_b{s}_{b1}\mathrm{.Math.}{i}_b{s}_{{\mathrm{bn}}_b}{i}_c{s}_{c1}\mathrm{.Math.}{i}_c{s}_{{\mathrm{cn}}_c}\right]t\text{=}\text{:}-V_{2}t,& \left(40\right)\end{array}$

(111) where the time dependency from the phase currents has been dropped to simplify the notation.

(112) To obtain a three-dimensional formulation that enables control of the instantaneous neutral point potential, the neutral point potential is used as the third variable (=v.sub.np), which results in the following third variable correction term in the cost function J:
J.sub.2(t){tilde over (v)}.sub.npv.sub.np,corr(t).sub.2⁽⁴¹⁾

(113) where {tilde over (v)}.sub.np refers to the error in the neutral point potential at time t,v.sub.np,corr(t) is the correction function for the neutral point potential applied from t to t+T.sub.p, and is a non-negative scalar weight. The three-dimensional problem is then formulated as:

(114) $\begin{array}{cc}\underset{t}{\min }J\left(t\right)={.Math.{\stackrel{~}{}}_{}-{}_{,\mathrm{corr}}\left(t\right).Math.}_2^2+{.Math.{\tilde{v}}_{\mathrm{np}}-v_{\mathrm{np},\mathrm{corr}}\left(t\right).Math.}_2^2+t^{T}Rt.& \left(42a\right)\\ s.t.& \\ {\mathrm{kT}}_s{t}_{a1}{t}_{a2}\mathrm{.Math.}{t}_{{\mathrm{an}}_a}{t}_{a\left(n_a+1\right)}^{*}& \left(42b\right)\\ {\mathrm{kT}}_s{t}_{b1}{t}_{b2}\mathrm{.Math.}{t}_{{\mathrm{bn}}_b}{t}_{b\left(n_b+1\right)}^{*}& \left(42c\right)\\ {\mathrm{kT}}_s{t}_{c1}{t}_{c2}\mathrm{.Math.}{t}_{{\mathrm{cn}}_c}{t}_{c\left(n_c+1\right)}^{*}.& \left(42d\right)\end{array}$

(115) Here, , 0, and R.sup.(n.sup.a.sup.+n.sup.b.sup.+n.sup.c.sup.)x(n.sup.a.sup.+n.sup.b.sup.+n.sup.c.sup.), R0 , are weighting factors. The corrections of switching instants are aggregated in the vector
t=[t.sub.a1t.sub.a2 . . . t.sub.an.sub.at.sub.b1 . . . t.sub.bn.sub.bt.sub.c1 . . . t.sub.cn.sub.c].sup.T.(43)

(116) Since the model of the correction function v.sub.np,corr in (40) has the same form in (19), the three-dimensional problem can be cast in the form (26). The output error is defined as
{tilde over (y)}=[{tilde over ()}.sub.{tilde over ()}.sub.{tilde over (v)}.sub.np].sup.T (44)
and the matrix V as

(117) $\begin{array}{cc}V={\left[\begin{array}{ccc}\frac{V_{dc}}{6}& 0& 0\\ 0& \frac{V_{dc}}{6}& 0\\ 0& 0& -\frac{1}{C_{dc}}\end{array}\right]\left[\begin{array}{ccc}2u_{a1}& 0& i_a{s}_{a1}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ 2u_{{\mathrm{an}}_a}& 0& i_a{s}_{{\mathrm{an}}_a}\\ -u_{b1}& \sqrt{3}{u}_{b1}& i_b{s}_{b1}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ -u_{{\mathrm{bn}}_b}& \sqrt{3}{u}_{{\mathrm{bn}}_b}& i_b{s}_{{\mathrm{bn}}_b}\\ -u_{c1}& -\sqrt{3}{u}_{c1}& i_c{s}_{c1}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ -u_{{\mathrm{cn}}_c}& -\sqrt{3}{u}_{{\mathrm{cn}}_c}& i_c{s}_{{\mathrm{cn}}_c}\end{array}\right]}^T.& \left(45\right)\end{array}$

(118) The definitions (24), (14) and (17) also apply to this formulation.

