**3.3 Distribution function**

Establishing a law distribution used in reliability implies good knowledge of the physical bases of the phenomenon of wear, of the specific ways in which these phenomena manifest themselves and of the type of wear to which each OEL component and the entire system has been subjected. Considered as an EDS component, the OEL contains, at its turn, components of a mechanical character, whose operations are directly influenced by mechanical actions, electrical ones (e.g. overvoltages, over currents), temperature, environmental pollution, etc. We can say with certainty that the OEL failures are due to wear and slow aging and, from the standpoint of their reliability, they are treated as IFR and NBU type, with an increasing failure rate.

Given these findings, *the law of Weibull distribution* is adopted as *theoretical law* for modelling such survival processes. This law is specific for positive wear systems, being also characteristic for overhead electrical lines.

In the case of an OEL in operation whose components are characterized by the absence of hidden defects, but show a striking phenomenon of aging in time while the intensity of failures increases monotonically, *the law of Weibull distribution* is adopted as theoretical law, which is specific for positive wear systems, being also characteristic for overhead electrical lines.

Of all the known forms of the Weibull distribution law (two and three parameters, normalized) let us accept the form with two parameters for modeling the reliability of the electrical line in study. This form has the mathematical expression (Baron et al., 1988), (IEC 61649, 2008):

$$\mathbf{R}\left(\mathbf{t}, \alpha, \beta\right) = \mathbf{e}^{-\alpha \cdot \mathbf{t}^{\delta}} \tag{6}$$

Where:

α > 0 - is a scale parameter;

β > 0 - is a shape parameter, β > 1 for components of IFR type;

t (0, +∞) - time variable.

The relationship (6) expresses the probability that the event will occur in time interval (0, t) or as they say in the theory of reliability is the probability of the OEL functioning without fault until *t* moment.

### **3.4 The parameters of Weibull distribution function**

There are several studies which present a lot of techniques and methods to evaluate the Weibull function parameters, depending on: the number of function parameters chosen, the scope of the application, the available statistical data, etc.

In (Dickey, 1991), (IEC 61649, 2008), are mentions statistical methods for assessing the parameters for Weibull distribution of failures type and constant repair time. The Statistics Toolbox MATLAB programming environment allows the evaluation of parameters of the Weibull function, with the instructions, (Blaga, 2002):

[parmhat,parmci]=wblfit (t)

$$\text{paramhat (1): } \text{paramhat (2): }$$

A synthesis of the calculation methods and accuracy of the Weibull parameters is presented in (IEC 61649, 2008).

In the paper (Baron et al.,1988), the authors presents a practical mathematical method of setting the *α* and *β* parameters with two parameters of Weibull law by the form of relationship (6), based on statistical data obtained from the analysis of operating arrangements of the OEL, as well as in the following assumptions:


The value of the parameters 'α' and 'β' is derived from the relationship (6), which by logarithm application, results as:

$$\log \mathcal{R}\_{\mathbb{N}} \left( \mathbf{t}\_{\mathbb{h}} \right) = -\boldsymbol{\alpha} \cdot \mathbf{t}\_{\mathbb{h}}^{\mathbb{B}} \cdot \log \mathbf{e} \quad \text{or: } \lg \left[ \mathbf{1} \,/ \, \mathrm{R}\_{\mathbb{N}} \left( \mathbf{t}\_{\mathbb{h}} \right) \right] = \boldsymbol{\alpha} \cdot \mathbf{t}\_{\mathbb{h}}^{\mathbb{B}} \cdot \log \mathbf{e} \tag{7}$$

repeating the operation of logarithms, result:

$$\log\left\{\lg\left[\mathbf{1}\left(\mathcal{R}\_{\mathrm{N}}\left(\mathbf{t}\_{\mathrm{f}}\right)\right)\right]\right\} = \log\left(\log\mathbf{e}\right) + \log\alpha + \beta\log\mathbf{t}\_{\mathrm{f}}\tag{8}$$

