Design and Implementation of a Kalman Filter-Based Time-Varying Harmonics Analyzer

Transcription

1 Design and Implementatin f a Kalman Filter-Based Time-Varying Harmnics Analyzer Abstract: Nwadays with increasing use f numerus nnlinear lads, vltage and current harmnics in pwer systems are ne f the mst imprtant prblems pwer engineers encunter. Many f these nnlinear lads, because f their dynamic natures, inject time-varying harmnics int pwer system. Cmmn techniques applied fr harmnics measurement and assessment such as FFT have significant errrs in present f time-varying harmnics due t the time-windws applied. In this paper, a Kalman filter-based algrithm is develped and implemented fr measuring time-varying harmnics. The efficiency f the prpsed algrithm has been successfully tested in ff-line mde by varius cmputer simulatins. Based n this algrithm, a prttype harmnic analyzer has been designed, fabricated and used fr n-line harmnic mnitring and assessment studies. T assess the severity f time-varying harmnics, cumulative time indices that are cmputed in real-time using the utput f the analyzer are prpsed. In additin, new cumulative time curves fr ttal harmnic distrtin and individual harmnic distrtins are presented. S. H. Hsseini K. Mhammadi Electrical Engineering Department Sharif University f Technlgy Tehran, Iran 35 Als, an aut-synchrnizatin algrithm is prpsed t accmpany the Kalman filter algrithm in rder t eliminate the errrs ccurred due t the variatins in the incming signal frequency. Keywrds:Time-Varying harmnics, Kalman filter, Pwer quality, Fast Furier transfrm.. INTRODUCTION The widespread develpment f pwer electrnics devices and applicatin f nnlinear lads, have increased the harmnic distrtin in pwer system vltage and current wavefrms. Thus, the cnventinal methds used fr measuring and assessment f pwer system electrical parameters are nt qualified anymre. In additin t the higher level f distrtin in vltage and current wavefrms, the new nnlinear lads have als changed the nature f the distrtin. As an example, gd number f mtrs used in industry have start-stp cycles and therefre the mtr speed cntrller has t change its wring مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز وز مستان 385 Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and Winter 2006

2 Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and winter 2006 pint peridically, the result f which is the timevarying harmnics injected int the system []. Accurate measurement f pwer system harmnics is essential in rder t evaluate harmnic distrtin in bth current and vltage wavefrms. Several methds have been applied fr harmnics measurement such as, multi filter analyzers, frequency sweepers, quadratic transfrmers and different frms f Furier transfrm amng which the fast Furier transfrm (FFT) is the mst cmmn apprach applied in harmnic analyzer equipments [2-5]. When FFT is used, time windwing must be cnsidered fr the real-time signals t btain twdimensinal spectra in time-frequency plane. Sme drawbacs arise because f this time windwing. These drawbacs are even severer in present f timevarying harmnics. Theretically, fast Furier transfrm will be accurate nly if the fllwing assumptins can be made: -Wavefrm is peridic and statinary 2- Sampling frequency is greater than twice f the highest frequency in the signal 3- Number f perids in each time windw is an integer and 4- Each frequency in the signal is an integer multiple f the frequency reslutin dictated by the time windw. When these assumptins are satisfied, the result f FFT is accurate. But in many cases, even in absence f time-varying harmnics, they are nt satisfied [6-8]. Fr example, the sampling frequency usually is cnstant but the pwer system fundamental frequency has deviatins frm the nminal value. Thus, the third assumptin is nt justified. This leads t an effect called spectral leaage [9]. In this case, fr a pure sinusidal wavefrm with % frequency deviatin, the FFT measures a ttal harmnic distrtin f 5%. As anther example, in absence f the furth assumptin, the picet-fence effect ccurs that leads t errrs in the characteristic harmnics