A new frequency approach for light flicker evaluation in electric power systems
 Luigi Feola^{1}Email author,
 Roberto Langella^{1} and
 Alfredo Testa^{1}
https://doi.org/10.1186/s1363401502070
© Feola et al.; licensee Springer. 2015
Received: 15 December 2014
Accepted: 16 February 2015
Published: 24 March 2015
Abstract
In this paper, a new analytical estimator for light flicker in frequency domain, which is able to take into account also the frequency components neglected by the classical methods proposed in literature, is proposed. The analytical solutions proposed apply for any generic stationary signal affected by interharmonic distortion. The light flicker analytical estimator proposed is applied to numerous numerical case studies with the goal of showing i) the correctness and the improvements of the analytical approach proposed with respect to the other methods proposed in literature and ii) the accuracy of the results compared to those obtained by means of the classical International Electrotechnical Commission (IEC) flickermeter. The usefulness of the proposed analytical approach is that it can be included in signal processing tools for interharmonic penetration studies for the integration of renewable energy sources in future smart grids.
Keywords
1 Introduction
Light flicker (LF) phenomenon is still considered one of the most important power quality (PQ) problems due to its ability to be directly perceived by customers, producing complaints from them.
LF is caused by the modulation of the supply fundamental voltage, which produces modulated light emissions whose severity, in terms of annoying effects on humans, depends on modulation amplitudes and frequencies as well as on lamp technologies [1]. LF is commonly measured by means of the International Electrotechnical Commission (IEC) flickermeter [2] that, for historical reasons, was designed and tested only with reference to voltage amplitude modulation (AM), which was the first source of LF identified and referring only to 60W incandescent bulbs, which were the most diffused lamps all over the word at that time.
Today, incandescent lamps are going to be banned, in particular in Europe, Australia and North America, but the IEC flickermeter is still the only instrument used also because international standards are based on it.
The main drawbacks of the IEC flickermeter are as follows: i) it is based on the incandescent bulb model; ii) it requires 10 min of time domain signals to give the shortterm flicker sensation index output, P _{st}; and iii) the output data cannot be used to study LF propagation effects in distribution and transmission networks.
Basic literature demonstrates the perfect equivalence of amplitude modulation to the summation of interharmonic tones of proper amplitudes and phase angles superimposed to the fundamental [3].
Starting from the beginning of the last decade, several papers aimed to model the IEC flickermeter in the frequency domain have been written [413]. Some of them [4,7,10,11] are pure frequency domain methods. Some others [6,9,12] are hybrid timefrequency domain methods.
Mayordomo et al. obtained very accurate analytical formulas that were applied to the voltages of DC and AC electrical arc furnace (EAF) measurements. In [13], the analytical formulas have been used to evaluate the propagation in the network of flicker produced by rapidly varying loads. In [11], a spectral decompositionbased approach is proposed to estimate LF caused by EAFs where the system frequency deviates significantly due to the EAF operation. Both methods start from the discrete Fourier transform (DFT) performed over 200 ms, which is perfectly compatible with the IEC Standard 6100047 [14] that defines harmonic and interharmonic measurement techniques.
All of the abovementioned methods are based on simplified assumptions essentially based on the concept that, due to the design specifications of the filters of the IEC flickermeter, the interharmonic components below 15 Hz and above 85 Hz can be neglected, with reference to 50 Hz systems. This assumption is demonstrated to be valid when the interharmonic source is an EAF which mainly produces modulation of the fundamental voltage in the frequency range from 0 to 20 Hz, that is to say modulations produced by interharmonics in the frequency range from 20 to 80 Hz. Recent studies have demonstrated that modern distributed energy resources, in particular wind turbines, are able to produce interharmonics in a wide range of frequency from DC to some kilohertz [15]. Moreover, in [1618], it was demonstrated that interharmonic components produced by adjustable speed drives can cover all the frequency range from DC.
In this paper, the abovementioned simplified assumption is overcome, leading to analytical solutions, of different complexities, able to take into account also the frequency components neglected by the classical methods. The analytical solutions proposed apply for any generic stationary signal affected by interharmonic distortion. The LF analytical estimator proposed is applied to numerous numerical case studies with the goal of showing i) the correctness and the improvements of the analytical approach proposed with respect to the other methods proposed in literature and ii) the accuracy of the results compared to those obtained by means of the classical IEC flickermeter. The usefulness of the proposed analytical approach is that it can be included in signal processing tools for interharmonic penetration studies for the integration of renewable energy sources in future smart grids.
2 Analytical assessment of IEC flickermeter response due to interharmonics
The analytical solutions proposed apply for any generic stationary signal affected by interharmonic distortion. The generic signal is decomposed into N interharmonic pairs. Each pair is constituted of two tones in symmetrical frequency positions with respect to the fundamental frequency. Moreover, each of two components of the pair has generic amplitude and generic phase angle. Obviously, the case of single interharmonic components can be easily obtained assuming the amplitude of one of the two components of the pair equal to zero.
Here, explicit reference is made to 50Hz systems, but the considerations developed may also be applied to 60Hz systems by changing the constants and parameters.
2.1 Single pair
with û representing the half cycle rms value processed through a firstorder filter with a time constant of 27.3 s; a _{0}, ω _{1} and φ _{1}, respectively, representing the relative amplitude, the angular frequency and the phase angle of the fundamental tone; a _{1_L} and a _{1_U} representing the relative amplitudes of the interharmonic tones and φ _{1_L} and φ _{2_L} representing their phase angles, respectively.
Design parameter of X ( s )
ω HPF  n _{ 0 }  d _{ 0 }  d _{ 1 }  d _{ 2 }  d _{ 3 }  d _{ 4 }  d _{ 5 } 

