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Quantitative and qualitative data analysis,

Quantitative and qualitative data analysis,

Condition based maintenance,

Evidential reasoning


Franjo JOVIĆ[1]



The article describes numerical techniques used for translating quantitative data collected by power distribution system's measuring equipment in to a qualitative domain.

Plausible qualitative degrees are essential in process of assessing condition state of power distribution system.

Entire aggregation process is based on evidential reasoning algorithm and it relays upon accuracy of the qualitative degrees. Special effort must be engaged in the translation process to ensure data integrity.

Techniques for quantitative to qualitative conversion of data are proposed.

Condition based maintenance of power distribution system can be performed after successful translation process and object decomposition based on Dampster-Shaffer theory.

The major obstacle is one – to – many correspondences of real-time measurement data and object condition based decomposition.

Results for quantitative to qualitative data conversion and corresponding aggregation processes are presented. 


Condition based maintenance is one of several types of technical maintenance. This kind of maintenance is based on the state of an object or system to be maintained. In spite of condition based maintenance, preventive maintenance is performed by a specific schedule with intend to avoid functional errors and failures. The basic advantage of this kind of maintenance is guaranteed high availability of maintained object or system. Basic disadvantage is partial and not adequate use of an object lifetime. Maintenance after failure is the next kind of maintenance where in opposite of the preventive maintenance entire lifetime of an object is used. Major disadvantage of this approach is the fact that failure needs to happen for the maintenance to begin with. Also it is not possible to predict time or expenses needed for failure recovery. This approach demands certain supply of the spare parts and/or adequate substitutes. The type of maintenance to be considered here is the condition based maintenance. It is a demanding approach because of the need for frequent inspection and monitoring of an object, part of the system or entire maintained system, but it offers an optimal usage of the objects lifetime. Experience with degradation of the object condition and analytic skills are required for this approach. Relevant information needs to be gathered frequently or even constantly. If this information can also be easily measured, then they are suitable for online intelligent monitoring. This means that this kind of information are gathered, locally processed and transferred to the central part of the monitoring system where it can be further processed by the use of complex algorithms, analyzed and stored. In spite of modern and powerful monitoring equipment there will still be information that cannot be online monitored due to complex measuring procedure or information nature (oil chromatography, assessment of objects general state, etc.). This kind of information requires a trained professional to provide measurement or assessment. Gathered information should be adequately processed and interpreted.

This article describes numerical techniques used for translating gathered information in to qualitative domain. The nature of gathered information (measured values) determines translation techniques to be used. There are several different types of information and adequate techniques for translating them. Procedures used for translation of continue values in to qualitative degrees are revised in the next section. In section 3, techniques used for assessing component condition state based upon component or object's age are presented. Translation of discrete values is covered in section 4, while conclusion is given in section 5.

After the translation process the necessary input values for multiple attribute decision analysis (MADA) [1,3] and evidential reasoning (ER) [2,8] approach are obtained. MADA and ER approach used in condition based maintenance of power distribution station (PDS) and power distribution system (PDSY) are in detail described in [4] and [13].


Let us assume that object condition state assessment is based on measurement of continuous value x (transformers oil humidity level, circuit breaker time off, etc.). Due to different variables in measuring process one or more measured values will have different values. Such gathered data are distributed according to certain statistical distribution. In most cases that distribution can be represented by normal or Gaussian distribution with following parameters: and 2 (mean and standard deviation). By the use of mean and standard deviation measured values are transformed in to qualitative values.

According to the manufacturers recommendation and users experience observed objects are grouped into classes with assigned qualitative degrees. Classes are defined according to gathered data discrepancy. Boundaries between classes are not strictly defined and degree overlapping is present. Instead of firm boundaries definition, mean and standard deviation are calculated for each class of data (shown in Figure 1.).

To each data set class n = 1, … N, qualitative degree Hn, mean  and standard deviation are assigned. Parameters and are assessed on different ways according to type of measured value, type of an object, manufacturers recommendation, malfunction statistic and experience of an assessor. 

Fig. 1. Division of measured value x into five qualitative classes with assigned values for bn

Qualitative degree Hn with degree of belief n is defined by normal distribution with and parameters as follows:

������������� ������������� ������������� ������������� ������������� (1)

It is reasonable to assume that measurement of physical value is exact procedure and that involved uncertainty does not exist. According to the above, degree of belief n is normalized as follows:

������������� ������������� ������������� ������������� ������������� ������������� (2)

������������� ������������� ������������� ������������� ������������� ������������� (3)

Normal distribution of data suggests certain probability for each measured value, therefore each qualitative degree would have certain probability. To simplify the procedure we can discard degrees with low probability (less then 0.05). Figure 1. and Figure 2. represent translating process and degree assessment of transformers oil according to the measurement of humidity level in ppm.

