Frequency estimation for grid-connected converters undertime-varying harmonics and inter-harmonicsCitation for published version (APA):Kapisch, E. B., Duque, C. A., Tibola, G., & Duarte, J. L. (2018). Frequency estimation for grid-connectedconverters under time-varying harmonics and inter-harmonics. In 2018 IEEE 23rd International Workshop onComputer Aided Modeling and Design of Communication Links and Networks, CAMAD 2018 [8514941] Instituteof Electrical and Electronics Engineers. https://doi.org/10.1109/CAMAD.2018.8514941

DOI:10.1109/CAMAD.2018.8514941

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Frequency Estimation for Grid-connected Convertersunder Time-varying Harmonics and Inter-harmonics

Eder B. Kapisch, Carlos A. DuqueFederal University of Juiz de Fora

Juiz de Fora – MG, Brazileder.kapisch@engenharia.ufjf.br

Gabriel Tibola, Jorge L. DuarteEindhoven University of Technology

Eindhoven - The Netherlandsj.l.duarte@tue.nl

Abstract—In order to allow the power flow control of grid-connected converters, this paper presents an improved schemefor grid fundamental frequency tracking. A comparison un-der frequency variation and time-varying harmonic and inter-harmonic condition is also discussed in order to show the methodrobustness. The scheme is based on a frequency-locked loopoperating integrated with a decomposition core, which utilizesa recursive implementation of a modified Hanning filter. Thefrequency value information is used to perform the modulationand demodulation required to make de decomposition into thecore. The compared frequency estimation scheme is based on azero-crossing with an interpolation correction. Simulation resultsshow that the proposed scheme is able to estimate the fundamen-tal component, including relative quadrature companion signaland the frequency value more accurately and stably than thescheme used to perform the comparison.

Index Terms—Fundamental component estimation, time-varying harmonics and inter-Harmonics, frequency-locked loop,zero-crossing.

I. INTRODUCTION

The growth in electricity demand has been, evidently, oneof the main reasons for the expansion of electric powersystems (EPSs) around the world [1]–[5]. Throughout theyears, EPSs have become increasingly complex and morebranched, feeding multiple types of loads, which drain powerfrom the grid in more peculiar ways [6]. Characteristics oftenpresent in many of the current loads are the time-varyingbehavior and high nonlinearity, which are responsible for mostcauses of deterioration in power quality (PQ). Among theseseveral issues, the presence of harmonics in voltage and currentwaveforms is a major fact [7].

Examples of loads generating waveform distortion are flexi-ble alternating current transmission systems (FACTs) devices,saturated magnetic core, arc furnaces, and variable frequencydrives (VFDs) [8]. In the current state of the EPSs, sustainableenergy sources have become increasingly important, and theintegration of these energy sources to the main grid is donethrough power electronic interfaces. This is another majorcause of harmonic distortions [9]. Indeed, grid interfacingequipment, such as voltage source inverters, could bring har-monic distortion into the voltage and current grid signals dueto non-ideal characteristics [10].

Adverse effects can result from harmonic distortion. Forinstance, one can cite the reduction in devices lifespan, er-roneous operation of control circuits, interference with com-

munication lines, and increased losses. These effects becomeeven more important within the smart grids (SGs) scenario,where the operation of power processing converters must beperformed in an integrated way through a real-time commu-nication between control and protection elements [11], [12].Therefore, accurate and fast-tracking parameter estimation andreliable communication are key aspects for a SG [13], [14].

The efficiency of harmonic mitigation techniques dependson an accurate estimation of the fundamental component ofthe waveform signals [9]. These techniques are becomingindispensable, especially in the new context of SGs and itsapplications, such as the control of power converters.

A crucial aspect in the control of power converters con-nected to the network is the detection of the fundamentalcomponent of the voltage and either the fundamental frequencyvalue or the phase angle [15]. The frequency estimationinformation is used for synchronization of the output variablesof the converters, power flow calculation, among other pur-poses. Frequency information is also essential for the controlof distributed generation systems, energy storage systems,FACTS [16], power line conditioners, uninterruptible powersupplies (UPS) [17], frequency inverters, and other powerconditioning equipment.