(119) The three-dimensional control problem for instantaneous neutral point potential control may be formulated as a deadbeat control method or as a full quadratic program as described above.

(120) Preventing Drift of Neutral Point Potential

(121) When a control objective in a neutral point clamped converter 12a is to control drift and large fluctuations of the neutral point potential, the neutral point potential may be filtered or higher harmonic contributions may be otherwise removed from the reference before the cost function is evaluated.

(122) For example, the estimated neutral point potential may be filtered. Let v*.sub.np(t) denote the reference of the DC component of the neutral point potential, which is typically zero. Correspondingly, the DC component of the neutral point potential at time t is given by v.sub.np(t). To only include the DC component as third variable, the estimated neutral point potential v.sub.np(t) may be low pass filtered or averaged over an appropriate window, such as a sixth of the fundamental period.

(123) Furthermore, the neutral point ripple, i.e. the higher order contributions to the neutral point potential, may be precomputed based on optimized pulse patterns. Based on the overall system model and precomputed optimized pulse patterns, one can determine the neutral point potential ripple that arises due to a switching as a function of the voltage phase angle. This precomputed neutral point potential ripple may then be added to the DC component of neutral point potential reference, which is typically zero, to form the full neutral point potential reference: v*.sub.np=v*.sub.np+v*.sub.np,ripple. In such formulation, only the unwanted part of the neutral point potential, which occurs for the reasons other than nominal switching, may be controlled.

(124) Control of Averaged Neutral Point Potential

(125) In this embodiment, the third variable is the third axis -flux, i.e., =.sub..

(126) The instantaneous neutral point potential V.sub.np(t) is known to oscillate with fundamental frequency 3.sub.1. Therefore, the instantaneous value of v.sub.np(t) alone may provide little information about the averaged neutral point potential that we wish to control.

(127) A control of an averaged neutral point potential in a neutral point clamped converter 12a may comprise the following three steps:

(128) In a first step, the necessary {tilde over ()}.sub. correction to drive the average neutral point potential to the desired value may be determined.

(129) In this step, the neutral point potential is averaged over a fundamental period, where T is the period of the fundamental frequency. Let v.sub.np(t) denote the neutral point potential at the current time t and v.sub.np.sup.des(t+T) be the desired neutral point potential at time t+T. The .sub. that achieves this is given by

(130) $\begin{array}{cc}{}_{}={\sin }^{-1}\left(\frac{C_{dc}{}_1\left(v_{\mathrm{np}}^{\mathrm{des}}\left(t+T\right)-v_{\mathrm{np}}\left(t\right)\right)}{12I\cos \left(\right)}\right)\frac{v_{dc}}{L-1},& \left(46\right)\end{array}$

(131) where C.sub.dc is the DC link capacitance, .sub.1 is the electrical angular speed, v.sub.dc is the total DC link voltage, I is the peak current at the converter terminals, cos() is the power factor, L is the number of levels and is a constant given by the optimized pulse pattern used. More specifically, for an optimized pulse pattern with N switches in the first 90 degrees, and nominal switching times t.sub.n, n=1,2, . . . ,N

(132) $\begin{array}{cc}=\underset{n=1}{\overset{N}{.Math.}}\sin \left(t_n\right).& \left(47\right)\end{array}$

(133) For obtaining (46) it is assumed that the angular corrections caused by {tilde over ()}.sub. are given by

(134) $\begin{array}{cc}\left({\stackrel{~}{}}_{}\right)={\stackrel{~}{}}_{}\frac{L-1}{v_{dc}}.& \left(48\right)\end{array}$

(135) This assumption is satisfied for deadbeat control. For a quadratic program, the error caused by this assumption is negligible.

(136) In this embodiment, the optimal third variable * is provided by (46) evaluated for the neutral point potential derived from the optimized pulse pattern and the estimated third variable is the corresponding flux determined based on measurements in the converter 12a.

(137) In a second step, an adjusted modulation index may be determined, since the introduction of a {tilde over ()}.sub. may change the magnitude of the fundamental component of the converter's output voltage. When a nonzero .sub. is used, the amplitude of the fundamental phase voltage is changed. This change may be quantified so it can be taken into account when selecting the modulation index.