Similar graph can be generated for all assessment attributes where qualitative degree is proportional to measured value (oil's gas level, joint temperature, load, circuit breaker's time off, etc.)

Fig. 2. Normalized degrees of belief  for each qualitative deegre

For assessment of attributes where the observed value is positive or negative deviation from optimal or ideal value, qualitative degree is proportional to absolute value of difference between optimal and measured value. For this case typical example is measurement of synchronous circuit breaking of three-pole circuit breaker (shown in Figure 3.). Ideal latency of other two poles over first one is t0 = 2 ms. Observed value is absolute value between measured time t and ideal value t0:

������������� ������������� ������������� ������������� ������������� ������������� (4)

Fig. 3. Assessment of synchronous circuit breaking of three-pole circuit breaker


In case when object's condition is performed on the basis of objects age attribute, the reliability time function is determined first. To determine this function, knowledge of failure statistic for observer or similar object is necessary. One of the major parameters is failure intensity (t), witch represents probability for failure to happen in certain point of time. Reliability function is given as follows:

������������� ������������� ������������� ������������� ������������� ������������� (5)

In most cases failure intensity is constant value for considerable objects exploitation time. Often it is expressed as Mean Time To Failure (MTTF):

������������� ������������� ������������� ������������� ������������� ������������� (6)

If  is a constant then expression (5) can be written as follows:

������������� ������������� ������������� ������������� ������������� ������������� (7)

Reliability interval R(t) [0,1] is divided into N intervals with corresponding qualitative degrees Hn, n=1,…,N. Division of reliability interval can be uniform (shown on Fig. 4) or different according to the assessors decision.

Fig. 4. Assessment of qualitative degrees based on age of observed object

Each data class is represented by its mean Rn , witch is mirrored to time line as follows:

������������� ������������� ������������� ������������� ������������� (8)

������������� ������������� ������������� ������������� ������������� ������������� (9)

For known age level t qualitative degrees Hn are assigned to observed attributes. The degree reliability is increasing with variable t closing to the middle of tn interval. For this reason each interval n = 1,…,N is assigned normal distribution  with mean n = tn:

������������� ������������� ������������� ������������� ������������� (10)

Standard deviation is chosen as follows:

������������� ������������� ������������� ������������� ������������� ������������� (11)

������������� ������������� ������������� ������������� ������������� (12)

Overall reliability of all degrees is normalized according to following expression:

������������� ������������� ������������� ������������� (13)

where H is measurement uncertainty. Mentioned measurement uncertainty can occur by the influence of several different factors like lack of knowledge of exact object age level or failure intensity of such devices. Figure 6. represents reliability for each five qualitative degrees normalized to 100 % for disconector with mean time to failure MTTF  = 12 months.

Fig. 5. Normalized reliability of qualitative degrees for disconector with MTTF=12 months


In general discrete value x can obtain an finite number of values K :

������������� ������������� ������������� ������������� ������������� (14)

Therefore xk can be:

- Specific numeric value (surge arrester count number).

- One value from a finite set of values (for example in assessment of Buholtz relay condition state three states are possible: A – functional state; B – warning; and C – failure/shutdown state).

- Descriptive value (for example good, bad, average).

In most cases number of values K is relatively small, meaning that K is lower than 10. For values greater than 10, methods for continuous variables mentioned above are appropriate. For each value xk set of qualitative degrees of belief is assigned upon following expression:

������������� ������������� ������������� ������������� ������������� (15)

The simplest way for degree assignment is by the use of a following lookup table.


























Table 1. Lookup table for transformation of discrete data into qualitative degrees

In practical assessment each state of xk is assigned with one or two degrees so that most cells of the table will be zeros. Uncertainty of a single state is analytically determined from the uncertainty of all states as follows:

������������� ������������� ������������� ������������� ������������� (16)

Expert assessor or a group of experts for specific device type assigns values of parameters nk. Methods used for uncertainty definition are mostly experience based and defined for each assessment separately. 

Let us have a closer look at the Buholtz relay:

- Buholtz relay is component with strong influence on overall assessment of transformer. (Especially state C – failure/shutdown possess a strong negative influence.) Therefore the weight of its attribute 1142 is relatively high.

- When Buholtz relay is in state C – failure/shutdown, transformer and corresponding elements of power distribution station are shutdown. In such case the qualitative degree 13 is strictly negative meaning 13=1.

- Desirable state for Buholtz relay s state A – witch indicates normal functional state of a device. In such case state A is neutral meaning that it does not give us any specific information about transformers condition, so this degree should not have strong influence on aggregation process and overall qualitative degree of transformer. Because the defined weight 1142 cannot lowered because state C, average qualitative degree for its condition state is assigned. It is possible for relay to be malfunction and in spite of a failure it signals A state. Therefore certainty of this degree is 70%.

