Oversampling (resampling) data in harmonic data processing

When using a rectangular window, it is important to synchronize the measurement window accurately with the power system frequency to achieve integer multiple of periods in analysed time series (i.e. window period is and integer multiple of analysed frequency component natural period). For example, if the power system frequency is 50.2 Hz whereas the window size is 200 ms, the fundamental frequency spectral line of the discrete Fourier transform is no longer projected (represented) by one complex vector in the orthogonal basis but spanned over the whole basis. One can even say that the power system frequency in the estimated spectrum becomes an interharmonic with spectral leakage as a consequence. Therefore appropriate data should be prepared before the projection into frequency domain.

In order to limit possible data processing errors the signal should be adjusted before spectral analysis. Power system frequency variation can be limited by application of various interpolation methods. It is strongly recommended to always oversample the analysed signal. In this particular case acquired waveforms with sample rate of 44.1 kS/s/ch are oversampled up to 51.2 kS/s/ch. According to this procedure 1024 samples per cycle is obtained, which is an integer power of 2 and can be used to apply fast Fourier transform. Please note that for 10-cycle data blocks discrete Fourier transform should be applied.

Frequency estimation
Figure 1 Single tone frequency. (a) Measured power system frequency variation. (b) Frequency variation after resampling (spline interpolation).

Oversampling of each 10-cycle data blocks before spectrum estimation significantly limits fundamental frequency variation over analysed window and therefore the stationarity assumption is strengthened for frequency component of interest (i.e. linked with the fundamental frequency). As it can be seen in Figure 1 resampling with high precision definitely improves signal quality. The figure presents analysed one minute of acquired voltage waveform and each point is an estimated fundamental frequency over 10-cycle rectangular window. The discrepancy as a difference between two measured values, given in Figure 1(b), is acceptably small and the sample standard deviation is 4.3548•10-6. Please note that scale 0.2s used on abscissa (i.e. horizontal axis) in Figure 1 denotes the step size between neighbouring samples. If there are 300 samples in presented results (as in Figure 1) the total duration is 1 minute (i.e. 0.2s∗300=60s).

It can be seen that both factors such as main tone detection algorithm as well as interpolation algorithm are crucial in appropriate harmonic magnitude and phase estimation. Various interpolation techniques can give different results. The most sophisticated unfortunately are usually the most time consuming. Therefore taking into consideration enormous amount of data to process from few months of multipoint measurements also the interpolation should be optimized and agreement between acceptable accuracy and calculation speed should be found.

Roughly speaking, during interpolation process new samples between existing samples are computed. Different methods were used in data processing such as nearest (coerce) method, linear method, spline method, cubic method, and finite impulse response filter method [1]. The nearest method finds the point nearest to xi  in an X array and then assigns the corresponding y value in an interpolated Y to yi. The linear method interpolates yi on the line which connects the two points (xj,xj+1) when the corresponding location xi of interpolated value yi which is located between the two points in an X array. The spline method known also as cubic spline method is almost always the most suitable solution. Within the method the third-order polynomial for each interval between two adjacent points in X (xj,xj+1) is derived. The polynomials have to meet the following conditions: the first and second derivatives at xj and xj+1 are continuous, the polynomials pass all the specified data points, the second derivatives at the beginning and end are zero. The cubic Hermitian spline method is the piecewise cubic Hermitian interpolation. This method is similar to cubic spline interpolation and derives a third-order polynomial but in Hermitian form for each interval (xj,xj+1) and ensures only the first derivatives of the interpolation polynomials are continuous. Compared to the cubic spline method, the cubic Hermitian method is characterized by better local property. The cubic Hermite interpolation method guarantees that the first derivative of the interpolant is continuous and sets the derivative at the endpoints in order to preserve the original shape and monotonicity of the data [2]. Also interpolation based on finite impulse response filter is popular [3], [4]. This method provides excellent results also for frequency analysis, although it is more intensive computationally [2].

