Chapter 11

Probabilistic Harmonic Indices

P. Caramia, G. Carpinelli, A. Russo, P. Verde, P. Varilone

11.1 Introduction

When the time-varying nature of harmonics is considered, the most adequate indices to represent it have to be chosen.

The indices can refer either to a single customer point-of-common coupling or to a segment of the distribution system or, more generally, to the utility’s entire distribution system.

While the single site index probabilistic treatment requires the collection of different observations in time, for system indices it is needed to collect also observations at different sites. These last indices – well known to characterize other aspects of service quality as sustained interruptions – only recently have been considered for harmonic purposes [1]; the system indices can serve as a metrics only and are of particular interest in the new liberalized market frame, because they can be used as a benchmark against which the quality level of different distribution systems, or of parts of the same distribution system, can be compared.

In the next, some single site and system probabilistic harmonic indices are analyzed and considerations about their statistical characteristics are presented, also on the basis of measurements on actual distribution systems; particular attention is paid to ascertain the statistical measures adequate to synthetically describe their probability distribution functions (mean value, standard deviation, percentiles, etc.) and to be included in a harmonic Standard or in a power quality contract.

On the basis of the statistical characteristics of the harmonic index probability density functions, it is shown that the expected value of peak factor on single site is a suitable statistical measure to be selected for inclusion in Standards or in a power quality contract.

Furthermore, different system indices can be introduced in qualifying a distribution system or a part of it, particularly in the frame of the new liberalized market. Proper selection among those here proposed depends on the particular aim of the analysis.

11.2 Single Site Indices

The harmonic distortion can be described by several indices (individual harmonic distortion, total harmonic distortion, the peak factor, and so on). In the next, without loss of generality, we will refer to the total harmonic distortion (THD) and peak factor (kp); on the other hand, they are very useful indices [2-6] and also easily transferable in the probabilistic field, since only the marginal probability density function is generally required to fully describe their random properties. In any case, considerations about the probabilistic behavior of individual harmonic distortion are reported in [7, 8]. Moreover, the Joint Working Group CIGRE C4.07/CIRED in [9] has collected available measurement data and existing probabilistic indices for MV, HV and EHV systems and recommends a set of internationally relevant harmonic indices and objectives.

The total harmonic distortion of Y (current or voltage) is given by:

,(11.1)

with Yh harmonic superimposed to the fundamental Y1.

The peak factor is defined as:

,(11.2)

where Yp is the peak value of the distorted waveform Y, and Yp1n the rated value of the fundamental.

The peak factor can be separated into two aliquots kpsY and kphY, the first related to the effects of fundamental variations and the second one related to the effects of harmonics superimposed to fundamental Yp1; it results:

.(11.3)

Both total harmonic distortion and peak value can be applied to voltage and current distortions. The peak factor of voltage given by (11.2) and its aliquots given by (11.3) have been considered in [5, 10] for translation in the probabilistic field while the current indices have been considered in [6].

The probability density functions (pdf’s) of the above mentioned indices exhibit different statistical characteristics, as clearly revealed by the analysis of several on site measurements and simulation results.

First of all, the pdf’s of all above indices can have several shapes and there is not a general rule to choice the probability density function which best fits one of them; in fact, various basic pdf’s (gaussian, uniform, linear increasing, and so on) have been tested to ascertain the best approximating the harmonic index pdf behavior, but any of them can’t be generally applied.

From the analysis of several daily or weekly on-site measurements and of numerical simulations on low voltage and medium voltage distribution systems, the following considerations on their probabilistic behavior can be provided.

The pdf’s of the total harmonic distortion of the voltage (THDV) and current (THDI) can have both monomodal and multimodal (usually bimodal) pattern. Moreover, mainly when the harmonic data refer to a week, they are generally characterized by significant values of standard deviations, due to the variability of harmonics in large ranges.

As an example, Figs. 11.1 a) and b) report the pdf’s of the THDV (a) and of the THDI (b) obtained processing over one week recorded data referred to a medium voltage distribution system.

Starting from the considerations about both the pdf shapes and statistical measures of THDV and THDI, it follows that to impose reasonable probabilistic limits on these indices is very difficult. In fact, to directly set limits on the whole pdf’s is neither realistic nor a reasonable solution, as a consequence of the obvious difficulties in the successive implementation in the Standards. On the other hand, to impose limits on one or more statistical measures (minimum value, maximum value, mean value, standard deviation, 95th percentile, and so on) can be practical but can lead to unfair decisions. In fact, if we refer to the 95th percentile or maximum value, as proposed by the IEC 61000-3-6 [3] in its assessment procedure, and we account for all the harmonic effects, disagreeable consequences can follow, since THDV and THDI pdf’s with the same 95th percentile and maximum value can have very different effects on electrical component behavior. As a very easy example, we can refer to the joule losses in a conductor (in percent of the losses at fundamental frequency) associated to the two different THDI pdf’s of Fig.11.2, characterized by the same 95th percentile and maximum. Assuming the conductor resistance independent on the frequency, the joule loss expected values in the two cases are equal to 0.095 (Fig.11.2a) and 0.035 (Fig.11.2b), that are far each other.