- State B indicates a warning, meaning failure or decreased oil level. Described state usually demands maintenance to occur so it is necessary that qualitative degrees are relatively poor,  12 = 0.4 and 22 = 0.3. For the same reasons mentioned above uncertainty of a given degree is relatively high H = 0.3.Asigned qualitative degrees and related uncertainty are given in Table 2.








A – functional







B – warning







C –failure/shutdown







Table 2. Assessment of transformer based on Buholtz relay condition state

  Assessment of qualitative degrees for surge arrester, based upon the surge count, is described in following example. Observed variable is surge count that can be rather high:

. ������������� ������������� ������������� ������������� ������������� (17)

Rough assessment of average surge count for local power grid company is estimated to 0.1 counts per surge arrester per year. During the average surge arrester's lifecycle (approximately 15 to 20 years) average count is 2. Surge counter counts only significant cases where high-energy surges are involved. More sophisticated surge arrestor design tends to rank current impulse of a surge, since it is crucial information for condition assessment.

Because of very small set of data x is available these attributes cannot be statistically analyzed attributes analyzed in previous sections.  For this attribute degree assessment following graphic representation is used as shown at Fig. 6.

Fig. 6. Assessment of qualitative degrees for surge arrestor based upon surge count number

Assessment of attributes degrees is proportional to 1/surge count number. Uncertainty of qualitative degrees depends upon uncertainty of component it self witch is subject to manufacturer, type and construction of surge arrestor.


Once the decomposition of power distribution station (or any other observed object) is performed and actual measurement has been accomplished, techniques described here are used to prepare measured variables for aggregation process. Methods for translating gathered data (quantitative and qualitative) are rather complicated as seen. Different types of data demand different translation techniques. Lots of experience knowledge is needed for a plausible translation.

Decomposition of power distribution station and qualitative degrees obtained by the use of described techniques are given in Table 3.

General attribute

Basic attribute

Assessment of PDS

Power distribution station

Primary equipment




Transformer oil 111

Gas level 1111

A(0.7), G(0.2)

Humidity level 1112

A(0.5), G(0.5)

Age state 1113


Coil 112

Winding temperature 1121

G(0.5), E(0.5)

Load 113

G(0.4), E(0.6)

Measuring and protection equipment 114

Temperature sensor 1141


Buholtz relay 1142


Cooling system 115


Tap changer 116

A(0.3), G(0.6)

Circuit breaker 12

A(0.4), G(0.6)

Disconnector 13


Busbar 14

Vibration 141


Joint temperature 142

A(05), G(0.4)

Instrument transformers 15

G(0.7), E(0.3)

Surge arrester 16

Surge counter 161


Leakage current 162

A(0.3), G(0.7)

Secondary equipment


Measuring equipment 21


Power supply 22


Protection 23


Communication equipment 24


Table 3. Decomposition of power distribution station and qualitative degrees

Data shown in Table 2. is just an input for aggregation process described in detail in [4] and [13].  This paper in addition to [4] and [13] represent set of procedures needed for Condition based maintenance of power distribution station or a power distribution system consisted of several distribution stations.

If procedures described here, in [4], and [13] are performed continuously, or at regular time intervals, then we have appropriate data to describe condition state of a power distribution station as a time function. Based on analysis of this time function it is possible to make decisions concerning maintenance. In case when we have condition state of power distribution station described as a time function and appropriate knowledge base system, prediction of failures may be achieved.

Example calculations in assessment of the power distribution system are performed by the use of windows based System assessor software (SAS) with implemented evidential reasoning algorithms for aggregation process.��

This tool gives us ability to possess an on demand insight of the condition state of a power distribution system and it’s degradation in operation and improvement after maintenance, as well.


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[4] F. Jovic, M. Filipovic, D. Blazevic, N. Slavek, Condition Based Maintenance in Distributed Production Environment, Machine engineering, 2004. 

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[7] G. Shafer, Mathematical Theory of Evidence. Princeton, NJ; Princeton Univ. Press, 1976.

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[10] J. B. Yang, Rule and utility based evidential reasoning approach for multiple attribute decision analysis under uncertainty, Eur. J. Oper. Res., 2001, 131/1, 31-61.

[11] J. Yen, Generalizing the Dempster – Shafer Theory to Fuzzy Sets, IEEE Transactions on Systems, Man, and Cybernetics,1990, 20/3, 559-570.

[12] Z. J. Zhang, J. B. Yang, & D. L. Xu , A hierarchical analysis model for multiobjective decision making, Analysis, Design and Evaluation of Man-Machine Systms, Oxford, U.K., 1990.

[13]  Ž. Jagnjić, N. Slavek, D. Blažević, Condition Based Maintenance of Power Distribution System, The 5th EUROSIM Congress on Modeling and Simulation, ESIEE, Paris, 2004.

[1] Faculty of Electrical Engineering Osijek


[2] HEP – Croatian National Grid Company

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