Determining a fundamental component of an AC voltage

Abstract

A method for estimating a fundamental component of an AC voltage includes receiving a timely varying measurement signal of the AC voltage; parametrizing a fundamental component of the AC voltage; and determining parameters of the fundamental component based on minimizing a cost function. The fundamental component has a rated frequency, a variable amplitude and a variable phase shift. The cost function is based on an integral of a norm of a difference between the measurement signal and the parametrized fundamental component via a time horizon. The time horizon starts at an actual time point and goes back via a predefined length. The cost function includes a term based on a norm of the difference between a value of the fundamental component at the actual time point and a value of a previously estimated fundamental component at the actual time point, where the previously estimated fundamental component has been determined for a previous time point.

Claims

1. A method for estimating a fundamental component ( {circumflex over (V)}.sub.pcc.sup.1) of an AC voltage (V.sub.pcc), the method comprising: receiving a timely varying measurement signal of the AC voltage (V.sub.pcc); parametrizing a fundamental component ({circumflex over (V)}.sub.pcc.sup.1) of the AC voltage (V.sub.pcc), the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) having a rated frequency, a variable amplitude and a variable phase shift; determining parameters ({circumflex over (x)}(kT.sub.s)) of the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) based on minimizing a cost function (J), wherein the cost function (J) is based on an integral of a norm of a difference between the measurement signal and the parametrized fundamental component via a time horizon (h), the time horizon (h) starting at an actual time point and going back via a predefined length, wherein the cost function (J) comprises a term based on a norm of the difference between a value of the fundamental component({circumflex over (V)}.sub.pcc.sup.1) at the actual time point and a value of a previously estimated fundamental component ({circumflex over (V)}.sub.pcc.sup.1*) at the actual time point, wherein the previously estimated fundamental component has been determined for a previous time point.

2. The method of claim 1, wherein the parameters ({circumflex over (x)}(kT.sub.s)) are functions of the amplitude and the phase shift.

3. The method of claim 1, wherein the parameters ({circumflex over (x)}(kT.sub.s)) are chosen, such that the cost function (J) is a quadratic function of the parameters and the cost function (J) has a term quadratic in the parameters defined by a cost function matrix (H(kT.sub.s)) and a term linear in the parameters defined by a cost function vector ((kT.sub.s)).

4. The method of claim 3, wherein the cost function matrix (H(kT.sub.s)) is based on an integral of products of trigonometric functions of a rated fundamental frequency; and/or wherein the cost function vector ((kT.sub.s)) is based on an integral of products of the measurement signal with trigonometric functions of a rated fundamental frequency.

5. The method of claim 3, wherein equations for the cost function matrix (H(kT.sub.s)) and the cost function vector ((kT.sub.s)) have been determined offline by minimizing the cost function (J) analytically; and/or wherein the parameters ({circumflex over (x)}(kT.sub.s)) are calculated from the equations by determining the cost function matrix (H(kT.sub.s)) and the cost function vector ((kT.sub.s)) from the measurement signal; and/or wherein the parameters ({circumflex over (x)}(kT.sub.s)) are calculated based on the inverse of the cost function matrix (H(kT.sub.s)).

6. The method of claim 1, wherein the cost function (J) is minimized online.

7. The method of claim 1, wherein at least two measurement signals for AC voltage components of a multi-phase voltage are received; wherein at least two fundamental components for the at least two measurement signals are parametrized and their parameters are determined based on minimizing one cost function for the at least two fundamental components.

8. The method of claim 1, wherein the at least two measurement signals are Clarke transformed and the parameters are determined in a Clarke transformed reference frame.

9. The method of claim 1, wherein the horizon (h) has a length of more than 0.01 the period of the rated frequency of the fundamental component.

10. The method of claim 1, further comprising: calculating an actual amplitude and/or an actual phase shift of the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) from the parameters.

11. The method of claim 1, further comprising: calculating an actual phase angle of fundamental components ({circumflex over (V)}.sub.pcc.sup.1) of a multi-phase AC voltage from the parameters.