Order Interpolation method Computation time [ms] Marker
1 Linear 110 x
2 Coerce 70 *
3 Cubic spline 130
4 Hermitian spline 430 +
5 FIR filter 290

Table above shows the computation time of data interpolation with different algorithms. In each of the cases the presented time is measured for resampling of 10 cycles of measured data. It can be seen that the simplest algorithms exhibit less computation burden. On the other hand in many cases simple algorithms cannot give satisfactory interpolation results and thus affect the harmonic calculation results. Figure 2 shows how different interpolation techniques affect harmonic magnitude estimation from measurements carried out at the LV side of the wind turbine transformer. It can be seen that the coerce method is affected the worst while other algorithms give comparable results.

Oversampled measurements
Figure 2 Different interpolation techniques used in oversampling of waveforms measured at LV side of the wind turbine transformer.

It was observed that in case of more distorted waveforms (e.g. gird-side converter output voltage) interpolation method choice plays a crucial role. In Figure 3 one can see that also linear interpolation and cubic Hermitian spline interpolation do not give satisfactory results. Therefore the most suitable for harmonic components estimation appear to be cubic spline interpolation as well as finite impulse response filter interpolation. Since the spline method (as presented in table above) is less time consuming, this method was used in further data processing. The selected measurement period was selected when the power system frequency were varying below 50 Hz or above 50 Hz.

Oversampled measurements
Figure 3 Different interpolation techniques used in oversampling of waveforms measured at the grid-side converter AC terminals.

[1] S. C. Chapra and R. Canale, Numerical Methods for Engineers: With Software and Programming Applications, 6th ed. McGraw-Hill Science, 2009.
[2] National Instruments, "LabVIEW 2011 Help," National Instruments Manual, 2011.
[3] R. A. Losada, "Digital Filters with MATLAB," The Mathworks, 2008.
[4] R. W. Schafer and L. R. Rabiner, "A digital signal processing approach to interpolation," Proceedings of the IEEE, vol. 61, no. 6, pp. 692-702, Jun. 1973.


Commercial power quality meters

The use of a rectangular window requires that the measurement window is synchronized with the actual power system frequency, hence the use of a 10-cycle window instead of a window of exactly  200 ms. The IEC standard [1] requires that 10 cycles correspond with an integer number of samples within 0.03%. To ensure synchronism between the measurement window and the power system frequency, most power quality meters use a phase-locked loop generating a sampling frequency that is an integer multiple of the actual power system frequency. One of the commonly used in the company power quality meters is Elspec G4500 which provides full functionality regarding measurements of power quality. An exemplary connection of such measurement equipment in there phase system is presented in Figure 1 [2].

Elspec G4500 connection
Figure 1 Exemplary connection of Elspec equipment in wind turbines.

Before power quality indices are calculated by the Elspec equipment, acquired data is processed. Processing stage comprises of fundamental frequency detection, sample rate adjustment according to the detected frequency, and Fourier decomposition applied. The approach of sample rate adjustment is to adjust the finite orthogonal Hilbert basis in order to express each of frequency components in the Fourier space only by one vector from the Hilbert basis. The whole data acquisition, processing and logging process is briefly presented in Figure 2.

A synchronization (i.e. sample rate adjustment according to the fundamental frequency) error leads to cross-talk between different harmonic frequencies. The 50 hz component is by far the dominating component in most cases so that the main concern is the cross-talk from the 50 hz component to higher order components. From the other side, resampling process can affect frequency components which are not integer multiple of the fundamental frequency.

This phenomenon can be easily seen in the spectrum of pulse width modulated voltage source converters with fixed frequency ratio. Since generated output voltage of the wind turbine is a function of the fundamental frequency (fo) and the carrier frequency (fc), results obtained by the Elspec system are incorrect to some extent. The magnitudes of Fourier transform harmonic components are the more inaccurate the higher significant is the fundamental variation. Even if the result of applying the discrete Fourier transform to the basic window is a spectrum with 5-hz spacing between frequency components and the spectrum thus contains both harmonics and interharmonics, the results can be sometimes significantly inaccurate.