measurement. This effect, in bth cases f statinary signals with fundamental frequency deviatin and signals with time-varying harmnics can ccur. The abve fur assumptins ften can t be satisfied fr time-varying signals, therefre the analysis f these signals by means f Furier transfrm in many cases leads t significant errrs. Regardless f thse assumptins, Furier transfrm is nt suitable fr time-frequency studies because f the time-reslutin and frequency-reslutin trade ff, i.e. decreasing time windw size imprves the time reslutin but frequency reslutin is decreased and vise versa. In ther wrds, all harmnic rders have t be studied under a cmmn time reslutin in FFT, 36 مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 cntradicting the requirement that higher rder harmnics must be presented in higher time reslutins. Many attempts have been made t apply different measurement methds with the ability f timevarying harmnics tracing such as, using grup energy cncept with FFT [6], applying nn-square time windws in FFT implementatin [7], applying artificial neural netwrs fr harmnic tracing [0, ] and, applying wavelet transfrm fr harmnic measurement [2, 3]. Each f these methds has been successful in sme cases but still sme prblems exist with applying them in practice. Fr example, very high cmputatinal requirements fr wavelet based methds, limited applicatin f neural netwr based methds in different harmnic patterns and, lac f prper peratin f FFT based methds when encuntering fast changing harmnics are f the main prblems. The challenges f the multimde RF frnt-end design are set by the standards and are related t different receptin bands as shwn in Table. The GSM perates at 900 MHz, the DCS perates at 800 MHz, and the WCDMA perates at 200 MHz. In additin, these systems have different channel spacing and symbl rates. In the past, multimde receivers have been realized as a multiple number f single-mde receivers thus ccupying a large area and being rather cstly. A direct cnversin receiver (DCR) is cnsidered as feasible architecture fr multimde mbile wireless applicatins []-[9]. A DCR is amenable t mnlithic integratin, prvides 2. Kalman Filter-Based Time-Varying Harmnics Measurment Algrithm The state variable representatins f signals including harmnics suitable fr Kalman filter-based apprach are well described in [4-6]. In this sectin, the derivatin f equatins fr Kalman filter algrithm (KFA) fr signals including time-varying harmnics is briefly stated. Cnsider a sinusidal signal with frequencyω, phase angle θ and amplitude A (, where A ( represents a cmbinatin f a cnstant value plus a time-varying cmpnent. This nise free signal may be expressed as: s( = A( cs( ωt + θ ) = A( cs θ cs ωt A( sin θ sin ωt State variables at time t are defined as: = A( t ) cs( ω t + θ ) (2) 2 A( t )sin( ω t + θ ) (3) =

3 T write the state equatins let us cnsider the signal at time t + : + = A( t+ )cs( ωt + ω t + θ) = cs( ω t ) 2 sin( ω t ) (4) and fr secnd state variable 2 : = A( t + ) sin( ωt + ω t + θ ) = sin( ω + 2 cs( ω (5) Frm (4) and (5), the state variable equatins will be: cs( ω sin( ω t ) w = + (6) t t w 2 sin( ω ) cs( ω ) In equatin (6), the time variatins f state variables are described by randm variables w and w 2. Measurement is made at time t accrding t the relatin given by the equatin: Z = [ 0] + v (7) 2 where the scalar v represents the measurement nise and its variance will be dented by R. Nw, cnsidering a signal with n different frequencies as: s n () t = i= A ( cs( iω t + θ ) (8) By selecting 2n state variables as: = A ( t ) cs( iω t + θ ) (9) 2 = A ( t )sin( iω t + θ ) (0) the state equatins can be presented in matrix frm: M w w = O N () M 0 O 0 M M 0 N O 0 w n M n n ( ) ( ) ( ) w (i) The submatrices M are similar t: ( ) M i cs( iω = sin( iω sin( iω cs( iω (2) The measurement equatin then will be 2 Z [ ] (3) = 0 L 0 M + v 2 With this selectin f state variables, the amplitude and phase angle f each harmnic can be cmputed by: ( ) 2 ( ) ( 2 ) 2 i A = + (4) - 2 θ = tan (5) The abve calculatins impse increased cmputatinal burden due t the special frm f state variables