Determining Thevenin equivalent model for a converter system

Abstract

A method for determining a converter Thevenin equivalent model for a converter system, includes: receiving measurement values of a coupling point voltage and a coupling point current measured at a point of common coupling between a grid emulator system and the converter system, wherein the grid emulator system supplies the converter system with a supply voltage; and determining a converter Thevenin impedance and a converter Thevenin voltage source of the converter Thevenin equivalent model by inputting the measurement values of the coupling point voltage and of the coupling point current into a coupled system model, which includes equations modelling the converter system and the grid emulator system and from which the converter Thevenin impedance and a converter Thevenin voltage source are calculated.

Claims

1. A method for determining a converter Thevenin equivalent model for a converter system, the method comprising: receiving measurement values of a coupling point voltage and of a coupling point current measured at a point of common coupling between a grid emulator system and the converter system, wherein the grid emulator system supplies the converter system with a supply voltage; determining a converter Thevenin impedance and a converter Thevenin voltage source of the converter Thevenin equivalent model by inputting the measurement values of the coupling point voltage and of the coupling point current into a coupled system model, which comprises equations modelling the converter system and the grid emulator system and from which the converter Thevenin impedance and the converter Thevenin voltage source are calculated; and using the converter Thevenin impedance and the converter Thevenin voltage in the converter Thevenin equivalent model to optimize operation of a large-scale electrical grid; wherein the grid emulator system is modelled with a set of grid emulator elements, each of which has grid emulator parameters; wherein a reduced coupled system model is calculated by determining a first coupled system model with grid elements having a first set of grid emulator parameters and a second coupled system model with grid elements having a second set of grid emulator parameters and by analytically eliminating the grid emulator parameters by putting equations of the second coupled system model into equations of the first coupled system model; and wherein the measurement values are input into the reduced coupled system model.

2. The method of claim 1, wherein the grid emulator parameters comprise at least one of: a grid voltage source; a grid series impedance interconnecting the grid voltage source with the point of common coupling; and a grid shunt impedance connected to the point of common coupling.

3. The method of claim 1, wherein the grid emulator system comprises an adjustable electrical component, such that the grid emulator parameters are changed, when the adjustable electrical component is adjusted; wherein two sets of measurement values of the coupling point voltage and of the coupling point current are determined with the adjustable electrical component adjusted to different settings; wherein the two sets of measurement values are input into the reduced system model.

4. The method of claim 3, wherein the adjustable electrical component comprises at least one of: a filter circuit with an exchangeable capacitor and/or exchangeable inductor; and a converter with an adjustable modulation scheme.

5. The method of claim 4, wherein at least three sets of measurement values of the coupling point voltage and of the coupling point current are determined with the adjustable electrical component adjusted to at least three different settings; wherein pairs of sets of measurement values are generated by combining two different sets of measurement values; wherein the two different sets set of measurement values of each pair are input into the reduced system model to produce a converter Thevenin impedance and a converter Thevenin voltage source for each pair; and wherein the converter Thevenin impedance and the converter Thevenin voltage source for the converter Thevenin equivalent model is determined by averaging the converter Thevenin impedance and the converter Thevenin voltage source for each pair.

6. The method of claim 5, wherein in the coupled system model, the grid emulator system is modelled with a grid Thevenin equivalent model, which comprises a grid Thevenin impedance and a grid Thevenin voltage source.

7. The method of claim 6, wherein the grid Thevenin impedance and the grid Thevenin voltage source are determined from known parameters of electrical components of the grid emulator system.

8. The method of claim 7, wherein the grid emulator system comprises an adjustable electrical component, such that the grid emulator parameters are changed, when the adjustable electrical component is adjusted; wherein a plurality of sets of measurement values of the coupling point voltage and of the coupling point current are determined with the adjustable electrical component adjusted to different settings; wherein from the plurality of sets of measurement values, a plurality of intermediate converter Thevenin impedances and intermediate converter Thevenin voltage sources are determined; and wherein a final converter Thevenin impedance is determined by eliminating outlier values from the intermediate converter Thevenin impedances at different frequency values and by averaging the intermediate converter Thevenin impedances.

