Tutorial on 3-s interval rms value assessment
Several methods can be used to assess the 3 s interval rms value. Three methods are explained in this appendix; consecutive-interval, sliding-interval and resynchronised-interval.
The consecutive-interval method (see Fig. III.1) can give a level lower than other methods since the interval used for assessment is normally not synchronised with the 3 s interval that contains the maximum Uu level.
Figure III.1: Consecutive-interval method
Figure III.2: Illustration of 3 Uu assessments using a sliding-interval method
The sliding-interval, or other similar methods such as moving average or rolling methods, can be used to find the maximum Uu that occurred during the survey period (> week). As illustrated in Fig III.2, this method consists of calculating a 3 s interval rms value for every instantaneous value recorded; all the m previous records are taken into account in this calculation. Therefore, with the sliding-interval method, the instrument assesses as many 3 s interval rms values as instantaneous values recorded.
Figure III.3: Rsynchronised interval to assess from windows 15 to 30
Field experimentation has demonstrated that the sliding-interval method, as well as the consecutive-interval method, show some error. A combined sliding and consecutive method, a so -called resynchronised-interval method, was developed to satisfy the three most important criteria:
The resynchronised-interval method [27] is based on the consecutive-interval explained previously. However, the resynchronised-interval method allows overlapping with the previous and the subsequent 3 s intervals allowing 9 s of data to be studied for each 3 s interval assessed. The maximum rms level over 3 s intervals is found using the sliding-interval method on 9 s of data at a time (see Fig III.3). The analysis of the next consecutive interval overlaps with the unused portion of the previous consecutive interval and the sliding interval is again used up to the subsequent interval (see Fig III.4). In this way, each consecutive interval is studied in combination with the previous and the subsequent interval.
Figure III.4: Resynchronised interval to assess from windows 30 to 45
The following 8 step procedure explains in detail the operation of the resynchronised-interval method [27].
III.1 Calculate the uu factor according one of the three options of annex II.
III.2 Calculate the 3 s instantaneous value uu-3s,inst according the following equation for each uu: uu,i are unbalance voltages over the M previous windows in the 3 s interval (similar calculations can also be performed to assess the 10 min interval using the 200 previous uu-3s,inst assessed)
III.3 Generate a clock as soon as the measurement is initiated to impose consecutive 3 s intervals on the reference point (Fig III.5) (the clock is adjusted to 10 min when the 10 min interval is assessed)
III.4 Find the maximum uu-max value among the uu-3s,inst values assessed after the first 3 s that follow the end of the interval containing the previous maximum uu-max value (point A, Fig. III.5) up to the end of the following reference interval (point B). If the interval containing the maximum uu-max value does not exceed the end of the reference interval selected for study (point C), proceed to step III.8.
III.5 Assess the width of the interval between the end of the interval containing the previous maximum uu-max value (point F, Fig. III.6) and the beginning of the interval containing the maximum uu-max value obteined in step III.4 (point E). If this interval is less than 3 s, proceed to step III.7.
Figure III.5: Illustration of step
Figure III.6: Illustration of the interval to be analyzed
III.6 Find the maximum uu-max value among the uu-3s,inst values determined in the interval used in step III.5 and go to step III.8.
III.7 Assess the uu-3s,inst value using the values of the interval less than 3 s that was found in step III.5. If it exceeds the maximum uu-max value of the interval found in step III.4, set the uu-3s,inst value equal to the mean of the two adjacent maximum uu-max values, i.e. that of the previous interval and the found in step III.4.
III.8 Record the maximum value found in step III.4, III.6 or III.7 as the uu-rms value of the reference interval selected for study.
Comparison between the methods for assessing the intervals
Field experimentation has demonstrated that the consecutive-interval method gives values related to the instant that the instrumentation start to record and may never give the highest voltage unbalance occurring on the power system.
As far as the sliding-interval method is concerned, one of the values is the actual highest Uu that has occurred. However, the assessments can be very misleading if they are used for statistical analysis.
The resynchronised-interval method shows very good performance to assess values that have a high probability to not be exceeded.
Figure III.7 shows a signal used to explain the difference between the three methods. From 15 s to 300 s, the signal remains at 1.
Figure III.7: Example of a signal to analyse
Both the consecutive-interval and the resynchronised-interval methods give 100 values over 300 s. If the probability for each value to occur is equally distributed, the probability level of each value is 1%. The probability function for 100 values equally distributed ranges from 0.5% to 99.5%. Therefore, the maximum-value recorded has 99.5% probabilty not to be exceeded.