selectin. Hwever, the frm f state variables is quite useful fr cmputing the frequency needed fr the aut- synchrnizatin algrithm described later. In additin, with this selectin f state variables, the cmpnents f the Kalman gain vectr becme cnstants and can be cmputed ff-line. The recursive KFA is shwn in Fig.. In this figure, Q represents the cvariance matrix f the prcess nise vectr w ). The state transitin matrix φ and the cnstant matrix H are the cefficient matrices in equatins () and (3), respectively. Fig. : Kalman Filter Algrithm 3. Aut-Synchrnizatin Algrithm In rder t eliminate the errrs ccurred due t variatins in the incming signal frequency, we decided t use the utput f the Kalman filter t estimate the actual frequency f the incming signal. This idea was initiated when it was realized, Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and Winter مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز وز مستان 385

4 Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and winter 2006 fllwing numerus simulatins, that even thugh there were errrs in the estimated fundamental frequency state variables, hwever the frequency f these variables well fllwed the variatins in the frequency f the incming signal. Thus, t estimate the frequency, the instantaneus frequencies f the fundamental frequency state variables were averaged ver five cycles with the instantaneus frequency being prprtinal t the rati f the derivative f inphase fundamental frequency cmpnent t the quadrature-phase fundamental frequency cmpnent. Nw, the KFA culd be synchrnized by updating the Kalman filter equatins using the estimated frequency having in mind that bth the Kalman gain and the state transitin matrices are functins f fundamental frequency. The gain matrix is cmputed ff-line thus, updating it with the estimated frequency maes the entire algrithm very slw and useless fr ur purpses. Frtunately, it was discvered that if nly the transitin matrix were updated using the estimated frequency and the gain matrix were cmputed at the fundamental 50 Hz frequency, the results wuld be cmpletely satisfactry. Thus, the KFA aut-synchrnizatin was dne by just updating the state transitin matrix. 4. Simulatin Results In this sectin, the perfrmance and efficiency f the KFA is studied by means f simulatins implemented in MATLAB. The Kalman filter parameters were selected as bellw: R = Measurement nise variance was selected t be 2 cnstant (0.05 p.u. ); Q = Prcess nise cvariance matrix (a diagnal 2 matrix with diagnal elements equal t 0.05 p.u. ); x ) 0 = Initial state vectr (a zer vectr); P 0 = Initial errr cvariance matrix (a diagnal 2 matrix with diagnal elements equal t 0 p.u. ); f c = Pwer system fundamental frequency (50 Hz) and f s = Sampling frequency (3200 Hz). In rder t study the ability f the algrithm in tracing time-varying harmnics, a sample test signal f the frm: s( s( = s2 ( 0 < t < < t < 0.2 مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 s s (6) 38 s( = cs( t + 0 ) + 0.3cs(3ω t + 20 ) + 0.cs(ω t + 30 ) cs(9ωt + 40 ) ω (7) s = cs( t + 0 ) + 0.8cs( 3ωt + 20 ) + 0.2cs( ω t + 30 ) + 0.cs( 9ω t + 40 ) 2 ( ω (8) is cnsidered. Amplitude and phase angle estimated by the KFA simulatin are shwn in Fig 2. There is a cnvergence perid at the start f estimatin that lasts abut 0 ms depending n the initial values selected fr recursive algrithm. Several studies shwed that the length f this cnvergence perid is cnstant when the equatins are written fr 50 Hertz system. Frtunately, this transient perid nly appears at the algrithm starting sectin and is referred t as the initiatin perid. Anther nticeable fact seen in Fig. 2 and cnfirmed by the results f numerus simulatins perfrmed is that the KFA has a delay in respnding t sudden changes in the signals. This delay is an unavidable characteristic f KFA. The delay perid is abut 0 ms and it depends n the fundamental frequency and the prcess nise matrix Q. Althugh this delay decreases when Q increases, hwever it will lse ne imprtant factr f the KFA which is the nise rejectin prperty. Therefre, the KFA has a limited speed fr tracing very fast changing harmnics.