9. The method of claim 1, wherein the grid emulator system comprises an electrical converter connected to an electrical grid, which electrical converter is adapted for converting a grid voltage from the electrical grid into the supply voltage to be supplied to the converter system; wherein the grid emulator system comprises a transformer connected between an output of the electrical converter and the point of common coupling; and wherein the grid emulator system comprises an electrical filter connected to the point of common coupling.

10. The method of claim 1, wherein the measurement values of the coupling point voltage and of the coupling point current are Fourier transformed before being input into the coupled system model; wherein the converter Thevenin impedance and the converter Thevenin voltage source are calculated with respect to a set of frequencies; and wherein a grid Thevenin impedance and a grid Thevenin voltage source are provided with respect to a set of frequencies.

11. The method of claim 3, wherein at least three sets of measurement values of the coupling point voltage and of the coupling point current are determined with the adjustable electrical component adjusted to at least three different settings; wherein pairs of sets of measurement values are generated by combining two different sets of measurement values; wherein the two different sets set of measurement values of each pair are input into a reduced system model to produce a converter Thevenin impedance and a converter Thevenin voltage source for each pair; and wherein the converter Thevenin impedance and the converter Thevenin voltage source for the converter Thevenin equivalent model is determined by averaging the converter Thevenin impedance and the converter Thevenin voltage source for each pair.

12. The method of claim 11, wherein in the coupled system model, the grid emulator system is modelled with a grid Thevenin equivalent model, which comprises a grid Thevenin impedance and a grid Thevenin voltage source.

13. The method of claim 1, wherein in the coupled system model, the grid emulator system is modelled with a grid Thevenin equivalent model, which comprises a grid Thevenin impedance and a grid Thevenin voltage source.

14. The method of claim 13, wherein the grid Thevenin impedance and the grid Thevenin voltage source are determined from known parameters of electrical components of the grid emulator system.

15. The method of claim 12, wherein the grid Thevenin impedance and the grid Thevenin voltage source are determined from known parameters of electrical components of the grid emulator system.

16. The method of claim 1, wherein the grid emulator system-comprises an adjustable electrical component, such that the grid emulator parameters are changed, when the adjustable electrical component is adjusted; wherein a plurality of sets of measurement values of the coupling point voltage and of the coupling point current are determined with the adjustable electrical component adjusted to different settings; wherein from the plurality of sets of measurement values, a plurality of intermediate converter Thevenin impedances and intermediate converter Thevenin voltage sources are determined; and wherein a final converter Thevenin impedance is determined by eliminating outlier values from the plurality of intermediate converter Thevenin impedances at different frequency values and by averaging the plurality of intermediate converter Thevenin impedances.

17. The method of claim 12, wherein the grid emulator system-comprises an adjustable electrical component, such that the grid emulator parameters are changed, when the adjustable electrical component is adjusted; wherein a plurality of sets of measurement values of the coupling point voltage and of the coupling point current are determined with the adjustable electrical component adjusted to different settings; wherein from the plurality of sets of measurement values, a plurality of intermediate converter Thevenin impedances and intermediate converter Thevenin voltage sources are determined; and wherein a final converter Thevenin impedance is determined by eliminating outlier values from the plurality of intermediate converter Thevenin impedances at different frequency values and by averaging the plurality of intermediate converter Thevenin impedances.

