reading / British practice / Vol D - 1990 (ocr) ELECTRICAL SYSTEM & EQUIPMENT

•The discrete states in which the components of a system can reside.

•The transition paths between the various states.

Events TYPES 1 and 2

Failure events TYPES I and 2, involving only component passive failures, are represented by relatively si mple two state models.

In the modelling of common mode failures in the analysis of SESs, the assumption is made that, where sets of components fail in common mode, they are returned to service independently after repair. The repair process on all failed components (independent and common mode) is conducted simultaneously and

each is returned separately to service as soon as it has been repaired.

Two state models for single and two component systems (components represented by i and j) are shown in Fig 2.12, where (u) represents 'up', the normal operating state, and (d) 'down', the failed state during repair. The two-component model (Fig 2.12 (b)) in-

dudes a second order CMF. Figure 2.13 represents a two-component, two-state system, including maintenance outages overlapping a forced outage:

•REPAIR RATE (pt) for a component is the reciprocal of its repair time.

•MAINTENANCE TRANSITION RATE for a component is the reciprocal of its maintenance duration.

•FORCED OUTAGE is any outage which is unexpected.

A two-state model for a three-component system which includes transitions due to two-component (second order) CMFs is shown in Fig 2.14.

A similar two-state model for a three-component system with third order CMF transition is shown in Fig 2.15.

A Markov model for a three-component system, including CMF and maintenance outages, is shown in Fig 2.16. Since the analysis techniques used for SESs are limited to events up to and including third

order, it will be noticed that only third order events which include two commonly failed components can include a failure due to a maintenance outage overlapping a CMF.

In developing all these Markov models to represent a SES, it is assumed that a maintenance operation on any component of a minimal cut set would not be commenced if one or more components of the same set are on forced outage or have actively failed. It is also assumed that only one component of each minimal cut set is on maintenance at any one time.

Events TYPES 3 and 4

For failure events involving component active failures (TYPES 3 and 4) a three-state model is used.

Three-state models for single and two component systems are shown in Fig 2.17, where (u) represents the normal operating state, (s) the state after an active failure but before switching (switching state) and (r) the state after switching but during repair.

It can be seen that a small increase in the number of system components gives rise to a significant increase in the number of system states, particularly for three- state models. The Markov models become very involved, particularly when second and third order CMFs are included.

To reduce the number of possible combinations of component failures leading to state (s), the following simplifying assumptions are made:

•The probability of two simultaneous active faults occurring during switching is negligible. This assumption is justified because the component switching times are relatively small. Hence the exposure ti me of the system to a second active failure is small and the probability of two overlapping active failures during switching is negligible.

•The probability of two simultaneously stuck breakers in the system is also negligible. This is justifiable

since the individual stuck breaker probability is normally very small, of the order 10 -3 .

Such states as j(s) in Fig 2.17 are thus excluded from

i(s)

all three-state Markov models used in the analysis. A three-state Markov model for three-component systems is shown in Fig 2.18. The model represents an active failure of component 1 and overlapping forced outages of components 2 and 3 due to independent and common mode failures. Similar models can be drawn for active failure of component 2 overlapping forced outages of components 1 and 3, and for active failure of component 3 overlapping forced outages of

components I and 2.

Event TYPE 5

The Markov model for second order TYPE 5 events, including overlapping forced outages (independent and

(a) Two-state. one component

II

Id)

(b)Two-state, two components

KEY

=PASSIVE FAILURE RATE COMPONENT PASSIVE FAILURE RATE COMPONENT j

p.= REPAIR RATE COMPONENT i

Ci = REPAR RATE COMPONENT j

= CMF RATE FOR COMPONENTS i AND1

.u.

d. - con n SYSTEM UP/HEALTHY STATE

system. Failure/restoration events of TYPE 1 and TYPE 2 are assessed.

Part 2 Analysis considering only active failure and active failures overlapping passive failures. A three-state model of the system is used and si mulation of active failures and active failures in conjunction with a stuck breaker condition are performed for each component, the system configuration being changed for each simulation according to the deduced switching effects. Failure/restoration events of TYPE 3 and TYPE 4 are assessed.