2π0.05  1.13 10^{14}  1.13 10^{14}  1.99 10^{12}  1.75 10^{10}  9.72 10^{7}  3.61 10^{5}  849.7 
Design parameter of F ( s )
k  λ  ω _{ 1 }  ω _{ 2 }  ω _{ 3 }  ω _{ 4 } 

1.74802  2π 4.05981  2π 9.15494  2π 2.27979  2π 1.22535  2π 21.9 
where \( H=\sqrt{1238400} \) is a gain factor related to the normalization of the weighting curve, and four different magnitude and phase gains \( {G}_{\varDelta {\omega}_1},{G}_{2\varDelta {\omega}_1},{G}_{2{\omega}_1\varDelta {\omega}_1} \) and \( {G}_{2{\omega}_12\varDelta {\omega}_1} \) are introduced for the five components depending on their angular frequency distances.
where PU_{1_DC} and PU_{1_AC}(t) are the DC and the residual AC component of the instantaneous flicker sensation, respectively.
As it will be shown in a more comprehensive manner in the following sections, the first component (9) assumes prevalent values for all the values of Δω _{1} while the second (10) assumes values increasingly larger how close is the frequency of the lower interharmonic tone to zero.
It is worth noting that (9) corresponds to the analogous expressions reported in [7] and in [10]. The difference is that in cited references, the filter X(s) was considered ideal, that is to say that G(s) = X(s)F(s) = F(s); so (9) considers the red curve of Figure 2 instead of the green one considered in [7] and in [10].
PU_{1_AC1} and PU_{1_AC2} are oscillating components due to the summation of sinusoidal signals with different angular frequencies (the angular frequency of each summand is indicated in the subscript). In particular, PU_{1_AC1} assumes higher values how close Δω _{1} is to zero, while PU_{1_AC2} assumes higher values how close Δω _{1} is to the fundamental angular frequency.
For the sake of brevity, the analytical assessment of the summands of (12) and (13) are reported extensively in Appendix A.
2.2 Two pairs
From the previous formulas, it should be noted that in case of two interharmonic couples superimposed to the fundamental signal, the PU is not only due to the sum of the contributions to the instantaneous flicker sensation of the single interharmonic pair, but there is also a component due to the combination effect of the interharmonic pairs. The entity of this effect will be evaluated in the following sections.
2.3 N pairs
3 Numerical case studies
 1.
A time domain signal with a length of 10 min with different characteristics, according with the case study, is generated.
 2.
The signal is analysed by means of the IEC flickermeter, which returns both the reference of the instantaneous flicker sensation and of the shortterm flicker severity value, PU_{ref} and P _{st_ref}, respectively.
 3.
A DFT is performed only on the Fourier period of the signalgenerated [1] since the signal is stationary in terms of fundamental and interharmonic signals in all the 10 min.
 4.
The output of the DFT is analysed by means of the analytical formulas of Section 2, according with the case study, to obtain the value of instantaneous flicker sensation (PU).
 5.
The instantaneous flicker sensation is used as input to the statistical evaluation (like that described in the Standard IEC 61000415) to obtain the shortterm flicker severity (P _{st}).
 6.
The values of PU_{ref}, P _{st_ref}, PU and P _{st} so obtained have been postprocessed to calculate the error of the analytical estimation.
3.1 Single pair producing AM
 I)$$ {\mathrm{PU}}_1^{\mathrm{I}}={\mathrm{PU}}_1={\mathrm{PU}}_{1\_\mathrm{D}\mathrm{C}1}+{\mathrm{PU}}_{1\_\mathrm{D}\mathrm{C}2}+{\mathrm{PU}}_{1\_\mathrm{AC}}; $$(17)
 II)$$ {\mathrm{PU}}_1^{\mathrm{II}}={\mathrm{PU}}_{1\_\mathrm{D}\mathrm{C}}={\mathrm{PU}}_{1\_\mathrm{D}\mathrm{C}1}+{\mathrm{PU}}_{1\_\mathrm{D}\mathrm{C}2}; $$(18)
 III)$$ {\mathrm{PU}}_1^{\mathrm{III}}={\mathrm{PU}}_{1\_\mathrm{D}\mathrm{C}1}. $$(19)
Obviously, the corresponding values of \( {P}_{\mathrm{st}1}^{\mathrm{I}},{P}_{\mathrm{st}1}^{\mathrm{I}\mathrm{I}} \) and \( {P}_{\mathrm{st}1}^{\mathrm{III}} \) have been calculated. Moreover, the results obtained implementing the analytical assessment proposed in [7] are reported and referred to as \( {P}_{\mathrm{st}1}^{\mathrm{BASE}} \).