18. A computer program stored on a non-transitory medium for determining a converter Thevenin equivalent model for a converter system, which, when being executed on a processor, is adapted for: receiving measurement values of a coupling point voltage and of a coupling point current measured at a point of common coupling between a grid emulator system and the converter system, wherein the grid emulator system supplies the converter system with a supply voltage; determining a converter Thevenin impedance and a converter Thevenin voltage source of the converter Thevenin equivalent model by inputting the measurement values of the coupling point voltage and of the coupling point current into a coupled system model, which comprises equations modelling the converter system and the grid emulator system and from which the converter Thevenin impedance and the converter Thevenin voltage source are calculated; and using the converter Thevenin impedance and the converter Thevenin voltage in the converter Thevenin equivalent model to optimize an operation of a large-scale electrical grid; wherein the grid emulator system is modelled with a set of grid emulator elements, each of which has grid emulator parameters; wherein a reduced coupled system model is calculated by determining a first coupled system model with grid elements having a first set of grid emulator parameters and a second coupled system model with grid elements having a second set of grid emulator parameters and by analytically eliminating the grid emulator parameters by putting the equations of second coupled system model into equations of the first coupled system model; and wherein the measurement values are input into the reduced coupled system model.

19. An evaluation device for determining a converter Thevenin equivalent model for a converter system, wherein the evaluation device is adapted for: receiving measurement values of a coupling point voltage and of a coupling point current measured at a point of common coupling between a grid emulator system and the converter system, wherein the grid emulator system supplies the converter system with a supply voltage; determining a converter Thevenin impedance and a converter Thevenin voltage source of the converter Thevenin equivalent model by inputting the measurement values of the coupling point voltage and of the coupling point current into a coupled system model, which comprises equations modelling the converter system and the grid emulator system and from which the converter Thevenin impedance and the converter Thevenin voltage source are calculated; and using the converter Thevenin impedance and the converter Thevenin voltage in the converter Thevenin equivalent model to optimize an operation of a large-scale electrical grid; wherein the grid emulator system is modelled with a set of grid emulator elements, each of which has grid emulator parameters; wherein a reduced coupled system model is calculated by determining a first coupled system model with grid elements having a first set of grid emulator parameters and a second coupled system model with grid elements having a second set of grid emulator parameters and by analytically eliminating the grid emulator parameters by putting equations of the second coupled system model into equations of the first coupled system model; and wherein the measurement values are input into the reduced coupled system model.

20. A test system, comprising: a grid emulator system for supplying the converter system with a supply voltage; and an evaluation device according to claim 14.

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 test system according to an embodiment of the invention.

(3) FIG. 2 shows an equivalent circuit for the test system of FIG. 1.

(4) FIG. 3 shows a diagram with curves for a Thevenin equivalent model of the converter system of FIG. 1.

(5) FIG. 4 shows a simplified equivalent circuit for the test system of FIG. 1.

(6) FIG. 5 shows a flow diagram for determining a Thevenin equivalent model according to an embodiment of the invention.

(7) FIG. 6 shows a diagram with curves of a Thevenin equivalent model determined with the method of FIG. 5.

(8) FIG. 7 shows a diagram with curves of a further Thevenin equivalent model determined with the method of FIG. 5.

(9) 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

(10) FIG. 1 shows a test system 10 comprising a converter system 12 and a grid emulator system 14. The converter system 12 comprises an electrical drive 16, which is composed of a converter 18 and a rotating electrical machine 20. The electrical drive 16 is mechanically connected to a prime mover 22, such as a turbine. Fluctuating power may be exchanged between the prime mover 22 and the electrical drive 16. However, the converter system 12 also com comprise other types of converters and/or may be used for other purposes, such as interconnecting two electrical grids or being feed by a photovoltaic system.

(11) Due to test reasons, the converter system 12 and in particular the electrical drive 16 are supplied by the grid emulator system 14 with an AC supply voltage v.sub.m and an AC supply current i.sub.m. The grid emulator system 14 and the converter system 12 are connected via a point of common coupling 24.