with notations: ( ) ( ) { ( ) } max R t 1 t / t ; a lg lg e lg ; b lg lg 1 /R t N fi =− = + α = fi fi <sup>i</sup> N fi (9)

is obtained from (8) equation of a straight line: ba l i f g <sup>i</sup> = + β t (10)

for which using the method of least squares system is obtained the following equations system is obtained:Citiţi fonetic

$$\begin{aligned} \sum\_{i=1}^{n} \mathbf{b}\_{i} &= \mathbf{n} \cdot \mathbf{a} + \boldsymbol{\mathfrak{B}} \cdot \sum\_{i=1}^{n} \mathbf{l} \cdot \mathbf{t}\_{\text{fl}} \\ \sum\_{i=1}^{n} \mathbf{b}\_{i} \log \mathbf{t}\_{\text{fl}} &= \mathbf{a} \sum\_{i=1}^{n} \mathbf{l} \cdot \mathbf{t}\_{\text{fl}} + \boldsymbol{\mathfrak{B}} \sum\_{i=1}^{n} \left( \mathbf{l} \otimes \mathbf{t}\_{\text{fl}} \right)^{2} \end{aligned} \tag{11}$$

$$\begin{aligned} \mathbf{A} &= \mathbf{n} \cdot \mathbf{a} + \mathbf{\beta} \cdot \mathbf{B} \\ \mathbf{C} &= \mathbf{B} \cdot \mathbf{a} + \mathbf{\beta} \cdot \mathbf{D} \end{aligned} \tag{12}$$

or :

90 Electrical Generation and Distribution Systems and Power Quality Disturbances


Nr 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ti 62 68 80 200 210 212 254 290 407 418 483 497 510 539 tri 213 118 352 209 339 640 165 305 229 215 154 1970 204 303

... 45 46 47 48 49 50 51 52 53 54 ... 1272 1370 1399 1483 1578 1596 1597 1608 1638 1712 ... 482 1350 270 150 343 270 200 403 178 841

Establishing a law distribution used in reliability implies good knowledge of the physical bases of the phenomenon of wear, of the specific ways in which these phenomena manifest themselves and of the type of wear to which each OEL component and the entire system has been subjected. Considered as an EDS component, the OEL contains, at its turn, components of a mechanical character, whose operations are directly influenced by mechanical actions, electrical ones (e.g. overvoltages, over currents), temperature, environmental pollution, etc. We can say with certainty that the OEL failures are due to wear and slow aging and, from the standpoint of their reliability, they are treated as IFR and NBU type, with an increasing

Given these findings, *the law of Weibull distribution* is adopted as *theoretical law* for modelling such survival processes. This law is specific for positive wear systems, being also

In the case of an OEL in operation whose components are characterized by the absence of hidden defects, but show a striking phenomenon of aging in time while the intensity of failures increases monotonically, *the law of Weibull distribution* is adopted as theoretical law, which is specific for positive wear systems, being also characteristic for overhead electrical

Of all the known forms of the Weibull distribution law (two and three parameters, normalized) let us accept the form with two parameters for modeling the reliability of the electrical line in study. This form has the mathematical expression (Baron et al., 1988), (IEC

The relationship (6) expresses the probability that the event will occur in time interval (0, t) or as they say in the theory of reliability is the probability of the OEL functioning without

( ) <sup>t</sup> R t, , e <sup>β</sup> −α⋅ αβ = (6)

the values are presented in Table 1.

Table 1. Incidents database

**3.3 Distribution function** 

characteristic for overhead electrical lines.

failure rate.

lines.

61649, 2008):

α > 0 - is a scale parameter;

t (0, +∞) - time variable.

fault until *t* moment.