The blue curve \( \left({P}_{\mathrm{st}1}^{\mathrm{I}}\right) \), in all the frequency range analysed, gives the best results and only for the pair with the frequency of 49.9 Hz shows an error greater than 5% (black line). For this frequency, the error is caused mainly by the transient behaviour of the filters contained in the IEC flickermeter, which becomes nonnegligible for interharmonic frequencies very close to the fundamental. This transient behaviour is not taken into account by pure frequency domain methods differently from hybrid methods [12].

The green curve \( \left({P}_{\mathrm{st}1}^{\mathrm{II}}\right) \) and the red curve \( \left({P}_{\mathrm{st}1}^{\mathrm{III}}\right) \) are virtually indistinguishable for frequencies greater than 15 Hz. Below this frequency (as mentioned in Section 2), the impact of PU_{1_DC2} on the total value of PU_{1} is nonnegligible, and for this reason, in this frequency range, the error committed by the analytical estimation \( {P}_{\mathrm{st}1}^{\mathrm{II}} \) is less than that produced by \( {P}_{\mathrm{st}1}^{\mathrm{III}} \).

The \( {P}_{\mathrm{st}1}^{\mathrm{II}} \) estimation makes an error greater than 5% (black line) for the pairs with frequencies higher than 48 Hz and lower than 0.8 Hz. The reason of these errors (in addition to the previously mentioned transient behaviour of the IEC flickermeter for interharmonic frequencies very close to the fundamental) is due to the AC component of PU_{1}, which is more consistent for interharmonic tones with Δω _{1} both close to the fundamental frequency or close to zero.

The \( {P}_{\mathrm{st}1}^{\mathrm{III}} \) estimation makes an error greater than 5% (black line) for frequencies lower than 6.2 Hz and higher than 48 Hz. The reasons of the different trend of the red curve with respect to the other two curves are the same as the ones mentioned in the previous point.

The \( {P}_{\mathrm{st}1}^{\mathrm{BASE}} \) (cyan curve) makes an error almost equal to the \( {P}_{\mathrm{st}1}^{\mathrm{II}} \) and the \( {P}_{\mathrm{st}1}^{\mathrm{III}} \) for frequencies higher than 25 Hz, but for lower frequencies, as a result of the simplifying assumptions, the error diverges, rapidly reaching the 40% already for a frequency of 15 Hz.
3.2 Two pairs producing AM
The amplitude of the single pair of interharmonic tones has been chosen to produce singly a P _{st} equal to 1 (Figure 4). The frequency modulation of the two pairs, which are defined Δω _{1} and Δω _{2} in (14), has been chosen to vary between 1 and 49 Hz, with steps of 1 Hz, and all their possible combinations have been evaluated. Since the Fourier period of the input signal is equal to 1 s, a DFT with a spectral resolution of 1 Hz is used.
 I)$$ {\mathrm{PU}}_{1+2}^{\mathrm{I}}={\mathrm{PU}}_{1+2}={\mathrm{PU}}_{1+2\_\mathrm{D}\mathrm{C}}+{\mathrm{PU}}_{1+2\_\mathrm{AC}}. $$(20)
 II)$$ {\mathrm{PU}}_{1+2}^{\mathrm{II}}={\mathrm{PU}}_{1+2\_\mathrm{D}\mathrm{C}}. $$(21)
 III)$$ {\mathrm{PU}}_{1+2}^{\mathrm{III}}={\mathrm{PU}}_1+{\mathrm{PU}}_2. $$(22)
Obviously, the corresponding values of \( {P}_{\mathrm{st}1+2}^{\mathrm{I}},{P}_{\mathrm{st}1+2}^{\mathrm{I}\mathrm{I}} \) and \( {P}_{\mathrm{st}1+2}^{\mathrm{III}} \) have been calculated. Moreover, the results obtained implementing the analytical assessment proposed in [7] are reported and referred to as \( {P}_{\mathrm{st}1+2}^{\mathrm{BASE}} \).