5 Fig. 2: Step respnse f Kalman filter in amplitude estimatin f harmnic rders, 3, and 9 In rder t study the algrithm ability fr tracing cntinuus variatins in harmnics, the fllwing test signal is applied: s ( = B( s( (9) s( = cs ( ω + 0.cs (3ω t + 0 ) + 0.cs (5ω t + 20 ) cs (7ω t + 30 ) + (20) 0.08cs ( ω t + 40 ) cs (3ω t + 50 ) cs (9ω t + 60 ) 0 < t < 0.2 s 5( t 0.2) 0.2 < t < 0.4 s + (2) B( = < t < 0.6 s 2 5( t 0.6) 0.6 < t < 0.8 s 0.8 < t < s This signal is shwn in Fig. 3 and as seen in Fig. 4, the cntinuus variatins are well fllwed by the KFA. Fig. 3: Test signal wave shape Fig. 4: Amplitude and phase angle f the 5 th harmnic 5. Practical Results Based n the KFA described, a prttype harmnic analyzer was designed and cnstructed. The analyzer has tw input prts; ne fr signals having peas f 5 vlts and less used fr testing and calibrating the analyzer and anther input prt fr signals having peas f 600 vlts maximum. The prttype device cnsists f a hardware and a sftware mdules. The hardware mdule includes anti-aliasing filter, input buffer and surge/vervltage prtectr, sample and hld and A/D cnverter, main and micrcntrller pwer supplies, RS232 cnverter and utput ptislatr. Data sampling fllwed by data transferring Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and Winter مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز وز مستان 385

6 Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and winter 2006 t a persnal cmputer serial prt is perfrmed by the micrcntrller-based hardware mdule in real-time. Then, the sftware mdule running n a PC, perfrms the KFA t estimate the individual harmnic distrtins (IHD s) and THD f the incming signal nline. In rder t analyze the perfrmance f the fabricated prttype analyzer, several distrted signals with different nwn characteristics were generated in MATLAB and exprted in analgue frmat frm PC sund card. These signals then were applied as inputs t the hardware mdule. As an example, the fllwing signal is cnsidered: s ( t < 2.3 s s( = (22) s2 ( t > 2.3 s s ( = cs( ω + 0.3cs(3ωt ) + 0.2cs(7ω ) (23) t s ( = cs( ω cs(3ωt ) + 0.3cs(7ω ) (24) 2 t The third harmnic amplitude f the abve signal estimated by the prttype analyzer is shwn in Fig. 5. The algrithm culd identify the crrect amplitude in less than ne-half the fundamental cycle. Fig. 5: Third harmnic measured by analyzer Figure 6 shws the results f a real-time measurement made by the prttype analyzer at a factry nearby a 6-pulse rectifier. Table shws harmnics amplitudes f the same signal that were measured by a cmmercial harmnic analyzer (LEM Analyst 2060). The differences in the tw measurements related t the 2nd, 29th and 3st harmnics are due t the % measurement errr caused by the micrcntrller (ATmega32) built-in 0-bit A/D cnverter. 40 مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 Fig. 6: Practical measurement frm utility main vltage Table : Harmnic amplitude which measured by a cmmercial harmnic analyzer 6. Cumulative Harmnic Curves After accurate n-line measurement f time-varying harmnics and ttal harmnic distrtin, it is necessary t emply a statistical apprach, because f the variable nature f distrtin level, in rder t evaluate the severity f time-varying harmnics. Mst standards cnsider harmnics as steady state phenmena. Therefre, the limitatins set by thse standards fr the harmnic cmpnents and THD cannt be used directly fr the time-varying harmnics cases. T slve this prblem in a way that the steady state harmnics limits culd be applied t the time-varying harmnics and als t present a way t handle shrt time excursins abve steady state harmnics limits [7], cumulative time curves were intrduced. Fr FFT-based instruments, IEC standard [8] recmmends three windw widths fr different categries f harmnics. Mrever, this standard recmmends five time intervals fr the purpse f data cmpressins. These recmmendatins lead t what the IEC standard calls it "cumulative prbability functin (prbability f nt exceeding the crrespnding value)". We extended this idea t the KFA utput having in mind that the gaps between the windw times and between time intervals are f n cncern with regards t KFA since the KFA estimates harmnic cmpnents cntinuusly in real-time. In additin, the lw