Power quality meter data processing
Figure 2 Data processing algorithm used by Elspec equipment.

As a good example of this is one of the most significant sideband harmonic components from the first carrier group. Exemplary results of measurements form the LV side of the wind turbine transformer can be seen in Figure 3. As it was mentioned previously the wind turbine frequency ratio is mf=49 and the analyzed sideband harmonic component is of frequency fc+2fo. Three scatter plots present the same harmonic component measured during the same period using different data acquisition devices and processing techniques.

Sideband harmonic - not resampled signal Sideband harmonic - resampled signal
(a) (b)
Sideband harmonic - Elspec G4500

Figure 3 Sideband harmonic component affected by different processing techniques: (a) sideband harmonic calculated in post-processing from resampled signal, (b) sideband harmonic calculated in post-processing from original signal, (c) sideband harmonic calculated on-line by power quality meter.

Data processing results presented in Figure 3 show how easily inappropriately applied processing techniques can provide wrong results. Results from Figure 3(a) show sideband harmonic component measured using results obtained from the measurement campaign with direct Fourier decomposition (i.e. without sample rate adjustment). From Figure 3(b) presents the same data but resampled and later discrete Fourier transform is applied, Figure 3(c) describe sideband harmonic component magnitude obtained by the Elspec measurement system.

It can be observed that results using various processing approach provide different results. Due to the fact that the frequency of sideband harmonic components generated by modulation with constant carrier frequency does not vary significantly and Fourier decomposition without earlier sample rate adjustment gives the most appropriate results. This is presented in Figure 3(a) and only small magnitude variation affected by nonlinear relation between the modulation index () and the sideband harmonic components as well as measurement and data processing (i.e. small spectral leakage) errors. Completely different and unacceptable results are seen in Figure 3(b) where analysed waveform is resampled. One can observe that due to significant spectral leakage the estimated magnitude sometimes can be even equal to zero. Therefore sometimes power quality meters can provide values significantly affected by processing errors. This behaviour is present in the scatter plot from Elspec measurements (Figure 3(c)). It is important to emphasize that the algorithm in the power quality meter applies lossy compression which also determines estimated magnitudes. Estimated harmonic components are assumed to be insignificant and not saved (i.e. set to zero) if the magnitude is lower than a certain threshold which is defined based on measured waveform distortion and maximum allowed database storage capacity per month. Such limitations provides scatter plots as in Figure 3(c) which is similar to Figure 3(b) but modified due to averaging and magnitudes below the threshold artificially set to zero.

[1] "Electromagnetic compatibility (EMC) - Part 4-30: Testing and measurement techniques - Power quality measurement methods," International Electrotechnical Commission Standard IEC 61000-4-30, 2008.
[2] P. Nisenblat, A. M. Broshi, and O. Efrati, "Methods of compressing values of a monitored electrical power signal," U.S. Patent 7,415,370 B2, Aug. 19, 2008.

Harmonics Wind Farms

Harmoniske svingninger i store havmølleparker

This time Danish abstract of the PhD report entitled "Harmonics in Large Offshore Wind Farms (Harmoniske Svingninger i Store Havmølleparker)". The project was defended on the 2nd of February in 2012 at Aalborg University, Denmark.

Antallet af vindmøller med frekvensomformer til nominel effekt i mw-klassen, der anvendes til store havmølleparker, er stærkt stigende. De er tilsluttet et udbredt og forgrenet mellemspændingskabelnet stort set uden egetforbrug og er tilsluttet transmissionsnettet ved hjælp af lange højspændingskabler. Det stiller vindmølleindustrien og netselskaberne over for nye udfordringer i forhold til at forstå harmoniske svingningers  karakter, udbredelse og virkning. Vindmøllebranchen udvikler sig hastigt. Det stiller branchen over for nye udfordringer, hvilket har medført gennemførelse af flere og flere forskningsprojekter, der omhandler analyse af harmoniske svingninger med særligt fokus på vindenergi, og det er grunden til, at dette projekt blev påbegyndt og gennemført med et positivt resultat. Virksomhedens erfaring fra tidligere havmølleprojekter i forbindelse med forskellige harmoniske aspekter har medført et behov for at udføre omfattende undersøgelser af harmoniske svingninger.