12. The method of claim 11, further comprising: calculating an actual frequency of fundamental components ({circumflex over (V)}.sub.pcc.sup.1) of a multi-phase AC voltage based on a difference between the actual phase angle and a previous phase angle calculated from a previous fundamental component.

13. A method for controlling an electrical converter, the method comprising: determining a fundamental component ({circumflex over (V)}.sub.pcc.sup.1) of an AC voltage supplied to the converter according to: receiving a timely varying measurement signal of the AC voltage (V.sub.pcc); parametrizing a fundamental component ({circumflex over (V)}.sub.pcc.sup.1) of the AC voltage (V.sub.pcc), the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) having a rated frequency, a variable amplitude and a variable phase shift; determining parameters ({circumflex over (x)}(kT.sub.s)) of the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) based on minimizing a cost function (J), wherein the cost function (J) is based on an integral of a norm of a difference between the measurement signal and the parametrized fundamental component via a time horizon (h), the time horizon (h) starting at an actual time point and going back via a predefined length, wherein the cost function (J) comprises a term based on a norm of the difference between a value of the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) at the actual time point and a value of a previously estimated fundamental component ({circumflex over (V)}.sub.pcc.sup.1*) at the actual time point, wherein the previously estimated fundamental component has been determined for a previous time point; controlling a reference voltage for the converter based on the fundamental component ({circumflex over (V)}.sub.pcc.sup.1).

14. A controller for a converter, wherein the controller is adapted for performing a method of: receiving a timely varying measurement signal of the AC voltage (V.sub.pcc); parametrizing a fundamental component ({circumflex over (V)}.sub.pcc.sup.1) of the AC voltage (V.sub.pcc), the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) having a rated frequency, a variable amplitude and a variable phase shift; determining parameters ({circumflex over (x)}(kT.sub.s)) of the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) based on minimizing a cost function (J), wherein the cost function (J) is based on an integral of a norm of a difference between the measurement signal and the parametrized fundamental component via a time horizon (h), the time horizon (h) starting at an actual time point and going back via a predefined length, wherein the cost function (J) comprises a term based on a norm of the difference between a value of the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) at the actual time point and a value of a previously estimated fundamental component({circumflex over (V)}.sub.pcc.sup.1*) at the actual time point, wherein the previously estimated fundamental component has been determined for a previous time point.

15. The method of claim 4, wherein equations for the cost function matrix (H(kT.sub.s) and the cost function vector ((kT.sub.s)) have been determined offline by minimizing the cost function (J) analytically; and/or wherein the parameters ({circumflex over (x)}(kT.sub.s)) are calculated from the equations by determining the cost function matrix (H(kT.sub.s)) and the cost function vector ((kT.sub.s)) from the measurement signal; and/or wherein the parameters ({circumflex over (x)}(kT.sub.s)) are calculated based on the inverse of the cost function matrix (H(kT.sub.s)).

16. The method of claim 2, wherein the parameters ({circumflex over (x)}(kT.sub.s)) are chosen, such that the cost function (J) is a quadratic function of the parameters and the cost function (J) has a term quadratic in the parameters defined by a cost function matrix (H(kT.sub.s)) and a term linear in the parameters defined by a cost function vector ((kT.sub.s)).

17. The method of claim 2, wherein the cost function (J) is minimized online.

18. The method of claim 7, wherein the at least two measurement signals are Clarke transformed and the parameters are determined in a Clarke transformed reference frame.

19. The method of claim 2, further comprising: calculating an actual amplitude and/or an actual phase shift of the fundamental component ({circumflex over (V)}.sub.pcc.sup.1) from the parameters.

20. The method of claim 3, further comprising: calculating an actual phase angle of fundamental components ({circumflex over (V)}.sub.pcc.sup.1) of a multi-phase AC voltage from the parameters.

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 for a method for determining a fundamental component of an AC voltage according to an embodiment of the invention.

(4) FIG. 3 schematically shows a controller for determining a fundamental component of an AC voltage according to an embodiment of the invention.

(5) FIG. 4 schematically shows a controller for controlling an electrical converter according to an embodiment of the invention.