Fig. 1 shows this multiple use of frequency estimationin various applications of the EPS. In all of these appli-cations, frequency estimation must be done as quick andaccurate as possible. This becomes quite challenging whenEPS operates under harmonic distortion condition. Severeadverse conditions, such as the presence of sub-harmonics ora marked variation in frequency, may interfere significantly onthe operation of conventional estimators. Therefore, a robustfundamental frequency estimation technique able to cope withthe aforementioned PQ disturbances has fundamental impor-tance.

Many of the algorithms used to estimate the frequency valueof EPSs are based on the assumption that the fundamentalfrequency is practically constant or that it experiences smallvariations, especially those based on the fast fourier transform(FFT) [18]. This premise has become increasingly far fromreality as new sources with intermittent features are incorpo-rated into the grid. For instance, the frequency of a distributionnetwork can fast change during transient events, becomingchallenging to accurately track [19]. Thus, the constant or

Fig. 1. Multiple usages of EPS frequency estimation information in thecontext of smart grids.

nearly constant frequency model is no longer appropriate forthe current state.

On account of that, this paper presents the implemen-tation and comparison between two fundamental frequencyestimation schemes. The initial stage for both schemes isthe same and consists of a recursive implementation of amodified version of the Hanning filter [20]. The second part isresponsible for providing the frequency value information andon this point the contribution of this work relies. While [20]uses a zero-crossing, here the use of a frequency-locked loop(FLL) seems to be more attractive in terms of stability andinter-harmonics immunity. In order to perform the comparison,a first scheme is implemented based on zero-crossing with alinear interpolation correction to improve the precision [21].A second one is the proposed scheme, and it is based onthe adaptive quadratic signals generator - phase locked loop(AQSG-PLL) [15].

This paper is organized as follows. Section II describesthe recursive implementation of the modified version of theHanning filter, the fundamental frequency estimators modelsand the proposed use of the FLL algorithm (AQSG-PLL) tomake the system suitable to operates in time-varying distortedconditions and its discretization. Section III presents simula-tion tests and validation results. Finally, Section IV concludeswith some remarks.

II. MODELING DESCRIPTION OF THE SCHEME PARTS

A. Decomposition core based on a recursive modified Han-ning filter

This is the first part of both schemes in the comparativestudy and it can be seen in Fig. 2. In this scheme, L isthe length of the moving average filter (MAF ) in samples,implemented in its recursive form [22]. In the H2 block,

Fig. 2. Fundamental component and frequency estimator.

θ = 2π/L, b0 = cos(θ), a0 = 1 − b0, and 0 < r < 1 isthe quantity responsible to locate the poles of the H2 transferfunction out of the unit circle to inward. This allows thecanceling of the first harmonic component with a length Lof just one fundamental cycle. A more detailed description ispresented in [20].

The input single-phase signal Sin[n] is connected to theinput on the scheme (see extreme left of Fig. 2). Then, thesignal is modulated in sine and cosine components usingthe fundamental frequency information (f0) provided by thefrequency estimation method (AQSG-PLL or zero-crossingin this work). The blocks MAF and H2 in series are theimplementation of the modified version of the Hanning filterin its recursive form [20]. After this filtering stage, which re-moves virtually all harmonics and attenuates inter-harmonics,both components sine and cosine of the signal are submittedto the demodulation step to recover the original fundamentalfrequency spectrum. Finally, signals are summed to generatethe fundamental component of the input signal.

This part of the scheme is responsible for delivering a re-constructed version of the fundamental component of the inputsignal Sin[n]. Moreover, in order to do that with precision itneeds an accurate value of the fundamental frequency appliedto the modulators and demodulators. For this reason, the nextpart is dedicated to accomplish this task.