Forskningsprojektet blev til på branchens foranledning, og blev gennemført i et  i samarbejde med institut for Energiteknik, Aalborg Universitet.  I forbindelse med planlægningen af projektforløbet blev rammerne for projektet lagt ud fra en traditionel rationalistisk tilgang for at kunne levere viden og en dybere forståelse for forskellige aspekter (f.eks. målinger, databehandling, dataanalyse, modellering, modelanvendelse) i studier af harmoniske svingninger. På baggrund af disse rammer, blev rapportens opbygning fastlagt. Læseren kan dermed følge alle projektforløbets stadier startende med målinger, databehandling og –analyse og sluttende med modellering og modelanvendelse. Forskellige aspekter af tidsdomænevalidering, frekvensdomæne og af brugen af statistiske metoder nævnes i forbindelse med specifikke problemer.

Målinger udgør en vigtig del af industriel forskning. Derfor er dette projekt unikt samtidig med, at det tilfører den akademiske verden vigtig praksis-orienteret indsigt og vice versa. Det er bevist, at analyse af systemer som store havmølleparker indebærer mange aspekter, der omhandler udvidede og mere præcise modeller, komplekse målekampagner og selvfølgelig bedre og mere anvendelige databehandlingsmodeller. Før de ovennævnte aspekter kan behandles, er det nødvendigt at have et pålideligt og robust målesystem til rådighed. Dette opnås gennem grundigt design af målesystemets hardware- og softwarelag.

I rapporten forklares det, at det er meget vigtigt at kende typen af de harmoniske svingninger, der genereres i store havmølleparker for at kunne anvende de rigtige databehandlingsteknikker. Tids-/frekvensanalyse baseret på multiresolution wavelettransformation bruges til at udføre tids-/frekvensdomæneanalyser, som kan bidrage til at definere de harmoniske svingningers oprindelse og observere korttidsvariationer. Ikke-parametrisk spektralanalyse anvendes på interpolerede signaler tilpasset de varierende elsystemfrekvenser. Forskellige databehandlingsteknikker er præsenteret og anvendt afhængig af signalet (dvs. om det er stationært eller ikke-stationært) eller typen af harmoniske svingninger (dvs. spline resampling eller direkte spektralanalyse). På baggrund af grundig analyse af målinger ses det, at visse harmoniske komponenter, der dannes på netsiden af omformeren i vindmøllen påvirkes af to faste frekvenser, dvs. af elsystemets grundfrekvens og basisbærefrekvenssignalet. Derfor er målinger af harmoniske svingninger udført primært med kommercielle spændingskvalitetsmålere i nogen grad utilstrækkelige, og den efterfølgende vurdering af resultaterne kan derfor være misvisende.

Forskellige statistiske værktøjer er anvendt til at analysere oprindelsen og karakteren af forskellige harmoniske komponenter. En omfattende sammenligning af harmoniske spændinger og strømme baseret på en vurdering af den sandsynlige fordeling samt passende statistiske beregninger (f.eks. middel, varians, sandsynlig tæthedsfunktion mv.) anvendes. En sådan tilgang giver et bedre overblik og en bedre sammenligning af harmoniske komponenters variationer og forekomst.

Flere frekvensdomænemetoder til beskrivelse af vindmølleparker bestående af flere komponenter såsom vindmøller, transformere, kabler mv. beskrives og sammenlignes. Det forklares, at store havmølleparker kan producere yderligere uønskede resonanser i lavfrekvensområdet. Dette kan have en betydelig indflydelse på systemets generelle stabilitet. Derfor er analyse og designoptimering af store havmølleparker mere komplekst end analyse og designoptimering af små landmølleparker.