(12) The grid emulator system 14 itself is supplied by an electrical grid 26. The grid emulator system 14 comprises an optional input transformer 28 connected to the grid 26, a converter 30 and an output transformer 32, which are all series connected between the electrical grid 26 and the point of common coupling 24. The point of common coupling 24 is earthed via a filter 34, which may comprise capacitors 36 and/or inductors 38.

(13) The converter 30 may comprise an AC-DC converter 40, a DC link 42 and a DC-AC converter 44. The converter 30 may be of two level, three level or multilevel type. The converter 30 may be controlled with a controller 46, which may be adapted for generating switching signals for semiconductor switches of the converter 30. An evaluation device 48 may receive measurement values of the supply voltage v.sub.m and the supply current i.sub.m at the point of common coupling 24 and may determine a Thevenin equivalent model of the converter system 12 as described above and below.

(14) FIG. 2 shows an equivalent circuit 50 for the test system 10. Here, the left side corresponds to the grid emulator system 14 modelled by the series element z.sub.s(ω), which models the transformer 32, and a shunt element z.sub.f(ω), which models the filter 34 and the voltage source v.sub.s(ω), which models the converter 30.

(15) The right side corresponds to the converter system 12 modelled with a Thevenin equivalent model 52, which is composed of a Thevenin voltage source v.sub.d(ω) and a Thevenin impedance z.sub.d(ω).

(16) Note that all these parameters are provided in the frequency domain with respect to the frequency ω.

(17) FIG. 3 shows a diagram with the absolute values of a converter Thevenin voltage source v.sub.d(ω) and a converter Thevenin impedance z.sub.d(ω), expected for a typical converter system 12. The converter Thevenin impedance z.sub.d(ω) is a continuous curve peaked at a specific value, wherein the Thevenin voltage source v.sub.d(ω) comprises values different from 0 at discrete frequencies, which correspond to the higher order harmonics of the converter system 12. The Thevenin equivalent model 52 is a commonly used model for the analysis of the steady state impact of converter harmonics on the grid 26.

(18) In general, the Thevenin voltage source v.sub.d(ω) and the Thevenin impedance z.sub.d(ω) may be modelled for a discrete set of frequencies and by a complex number for each frequency. The Thevenin voltage source v.sub.d(ω) may be seen as a harmonic injection at various frequencies with different magnitudes.

(19) The Thevenin voltage source v.sub.d(ω) may strongly depend on the modulation strategy, converter topology of the converter 18 but also may be influenced by passive filter components, such as 34.

(20) Also, the Thevenin impedance z.sub.d(ω) may be strongly influenced by passive filter components, such as 34, but also may be influenced by the control system and sampling delays introduced. With the method as described above and below, an accurate determination of the parameters v.sub.d(ω), z.sub.d(ω) of the Thevenin equivalent model 52 of the converter system 12 may be performed with the test system shown in FIG. 1.

(21) As the hardware of the grid emulator system 14 is known, the parameters z.sub.s(ω), z.sub.f(ω) and v.sub.s(ω) of FIG. 2 may be considered known and an equivalent Thevenin model 54 of the grid emulator can be derived as follows, resulting in the simplified equivalent circuit 56 shown in FIG. 4

(22) $v_{th}^s\left(\omega \right)=\frac{z_f\left(\omega \right)}{z_f\left(\omega \right)+z_s\left(\omega \right)}{v}_s\left(\omega \right)z_{th}^s\left(\omega \right)=\frac{z_f\left(\omega \right)z_s\left(\omega \right)}{z_f\left(\omega \right)+z_s\left(\omega \right)}$

(23) Here, the Thevenin voltage source v.sub.th.sup.s(ω) and the Thevenin impedance z.sub.th.sup.s(ω) of the grid emulator system 14 have been introduced.

(24) The measurements of the voltage v.sub.m(t) and the current i.sub.m(t) at the point of common coupling 24, which are time-dependent signals, can be transformed into the Frequency domain through a discrete Fourier transform, resulting in the frequency dependent voltage v.sub.m(ω) and current i.sub.m(ω). A sample rate for the measurements may be of at least twice but preferably 5 times the desired frequency range of the models, such as a sample rate of 10-50 kHz. It also may be beneficial to perform the measurement during stationary operating conditions.