For failure events TYPES 1 and 2, the reliability indices of the load point are obtained by reducing all second and third order minimal cut sets to an equivalent first order cut set. All real and equivalent first order minimal cuts are then combined 'in series' to give the overall load point indices.

Greater precision in the evaluation of the busbar indices can be achieved if the probability of failure of the (NO) paths available as the means of restoring supply to the load point being evaluated (i.e., TYPE 2 events), during the period they are required to function, is taken into consideration.

The GRASP2 computer program can include this facility by the use of study control parameters I and 4 (see Fig 2.6).

Inclusion of the failure probability of the (NO) paths requires the calculation of auxiliary indices involving the simulation of failure events associated with the (NO) paths up to the order specified in the study control parameters. These auxiliary results are then combined with the main indices for the associated TYPE 2 failure event, using the appropriate minimal cut set equations.

The method of deriving the equations for each event type is fully discussed in [2].

Second order failure events (TYPE 1 and TYPE 2)

1-1 SYSTEM DOWN/UNHEALTHY STATE

I) 2 12 I N\o-state models for single and mo component systems

common mode) and maintenance outages overlapping forced outages, is shown in Fig 2.19.

2,5.11 Evaluation techniques (busbar indices)

Since passive and active failures are independent failures, the analysis of a system is divided into the foliov.ine parts:

Part 1 Analysis considering only overlapping passive failures and using a two-state model of the

There are three failure sequences for a second order failure event:

(a)Failure or maintenance outage of component 1 followed by the independent failure of component 2.

(b)Failure or maintenance outage of component 2 followed by the independent failure of component 1.

(c)A common mode failure of components 1 and 2.

The equations for the evaluation of TYPE I and TYPE 2 failure events are shown in Tables 2.1 and 2.2.

Events (a) and (b) involve sequential independent failures of each component and (c) refers to common failure of both components in the minimal cut set.

The equations for TYPE 2 assume that supply can be restored by either closing a (NO) path in a time t,

or by completing a repair process that has already commenced. A repair process will not be started in preference to closing a (NO) path. This is a realistic assumption for two reasons. First, the greatest effort will be placed on recovering supply by the easiest achievable means, which is normally by switching. Secondly, repair times are generally much greater than switching times. Consequently, the outage time of sequence (a) for TYPE 2 is the overlapping time between repair of component I (already started) and the switching time of the (NO) path, whereas that of sequence (c) is only the switching time.

Third order failure events (TYPE I and TYPE 2)

The equations for evaluating third order TYPE 1 and TYPE 2 failure events which include second and third order CMFs are listed in Tables 2.3 and 2.4. They are based on the Markov model of Fig 2.14, where it is assumed that components 2 and 3 can fail in common mode, and on the Markov model of Fig 2.15, where it is assumed that all three components can fail in common mode.

Events (a) to (f) involve sequential independent failures of components 1, 2 and 3. Event (g) involves

the independent failure of component 1, followed by CMF of components 2 and 3. Event (h) refers to CMF of components 2 and 3, followed by the independent failure of component 1, and event (1) refers to CMF of all three components.

Equations for the evaluation of third order TYPE 1 and TYPE 2 failure events involving overlapping maintenance outages are shown in Tables 2.5 and 2.6. Only second order component CMFs can be involved in these events.

Second order failure events (TYPE 3)

The equations for the evaluation of second order TYPE 3 failure events, involving component active failures overlapping with forced outages, are shown in Table 2.7. There are obviously no events involving component CMFs.

Since it is assumed that a maintenance outage would not be commenced if one or more of the components of a minimum cut set have already actively failed, there are only two event sequences involving maintenance that need be considered. The necessary equations are shown in Table 2.8. Again, there are no events involving component CMF.

Third order failure events (TYPE 3)

A third order event of TYPE 3 involves the outage of one component due to active failure overlapping with the forced outage of the other two components of each minimal cut set. The forced outage of the other two components can include a second order component CMF, if applicable. Equations for the evaluation of these events are shown in Table 2.9 and are based on the Markov diagram of Fig 2.18.