I dag er vindmøller komplekse anlæg udstyret med den nyeste teknologi. Derfor er analyse af harmoniske svingninger i sådanne anlæg ikke så ligetil. På grund af vindmøllernes kompleksitet kan man ved studier af harmoniske svingninger fokusere på flere forskellige aspekter såsom reguleringsstrategi, moduleringsteknik, omformerdesign og hardwareimplementering.

Forskellige reguleringsstrategier er blevet overvejet sammen med deres indflydelse på dannelsen af harmoniske svingninger og generel systemstabilitet. Analyser er hovedsaglig udført i frekvensdomænet. En analyse går ud på at finde ud af, hvordan forskellige komponenter i reguleringskonceptet (f.eks. filtre, kontrolenheder mv.) kan påvirke styringen og dens evne til at udkompensere harmoniske svingninger. Reguleringsstrategiernes indflydelse på mølleparkens generelle stabilitet er ligeledes blevet grundigt undersøgt. Egnede stabilitetsindeks er foreslået og anvendt i flere konkrete cases.

Omhyggeligt modelerede ækvivalenter af store vindmølleparker i frekvensdomænet sammen med møllernes frekvensrespons giver et godt overblik over, hvordan store havmølleparker reagerer ved forskellige frekvenser. En sådan tilgang har vist gode resultater i forbindelse med studier af eksisterende mølleparker.

Da harmoniske svingninger i vindmøller og vindmølleparker har forskellig oprindelse og er af forskellige typer, kan det være problematisk at sammenligne dem. Derfor er selektiv validering af specifikke frekvenskomponenter til tider mere anvendelig. Det blev observeret, at sammenligning af resultater i frekvensdomænet og tidsdomænet og anvendelse af statistiske metoder er nøglen til forståelse af resultaterne.

På baggrund af de præsenterede studier kan det ses, at store havmølleparker sammenlignet med typiske landmølleparker kan generere flere uønskede resonansscenarier. Uønskede resonanser kan påvirke mølleparkens generelle stabilitet og ydelse (f.eks. kan harmonisk resonans anslåsog forstærkes). Derfor er det meget vigtigt at analysere mølleparker grundigt, især store havmølleparker, også ud fra et harmonisk perspektiv.

Denne erhvervsPhD fokuserer på at finde frem til de bedst mulige metoder til at gennemføre forskellige harmoniske studier af havmølleparker, herunder en række forhold som ikke før er blevet overvejet. Anvendelse af nye metoder og en udvidelse af rækken af modeller bidrager til at opnå den højere rådighed, der er nødvendig på havmøllerparker, hvis de skal fungere som store kraftværker i det elektriske system.

Harmonics Wind Farms

Harmonics in large offshore wind farms

English abstract of the PhD report entitled "Harmonics in Large Offshore Wind Farms". The project was defended on the 2nd of February in 2012 at Aalborg University, Denmark.

The number of wind turbines with full converters in the MW range used in large offshore wind farms is rapidly increasing. They are connected through a widespread MV cable network with practicably no consumption and connected to the transmission system by long HV cables. This represents new challenges to the industry in relation to understanding the nature, propagation and effects of harmonics. Recently, the wind power sector is rapidly developing. This creates new challenges to the industry, and therefore more and more research projects, including harmonic analyses especially focused on wind power applications, are conducted and that is why the project was initiated and successfully developed. Also experience from the past regarding offshore projects developed in the company and various harmonic aspects causes a need to carry out extensive harmonic research.