(25) The Fourier transformed measurements v.sub.m(ω), i.sub.m(ω) will have contributions from both the grid emulator system 14 and from the converter system 12.

(26) $v_m\left(\omega \right)=\stackrel{\stackrel{\mathrm{grid}\mathrm{emulator}\mathrm{contribution}}{︷}}{v_{th}^s\left(\omega \right)\frac{z_d\left(\omega \right)}{z_{th}^s\left(\omega \right)+z_d\left(\omega \right)}}+\stackrel{\stackrel{\mathrm{converter}\mathrm{contribution}}{︷}}{v_d\left(\omega \right)\frac{z_{th}^s\left(\omega \right)}{z_{th}^s\left(\omega \right)+z_d\left(\omega \right)}}$$i_m\left(\omega \right)=\underset{\underset{\mathrm{grid}\mathrm{emulator}\mathrm{contribution}}{︸}}{-\frac{v_{th}^s\left(\omega \right)}{z_{th}^s\left(\omega \right)+z_d\left(\omega \right)}}+\frac{v_d\left(\omega \right)}{\underset{\underset{\mathrm{converter}\mathrm{contribution}}{︸}}{z_{th}^s\left(\omega \right)+z_d\left(\omega \right)}}$

(27) Unless compensated for, the harmonics introduced by the grid emulator system 14 usually will introduce bias and other inaccuracies in an estimation of the unknown Thevenin equivalent parameters v.sub.d(ω), z.sub.d(ω).

(28) With respect to FIG. 5, a method is described, with which the converter Thevenin voltage source v.sub.d(ω) and the converter Thevenin impedance z.sub.d(ω) may be simultaneously and accurately determined. Background harmonics introduced by the grid emulator system 14 are compensated. The method may be automatically performed by the evaluation device 48.

(29) In step S10, measurement values of the coupling point voltage v.sub.m and of the coupling point current i.sub.m measured at the point of common coupling 24 are determined, wherein the grid emulator system 14 supplies the converter system 12 with a supply voltage. The grid emulator system 14 may emulate an electrical grid by generating a specific frequency and a specific output voltage magnitude. The measurement values of the coupling point voltage v.sub.m and of the coupling point current i.sub.m may be Fourier transformed into Fourier transformed quantities v.sub.m(ω), i.sub.m(ω) before being further processed in step S14 and/or step S16.

(30) For the steps S14 and S16, several sets of measurement values may be needed, which may be generated with respect to different operating points of the grid emulator system 14. Therefore, in step S12, the grid emulator system 14 may be changed, such that it operates at a different operation point. This may be done automatically by the evaluation device 48, which may instruct the converter 30 to change its modulation scheme, for example. It also may be that the grid emulator system 14 is modified manually, by exchanging a component, such as 36, 38.

(31) The grid emulator system 14 may comprise for this an adjustable electrical component, such as the converter 30 and/or the filter 34, such that at least some of the grid emulator parameters z.sub.s(ω), z.sub.f(ω) and v.sub.s(ω) are changed, when the electrical component 30, 34 is adjusted. For example, the adjustable component may be at least one of a filter circuit 34 with an exchangeable capacitor 36 and/or exchangeable inductor 38. The adjustable component also may be a converter 30 with an adjustable modulation scheme.

(32) After the adjustment of the grid emulator system 14, the method may continue in step S10 and a further set of measurement values may be generated. It may be that a plurality of sets of measurement values of the coupling point voltage and of the coupling point current are determined with the electrical component adjusted to different settings.

(33) In the steps S14 and S16, which may be performed both or only one of them, from the measurement values at least one converter Thevenin impedance v.sub.d(ω) and at least one converter Thevenin voltage source v.sub.d(ω) is determined.