(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) FIG. 1 shows a converter system 10, which comprises an electrical converter 12 controlled by a controller 14. The electrical converter 12 is connected to an electrical grid 16 or equivalent power source and comprises a transformer 18, a rectifier 20 and a DC link 22, which may comprise a bank of capacitors. The DC link 22 may be connected to a load, like a combination of an inverter with an electrical motor. It also may be possible that the DC link is connected via an inverter to an electrical generator, i.e. it is possible that there is a power flow in the converter system 10 from the grid 16 to the DC link 22 or vice versa from the DC link 22 to the grid 16.

(8) The electrical grid 16 may supply the electrical converter 12 and in particular the transformer 18 with a three-phase AC voltage V.sub.pcc, which (after transformation) is supplied to the rectifier 20, which produces a DC voltage supplied to the DC link 22.

(9) The controller 14 may be adapted for controlling the electrical converter 14 and in particular the rectifier 20, which may be an active rectifier. Furthermore, the controller 14 may be adapted for controlling an inverter connected to the DC link 22.

(10) In particular, the controller 14 may be adapted for stabilizing the stored energy in the DC link 22 at a prespecified level. The controller 14 may consider to draw limited power from the DC link 22 in the presence of various requests for power withdrawal from the DC link 22. This is, for example, the case of a full power converter in which the DC link 22 is connected to an inverter that converts the DC voltage to an AC voltage in order to drive an electric machine.

(11) In order to be able to maintain the energy storage at some prespecified level, the controller 14 may control the active power drawn from the grid/source 16. This latter functionality of the controller 14 may rely on the knowledge of the fundamental components V.sub.pcc.sup.1 of the voltage V.sub.pcc at the point of common coupling. The fundamental component is not measurable but must be estimated from the measured AC voltage V.sub.pcc, which is given by
V.sub.pcc=V.sub.pcc.sup.1+V.sub.pcc.sup.H

(12) where V.sub.pcc.sup.1 is the fundamental component of the measurement signal and V.sub.pcc.sup.H comprises all the harmonics that are present in the signal. Typically.

(13) $V_{\mathrm{pcc}}^H=\underset{k1}{.Math.}{V}_{\mathrm{pcc}}^k$

(14) where is V.sub.pcc.sup.k the k-th harmonic component. The signal to noise ratio

(15) $\mathrm{SNR}=\frac{.Math.V_{\mathrm{pcc}}^1.Math.}{.Math.V_{\mathrm{pcc}}^H.Math.}$

(16) is usually quite high in a medium voltage converter 12 due to the limited value of the switching frequency.

(17) The controller 14 is adapted for determining an estimate {circumflex over (V)}.sub.pcc.sup.1 of the fundamental component V.sub.pcc.sup.1, which will be described with respect to FIG. 2.

(18) FIG. 2 shows a flow diagram for a method for determining an estimated fundamental component {circumflex over (V)}.sub.pcc.sup.1 based on the measurement signal V.sub.pcc.

(19) It has to be noted that the AC voltage V.sub.pcc and the estimated fundamental component {circumflex over (V)}.sub.pcc.sup.1 may be vectors, for example three component vectors in the case of a three-phase voltage. The vectors V.sub.pcc and {circumflex over (V)}.sub.pcc.sup.1 may be seen as multiple single AC voltages and multiple single estimated fundamental components.

(20) In step S10, a timely varying measurement signal of the AC voltage V.sub.pcc is received. For example, the AC voltage V.sub.pcc may be measured at the point of common coupling of the converter 12 to the electrical grid/source 16.

(21) For example, at the Point of Common Coupling (PCC), a three-phase AC voltage V.sub.pcc may be measured, which may be then Clarke transformed into the xy or fixed frame through the Clarke transformation

(22) $T_{\mathrm{abc}}^{\mathrm{xy}}=\frac{2}{3}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\end{array}\right]$

(23) As such, and without any loss of generality, we shall pursue the derivations in the xy plane but the same holds for the abc components.

(24) The method for extracting/estimating the fundamental component {circumflex over (V)}.sub.pcc.sup.1 from the measured signal of the AC voltage V.sub.pcc is based on a cost function J.