The research project was initiated by the industry and carried out in cooperation with academia. In order to organize the project development process, the research development framework was suggested based on rationalistic tradition approach in order to provide knowledge and better understanding of different aspects (e.g. measurements, data processing, data analysis, modelling, models application) in harmonic studies. Based on the framework, also the structure of the report was organized. This allows the reader to go through all of the stages in project development starting from measurements, through data processing and analysis, and finally ending up on modelling and models application. Different aspects of validation in time domain, frequency domain, and by application of statistical methods are mentioned in relation to respective problems.

Measurements constitute a core part in industry-oriented research. Due to this fact, the research project owes its uniqueness and contributes new insight to the academia. It is proven that an analysis of such systems as large offshore wind farms considers many aspects related to extended and accurate models, complex measurement campaigns and of course appropriate and more suitable data processing methods. Before any of the above aspects could be seriously taken into consideration, a reliable and robust measurement system is needed. This is achieved by carefully designing the hardware and the software layers of the measurement system.

It is explained in the report that it is of great importance to know the nature of generated harmonics in large offshore wind farms in order to apply the most suitable data processing technique. Time-frequency analysis based on multiresolution wavelet transform is used in order to perform time-frequency domain analysis helpful to distinguish harmonic origin and observe short-term variation. Non-parametric spectrum estimation is successfully applied to interpolated signals adjusted according to the varying power system frequency. Different data processing techniques are presented and applied depending on the signal (i.e. stationary or non-stationary) or harmonic nature (i.e. spline resampling or direct spectrum estimation). Based on an in-depth investigation of measurements, it is observed that certain harmonic components generated by the grid-side converter in the wind turbine are affected by two driven frequencies, i.e. the power system fundamental frequency and the carrier signal fundamental frequency. Therefore, harmonic assessment made by major part of commercial power quality meters is to some extent inappropriate, and their measurements interpretation can be misleading.

Different statistical tools were used in order to analyse the origin and nature of various harmonic components. A comprehensive comparison of harmonic voltages and currents based on probability distribution estimation and appropriate statistics calculation (mean, variance, probability density function, etc.) is applied. Such approach gives a better overview and comparison of harmonic components variation and occurrence frequency.

Several frequency domain methods of describing wind farms comprising of various components such as wind turbines, transformers, cables, etc. are shown and compared. It is explained that large offshore wind farms can introduce additional unwanted resonances within the low frequency range. This can significantly affect overall system stability. Therefore, the analysis and design optimization of large offshore wind farms are more complex than smaller onshore wind farms.

Nowadays, wind turbines are complex devices equipped with the newest technologies. Therefore, also harmonic analysis of such devices is not a straightforward task. Harmonic studies, due to the complexity of the wind turbine structure, can be focused on several parts such as control strategy, modulation technique, converter structure, and hardware implementation.

Various control strategies are taken into consideration and their impact on possible harmonic emission and overall system stability. An analysis is performed mainly in the frequency domain. One analyses how particular components in the control structure (e.g. filters, controllers, etc.) can affect the control and its harmonic rejection capability. The influence of control strategies on overall wind farm stability is also deeply investigated. Appropriate stability indices are suggested and applied in several study cases.

Carefully modelled and aggregated large wind farms in frequency domain together with the wind turbines frequency response give a good overview about large offshore wind farm behaviour for different frequencies. Such approach is successfully used in studies of real-life existing wind farms.

Since harmonics in wind turbines and wind farms are characterized by different origin and nature, comparison of them may be problematic. Therefore, sometimes selective validation of particular frequency components is more suitable. It was observed that comparison of results in frequency domain and time domain, as well as application of statistical methods, is the core part of results understanding.

Based on presented studies, we see that large offshore wind farms, in comparison to typical onshore wind farms, can affect more unwanted resonance scenarios. Unwanted resonances can cause overall wind farm stability and performance (e.g. unwanted harmonic excitation and amplification). Therefore, it is of great importance to carefully analyse wind farms, especially large offshore wind farms, also from a harmonic perspective.