(34) In step S14, these quantities are determined with a model 58 of the coupled system, which comprises parameters of the grid emulator system 14. In step S16, these quantities are determined with a reduced model 60, where at least some of the parameters of the grid emulator system 14 have been eliminated.

(35) In general, the coupled system model 58 and the reduced coupled system model 60 comprise equations modelling the converter system 12 and the grid emulator system 14 and from which the converter Thevenin impedance z.sub.d(ω) and the converter Thevenin voltage source v.sub.d(ω) are calculated. Both models 58, 60 may comprise the functions implemented as software routines, in which the measurement values are input and the quantities z.sub.d(ω) and v.sub.d (ω) are output. Note that the converter Thevenin impedance v.sub.d(ω) and the converter Thevenin voltage source v.sub.d(ω) may be calculated with respect to a set of frequencies, such as the frequency bins determined during Fourier transform of the measurement values.

(36) For deriving the coupled system model 58, based on the simplified equivalent circuit 56 in FIG. 4, Kirchhoff's laws gives the following relations:

(37) $v_m\left(\omega \right)=v_d\left(\omega \right)+i_m\left(\omega \right)z_d\left(\omega \right)i_m\left(\omega \right)=\frac{v_d\left(\omega \right)-v_{th}^s\left(\omega \right)}{z_{th}^s\left(\omega \right)+z_d\left(\omega \right)}$

(38) From these equations, the expressions for the unknown quantities can be solved:

(39) $v_d\left(\omega \right)=\frac{v_m\left(\omega \right)+v_{th}^s\left(\omega \right)+i_m\left(\omega \right)z_{th}^s\left(\omega \right)}{2}{z}_d\left(\omega \right)=\frac{v_{th}^s\left(\omega \right)-v_m\left(\omega \right)+i_m\left(\omega \right)z_{th}^s\left(\omega \right)}{-2i_m\left(\omega \right)}$
Thus, the unknown quantities v.sub.d (ω), z.sub.d (ω) may be found from a single set of measurement values v.sub.m(ω) and i.sub.m(ω), provided the parameters v.sub.th.sup.s (ω), z.sub.th.sup.s(ω) of the grid emulator system Thevenin equivalent 54 known. The above equations for v.sub.d(ω), z.sub.d(ω) may be seen as the coupled system model 58.

(40) As the grid emulator system 14 may be part of the laboratory equipment, it may be manufactured using high quality components, such that its parameters may be considered nearly perfectly known. For the case of a grid emulator system 14, for example with an LC filter 34, as shown in FIG. 1, the Thevenin impedance z.sub.th.sup.s(ω) of grid emulator system 14 can be calculated as

(41) $z_{th}^s\left(\omega \right)=\frac{z_f\left(\omega \right)z_s\left(\omega \right)}{z_f\left(\omega \right)+z_s\left(\omega \right)}$

(42) Furthermore, the converter voltage v.sub.s(ω) can be measured in the grid emulator system 14, the Thevenin voltage source v.sub.th.sup.s(ω) can be calculated as

(43) $v_{th}^s\left(\omega \right)=\frac{z_f\left(\omega \right)}{z_f\left(\omega \right)+z_s\left(\omega \right)}{v}_s\left(\omega \right)$

(44) This may be sensitive to noise, and while it may provide satisfactory results, it may be beneficial to estimate the unknown parameters v.sub.d(ω), z.sub.d(ω) using multiple sets of measurement values, for example as discussed below.

(45) In summary, in the coupled system model 58, the grid emulator system 14 may be modelled with a grid Thevenin equivalent model 54, which comprises a grid Thevenin impedance z.sub.th.sup.s(ω) and a grid Thevenin voltage source v.sub.th.sup.s(ω). The grid Thevenin impedance z.sub.th.sup.s(ω) and the grid Thevenin voltage source v.sub.th.sup.s (ω) may be determined from known parameters of electrical components 30, 32, 34 of the grid emulator system 14. Note that the grid Thevenin impedance z.sub.th.sup.s(ω) and the grid Thevenin voltage source v.sub.th.sup.s(ω) may be provided with respect to a set of frequencies a.