(138) In a third step, a switching angle correction induced by the flux error {tilde over ()} in ,, may be determined using the deadbeat or quadratic program control method described above.

(139) It also may be possible to average the neutral point potential not over a fundamental period but over a time window of size W, which may be smaller than the time window of the fundamental period T. This averaged neutral point potential then may be used to compute the required .sub.. Notice that by making W<T, it is possible to drive the neutral point potential to a desired reference in less than one fundamental period.

(140) For example, the averaged neutral point potential between time tW and t may be defined as

(141) $\begin{array}{cc}{\stackrel{\_}{v}}_{\mathrm{np}}\left(t\right)=\frac{1}{W}{}_{t-W}^t{v}_{\mathrm{np}}\left(\right)d.& \left(49\right)\end{array}$
The signal v.sub.np(t) may be sampled at times t=0, W, 2W, 3W, . . . .sub.,(t) also may be maintained constant in the interval (k1) W<tkW, for kN.

(142) The required .sub.(t) to drive the averaged neutral point potential to zero within a time window W is then given by

(143) $\begin{array}{cc}{}_{}\left(t\right)=-\frac{v_{dc}}{L-1}{\sin }^{-1}\left(\frac{{\mathrm{TC}}_{dc}{}_1{\stackrel{\_}{v}}_{\mathrm{np}}\left(t\right)}{12\mathrm{WI}\cos \left(\right)}+\frac{\sin \left(\frac{L-1}{v_{dc}}{}_{}\left(t-W\right)\right)}{2}\right)& \left(50\right)\end{array}$

(144) Control of a Circulating Current

(145) FIG. 6 shows a modular multi-level converter 12b, which comprises three branches 44 each of which comprises a plurality of series-connected converter cells 46. The branches 44 are delta-connected and interconnect the three phases of an electrical grid 14.

(146) As shown in FIG. 7, each converter cell 46 comprises a cell capacitor 48. The semiconductor switches 40 of the cell 46, which are configured as an H-bridge, are adapted for connecting the cell capacitor in two directions to the branch 44 and to disconnect the cell capacitor 48 from the branch 44. The converter cell 46 is a bipolar cell. Each semiconductor switch 40 comprises an actively controlled power semiconductor and a freewheeling diode.

(147) FIGS. 6 and 7 furthermore show variables used in the following formulas, such as currents and voltages.

(148) As example, FIG. 8 shows an optimized pulse pattern 20 for a converter branch 44 of the converter 12b. The pattern 20 is an optimized pulse pattern for one phase of the modular multi-level converter 12b. The optimized pulse patterns 20 for other phases may be obtained by phase-shifting the optimized pulse patterns corresponding to the required modulation index.

(149) The modular multi-level topology is particularly well-suited for Statcom applications, because there is no permanent active power exchanged with the grid 12. Therefore, the module capacitors 48 may be precharged by taking the power from the grid 12, while during the Statcom nominal operation their average charge remains the same. This is possible because during nominal Statcom operation, the currents and voltages of each individual branch are phase-shifted by 90. Therefore, if the pulse pattern is symmetric, the mean voltage at the module capacitors remains constant during nominal operation.

(150) The Statcom control is required to generate sinusoidal currents i.sub.a, i.sub.b and i.sub.c that are generated/absorbed by the Statcom. Typically, the reference for the output current is given by the positive-sequence three-phase currents.

(151) However, the reference of the Statcom currents can include also the negative sequence currents. This is required if some plant connected to the grid generates a strong negative-sequence current, which can then be compensated by injecting the currents of the opposite phase by the Statcom.

(152) During imbalanced operating scenarios, the voltages and the currents in the branches 44 (computed directly from the line current requirements) are not phase-shifted by 90 in all branches 44. Consequently, active power is exchanged with the grid 14. As a result, the mean voltage of the cell capacitors 48 starts to drift, which may lead to inappropriate operation of the modular multi-level converter 12b.