(25) In particular, the optimization problem, on which the method is based is

(26) $\begin{array}{cc}\underset{{\hat{m}}_x,{\hat{m}}_y,{\hat{}}_x,{\hat{}}_y}{\min }J={}_{{\mathrm{kT}}_s-h}^{{\mathrm{kT}}_s}{.Math.V_{\mathrm{pcc}}\left(\right)-{\hat{V}}_{\mathrm{pcc}}^1\left(,{\mathrm{kT}}_s\right).Math.}^{2}d+{.Math.{\hat{V}}_{\mathrm{pcc}}^1\left({\mathrm{kT}}_s,{\mathrm{kT}}_s\right)-{\hat{V}}_{\mathrm{pcc}}^{1*}\left({\mathrm{kT}}_s,\left(k-1\right)T_s\right).Math.}^2\mathrm{subject}\mathrm{to}{\hat{V}}_{\mathrm{pcc}}^1\left(,{\mathrm{kT}}_s\right)=\left[\begin{array}{c}{\hat{m}}_x\left({\mathrm{kT}}_s\right)\sin \left(+{}_x\left({\mathrm{kT}}_s\right)\right)\\ {\hat{m}}_y\left({\mathrm{kT}}_s\right)\sin \left(+{}_y\left({\mathrm{kT}}_s\right)\right)\end{array}\right]& \left(1\right)\end{array}$

(27) where J is a quadratic cost function to be minimized, h is a horizon or window of time (in the range of few millisecond) over which we posit the cost function, 0 is a continuity/convexification weight factor, {circumflex over (m)}.sub.x, {circumflex over (m)}.sub.y, {circumflex over ()}.sub.x, and {circumflex over ()}.sub.y are the modulation indices and phase shifts in the x and y components of the estimated vector {circumflex over (V)}.sub.pcc,T.sub.s is the sampling time (usually few tens of millisecond), and {circumflex over (V)}.sub.pcc.sup.1* is the previously estimated fundamental component that was computed at the previous time step (k1)T.sub.s.

(28) The cost function J is based on an integral of a norm of a difference between the measurement signal V.sub.pcc and the fundamental component {circumflex over (V)}.sub.pcc.sup.1 via a time horizon h, the time horizon starting at an actual time point kT.sub.s and going back via a predefined length.

(29) It has to be noted that in the optimization problem (1), the fundamental component {circumflex over (V)}.sub.pcc.sup.1 has a rated frequency , variable amplitudes {circumflex over (m)}.sub.x, {circumflex over (m)}.sub.y and variable phase shifts {circumflex over ()}.sub.x, {circumflex over ()}.sub.y. However, for solving the optimization problem (1), not the variables {circumflex over (m)}.sub.x, {circumflex over (m)}.sub.y, {circumflex over ()}.sub.x, and {circumflex over ()}.sub.y but other parameters are used.

(30) In particular, the optimization problem (1) may be simplified by rewriting the fundamental component {circumflex over (V)}.sub.pcc.sup.1 using trigonometric identities as

(31) $\begin{array}{cc}\begin{array}{c}{\hat{V}}_{\mathrm{pcc}}^1\left(,{\mathrm{kT}}_s\right)=I_{22}.Math.\left[\sin \left(\right)\cos \left(\right)\right]\left[\begin{array}{c}{\hat{m}}_x\left({\mathrm{kT}}_s\right)\cos \left({\hat{}}_x\left({\mathrm{kT}}_s\right)\right)\\ {\hat{m}}_x\left({\mathrm{kT}}_s\right)\sin \left({\hat{}}_x\left({\mathrm{kT}}_s\right)\right)\\ {\hat{m}}_y\left({\mathrm{kT}}_s\right)\cos \left({\hat{}}_y\left({\mathrm{kT}}_s\right)\right)\\ {\hat{m}}_y\left({\mathrm{kT}}_s\right)\sin \left({\hat{}}_y\left({\mathrm{kT}}_s\right)\right)\end{array}\right]\\ \stackrel{}{=}{I}_{22}.Math.\left[\sin \left(\right)\cos \left(\right)\right]\hat{x}\left({\mathrm{kT}}_s\right)\end{array}& \left(2\right)\end{array}$