This industrial PhD project is focused on investigating the best possible way to perform various harmonic studies of offshore wind farms including some conditions not taken into consideration before. Application of new methods and widening the range of models contributes to achieve the necessary higher reliability of offshore wind farms as large power generation units in electrical power systems.


Harmonics in power systems and power electronics

Ideally, an electrical power system in every certain point should show a perfectly sinusoidal voltage signal. However, it is difficult to preserve such desirable conditions. The deviation of the voltage and current waveforms from sinusoidal is described in terms of the waveform distortion, often expressed as harmonic distortion.

Harmonics are created when nonlinear loads draw nonsinusoidal current from a sinusoidal voltage source or are generated by purpose by active components. Harmonic distortion is caused by nonlinear devices in the power system where driven frequency is the power system fundamental component fo. A nonlinear device is one in which the current is not proportional to the applied voltage. While the applied voltage is perfectly sinusoidal, the resulting current is distorted.

According to definition commonly used in power system studies [1] characteristic harmonic exist when analyzed three-phase electrical system is considered to be balanced, the voltages and currents waveforms have identical shape and current and voltage are separated by exactly ±1/3 of the fundamental period. In such case zero sequence harmonics are for orders n=3m where m=1,2,3,... , positive sequence harmonics are for orders n=3m-2 and negative sequence harmonics are for orders n=3m-1.

The harmonic emission of power electronic components can be categorized in characteristic and non-characteristic harmonics. The characteristic harmonic emissions are determined by the converter topology and the switching pattern applied. For instance, a typical configuration is a two-level, three-phase voltage source converter with sinusoidal pulse-width modulation. The modulation frequency ratio mf is defined as the switching frequency fc divided by the power system fundamental frequency fo. In double-edge naturally sampled pulse-width modulation significant sideband harmonics in this carrier group will occur at frequencies of ωct±2ωot, ωct±4ωot. And for the second carrier group, the significant sideband harmonics will occur at 2ωct±ωot, 2ωct±5ωot, 2ωct±7ωot. All triple sideband harmonics (e.g. 2ωct±3ωot, etc.) are canceled between legs because the phase angles of these harmonics rotate by multiples of 2π for all phase legs (i.e. common mode signals). [2], [3].

where Amn and Bmn are the Fourier coefficients.

Non-characteristic harmonics are not related to the converter topology, but are determined by the operating point and control scenario of the individual converter. Therefore, these are weakly correlated or even completely uncorrelated between different wind turbine generators [4].

Typical device with another driven frequency than fundamental component in the power system is voltage source converter with pulse width modulation which can be seen in nowadays wind turbines. If such converter is a grid-connected device the frequency components generated by it are dependent on both the power system fundamental component (i.e. modulated signal) and the carrier signal fundamental component (i.e. modulating signal). In this particular case harmonic components generated by the voltage source converter can be integer multiple of grid frequency, carrier frequency or a mixture of both of them.

Therefore in power systems where there is more than one driven frequency it is not straight forward to identify harmonic components and their origin. In fact the only occurrence of integer multiple frequencies of the power system fundamental frequency is the trivial case where a single sinusoid interacts with itself or its own harmonics. This situation was popular in the past when power systems comprised only passive components and synchronously rotating generators. Nowadays broadly used in modern power systems advanced power electronics contains also its own driven frequencies (not always equal or multiple integer of the power system frequency) which can affect generation of harmonic components of frequencies constituting a mixture of different driven frequencies.

[1] N. R. Watson and J. Arrillaga, Power System Harmonics. Wiley and Sons, 2003.
[2] N. Mohan, T. M. Undeland, and W. P. Robbins, Power electronics: Converters, Applications, and Design, 3rd ed. New York: Wiley and Sons, 2003.
[3] D. G. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters: Principles and Practice. IEEE Press, 2003.
[4] J. Verboomen, R. L. Hendriks, Y. Lu, and R. Voelzk, "Summation of Non-Characteristic Harmonics in Wind Parks," in Proc. Nordic Wind Power Conference, Bornholm, 2009.