(153) Energy balancing between the converter branches 44 may be done by means of additional current or voltage control. By controlling the circulating current i.sub.circ in the modular multi-level converter 12b, the imbalance can be compensated by sending the active power around the branches 44 of the converter 12b, instead of exchanging the active power with the grid 14. This enables the functional operation of the Statcom during imbalances. The carrier of the energy is the circulating current i.sub.circ, which is defined as
i.sub.circ=(i.sub.1+i.sub.2+i.sub.3),(51)

(154) where i.sub.1, i.sub.2 and i.sub.3 are the branch currents. The circulating current i.sub.circ flows inside the delta of the converter 12b.

(155) In Statcom applications, the control of the circulating current i.sub.circ may be used for the purpose of branch energy balancing in the following ways:

(156) The circulating current i.sub.circ may be eliminated during normal operation to prevent a large ripple of cell capacitor voltages and, in particular, a drift of the cell capacitor voltages. The unwanted circulating current i.sub.circ arises from any common-mode voltage that is generated by the converter 12b. Due to the inductance L.sub.b in the converter branches 44, the circulating current i.sub.circ depends on the integral of the common-mode voltage, which is the reason why the circulating current i.sub.circ is especially susceptible to drifting.

(157) During imbalanced operation, the circulating current i.sub.circ may be used to re-establish an energy balance between the branches 44. The circulating current reference is a sinusoid at fundamental frequency.

(158) A Statcom is often used in the so-called idle operation, i.e. it is connected to the grid 14 but the current flow between the grid 14 and the converter 12b needs to be zero. During the idle operation, one can inject a circulating current i.sub.circ in order to activate the cell balancing algorithm that will redistribute voltages among the capacitors 48 within one branch 44 and balance them. For this purpose, typically a sinusoidal circulating current i.sub.circ at triple fundamental frequency is injected.

(159) Furthermore, the control of the circulating current i circ may be to reduce a voltage ripple in nominal operation, for which 2nd and 4th current harmonics may have to be injected.

(160) In the following, a circulating current control for the converter 12b based on the method described with FIG. 2 is presented as a further embodiment.

(161) The voltage v.sub.123 is referred to as converter voltage. To simplify the exposition, it is assumed that the voltage of the cell capacitors 48 is constant. The converter voltage v.sub.123 obtained by modulation based on an optimized pulse pattern 20 is described as v.sub.123=u.sub.123v.sub.c, where u.sub.123 denotes the three-phase optimized pulse pattern for modular multi-level converter 12b. The variable u denotes the number of the converter cells 46 in a branch 44, in which the converter cell output is connected to the capacitor 48. The connection of a capacitor 48 to the output of the converter cell v.sub.m with the same polarity leads to positive u, while connecting it with opposite polarity leads to negative u.

(162) The voltages v.sub.g,abc denote the voltages at the point of connection to the grid 14 (usually at the point of common coupling) and are typically measured. The currents i.sub.g,abc are injected into the grid by the converter 12b. The current reference i*.sub.g,abc is obtained from the grid control level and needs to be tracked by the converter 12b. The inductances in the grid connection (which includes the transformer inductance) are denoted as L.sub.g and the inductors in the branches 44 of the modular multi-level converter 12b are denoted as L.sub.b. The branch currents are denoted as i.sub.123, while i.sub.circ denotes the circulating current.

(163) The tracking of grid currents i.sub.g,abc and, in particular, the circulating current i.sub.circ may be tracked by using the control method as described above, i.e. by applying time-shifts to the switching instants 28 determined from an optimized pulse pattern 20.

(164) Modelling the Fundamental Converter Flux Reference

(165) The circuit in FIG. 6 is described by the following equations that arise from Kirchhoff laws:

(166) $\begin{array}{cc}{v}_{123}-D_1{v}_{g,\mathrm{abc}}=L_b\frac{d}{\mathrm{dt}}{i}_{123}+L_g{D}_1\frac{d}{\mathrm{dt}}{i}_{\mathrm{abc}},& \left(52\right)\\ i_{g,\mathrm{abc}}=D_2{i}_{123},& \left(53\right)\end{array}$
where

(167) $D_1:=\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right]\mathrm{and}{D}_2:=\left[\begin{array}{ccc}1& 0& -1\\ -1& 1& 0\\ 0& -1& 1\end{array}\right].$
The equation (52) can be integrated over time to obtain, according to the definition of flux (2),
.sub.123D.sub.1.sub.g,abc=L.sub.bi.sub.123+L.sub.gD.sub.1i.sub.abc,(54)
where .sub.g,abc is the flux at the point of grid connection and .sub.123 is the converter flux.