(32) where {circumflex over (x)}(kT.sub.s) are another set of parameters of the fundamental component {circumflex over (V)}.sub.pcc.sup.1 non-linear related to the variables {circumflex over (m)}.sub.x, {circumflex over (m)}.sub.y, {circumflex over ()}.sub.x, and {circumflex over ()}.sub.y. In (2), for any two matrices A and B the Kronecker product is defined as

(33) $A.Math.B=\left[\begin{array}{cccc}{A}_{1,1}B& A_{1,2}B& \mathrm{.Math.}& A_{1,m}B\\ A_{2,1}B& A_{2,2}B& \mathrm{.Math.}& A_{2,m}B\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}\\ A_{n,1}B& A_{n,2}B& \mathrm{.Math.}& A_{n,m}B\end{array}\right].$

(34) With this parameterization, the optimization problem (1) may be transformed to the following (almost) equivalent problem (where the weight also has been redefined)

(35) $\begin{array}{cc}\mathrm{minimize}J={}_{{\mathrm{kT}}_s-h}^{{\mathrm{kT}}_s}{.Math.V_{\mathrm{pcc}}\left(\right)-I_{22}.Math.\left[\begin{array}{cc}\sin \left(\right)& \cos \left(\right)\end{array}\right]\hat{x}\left({\mathrm{kT}}_s\right).Math.}^{2}d+{.Math.\hat{x}\left({\mathrm{kT}}_s\right)-{\hat{x}}^{*}\left(\left(k-1\right)T_s\right).Math.}^2& \left(3\right)\end{array}$

(36) Therefore, in step S12, the fundamental component {circumflex over (V)}.sub.pcc.sup.1 of the AC voltage V.sub.pcc is parametrized with the parameters {circumflex over (x)}(kT.sub.s).

(37) When one defines a cost function matrix by

(38) $H\left({\mathrm{kT}}_s\right)={}_{{\mathrm{kT}}_s-h}^{{\mathrm{kT}}_s}{I}_{22}.Math.{\left[\begin{array}{cc}\sin \left(\right)& \cos \left(\right)\end{array}\right]}^T\left[\begin{array}{cc}\sin \left(\right)& \cos \left(\right)\end{array}\right]d+I_{44}$

(39) and a cost function vector by

(40) ${f\left({\mathrm{kT}}_s\right)}^T={}_{{\mathrm{kT}}_s-h}^{{\mathrm{kT}}_s}{V_{\mathrm{pcc}}\left(\right)}^T{I}_{22}.Math.\left[\begin{array}{cc}\sin \left(\right)& \cos \left(\right)\end{array}\right]d+{{\hat{x}}^{*}\left(\left(k-1\right)T_s\right)}^T$

(41) the cost function J can be rewritten as
J={circumflex over (x)}(kT.sub.s).sup.TH(kT.sub.s){circumflex over (x)}(kT.sub.s)2(kT.sub.s).sup.T{circumflex over (x)}(kT.sub.s)+c(kT.sub.s)

(42) where c is a term not depending on the parameters {circumflex over (x)}(kT.sub.s).

(43) In such a way, the optimization problem has been cast in a standard quadratic form or quadratic program (here with trivial constraints).

(44) In the presence of further constraints, this quadratic program may be solved online by the controller 14.

(45) Alternatively, the optimal (unconstrained) solution of the quadratic program is computed by setting

(46) 0$\frac{J}{\hat{x}\left({\mathrm{kT}}_s\right)}=0,$
and is given by

(47) $\begin{array}{cc}{\hat{x}}^{*}\left({\mathrm{kT}}_s\right)=\left[\begin{array}{c}{\hat{x}}_1^{*}\left({\mathrm{kT}}_s\right)\\ {\hat{x}}_2^{*}\left({\mathrm{kT}}_s\right)\\ {\hat{x}}_3^{*}\left({\mathrm{kT}}_s\right)\\ {\hat{x}}_4^{*}\left({\mathrm{kT}}_s\right)\end{array}\right]={H\left({\mathrm{kT}}_s\right)}^{-1}f\left({\mathrm{kT}}_s\right)& \left(4\right)\end{array}$

(48) It is important to note that the inverse cost function matrix H(kT.sub.s).sup.1 exists and becomes numerically more stable as we increase the values of h and .