7 cmputatinal requirements needed by the Kalman methd mae it pssible t btain cumulative severity factrs n-line giving mre infrmatin abut the distrtin severity that the IEC standard methd gives. The prttype harmnic analyzer perfrms data sampling at 3200 Hz. Fr each sample, THD and harmnics cntent up t the 3st harmnic are estimated in real-time. Every 64 cnsecutive THD and individual harmnic distrtins values are averaged and the results are ept temprarily. These averaged values are used fr cmputing the cumulative times explained later. The user defines three threshlds fr THD and three threshlds fr each IHD. These threshlds culd be (and better be) different fr different IHD s. We suggest the first set f THD and individual harmnic threshlds t be the maximum allwed steady state THD and individual harmnic distrtins limits and the secnd and the third sets f threshlds t be twice and three times f the first set, respectively. The user als defines three cumulative time percentage threshlds crrespnding t the three distrtin threshld sets. We suggest 5%, % and 0.% time threshlds fr the first, secnd and third sets f distrtin threshlds, respectively. Every ne secnd, temprarily ept THD and individual harmnic distrtins values are averaged and the results are displayed n the mnitr and stred n the hard dis. In additin, every ne secnd, the cumulative times, as percentage f the ttal measurement perid, f ccurrence f THD and each IHD that exceed the defined distrtin threshlds are cmputed, displayed and cmpared against the defined cumulative time percentage threshlds. Whenever any f the n-line cmputed cumulative times, referred t as cumulative time indices (factrs), related t THD r any f the IHD s exceeds the time threshlds, the clr f numbers shwing the cumulative times will change frm blac t red. By this, we allw the cumulative times f ccurrence f harmnics severer than the steady state limits, severer than twice the steady state limits and three times the steady state limits t be at mst 5%, % and 0.% f the ttal measurement perid, respectively. The cumulative curves can be pltted in ff-line mde fr any user specified time duratin f the ttal measurement perid. Fr each THD level n the hrizntal axis, the cumulative time f ccurrence f THD values exceeding that particular level is cmputed as percentage f the time duratin specified by the user, t plt a curve similar t that shwn in Fig. 7. On this curve, the distrtin and time threshlds are shwn as three steps. An acceptable harmnic distrtin is the ne fr which its cumulative time curve lies under the three-step limit. Similar cumulative curves can be pltted fr any individual harmnic distrtins. Fig. 7: Cumulative harmnic distrtin curve with its acceptance levels 7. CONCLUSIONS The need fr having a device capable f nline measuring, mnitring and assessing time-varying harmnics, mtivated us t design and fabricate a prttype Kalman filter-based harmnic analyzer. Against drawbacs f Furier transfrm-based methds fr harmnics measurement, specially in the present f time-varying harmnics, Kalman filter algrithm has utstanding advantages including gd nise rejectin prperty, ability t btain desired frequency reslutin independent f time reslutin, nt requiring synchrnized sampling, lw cmputatinal requirements, ability t trac timevarying harmnics and ability t perfrm n-line measurements. A pwerful sftware mdule accmpanies the micrcntrller-based hardware mdule. An autsynchrnizatin algrithm fr estimating the frequency f the incming signal, estimating and reprting THD and individual harmnic distrtins levels and als prviding cumulative times f ccurrence f THD and all IHD s exceeding user defined threshlds ( three threshlds fr THD and three threshlds fr each IHD) are n-line capabilities f the device. We prpsed cumulative time curves fr THD and all IHD s as means fr evaluating the severity f time-varying harmnics and as criteria fr judging whether the harmnic pllutin is acceptable. These Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and Winter مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز وز مستان 385