β > 0 - is a shape parameter, β > 1 for components of IFR type;

Where:

which solved in relation with a and β unknown, result:

$$\mathbf{a} = \frac{\mathbf{C} \cdot \mathbf{B} - \mathbf{A} \cdot \mathbf{D}}{\mathbf{B}^2 - \mathbf{n} \cdot \mathbf{D}} \; ; \; \mathbf{\beta} = \frac{\mathbf{n} \cdot \mathbf{C} - \mathbf{A} \cdot \mathbf{B}}{\mathbf{n} \cdot \mathbf{D} - \mathbf{B}^2} \quad \text{and} \; \alpha = 10^{\left[ \mathbf{a} - \mathbb{I}\_{\mathbb{B}}(\log \epsilon) \right]} \tag{13}$$

For the OEL 20 KV in study, by replacing the parameters from Table 3.1 in the previous relations, the following values of parameters of the Weibull function distribution result:

$$
\alpha = 1.2669 \cdot 10^{(-4)} \quad \text{and} \quad \beta = 1.2939 \tag{14}
$$

which allows modeling the reliability function *R(t)* of OEL the through the relationship:

$$\mathbf{R(t)} = \mathbf{e}^{-1.2669 \cdot 10^{l-4} \cdot t^{1.2669}} \tag{15}$$

Reliability Centered Maintenance Optimization of Electric Distribution Systems 93

The following values resulted are h=0, p=0.935. This means the acceptance of the null hypothesis of concordance between the observed data and the theoretical Weibull

Determining the reliability indices of an OEL facilitates the knowledge of the safety level in the operation of the OEL analyzed and the whole essembly which composes the OEL. The OEL 20 KV is studied as a reparable simple element, which regains its operating ability after failure, through repair, and then it can continue operation until the next failure. The evolution in time of such an element is a sequence of *tfi* operating times with *tri* and repair times, for which, the following indices of reliability can be defined and calculated (Dub,

[ ] [ ]

MTTR day ; MTTR 0.2506 day n 1 <sup>=</sup> = = <sup>−</sup>

− − λ = λ = (18)

− − μ = μ = (20)

M t PT ( ) α =⋅ ( ) (23)

M t QT 1 P T ( ) β( ) = ⋅= − ⋅ ( ) (24)

(17)

(19)

fi i i 1

MTBF day with t t t ; MTBF 31.7037 day

<sup>=</sup> = = − = <sup>−</sup>

n i i 1 tr

For the case of steady state, indices P and Q are defined. In our case, which chose the Weibull model for shaping the reliability function, and considering that (β = 1.2939), β ≅ 1 , we used the relationships of availability associated with the exponential distribution with





distribution of the parameters α, β, with the level of significance of 6.5%.





n fi i 1

operating times and repair times.

t

n

**3.6 Reliability indices** 

2008):

and the nonreliability function *F(t),* through the relationship: Ft 1 Rt () () = − (16)

and the graph presented in Figure 3.

Fig. 3. Variation of reliability functions and failure probability

#### **3.5 Concordance test**

The fundamental criterion in adopting the distribution law is the concordance between the theoretical law, which in our case is Weibull distribution and experimental data, which in this case is the recorded database.

For this study, the validation of the chosen distribution law it is imposed by concordance study between: Weibull distribution by the form (15), with two parameters α, β, calculated with formulae (13), with experimental data presented in Table 1.

This can be achieved using the chi square concordance test which, in MATLAB, Statistics Toolbox is achieved with the procedure, (Blaga, 2002) :

[h,p] = chi2gof(t,@cdfweib\_OEL)

where: - t is line matrix of experimental data, and


The following values resulted are h=0, p=0.935. This means the acceptance of the null hypothesis of concordance between the observed data and the theoretical Weibull distribution of the parameters α, β, with the level of significance of 6.5%.

