Power distribution systems does in general have several states of functionality (e.g. several

load points that can function separately), which makes it reasonable to model them as multistate

systems 38, i.e. systems that allow for several levels of function of for example

availability. The component reliability importance indices presented in subchapter 3.1 are

based on systems that are binary, i.e. either functioning or not (two states). This is an approach

which proves ambiguous for networks with for example more than one load point, as for

example shown in paper III, Table 3. One component might be crucial from the perspective of

one load point while virtually unnecessary from another load point’s perspective. This calls for

an approach that takes the whole network’s reliability performance into account in one

measure and relates this measure to the individual component. The concept of the developed

indices is to utilize customer interruption costs as a measure of system reliability performance.

Component reliability importance indices for power systems is identified as a topic of

increasing interest to the research community. This can be seen in that most of the publications

in the topic are relatively new (see references for this chapter). The increased interest is

probably explained by the reregulation of the electricity market, resulting in a higher interest in

good payoff of maintenance actions, and in increased possibilities to perform advanced

reliability calculations.

This chapter starts with a brief introduction to general component reliability importance

indices, followed by a survey on what has been done in this specific topic for power systems.

The chapter continues with a more detailed presentation of the indices developed within the

PhD project. The chapter ends by outlining an approach to component reliability importance

indices for transmission systems.

20

3.1 Traditional component reliability importance indices

This subchapter contains a short description of some of the most referred component reliability

importance indices, followed by a brief discussion on their potential use in transmission and

distribution systems.

3.1.1 Birnbaum’s reliability importance

Birnbaum’s measure of component importance is a partial derivative of system reliability with

respect to individual component failure rate 47. It can also be argued that this is a sensitivity

analysis of system reliability with respect to component reliability. This index gives an

indication of how system reliability will change with changes in component reliability.

i

B

i p

h I t

?

? = ( ) ( ) p (3.1)

where h is the system reliability depending on all component reliabilities p (and system

structure) and pi component i’s reliability. A drawback with this method is that the studied

component’s reliability does not affect the importance index (for the specific component).

Another issue regarding this index is that it cannot be used in order to predict the effect of

several changes at the same time, i.e. reliability changes in several components at a time 56.

This is, however, a drawback shared with most component reliability importance indices.

3.1.2 Birnbaum’s structural importance

Birnbaum’s structural importance does not take any reliability into account, and hence it can be

stated that this method is truly deterministic. The method defines component importance as the

component’s number of occurrences in critical paths, normalized by the total number of system

states.

Definition of structural importance in accordance with 47 and 57:

I?(i) = ??(i) / 2n-1 (3.2)

where ??(i) is the number of critical path vectors for component i and 2n-1 is the total of

possible state vectors. In other words; the number of critical paths a component is involved in

is proportional to its importance. The structural importance can be calculated from Birnbaum’s

reliability importance by setting all component reliabilities to ½ 47

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