The xaxis and the yaxis represent the frequency of lower interharmonic tone of the first and of the second interharmonic pair, respectively.

The most complete analytical estimation \( {P}_{\mathrm{st}1+2}^{\mathrm{I}} \) (Figure 6), for the most part of the cases analysed, makes a mistake lower than 5% (light green squares). The cases with an error greater than the 5% occur essentially when the interaction between the two interharmonic pairs produces one or more oscillating component of the PU_{1+2} with an angular frequency that the filter Y(s) (6) is not able to eliminate, which is not evaluated analytically.
 i)
Δω _{1} ≈ Δω _{2} (main diagonal);
 ii)
Both the lower interharmonic tones of the two pairs are lower than 5 Hz (in the top left corner);
 iii)
Δω _{1} ≈ 2(ω _{1} − Δω _{2}) (in the top right corner);
 iv)Δω _{2} ≈ 2(ω _{1} − Δω _{1}) (in the bottom left corner).

Comparing the results obtained for \( {P}_{\mathrm{st}1+2}^{\mathrm{I}} \) (Figure 6) and for \( {P}_{\mathrm{st}1+2}^{\mathrm{II}} \) (Figure 7), it should be noted that neglecting the effects of the AC components produced by the single interharmonic pair increases the number of nonlight green squares when

 i)
Δω _{1} ≈ Δω _{2} (main diagonal);
 ii)
Δω _{1} ≈ 2(ω _{1} − Δω _{2}) (in the top right corner) and when Δω _{2} ≈ 2(ω _{1} − Δω _{1}) (in the bottom left corner);
 iii)Δω _{1} = 1 (last column to the right) and when Δω _{2} = 1 (last column to the bottom).

Comparing the results obtained for \( {P}_{\mathrm{st}1+2}^{\mathrm{I}} \) (Figure 6) and for \( {P}_{\mathrm{st}1+2}^{\mathrm{III}} \) (Figure 8), it should be noted that neglecting the effects of the interactions between the two interharmonic pairs on the DC component of the PU_{1+2} are present with a nonnegligible entity when

 i)
Both the pairs have the lower interharmonic tone frequency lower than 15 Hz (in the top left corner);
 ii)Δω _{1} = 2(ω _{1} − Δω _{2}) (in the top right corner) or when Δω _{2} ≈ 2(ω _{1} − Δω _{1}) (in the bottom left corner).

In similar mode to the case of an interharmonic pair shown in Figure 5, the analytical assessment \( {P}_{\mathrm{st}1+2}^{\mathrm{BASE}} \) (Figure 9) makes errors very high (sometimes even more than 100%) when one or both the pairs have the lower interharmonic tone with frequencies lower than 20 Hz.

Finally, it is worthwhile to note that from the implementation point of view on a general PQ instrument, the proposed analytical approach requires only some manipulations, of different complexities depending on the level of approximation desired (see (20), (21) and (22)), of the spectra which are already evaluated by the PQ instrument for the harmonic and interharmonic analysis. On the other hand, the digital signal processing of the conventional IEC flickermeter requires the implementation of blocks 1 to 4 independently from the spectral analysis.

4 Conclusions
In this paper, a new analytical estimator for LF in the frequency domain, which is able to take into account also the frequency components neglected by the classical methods proposed in literature, has been proposed. The analytical solutions proposed apply for any generic stationary signal affected by interharmonic distortion. The LF analytical estimator proposed has been applied to numerous numerical case studies with the goal of showing i) the correctness and the improvements of the analytical approach proposed in respect with the other method proposed in literature and ii) the accuracy of the results compared to those obtained by means of the classical IEC flickermeter. The usefulness of the proposed analytical approach is that it can be included in signal processing tools for interharmonic penetration studies for the integration of renewable energy sources in future smart grids.

In the presence of interharmonic tones in the frequency range from DC to 15 Hz and from 85 to 100 Hz, the simplified assumptions made by classical methods proposed in literature can lead to very inaccurate results.

The analytical formulas can be used to perform interharmonic penetration studies in transmission and distribution networks.
Future development of the research will be aimed to generalize the methodology adapted to interharmonic components at frequencies higher than 100 Hz which is proven to affect modern lighting systems different from incandescent bulbs.
Declarations
Acknowledgements
The research activity discussed in this paper has been partially supported by the Project PON01_02582 “Command, control, protection and supervision integrated system for production, transmission and distribution (ColAdMin integrated SCADA) of renewable and non renewable electrical energy, with fielddeviceinterface, for rational use of electrical power” funded by the Italian Ministry for the Instruction, University and Research.
Authors’ Affiliations
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