(46) In step S16, the unknown quantities v.sub.d(ω), z.sub.d(ω) are determined with the reduced coupled system model 60. In this case, at least two sets of measurement values have to be determined in step S10 with respect to different operation points of the grid emulator system 14.

(47) When one assumes that the parameters v.sub.d(ω), z.sub.d(ω) of the converter Thevenin equivalent model 52 stay the same for the two sets of measurement values, Kirchhoff's equations for the equivalent circuit 50 in FIG. 2 can be used to solve for the unknown Thevenin parameters v.sub.d(ω) and z.sub.d(ω).
−v.sub.s1(ω)−z.sub.s1(ω)i.sub.c1(ω)+z.sub.f1(ω)(i.sub.m1(ω)−i.sub.c1(ω))=0
−z.sub.f1(ω)(i.sub.m1(ω)−i.sub.c1(ω))−i.sub.m1(ω)z.sub.d(ω)+v.sub.d(ω)=0
i.sub.m1(ω))−i.sub.c1(ω)−v.sub.m1(ω)/z.sub.f1(ω)=0
−v.sub.s2(ω)−z.sub.s2(ω)i.sub.c2(ω)+z.sub.f2(ω)(i.sub.m2(ω)−i.sub.c2(ω)=0
−z.sub.f2(ω)(i.sub.m2(ω)−i.sub.c2(ω))−i.sub.m2(ω)z.sub.d(ω)+v.sub.d(ω)=0
i.sub.m2(ω)−i.sub.c2(ω)−v.sub.m2(ω)/z.sub.f2(ω)=0

(48) The indices 1 and 2 refer to the two different measurement passes, i.e. different sets of measurement values and/or the two different operation points. Here, changes in the filter circuit would imply that z.sub.s1(ω)≠z.sub.s2(ω) or z.sub.f1(ω)≠z.sub.f2(ω), and changes in the modulation strategy, switching frequency, DC link chopper or AC chopper units of the grid emulator converter would imply that v.sub.s1(ω)≠v.sub.s2(ω). To make the equations linearly independent, at least some of these parameters need to be different between the different measurement passes, for every frequency value.

(49) Based on this extended set of equations, expressions for the unknown quantities v.sub.d(ω), z.sub.d(ω) can be solved, yielding

(50) $v_d\left(\omega \right)=\frac{i_{m1}\left(\omega \right)v_{m2}\left(\omega \right)-i_{m2}\left(\omega \right)v_{m1}\left(\omega \right)}{i_{m1}\left(\omega \right)-i_{m2}\left(\omega \right)}{z}_d\left(\omega \right)=\frac{v_{m2}\left(\omega \right)-v_{m1}\left(\omega \right)}{i_{m1}\left(\omega \right)-i_{m2}\left(\omega \right)}$

(51) Note that, compared to the case of step S14, the grid emulator Thevenin parameters do not occur in the equations and thus, this approach may be used also when those parameters are not perfectly known, which may be an advantage. The above equations for v.sub.d(ω), z.sub.d(ω) may be seen as the reduced coupled system model 60.

(52) In summary, the reduced coupled system model 60 is calculated by determining a first coupled system model with the grid elements having a first set of grid emulator parameters and a second coupled system model with the grid elements having a second set of grid emulator parameters and by analytically eliminating the grid emulator parameters by putting the equations of the second coupled system model into the equations of the first coupled system model. The two sets of measurement values, which are input into the reduced system model 60, are then determined with the electrical component adjusted to different settings.