(168) Now consider the transformation of the three phase system to stationary orthogonal coordinates (Clarke transformation). This transformation is bijective and therefore invertible:
().sub.=M.sub.c().sub.abc, ().sub.abc=M.sub.c.sup.1().sub.,(55)
where

(169) 0$M_c=\frac{2}{3}\left[\begin{array}{ccc}0& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].$
Note that according to this definition and (51) the -component of the branch currents i.sub. corresponds to the circulating current i.sub.circ:
i.sub.=(i.sub.1+i.sub.2+i.sub.3)=i.sub.circ.(56)

(170) In the following, the notions of circulating current and converter current is used interchangeably.

(171) Introducing the substitution ().sub.abc=M.sub.c.sup.1().sub. in (52)-(53) and multiplying the equations by M.sub.c from left leads to the following equations:
.sub.{circumflex over (D)}.sub.1.sub.g,=L.sub.bi.sub.+L.sub.g{circumflex over (D)}.sub.1i.sub.g,,(57)
i.sub.g,={circumflex over (D)}.sub.2i.sub.(58)
where

(172) ${\hat{D}}_1=M_c{D}_1{M}_c^{-1}=\left[\begin{array}{cc}\sqrt{3}{M}^T& \left[\begin{array}{c}0\\ 0\end{array}\right]\\ \left[\begin{array}{cc}0& 0\end{array}\right]& 0\end{array}\right],{\hat{D}}_2=M_c{D}_2{M}_c^{-1}=\left[\begin{array}{cc}\sqrt{3}M& \left[\begin{array}{c}0\\ 0\end{array}\right],\\ \left[\begin{array}{cc}0& 0\end{array}\right]& 0\end{array}\right].$

(173) Here,

(174) $M:=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]$
is a matrix that describes the vector rotation for 30 clockwise. Note that this matrix is regular and its inverse is given by M.sup.1=M.sup.T. In (57)-(58), .sub. stands for the transformed converter flux .sub.123,i.sub. is the transformed branch current, while .sub.t, and i.sub.g, are the transformed grid flux and output current.

(175) The equations (57)-(58) are equivalent to the following equations separated in - and -related equations:
.sub.{square root over (3)}M.sup.T.sub.g,=L.sub.bi.sub.+{square root over (3)}L.sub.gM.sup.Ti.sub.g,(59)
i.sub.g,={square root over (3)}Mi.sub.(60)
.sub.=L.sub.bi.sub.(.sub.61)
i.sub.g,=0.(62)

(176) From these equations, the following may be concluded:

(177) The control of the output currents of the Statcom i.sub.abc may be achieved by controlling the -components of the converter flux .sub.. The reference for the converter flux in is:

(178) $\begin{array}{cc}{}_{,\mathrm{ref}}=\sqrt{3}{M}^T\left({}_{g,}+\left(L_g+\frac{1}{3}{L}_b\right)i_{g,,\mathrm{ref}}\right),& \left(63\right)\end{array}$

(179) where .sub.g, is the estimate of the flux at grid connection and i.sub.g,,ref is the reference for the output current obtained from the grid controller. In case of the connection to a weak grid, in which case the grid flux is dynamically dependent on the output currents, or in the cases of unknown parameters, this feed-forward determination of the converter flux references may be replaced by feedback loop.

(180) The control of the circulating current i.sub.circ may be achieved by controlling only the -component of the converter flux. The reference for the -flux can be determined as:
i.sub.,ref=L.sub.bi.sub.circ,ref,(64)
where the circulating current reference i.sub.circ,ref is obtained from an outer control loop.

