**1. Introduction**

82 Electrical Generation and Distribution Systems and Power Quality Disturbances

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switching converter mode lings of a PV grid-connected system under islanding

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> A basic component of the power quality generally and electricity supply in particular is the management of maintenance actions of electric transmission and distribution networks (EDS). Starting from this fact, the chapter develops a mathematical model of external interventions upon a system henceforth, called Renewal Processes. These are performed in order to establish system performance i.e. its availability, in technical and economic imposed constraints. At present, the application of preventive maintenance (PM) strategy for the electric distribution systems, in general and to overhead electric lines (OEL) in special, at fixed or variable time intervals, cannot be accepted without scientifically planning the analysis from a technical and economic the point of view. Thus, it is considered that these strategies PM should benefit from mathematical models which, at their turn, should be based, on the probabilistic interpretation of the actual state of transmission and distribution installations for electricity. The solutions to these mathematical models must lead to the establishment objective necessities, priorities, magnitude of preventive maintenance actions and to reduce the life cycle cost of the electrical installations (Anders et al., 2007).

#### **1.1 General considerations and study assumptions**

Based on the definition of IEC No: 60300-3-11 for RCM: "method to identify and select failure management policies to efficiently and effectively achieve the required safty, availability and economy of operation", it actually represents a conception of translating feedback information from the past time of the operation installations to the future time of their maintenance, grounding this action on:


So, Reliability Based Maintenance (RCM) implies planning the future maintenance actions ( T<sup>+</sup> ) based on the technical state of the system, the state being assessed on the basis of the estimated reliability indices of the system at the planning moment ( <sup>0</sup> T ). At their turn, these reliability indices are mathematically estimated based on the *record of events*, that is, based on previously available information, related to the behaviour over period ( T<sup>−</sup> ), i.e. to the ( <sup>0</sup> T ) moment, concordant with Figure 1.

Reliability Centered Maintenance Optimization of Electric Distribution Systems 85


In theory of reliability, the concept of the *wear* has a wider meaning than in ordinary language. In this context, the wear includes any alteration in time of characteristics of reliability, for the purposes of worsening or improving them (Catuneanu & Mihalache 1983). Considering the function of the reliability of equipment for a mission with a duration *x,* 

( ) ( )

R t,t x R t



From a the mathematical point of view, we can express the wear through *failure rate (FR)* of the equipment. From the relationship (2) and from the defining relationship of failure rate

( )

If the failure rate of the equipment with positive wear increases in time, then these systems will be called IFR type systems (Increasing Failure Rate) and if the failure rate of the equipment with negative wear decreases in time, these systems will be called DFR type

An equipment is of *NBU degradable type* if the reliability function associated to a mission of the duration *x,* initialized at the *t* age of equipment, is less than the reliability function in

From the previous definition, results that a degradable equipment which was used is

The notion of degradable equipment is less restrictive than that of equipment with positive wear, which supposed the decreasingd character of the function *Rt t x* ( ) , + with *t* age of

x 0 x 0 R t R t x R t,t x z t lim lim → → xRt <sup>x</sup>

( ) () ( )

( ) Rt x

+ = (2)

( )

R t,t x R t ; t,x 0 ( ) () +< ≥ (4)

−+ + = = <sup>⋅</sup> (3)

+

reliability and of the notions of: *wear, degradable system and renewal actions*.

**1.3 Modelling of systems wear** 

we obtain:

equipment.

initialized at the *t* moment, of the type:

systems (Decreasing Failure Rate).

inferior that to a new equipment.

equipment, its reliability decreases with its age.

equipment, its reliability increases with its age.

interval *(0, t)*, regardless of the age and duration of *x* mission.

Fig. 1. Assessment and planning times of the RCM

Even if in case of the OEL, the two actions are apparently independent, as they take place at different times, they influence each other through the model adopted for each of them. Thus:


Accepting that the expression of the availability at a time *t,* the relationship given by the form (Baron et al., 1988):

$$\mathbf{A(t)} = \mathbf{R(t)} + \left\lfloor \mathbf{1} - \mathbf{R(t)} \right\rfloor \cdot \mathbf{M(t')} \tag{1}$$

where:

A(t) – Availability;

R(t) – Reliability;

M(t') – Maintainability

It can be inferred that the availability of an element or of a system is determined by two probabilities:


Thus, for components of EDS which are submissions by RCM actions, a distinctive study is required to model their availability under the aspect of modelling those two components:


As the mathematical models of the two components R(t) & M(t') interdependent, we will continue by exposing some considerations regarding the establishment of M(t'), and section 3 will be devoted to developing the component R(t) of the EDS.

#### **1.2 EDS-Degradable systems subjected to wear processes**

EDS in general and the OEL in particular, contain parts with mechanical and electrical character and their operation is directly influenced by different factors, being constantly subjected to the requirement of mechanical, electrical, thermal processes, etc. The literature shows the percentage of causes which determin interruptions of OEL, (Anders et al., 2007).