(49) In step S14 of the method, the optimal values {circumflex over (x)}.sup.*(kT.sub.s) of the parameters {circumflex over (x)}(kT.sub.s) of the fundamental component are determined based on minimizing the cost function J, either by solving the quadratic program online or by using the equations (4), which have been determined analytically.

(50) In step S18, after the estimation of the optimal values the fundamental component {circumflex over (V)}.sub.pcc.sup.1 and/or at least some specific quantities based on the fundamental component may be calculated from the optimal estimated values {circumflex over (x)}.sup.*. The definition of the estimated optimal values {circumflex over (x)}.sup.* in (2) of the parameters {circumflex over (x)}(kT.sub.s) may be used to obtain the amplitudes and/or phase shifts at time step kT.sub.s as

(51) ${\hat{m}}_x^{*}\left({\mathrm{kT}}_s\right)=\sqrt{{\left({\hat{x}}_1^{*}\left({\mathrm{kT}}_s\right)\right)}^2+{\left({\hat{x}}_2^{*}\left({\mathrm{kT}}_s\right)\right)}^2},{\hat{}}_x^{*}\left({\mathrm{kT}}_s\right)={\tan }^{-1}\left(\frac{{\hat{x}}_2^{*}\left({\mathrm{kT}}_s\right)}{{\hat{x}}_1^{*}\left({\mathrm{kT}}_s\right)}\right)$${\hat{m}}_y^{*}\left({\mathrm{kT}}_s\right)=\sqrt{{\left({\hat{x}}_3^{*}\left({\mathrm{kT}}_s\right)\right)}^2+{\left({\hat{x}}_4^{*}\left({\mathrm{kT}}_s\right)\right)}^2},{\hat{}}_x^{*}\left({\mathrm{kT}}_s\right)={\tan }^{-1}\left(\frac{{\hat{x}}_4^{*}\left({\mathrm{kT}}_s\right)}{{\hat{x}}_3^{*}\left({\mathrm{kT}}_s\right)}\right)$

(52) Furthermore, the estimated magnitude of the estimated vector {circumflex over (V)}.sub.xy.sup.* may be calculated by

(53) $.Math.{\hat{V}}_{\mathrm{pcc}}^{1*}.Math.=\sqrt{{\left({\hat{m}}_x^{*}\left({\mathrm{kT}}_s\right)\right)}^2{\sin }^2\left(\left({\mathrm{kT}}_s\right)+{\hat{}}_x^{*}\left({\mathrm{kT}}_s\right)\right)+{\left({\hat{m}}_y^{*}\left({\mathrm{kT}}_s\right)\right)}^2{\sin }^2\left(\left({\mathrm{kT}}_s\right)+{\hat{}}_y^{*}\left({\mathrm{kT}}_s\right)\right)}$

(54) Moreover, the corresponding estimated angle may be calculated by

(55) ${}_{\mathrm{PLL}}^{*}\left({\mathrm{kT}}_s\right)\stackrel{}{=}{\hat{V}}_{\mathrm{pcc}}^{1*}=\arctan \left(\frac{\left({\hat{m}}_y\left({\mathrm{kT}}_s\right)\right)\sin \left(\left({\mathrm{kT}}_s\right)+{\hat{}}_y\left({\mathrm{kT}}_s\right)\right)}{\left({\hat{m}}_x\left({\mathrm{kT}}_s\right)\right)\sin \left(\left({\mathrm{kT}}_s\right)+{\hat{}}_x\left({\mathrm{kT}}_s\right)\right)}\right)$

(56) In order to obtain the estimated speed of the rotation {circumflex over ()} or estimated frequency, i.e., the estimated fundamental rotational frequency, a discrete derivative of the angle may be taken, i.e.,