Similar equations can be written for TYPE 3 events involving active failure of component 2, and also of component 3, overlapping forced outages.

The equations shown in Table 2.10 are for third order TYPE 3 events involving active failure of component 1 overlapping with maintenance and/or forced outages of components 2 and 3. There can be no component CMFs included in these events.

Similar equations can be written in respect of active failure of component 2 overlapping with maintenance and/or forced outages of components 1 and 3, and also for active failure of component 3 overlapping with maintenance and/or forced outages of components 1 and 2.

Failure events (TYPE 4)

The equations to evaluate TYPE 4 failure events are not tabulated. They are basically similar to those for TYPE 3 (for first, second and third order), with the active FR of the actively failed component of each minimal cut set multiplied by the stuck probability of one of the circuit-breakers protecting the actively failed component.

Failure event TYPE 5

Standby plant is, by definition, always connected to the SES through a (NO) circuit-breaker. This applies so far as reliability evaluation is concerned, even if the standby system comprises, for example, a battery or inverter source, continuously float charged and connected to the system via a (NC) circuit-breaker (sometimes referred to as a 'No-break' supply system). To model the latter with the GRASP2 computer program, the (NC) circuit-breaker would be represented as a (NO) circuit-breaker with a zero switching time.

If the standby plant can be considered as having an unlimited capacity to supply energy to the system,

..■■••••

1, Three slates one comp° .enr

FIG. 2.17 Three-state models for single and two component systems

for example, a standby generator having an unlimited fuel supply or a battery having a storage capacity sufficiently large to meet the load requirements of the system for a duration greater than the longest component repair time, it would be modelled as such by the engineer and events of TYPE 2 would apply.

If, however, the standby source has a limited energy capacity (see the definition of an LES in Section 2.3.3 of this chapter), the failure events to be considered are of TYPE 5.

A conditional probability approach has been used in the development of the equations for TYPE 5 events. The method is fully described in [3).

The following assumptions have been made:

•The standby source (LES) is the only (NO) path available for supply restoration and it does not fail when required to operate.

•The failure modes involve either forced outages (independent and common mode) or forced outages overlapping with maintenance outages.

.• The LES time limit (t) is measured in hours.

Reliability evaluation of power systems

• Component repair rates are exponentially distributed with time.

The equations for second order TYPE 5 events are shown in Tables 2.11 and 2.12 and are based on the

Markov diagram of Fig 2.19.

Table 2.11 contains the equations for overlapping forced outages, which can include component CMFs,

derived in a similar way: those in Table 2.13 are for three independent overlapping forced outages, together with common mode failure of components 2 and 3. For third order events involving three independent overlapping forced outages, combined with a third order component CMF, the relevant equations are shown in Table 2.14.

Consideration of forced outages overlapping with maintenance outages results in the set of equations in Table 2.15, in which CMF of components 2 and 3 has been included.

2.5.12 Evaluation techniques (system indices)

Evaluation of the reliability of a SES in terms of the reliability of supply at its individual load point busbars is just one of the analysis techniques available.

If the emphasis is placed more on the overall performance of the SES, and it is required to determine the effect of its reliability on the availability of the associated boiler/turbine-generator unit, the use of busbar indices to compare different designs of SES objectively is not appropriate. Objective comparison becomes difficult, since the various busbars have different importances so far as the output of the unit is concerned and many failure events are common to more than one busbar.

To overcome these difficulties, an overall system approach is used. The independent failure events of the system busbars are considered as a set of events that include the unit independent failure events. Therefore this set of events represents the unit cut sets from which the unit minimal cut sets must be obtained.

The procedure is quite complicated because cut sets of different types have to be compared. Also, the simultaneous loss of two busbars may cause a derated state of operation of the associated unit which is different from the effect caused by the loss of each busbar separately. All these features must be taken into consideration.

To calculate the indices of each derated state of the unit due to the unreliability of the SES, the loss of each busbar is considered to cause one of the following states:

•Total loss of the unit.