#### **1.3 Modelling of systems wear**

84 Electrical Generation and Distribution Systems and Power Quality Disturbances

Even if in case of the OEL, the two actions are apparently independent, as they take place at different times, they influence each other through the model adopted for each of them. Thus: - the analytical expression of the reliability function adopted in while ( T<sup>−</sup> ), depends directly by the basis of specific physical phenomena (such as wear, failure, renewal,


It can be inferred that the availability of an element or of a system is determined by two

a. the probability that the product to be in operation without failure over a period of time

b. the probability that the element or the system, which fails over time interval (*t*), to be reinstated in operation in time interval (*t'*) ; M(t') *-* is called the *Maintainability function.*  Thus, for components of EDS which are submissions by RCM actions, a distinctive study is required to model their availability under the aspect of modelling those two components: 1. The *reliability function* R(t) of EDS, respectively of the studied component, in this case a

2. The *Maintainability function* M(t'), under the aspect of their specific constructive and

As the mathematical models of the two components R(t) & M(t') interdependent, we will continue by exposing some considerations regarding the establishment of M(t'), and section

EDS in general and the OEL in particular, contain parts with mechanical and electrical character and their operation is directly influenced by different factors, being constantly subjected to the requirement of mechanical, electrical, thermal processes, etc. The literature shows the percentage of causes which determin interruptions of OEL, (Anders et al., 2007).

A(t) R t 1 R t M t' = + () () ( ) − ⋅ (1)

etc.) of equipment during maintenance actions in time interval ( T<sup>+</sup> );

Fig. 1. Assessment and planning times of the RCM

(*t*); R(t) – is called the *reliability function*;

operation conditions, maintenance.

3 will be devoted to developing the component R(t) of the EDS.

**1.2 EDS-Degradable systems subjected to wear processes** 

form (Baron et al., 1988):

A(t) – Availability; R(t) – Reliability; M(t') – Maintainability

probabilities:

OEL;

where:

In theory of reliability, the concept of the *wear* has a wider meaning than in ordinary language. In this context, the wear includes any alteration in time of characteristics of reliability, for the purposes of worsening or improving them (Catuneanu & Mihalache 1983). Considering the function of the reliability of equipment for a mission with a duration *x,*  initialized at the *t* moment, of the type:

$$\mathbf{R}\left(\mathbf{t}, \mathbf{t} + \mathbf{x}\right) = \frac{\mathbf{R}\left(\mathbf{t} + \mathbf{x}\right)}{\mathbf{R}\left(\mathbf{t}\right)}\tag{2}$$


From a the mathematical point of view, we can express the wear through *failure rate (FR)* of the equipment. From the relationship (2) and from the defining relationship of failure rate we obtain:

$$\text{var}(\mathbf{t}) = \lim\_{\mathbf{x} \to \mathbf{0}} \frac{\mathbf{R}(\mathbf{t}) - \mathbf{R}(\mathbf{t} + \mathbf{x})}{\mathbf{x} \cdot \mathbf{R}(\mathbf{t})} = \lim\_{\mathbf{x} \to \mathbf{0}} \frac{\mathbf{R}(\mathbf{t}, \mathbf{t} + \mathbf{x})}{\mathbf{x}} \tag{3}$$

If the failure rate of the equipment with positive wear increases in time, then these systems will be called IFR type systems (Increasing Failure Rate) and if the failure rate of the equipment with negative wear decreases in time, these systems will be called DFR type systems (Decreasing Failure Rate).

An equipment is of *NBU degradable type* if the reliability function associated to a mission of the duration *x,* initialized at the *t* age of equipment, is less than the reliability function in interval *(0, t)*, regardless of the age and duration of *x* mission.

$$\mathbb{R}(\mathbf{t}, \mathbf{t} + \mathbf{x}) < \mathbb{R}(\mathbf{t}); \quad \mathbf{t}, \mathbf{x} \ge \mathbf{0} \tag{4}$$

From the previous definition, results that a degradable equipment which was used is inferior that to a new equipment.

The notion of degradable equipment is less restrictive than that of equipment with positive wear, which supposed the decreasingd character of the function *Rt t x* ( ) , + with *t* age of equipment.

A particular problem consist of identifying the type of wear that characterizes an equipment/system, which can be obtained from reliability tests or from the analysis of the moments of failure in operation, which mathematically modelled, give the wear function, denoted by Ts (F) .


$$\mathbf{F(t) = 1 - R(t)}\tag{5}$$

Reliability Centered Maintenance Optimization of Electric Distribution Systems 87

The classification of renewal actions can be performed on several criteria, of which we remind a few: the purpose, timing and costs of their occurrence, distribution and frequency

In this work, we analyze and study only preventive renewal processes, through their






The evolution of an equipment will thus be represented by the succession of renewal moments 1 2 , , ... , *<sup>n</sup> tt t* , and the intervals between them: 1 2 , , ... , ... *<sup>n</sup> xx x* presented in the

If it is considered a certain time interval (0, t), the number of PR denoted by *Nt* , performed during this time, is a discrete deterministic process, called preventive renewal process, as the basic component of preventive maintenance, which in turn determine planning, development and the effects of RCM on system. This, requires the development of PR strategies, enabling knowledge of behaviour of renewal equipment, used in the

Studies classify and model the renewal strategies according to technical and economic parameters, which determine these strategies, in two distinct categories: non-periodic and



and last but not least, the effects on the system safety, (Anders et al., 2007).

added to a random process of failure renewals.

modelling influence on the system in the following cases:

Fig. 2. The evolution of an equipment with renewal

periodic, (Catuneanu & Mihalache, 1983), (Andres et al., 2007).

received from operation;

this point of view;

economic ones;

the system as a whole.

development of PM programs.

**2.2 Renewal strategies** 

failure renewals.

renewal;

Figure 2.

their strategies. Thus, the random or deterministic strategy of preventive renewals is


According to system reliability T fFt <sup>S</sup> = ( ) ( ) , the graph function of the wear inscribed in a square with l side, can indicate the type of equipment wear by its form:


Thus, the literature allow for the study maintenance of OEL, the following assumptions:


We mention that all models specific to these assumptions/behaviours of the OEL are dependent on the reliability function of the system in study and their effects model this function.