(57) $\hat{}=\frac{{}_{\mathrm{PLL}}^{*}\left({\mathrm{kT}}_s\right)-{}_{\mathrm{PLL}}^{*}\left(\left(k-1\right)T_s\right)}{T_s}$

(58) It is important to note that there are methods that rely on the so-called virtual flux as a quantity for control, for such methods, one can easily create a flux estimate by the following operation

(59) ${\hat{}}^1\left(t\right)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]{\hat{V}}^1\left(t\right)$

(60) Furthermore, it has to be noted that the x and y components as well as the abc components of the fundamental component {circumflex over (V)}.sub.pcc.sup.1 may be treated as separate estimation problems, i.e. they may be optimized independently from each other. In this way, the multidimensional optimization problem is split into two or more identical lower dimensional optimization problems.

(61) FIG. 3 shows a control scheme that may be implemented by the controller 14 for determining an optimal estimate {circumflex over (V)}.sub.pcc.sup.1* of the fundamental component {circumflex over (V)}.sub.pcc.sup.1.

(62) The integrator block 24 may receive the measurement signals of the AC voltage V.sub.abc and may transform it as described with respect to step S10 above.

(63) Furthermore, the integrator block 24 may receive the values of the sine and cosine functions as used in the parametrized cost function matrix H(kT.sub.s) and cost function vector (kT.sub.s) defined above with respect to step S12 and may calculate the integral parts of the cost function matrix H(kT.sub.s) and the cost function vector (kT.sub.s) by performing a discrete integration on the received values.

(64) In the optimizer block 26, which receives the weight factor and the optimal estimated parameters x.sup.*((k1)T.sub.s) of the previous time step, the final results of the cost function matrix H(kT.sub.s) and the cost function vector (kT.sub.s) are calculated, the cost function matrix H(kT.sub.s) is inverted and multiplied by the cost function vector (kT.sub.s) to calculate the optimal estimated parameters x.sup.*(kT.sub.s) of the actual time step.

(65) The optimal estimated parameters x.sup.*(kT.sub.s) are received in the post-processing block 28, which determines further quantities such as the estimated magnitude {circumflex over (V)}.sub.pcc.sup.1* and/or the estimated angle {circumflex over (V)}.sub.pcc.sup.1* of the optimal estimated fundamental component {circumflex over (V)}.sub.pcc.sup.1*.

(66) FIG. 4 shows a control scheme that may be implemented by the controller 14 for controlling the active rectifier 18 based on the optimal estimate {circumflex over (V)}.sub.pcc.sup.1* of the fundamental component {circumflex over (V)}.sub.pcc.sup.1.

(67) Estimation block 30 may perform the steps of the method of FIG. 2 and/or may implement the control scheme of FIG. 3.

(68) The estimated magnitude {circumflex over (V)}.sub.pcc.sup.1* of the optimal estimated fundamental component {circumflex over (V)}.sub.pcc.sup.1* is used for modifying a reference voltage V.sub.aru.sup.* for the rectifier 20.

(69) The reference voltage V.sub.aru.sup.* is based on a difference between a DC link current, which is determined from an active current and I.sub.d.sup.* a reactive current I.sub.q.sup.*, and a shifted converter current, which is determined based on a converter current and the estimated angle {circumflex over (V)}.sub.pcc.sup.1* of the optimal estimated fundamental component.

(70) 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

(71) 10 converter system 12 electrical converter 14 controller 16 electrical grid/source 18 transformer 20 rectifier 22 DC link V.sub.pcc AC voltage V.sub.pcc.sup.1 fundamental component {circumflex over (V)}.sub.pcc.sup.1 fundamental component to be optimized J cost function h horizon {circumflex over (x)}(kT.sub.s) parameters of the fundamental component H(kT.sub.s) cost function matrix (kT.sub.s) cost function vector {circumflex over (V)}.sub.pcc.sup.1* estimated fundamental component {circumflex over ()} estimated speed of the rotation 24 integrator block 26 optimizer block 28 post-processing block 30 estimation block