(53) If a number n of measurement passes is made, those can be paired 1 by 1 in

(54) $\hspace{1em}\left(\begin{array}{c}n\\ 2\end{array}\right)$
combinations. For example, if 5 separate experiments with different parameters for the grid emulator system 14 have been made, those can be combined in

(55) $\left(\begin{array}{c}5\\ 2\end{array}\right)=10$
unique combinations of pairs. At least three sets of measurement values of the coupling point voltage v.sub.m and of the coupling point current i.sub.m may be determined with the electrical component 30, 34 adjusted to at least three different settings. Pairs of sets of measurement values may be generated by combining two different sets of measurement values. The two sets of measurement values of each pair then may be input into the reduced system model 60 to produce a converter Thevenin impedance z.sub.d(ω) and a converter Thevenin voltage source v.sub.d(ω) for each pair.

(56) After steps S14 and S16, a plurality of intermediate converter Thevenin impedances z.sub.d(ω) and intermediate converter Thevenin voltage sources v.sub.d(ω) are present.

(57) FIG. 6 shows an example of an intermediate converter Thevenin impedance z.sub.d(ω) and an intermediate converter Thevenin voltage source v.sub.d(ω), which may have been produced as described above. As can be seen, in particular, the intermediate converter Thevenin impedance z.sub.d(ω) is rather noise and may be further processed to receive a smooth curve.

(58) In step S18, the intermediate converter Thevenin impedances z.sub.d(ω) and intermediate converter Thevenin voltage sources v.sub.d(ω) may be statistically evaluated to generate the final converter Thevenin impedance z.sub.d(ω) and the final converter Thevenin voltage source v.sub.d (ω), such as shown in FIG. 7. With such statistical method, noise in the results may be reduced.

(59) Statistical evaluation may comprise averaging, outlier elimination and smoothing.

(60) The final converter Thevenin impedance z.sub.d(ω) and/or the final converter Thevenin voltage source v.sub.d(ω) may be determined by averaging the intermediate converter Thevenin impedances z.sub.d(ω) and the intermediate converter Thevenin voltage sources v.sub.d(ω), for example such as determined for each pair of measurement values in step S16. Averaging may be done for by averaging values at the same frequency.

(61) Alternatively or additionally, the final converter Thevenin impedance z.sub.d(ω) may be determined by eliminating outlier values from the intermediate converter Thevenin impedances z.sub.d(ω). This may be done by eliminating outlier values at the same frequency values.

(62) In a final step, the Thevenin impedance z.sub.d(ω) may be smoothed. For example, a Butterworth low pass filter may be applied to smoothen the Thevenin impedance z.sub.d(ω), such as shown in the lower diagram of FIG. 7. Bidirectional filtering may be used to ensure that there is no frequency shift of the Thevenin impedance as a result of smoothing.

(63) 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

(64) 10 test system 12 converter system 14 grid emulator system 16 electrical drive 18 drive converter 20 rotating electrical machine 22 prime mover v.sub.m AC supply voltage i.sub.m AC supply current 24 point of common coupling 26 electrical grid 28 input transformer 30 emulator converter 32 output transformer 34 electrical filter 36 capacitor 38 inductor 40 AC-DC converter 42 DC link 44 DC-AC converter 46 controller 48 evaluation device 50 equivalent circuit ω frequency z.sub.s(ω) series element for grid emulator model z.sub.f(ω) shunt element for grid emulator model v.sub.s(ω) voltage source for grid emulator model 52 converter Thevenin equivalent model v.sub.d(ω) converter Thevenin voltage source z.sub.d(ω) converter Thevenin impedance v.sub.th.sup.s(ω) grid emulator Thevenin voltage source z.sub.th.sup.s(ω) grid emulator Thevenin impedance v.sub.m(ω) Fourier transformed voltage measurements i.sub.m(ω) Fourier transformed current measurements 54 equivalent Thevenin model of the grid emulator 56 simplified equivalent circuit 58 coupled system model 60 reduced coupled system model