B. Zero-Crossing with linear interpolation correction

This scheme [21] is composed of a simple zero-crossingdetector followed by an interpolation correction stage per-formed to improve the accuracy of the estimation (Fig. 3). Thisinterpolation stage estimates the exact moment of the signalcrossing between the two samples, xc[n − 1] immediatelybefore and xc[n] after the zero-crossing. From the samplingperiod Ts it is possible to deduct the periods Na and Nb in

Na =xc[n− 1]

xc[n− 1]− x[n], and (1)

Nb =xc[n]

x[n]− xc[n− 1]. (2)

Fig. 3. Samples at zero crossing: linear interpolation.

C. AQSG-PLL: Continuous-time modeling

Common solutions for the task of tracking the fundamentalcomponent of the EPS signals, which are to a certain extentinsensitive to frequency variations, consist of the use of phase-locked loop (PLL) structures [23]. Most of these structuresdeliver the fundamental component voltage of the systemeither in normalized or scaled version. Besides, the estimationof the phase angle and the value of the fundamental frequencyis delivered. One of the most used PLL is the three-phase PLLbased on the synchronous reference frame (SRF) [23]–[25].

Although the SRF scheme was originally designed for three-phase systems, several authors have directed the principle tosingle-phase cases [26], [27]. For this, the single phase systemis considered as a balanced three-phase system, where the α-component (from the Clarke’s transform) is represented by thesingle-phase voltage of the network, while the β-component(or quadrature companion signal - QCG) is synthesized usingseveral approaches [28], [29]. Thus, these two quadrature sig-nals are equivalent to a “virtual” or an “emulated” three-phasesystem, in a fixed-frame representation to which the sameideas developed for three-phase SRF-PLLs can be directlyapplied [15].

The proposed single-phase frequency estimation scheme isbased on AQSG-PLL [15]. The AQSG-PLL is composed ofthree main blocks, the adaptive quadrature signals generator(AQSG), the fundamental frequency estimator (FFE), and thequadrature companion generator (QCG). These blocks areshown in Fig. 4 and described by the set of equations

AQSG:

˙vα,1 = Ω0ψ1 + γ1vα,1˙ψ1 = −$0vα,1

FFE: ˙Ω0 = λψ1vα (3)

QCG: vβ,1 = ω0ψ1

$0.

The notation x = dx(t)dt means the time derivative of the

variable x(t), and x is the estimation of x(t). Also, ψ = $0vβω0,

where $0 represents the nominal value of the fundamentalangular frequency, that is a positive constant. This constantcan be a simple approximation of the angular frequency ω0,and vβ is the quadrature companion signal of vα. The variable

Ω0 =ω20

$0(4)

is the square of the real frequency ω0 scaled by $0.In Eq. (3), Ω0 is the estimation of Ω0, ψ1 and vα,1 are

fundamental component estimations of vα and ψ, while ˙vα,1is the fundamental component estimation time derivative ofthe input vα, and vα,1 =

4vα − vα,1 is defined as an error

between the input signal vα and its fundamental componentestimation vα,1.

The variable ˙ψ1 is the fundamental component estimation

time derivative of ψ, and vβ,1 is the estimation of the fun-damental component of the quadrature companion signal vβ ,and γ1 > 0 is a design parameter used to introduce a dampingrequired by the model. Finally, λ is a design parameter knownas adaption gain.

D. Discretization of AQSG-PLL

In order to implement the system on a digital platform,using either DSPs or FPGAs, transformations from continuoustime domain (s) to the discrete time domain (z) are needed.The three integrators

(1s

)must be transformed. The Adams-

Bashforth rule

Hab(z) =Ts2

(3z − 1

z2 − z

)(5)

was applied to integrator A in Fig. 4. The trapezoidal integra-tion rule, described as

Htr(z) =Ts2

(z + 1

z − 1

)(6)

was adopted to the integrators B and C.This strategy was planned in order to avoid the “algebraic

loops” in the discrete system through the application of anexplicit integration method.