The pdf’s of the voltage peak factor (kpV) are characterized, in the most general case, by very low values of standard deviations. In all the results reported in literature and in our experience, the standard deviation shows values at least of very few percents, because the voltage harmonics have values lower than fundamental voltage that ranges in no particularly large intervals for system operation safety. Due to the very low values of standard deviations, it is useless to talk about the shape of the pdf (monomodal or multimodal). A similar behavior has been experimented with reference to the aliquot kphV, related to the effects of harmonics superimposed to the fundamental.

As an example, Fig. 11.3 shows the pdf’s of the kpV (a) and of kphV (b) obtained processing over one week recorded data referred to a medium voltage distribution system.

(a) / (a) / (a)
(b) / (b) / (b)
Fig. 11.1 – Probability density functions: a) THDV; b) THDI / Fig. 11.2 – Different THDI pdf’s with same 95th percentile and maximum values / Fig. 11.3 – Probability density functions: a) kpV; b) kphV

From the analysis of Fig.11.3 b) and from all the measurements on distribution systems we analyzed, it was also noted that the voltage harmonics in several cases cause increasing of the peak voltage with respect to the value that would have been in presence of only the fundamental component. In [10] it was instead experimented a lowering of the peak voltage.

Starting from the considerations about the kpV and kphV pdf shapes and their statistical measures, it follows that to impose probabilistic limits on these indices is now possible and can be a reasonable choice. In fact, the very low value of pdf standard deviations allow us to say that it can be enough to assign a limit on only the mean value of the pdf. Moreover, in practice only the mean value of the voltage peak factor[1]kpV seems to be suitable for inclusion in Standards or in power quality contracts. The aliquot kphV does not allow to compute all the harmonic effects on distribution system components, because to distinguish the effects between fundamental and harmonics is a very difficult matter[2].


As an example of setting limits on the mean value of peak factor, let’s consider the case of low voltage industrial systems. The limits can be assigned making use of the diagram in Fig. 11.4 [5], where the normalized expected value of life is plotted versus the mean value of peak factor; each point of Fig. 11.4 is obtained averaging the expected values of normalized useful life for different distribution system components (cables, transformers, capacitors and ac motors).

The pdf’s of the current peak factor (kpI) and of its aliquot kphI, related to the effects of harmonics superimposed to the fundamental can have both monomodal and multimodal pattern, once again usually bimodal. The variance of the kpI pdf is characterized, in the most general case, by significant values of standard deviations; on the contrary, reduced variances of kphI pdf’s were experimented.

As an example, Figs. 11.5 a) and b) show the pdf’s of the kpI (a) and of the kphI (b) obtained processing over one week recorded data referred to an actual medium voltage distribution system.

(a) / (b)
Fig. 11.5 – Probability density functions: a) kpI; b) kphI

From the analysis of Fig. 11.5 a) and from all the measurements we analyzed on distribution systems, it was also noted that the high variability of current peak factor kpI is due to the wide variation range of the fundamental component, as clearly highlighted in Fig. 11.6, which reports pdf of the aliquot kpsI corresponding to the same measurements of Fig. 11.5. This is a reasonable result, for the very large variability of connected loads.

From the analysis of Fig.11.5 b), it appears that the current harmonics usually cause decreasing of the peak current with respect to the value that would have been in presence of only the fundamental component. However, this is not a general rule, because in other on site measurements on distribution network it was experimented an increasing of the peak current, in some cases significant.

Finally, starting from the considerations about both the kpI pdf shapes and their statistical measures, it follows that to impose reasonable probabilistic limits on this index is a difficult task, for the same reasons evidenced in the case of THDV.

Fig. 11.6 – Probability density function of kpsI

On the contrary, if the trend evidenced by our measurements were confirmed, to impose probabilistic limits on kphI index would be possible and also a reasonable choice. In fact, the very low value of pdf standard deviation allows to say, once again, that it is enough to assign a limit only on the mean value. A possible criterion for assigning limits to the mean value of kphI could be to set the fundamental current at rated value and derive the limits on the basis of the component effects.

11.3 System Indices

As clearly pointed out in [1], the system indices serve as metrics and are not intended as an exact representation of the quality of service provided to each customer served from the assessed system. They can be used as a benchmark against which quality levels of parts of the distribution system and of different distribution systems can be compared.

The system indices here taken into account only refer to voltage distortion; they are:

  • the System Total Harmonic Distortion 95th percentile, STHD95;
  • the System Average Total Harmonic Distortion (Peak Factor) SATHD (SAkp);
  • the System Average Excessive Total Harmonic Distortion (Peak Factor) Ratio Index, SAETHDRITHD*, (SAEkpRIKP*).

The system indices referred to Total Harmonic Distortion have been defined in [1], the ones concerning the Peak Factor are introduced in [6].