•No influence on the output of the unit.

•A loss of x% of the output of the unit, where x may have any value and may be different for each busbar or combination of busbars.

Generally, the loss of two busbars is assumed to create a derated state that may or may not be equal to the derated states caused by loss of the busbars individually. It is assumed that the loss of three busbars causes total loss of the unit.

The algorithm developed for the evaluation of system indices and implemented in the GRASP interactive computer programs is described in [3]. It is based on the assumption that the independent failure events of each busbar are already known. The initial part of the reliability evaluation to calculate the system indices therefore follows basically the same procedure

as for the calculation of busbar indices, outlined in the previous section. Here, however, every busbar of the system that is known to affect the output capability of the associated unit must be included in the evaluation.

Having deduced all the feasible failure events for each busbar of the system, the following additional data is provided by the engineer from his detailed knowledge of the system and its operational characteristics:

•The consequences on the output power capability of the associated unit of losing the supply to each busbar.

•The consequences of simultaneously losing the supply to all combinations of two busbars, both of which separately lead to a derated state of the associated unit.

The algorithm uses these data to calculate the average rate of occurrence or encounter (ER), the average outage duration (AOD) and the average annual outage ti me (AOT) of each feasible derated state. The overall loss of generation (LOG) in terms of MWh/year of operation, due to the unreliability of the SES, is also calculated.

2.5.13 Presentation of results

The results of a reliability evaluation study are ini-

tially presented as VDU displays selected from the following:

(a)Failure rate (FR) for each load point busbar evaluated, displayed on a diagram of the selected subsystem alongside the appropriate busbar (Fig 2.20).

(b)Average outage duration (AOD) for each load point busbar evaluated, displayed on a diagram of the selected subsystem alongside the appropriate busbar (Fig 2.21).

(c)Annual outage time (AOT) for each load point busbar evaluated, displayed on a diagram of the selected subsystem alongside the appropriate busbar (Fig 2.22).

(d)A full list of the input data for the selected subsystem (Figs 2.23 and 2.24).

(e)A full list of the minimal paths (NC and NO), in the selected subsystem deduced for each load point busbar evaluated (Fig 2.25).

(f)A full list of minimal cuts, in the selected subsystem, deduced for each load point busbar evaluated (Fig 2.26).

(g)A full list of nodal failure events (with calculated indices) in the selected subsystem, for each load point evaluated. A sample from such a list is shown in Fig 2.27.

(h)A full list of the calculated busbar indices in the selected subsystem, for each load point evaluated.

The list shows separately the contribution due to each of the event types (Fig 2.28).

(i)A full list of the system failure events. A sample from such a list is shown in Fig 2.29.

(j)A full list of system indices (Fig 2.30).

Figures 2.20 to 2.30 are typical results displays for the small system of Fig 2.5 in respect of:

•Evaluation of busbar indices for load point S3.

•Overall system indices, assuming that the system is associated with a 660 MW unit and that each busbar/combination of busbars has the effects shown in Fig 2.29.

After selective viewing of the results on the VDU, hard copies of the required pages can be taken for use in reports, etc.

Alternatively, full paper printed output, which includes the results described in (d) to (j) above, can be obtained.

Obviously, the full results listing is much more

extensive for a larger system. The list of failure events is extremely lengthy, particularly for studies involving events up to third order. In practice, therefore, the most commonly used form of results output consists of hard copies of each of the busbar indices displayed on the system diagram, together with hard copies of the input component reliability data, the study control parameters, and the summary of busbar and/or system indices (results (h) and (j) above).

For important reliability evaluation work, for example, the assessment of the SESs for nuclear power stations, where there is a need for long term storage of results, it is normal practice to obtain a paper listing of the full detailed results, which is then microfilmed and archived.

2.6 Quality assurance

It is fairly straightforward to check by inspection of the system diagram that, for a small system such as that shown in Fig 2.5, the minimal paths and minimal cut sets of the appropriate order are correctly deduced by the interactive computer program.