8 Jurnal f Iranian Assciatin f Electrical and Electrnics Engineers - Vl.3- N.2- Fall and winter 2006 curves are pltted in ff-line mde depicting, fr any THD level amng the entire range f THD shwn n the hrizntal axis, the cumulative time f ccurrence f THD s that exceed that particular level f THD as the percentage f ttal mnitring time. Similar cumulative time cures can be pltted fr each IHD. We allwed the cumulative times f ccurrence f distrtins severer that the steady state harmnics limits, twice the steady state harmnics limits and three times the steady state harmnics limits t be at mst 0%, % and 0.% f the ttal mnitring time. These acceptance criteria appear as three steps and harmnic pllutins crrespnding t the curves underlying these steps are accepted. The harmnic and time threshlds presented in this paper are the authrs preferences. These threshlds can be defined by the use t be in agreement with any natinal r internatinal standard. Indeed, the idea f cumulative time curves is strng enugh t be used by internatinal cmmittees as a base fr establishing time-varying harmnics assessment prcedures REFERENCES [] Lazendby, H., and Zivanvic, R., Sme Observatins n the Time-varying Harmnics and Inter Harmnics, IEEE Transactins n Pwer Delivery, pp , 999. [2] Hlawatsch, F. and Budraux-Bartels, G.F., Linear and Quadratic Time-Frequency Signal Representatin, IEEE Signal Prcessing Magazine, pp. 2-76, April 992. [3] Pissn, O., Rival, P., and Meunier, M., New Signal Prcessing Tls Applied t Pwer Quality Analysis, IEEE Transactins n Pwer Delivery, pp , April 999. [4] Van den Keybus, J., Driesen, J., and Belmans, R., Using Furier Transfrm and Mdel Based Filters t Measure Time-Varying Harmnics, Prceedings f the IEEE Pwer Engineering Sciety General Meeting, Vl. 3, pp , June [5] Miegeville, L. and Guerin, P., Identificatin f Time-Varying Pattern f Peridic Harmnics, IEEE Transactins n Pwer Delivery, pp , April [6] M, C.S., Chang, N.Y., and M, P.P., A Digital Measurement Scheme fr Time-varying Transient Harmnics, IEEE Transactins n Pwer Delivery, pp , April 995. [7] Heydt, G. T., Liu, C. C., and Tu, L., Applicatin f the Windwed FFT t Electric Pwer Quality Assessment, IEEE Transactins n Pwer Delivery, pp. 4-46, Octber مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 [8] Heydt, G. T., and Gewell, W. T., Pitfalls f Electric Pwer Quality Indices, IEEE Transactins n Pwer Delivery, pp , April 998. [9] Lin, H. C., Fast Tracing f Time-Varying Pwer System Frequency and Harmnics Using Iterative-Lp Appraching Algrithm, IEEE Transactins n Industrial Electrnics, pp , April [0] Liew, A. C., An Adaptive Linear Cmbiner fr Online Tracing f Pwer System Harmnics, IEEE Transactins n Pwer Systems, pp , Nv [] Lai, L. L., Chen, W. L., Tse, C. T., and S, A. T. P., Real-Time Frequency and Harmnic Evaluatin Using Artificial Neural Netwrs, IEEE Transactins n Pwer Delivery, pp , Jan [2] Wang, J., and Ran, Q., Time-Varying Transient Harmnics Measurement Based n Wavelet Transfrm, Prceedings f the IEEE Instrumentatin and Measurement Technlgy Cnference, pp , June 996. [3] Lin, T., and Yamada, E., Wavelet Apprach t Pwer Quality Mnitring, Prceedings f the IEEE Industrial Electrnics Sciety Annual Cnference, pp , 200. [4] Brwn, R.G., and Hwang, Y.C., Intrductin t Randm Signal and Applied Kalman Filtering, Third Editin, Jhn Willy and Sns, 996. [5] Adly, A., Girgis, W., Chang, B., and Maram, E., A Digital Recursive Measurement Scheme fr On-line Tracing f Pwer System Harmnics, IEEE Transactins n Pwer Delivery, pp , July 99. [6] Kennedy, K., Lightbdy, G., and Yacamini, R., Pwer System Harmnic Analysis Using the Kalman Filter, Prceedings f the IEEE Pwer Engineering Sciety General Meeting, Vl., pp , July [7] Ortmeyer, T., u, W., Setting Limits n Time Varying Harmnics, Prceedings f the IEEE Pwer Engineering Sciety General Meeting, Vl. 2, pp 72-75, July [8] IEC Standard , Electrmagnetic Cmpatibility (EMC), Part 4. Testing and Measurement Techniques, Sectin 7. General Guide n Harmnics and Interharmnics Measurements and instrumentatin fr Pwer Supply Systems and Equipment Cnnected Theret, 99.