Let’s refer to a distribution system with N busbars. Even if the system indices can also refer to small segments of the entire distribution system, such as a single feeder or a single busbar, in the following, without loss of generality, we consider indices referred to the entire distribution system.

The STHD95 index is defined as the 95th percentile value of a weighted distribution; this weighted distribution is obtained collecting the 95th percentile values of the M individual index distributions, each distribution obtained with the measurements recorded at a monitoring site. The weights can be linked to the connected powers or the number of customer served from the area that monitor data represents, and so on; in the next, without loss of generality, we assume as weights the connected powers. Under these assumptions, the following relations compute STHD95:

,(11.4)

with CP95THDscalculated via the succeeding expression:

(11.5)

where:

s = monitoring site;

XTHDs,i = ith steady state THD measurement at the site s;

Ls= connected kVA served from the system segment where there is the monitoring site s;

fTHDs (XTHDs,i) = probability distribution function of the sampled THD values for monitoring site s;

CP95THDs = 95th cumulative probability value of THD for the monitoring site s;

ft(CP95THDs) = probability distribution function of the individual monitoring site CP95THDs values.

The system index STHD95 allows to summarize the measurements both temporally and spatially handling measurements at M sites of the system in a defined time period, assumed as significant for the characterization of system service conditions.

As an example, the value of STHD95 obtained processing over one week recorded data referred to the same medium voltage distribution system considered in Section 11.2 results equal to 2.2%.

The System Average Total Harmonic Distortion (Peak Factor) is based on the mean value of the distributions rather than the CP95 value:

(11.6)

(11.7)

being:

(THDs) = mean value of sampled THD values for monitoring site s;

(kp,s) = mean value of sampled kp values for monitoring site s;

LT= total connected kVA served from the system.

SATHD and SAkp give average indications on the system voltage quality and, thanks to the introduced weights, allow to assign different importance to the various sections of the entire distribution system.

The values of SATHD and of SAkp, again computed on the same distribution system before referred to, result equal to 1.16% and 1.05, respectively. It is worthwhile to note that SATHD is far from STHD95 due to the not negligible value of standard deviation.

The System Average Excessive Total Harmonic Distortion (Peak Factor) Ratio Index, SAETHDRITHD* (SAEkpRIkp* ) is related to the number of steady-state measurements that exhibit a THD (kp) value exceeding the THD* (kp*) threshold. It is computed, for each monitoring site of the system, counting the measurements exceeding the THD* (kp*) value and normalizing this number to the total number of the measurements effected at site s. They are defined by the following relations:

(11.8)

(11.9)

where:

NTHD*s (Nkp*s) = number of steady state measurements at monitoring site s that exhibit a THD (kp) value which exceeds the specified threshold THD* (kp*);

NTot,s= total number of steady state measurements effected at monitoring site s over the assessed period of time.

SAETHDRITHD* and SAEkpRIkp* are linked to the measure of the total portion of the time that the system exceeds the specified THD* and kp* thresholds, respectively.

Table 11.1 shows the values of SAETHDRITHD* and SAEkpRIkp* computed on the same distribution system already referred to and with two thresholds.

Table 11.1 - Values of SAETHDRITHD* and SAEkpRIKP*

SAETHDRI5% / 0
SAETHDRI2% / 0.016
SAEkpRI1.03 / 0.97
SAEkpRI1.05 / 0.41

Summarising, the introduced indices provide different global measures to qualify the system in terms of service quality. The availability of several indices for a phenomenon can be useful to give evidence to different aspects of the same disturbance. No general rule can be established to suggest the most adequate index to be used; it will depend on which aspect is judged more relevant.

For example, in the considered real distribution system, SATHD equal to 1.16 % indicates not only the average level of distortion, but also takes into account the sites serving larger loads. This can be more evident if we consider that the average value of THD on the same distribution system, without any weights, is greater of about 25%.

Further indication on the same system and on the same measurements concerns the highest levels of distortion. STHD95 value indicates the weighted 95th percentile of the 95th percentiles of the system sites and SAETHDRITHD* values give more details about the weighted relative frequencies of the records above specified threshold. Analogous considerations can be developed concerning the system indices referred to the peak factor.

An alternative way to compute the aforementioned indices concerns the statistical analysis of all the measurements effected at the M sites of the system. Assuming that centralized handling all the measurements of the M sites is not hard, this method seems to be interesting, since any statistical measure of a pdf is estimated with an accuracy that depends on the number of processed data. This extensive procedure gives indication on the global behavior of the system; on the contrary the first recalled procedure furnishes the indication on the behavior of the sites.

For example, with reference to THD, for each site s we compute the relative frequencies of the measured THD and we weight them with the ratio Ls/LT. Starting from the weighted relative frequencies at each site s, we evaluate the system weighted relative frequencies WFTHD and, then, compute the mean value and the 95th percentile that coincide with the indices SATHD and STHD95, respectively. The system weighted relative frequencies of THD can be also used to evaluate the indices SAETHDRITHD*.

An analogous procedure can be implemented to calculate the indices SAkp andSAEkpRIkp* referred to the peak factor.