The recording may start at any 200 ms time step during the first 3 s interval. Depending the starting time, the 3 s interval at 2 p.u. can be shared in two consecutive intervals, which explains whyt the consecutive-interval method generates a range of maximum values.
The sliding-interval method gives a value for each 200 ms time step. After 300 s, the method generates 1500 values each having a 1/15% probability of occurrence. In this case, the probability function ranges from 0.03% to 99.97%. Therefore, the maximum value recorded has a 99.97% probability not to be exceeded. This statistical interpretation is obviously artificially produced because actual signal sto analyse still comprise 100 3 s intervals.
The signal in Figure III.7 is easy to analyse. The 2 level has occurred only during three seconds over 300 s. This level therefore has a 99.5% probability to be exceeded. Probability levels ranging from 2.5% to 98.5% should match with 1 p.u.
As shown in Table III.1, the expected result is obtained with the resynchronised-interval method. The level at 99.5% probability that is obtained by the consecutive-interval method is starting time dependent in ranges between 1.46 to 2 p.u. At 98.5% however, the consecutive-interval method reports1.37, that is 37% higher than expected.
Table III.1: Comparison of the three methods to assess the interval
Probability |
Consecutive method |
Sliding method |
Resynchronized Method |
||||
Lowest possible value |
Highest possible value |
||||||
Value |
error |
Value |
error |
Value |
error |
Value |
|
99.5 |
1.46 |
-27.1% |
2.00 |
0.0% |
1.72 |
-14.2% |
2.00 |
98.5 |
1 |
0.0% |
1.37 |
37.1% |
1.03 |
3.0% |
1 |
97.5 |
1 |
0.0% |
1 |
0.1% |
1.01 |
0.7% |
1 |
96.5 |
1 |
0.0% |
1 |
0.0% |
1.01 |
0.5% |
1 |
As mentioned earlier, the sliding-interval method also gives a sliding function to the probability distribution. In this case, the highest value (2.00) matches a 99.97% probability not to be exceeded. This method can be exact for the specific condition that an unmeasured period of 1200 s is composed of the same data as those calculated with the sliding-interval method. This condition can be possible if the user knows exactly the time period of the process which produce the disturbances. However, without specific precautions, this condition has been very observed rarely because most processes have random start and stop time period mismatches with the recording period.
This example, illustrated in Fig. III.7, allows the error assessment if the fluctuation level is 2. Some field measurement have already shown higher differences between the methods due to the fluctuating nature of the disturbances.
The uu-3s values assessed over each 24 hour period are classified in histograms to provide the level uu3s-95% that matches the daily 95% probability of being the highest level that can occur.
The maximum of the daily uu3s-95% values and the maximum uu3s-max values of all assessments performed during at least one week should be compared with 1 and 1.5 times the planning level respectively, as recommended by CC02. However, the need for such comparisons in terms of equipment susceptibility has never been expressed or explained.
The uu3s-95% and uu3s-max values can give an indication of the ripple severity on the DC side of a three-phase converter but no problem has ever been reported, with the result that no compatibility level can be quoted.
CC02 also recommend measuring the rms level over a 10 min interval. The value to compare with the planning level is the maximum recorded during one week i.e. 99.9% probability of not being exceeded. This value could be useful to obtain an indication of the severity level of the disturbance on power electronics used in some power system.
References
Roger Bergeron, "Voltage Unbalance on Dstribution Systems - Phase I", Canadian Electrical Association, Project No. 231 D 488, Montréal, Québec, January 1989.
Westinghouse Electric Corporation. "Applied Protective Relaying", Silent Sentinels publication, Newark, New Jersey.
Cumming, P.G. "Protection of Induction Motors Against Unbalance Voltage Operation", IEEE-PCI, September 1983, IEEE-PIT, June 1984 and IEEE-IAS October 1984.
Roger Bergeron, "A Measurement Protocol for Power Quality Coordination", CIRED 1991, paper 2.17, April 1991.
IEC 27-1 (1992).- Letter Symbols To Be Used in Electrical Technology. Part 1: General.
IEC 146. Parts 1 to 6 .- Semiconductors Convertors.
- IEC 375 (1972).- Conventions Concerning Electric and Magnetic Circuits.
- Roger Bergeron, "Power Quality Measurement Protocol, CEA Guide to Performing Power Quality Surveys," CEA report 220 D 711, Canadian Electrical Association 1 Westmount Square, Suite 1600 Montréal, Québec, Canada H3Z 2P9, 1996, 216 pp.