E. Proposed fundamental component frequency estimationscheme

The proposal of this work is to couple the AQSG-PLLfrequency estimator to the decomposition core described insubsection II-A. In that way, the output of the decompositioncore is tied to the input of AQSG-PLL, providing a reliablefundamental component approximation, while the AQSG-PLLuses this approximation to calculate the frequency value andthe quadratic companion signal as well.

Fig. 4. Estimator AQSG-PLL of the angular frequency ω0.

III. SIMULATION TESTS AND RESULTS

In order to show the robustness of the proposed scheme,signals containing several disturbances were applied to thesystem. For the sake of simplicity, one of them is shown in thispaper. Disturbances include time-varying harmonics and inter-harmonics, variation in the fundamental frequency and ampli-tude. All tests were performed on MATLAB environment. Inall tests AQSG-PLL performed better than zero-crossing.

A sampling frequency of 7.68 kHz was used, which corre-sponds to 128 samples per cycle in a nominal f0 = 60 Hzfundamental signal. The nominal amplitude was 180 V. TheAQSG-PLL parameters γ1 and λ were set to 300 and 2.22according to [15].

A signal with harmonic, inter-harmonic and frequency vari-ation content was generated as

Sin1 =A sin(ω0t+ φ0) + 0.1A sin(2π17t+ φ1)

+0.1A sin(2π2f0t+ φ2)∀t ≤ 8.33 s,(7)

andA sin(2π(f0 + 1)t+ φ3)∀t > 8.33 s, (8)

where

A = 180V,

ω0 = 2πf0,

φk = 0 rad, and

k ∈ 0, 1, 2, 3 .

(9)

Fig. 5 depicts the estimation process using AQSG-PLLapproach, while Fig. 6 shows the same using zero-crossing.At t = 8.33 s, the fundamental frequency of the input signalsuffers a step change from 60 Hz to 61 Hz, and both harmonicand inter-harmonic components disappear at the same moment.

Comparing both results, one can note that, from the instantof change (8.33 s), the frequency estimation provided by

AQSG-PLL oscillates much less than the estimation providedby zero-crossing. After T = 8.42 s AQSG-PLL estimationbecomes stable, while zero-crossing is still bouncing untilt = 9.6 s, failing during several cycles after the change.Moreover, the zero-crossing overshot is higher than the onecaused by AQSG-PLL estimation.

IV. CONCLUSION

The improvement resulted from the insertion of thefrequency-locked loop frequency tracking method (AQSG-PLL) proposed in this work, as shown in the results, pro-vided very accurate response to the fundamental componentfrequency estimation. In addition to the frequency value, thescheme using the AQSG-PLL can deliver an estimation of bothfundamental component and quadrature companion signal ofthe input signal, having a slightly higher complexity whencompared to the zero-crossing method, which can deliver onlythe frequency value.

Once the natural behavior of the zero-crossing estimationis to update the estimation value every cycle of the signal,it leads to a waveform shape in which one can observediscontinuities in step-form. This does not happen with AQSG-PLL, which presents a smooth-tracking behavior ideal towork cooperatively with the modified Hanning filter baseddecomposition core.

ACKNOWLEDGMENT

This project has received funding from the Electronic Com-ponents and Systems for European Leadership Joint Undertak-ing under grant agreement No 737434. This Joint Undertakingreceives support from the European Union’s Horizon 2020research and innovation programme and Germany, Slovakia,Netherlands, Spain, Italy.

Fig. 5. Results using AQSG-PLL algorithm: (a) input signal Sin1 with time-varying frequency, harmonic and inter-harmonic content, (b) fundamentalcomponent waveform estimation, and (c) fundamental frequency estimation.

Fig. 6. Results using zero-crossing algorithm: (a) fundamental component waveform estimation applying Sin1 as input signal, (b) transition periodmagnification, and (c) fundamental frequency estimation.

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