(181) From what is shown in the above, the control of the output currents and circulating current i.sub.circ of the modular multi-level converter 12b may be achieved by controlling the - and -components of the converter flux. In particular, the optimal third variable * may be L.sub.bi.sub.circ,ref determined from the circulating current reference i.sub.circ,ref provided by an outer control loop and the estimated third variable may be L.sub.bi.sub.circ determined from the estimated circulating current i.sub.circ.

(182) Consideration of Common Mode Voltage

(183) With the control method for the modular multi-level converter 12b, one may need to track a fast evolving -flux reference. Therefore, it may be necessary that a ripple of the -component is taken into account. The nominal ripple of the -component is defined by the optimized pulse patterns 20 that are used and may be computed from the optimized pulse patterns 20 and the nominal system model.

(184) Two approaches for circulating current tracking are described in the following. Note that we denote the fundamental of the flux as .sub.,ref, while *.sub. denotes the optimal -flux with the nominal ripple included. This may be seen equivalent to the notation of .sub.,ref and *.sub..

(185) FIG. 9 shows a controller 16a, which may be used with design optimized pulse patterns 20 that do not have a common-mode voltage. With this approach, the setup of the online controller 16a may be simplified. A current controller 50 for the Statcom receives a Clarke transformed current reference i*.sub.g,abc and together with measurements from the system 10 determines a reference flux .sub.,ref from which a reference flux angle *, a modulation index m* and pulse number d* is determined. From these inputs, the pattern selector 18 selects appropriate optimized pulse patterns 20 and determines an optimal flux *.sub.. An estimated flux .sub. is subtracted from the optimal flux *.sub. to determine a flux error.

(186) From a DC link voltage V*.sub.DC and system measurements 52, a balancing controller 54 for the branch energy estimates an optimal circulating current i.sub.circ, which by a circulating current controller 56 is translated together with system measurements 52 in an optimal -flux reference *.sub.. The -flux error is determined by subtracting an estimated -flux .sub..

(187) The pattern optimizer 24 then determines a modified optimized pulse pattern 20 from both errors. As indicated in FIG. 9, the optimization procedure in the pattern optimizer 24 that is utilized to compute optimized pulse patterns 20 may be constrained only to the optimized pulse patterns 20 that exhibit no common mode in the three-phase system, which translates to zero -flux reference. Thus, the -flux may be determined independently from the selection of the optimized pulse pattern 20. However, in this case, the feasible set for optimized pulse pattern computation may be smaller than in the nominal case, which may lead to spectrally suboptimal optimized pulse patterns 20.

(188) FIG. 10 shows a controller 16b, which may be used with design optimized pulse patterns 20 that have a limited amount of common-mode voltage. The ripple of the common-mode voltage cannot be provisionally large because it would cause large charging and discharging of the cell capacitors 48. However, a certain amount of common-mode voltage may be allowed.

(189) Contrary to FIG. 9, in FIG. 10, the output of the circulating current controller 56 is input into a control logic 58, which also receives the output of the current controller 50 and determines the angle *, the modulation index m* and the pulse number d* on both outputs.

(190) After that, the pattern selector 18 determines a three-component optimal flux *.sub. from which a three-component error is determined, which is treated by the pattern optimizer 24 as two-component flux error and third variable error.

(191) In this case, the spectral suboptimality of the computed optimized pulse patterns 20 may be reduced. However, the nominal ripple of the -flux may still exist and may have to be taken into account (if it is significantly large) in deriving the reference for the -component of the flux *.sub..

(192) 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

(193) 10 converter system 12 electrical converter 14 electrical grid 16 controller 18 pattern selector 20 optimized pulse pattern 22 table with optimized pulse patterns 24 pattern optimizer 26 modified optimized pulse pattern .sub.,ref reference flux *.sub. optimal flux .sub. estimated flux * optimal third variable estimated third variable 28 switching instant 30 prediction horizon 12a neutral point clamped converter 32 DC link 34 phase circuit 36a positive point 36b negative point 38 neutral point 40 semiconductor switch 42 diode 12b modular multi-level converter 44 converter branch 46 converter cell 48 cell capacity 50 current controller 52 system measurements 54 balancing controller 56 circulating current controller 58 control logic