#### **3.6 Reliability indices**

92 Electrical Generation and Distribution Systems and Power Quality Disturbances

For the OEL 20 KV in study, by replacing the parameters from Table 3.1 in the previous relations, the following values of parameters of the Weibull function distribution result:

which allows modeling the reliability function *R(t)* of OEL the through the relationship:

and the nonreliability function *F(t),* through the relationship: Ft 1 Rt () () = − (16)

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>0</sup>

The fundamental criterion in adopting the distribution law is the concordance between the theoretical law, which in our case is Weibull distribution and experimental data, which in

For this study, the validation of the chosen distribution law it is imposed by concordance study between: Weibull distribution by the form (15), with two parameters α, β, calculated

This can be achieved using the chi square concordance test which, in MATLAB, Statistics

[h,p] = chi2gof(t,@cdfweib\_OEL)

⋅− ⋅ ⋅ − ⋅ = β <sup>=</sup> −⋅ ⋅ − and a lg( ) lg <sup>e</sup> <sup>10</sup> <sup>−</sup> α = (13)

( ) <sup>4</sup> 1.2669 10 and 1.2939 <sup>−</sup> α= ⋅ β= (14)

Rs Reliability function Fs Nonreliability function

( ) ( ) <sup>4</sup> 1.2939 1.2669 10 t Rt e <sup>−</sup> − ⋅⋅ = (15)

T[days]

2 2 CB AD nC AB a ; B nD nD B

and the graph presented in Figure 3.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

**3.5 Concordance test** 

this case is the recorded database.

Fig. 3. Variation of reliability functions and failure probability

with formulae (13), with experimental data presented in Table 1.


Toolbox is achieved with the procedure, (Blaga, 2002) :

where: - t is line matrix of experimental data, and

Determining the reliability indices of an OEL facilitates the knowledge of the safety level in the operation of the OEL analyzed and the whole essembly which composes the OEL. The OEL 20 KV is studied as a reparable simple element, which regains its operating ability after failure, through repair, and then it can continue operation until the next failure. The evolution in time of such an element is a sequence of *tfi* operating times with *tri* and repair times, for which, the following indices of reliability can be defined and calculated (Dub, 2008):


$$\begin{aligned} \text{MTBF} &= \frac{\sum\_{i=1}^{n} \text{t}\_{\text{fl}}}{\text{n}} \begin{bmatrix} \text{day} \end{bmatrix} & \text{with} \quad \mathbf{t}\_{\text{fl}} = \mathbf{t}\_{i} - \mathbf{t}\_{i-1}; \text{ MTBF} = \text{31.7037} \begin{bmatrix} \text{day} \end{bmatrix} \end{aligned} \tag{17}$$

$$\lambda \text{ : } \lambda \text{ , } \text{failure rate}, \qquad \lambda = \frac{1}{\text{MTBF}} \left[ \text{day}^{-1} \right]; \text{ : } \lambda = 0.031542 \left[ \text{day}^{-1} \right] \tag{18}$$

$$\frac{\sum\_{\text{tr}\_i}^{n} \text{tr}\_i}{\sum\_{\text{tr}\_i}^{n} \text{Mean Time Rep pair}}, \text{ MTTR} = \frac{\sum\_{\text{tr}\_i}^{n} \text{tr}\_i}{\text{n-1}} \left[ \text{day} \right]; \text{ MTTR} = 0.2506 \left[ \text{day} \right] \tag{19}$$

$$\text{--} \quad \mu \text{, repair rate,} \qquad \text{ $\mu = \frac{1}{\text{MTTR}}$  }\left[\text{day}^{-1}\right] \text{; } \mu = 3.9897 \left[\text{day}^{-1}\right] \tag{20}$$

For the case of steady state, indices P and Q are defined. In our case, which chose the Weibull model for shaping the reliability function, and considering that (β = 1.2939), β ≅ 1 , we used the relationships of availability associated with the exponential distribution with operating times and repair times.


$$\mathbf{M}(\alpha(\mathbf{t})) = \mathbf{P} \cdot \mathbf{T} \tag{23}$$


$$\mathbf{M}(\mathbf{\hat{\beta}(t)}) = \mathbf{Q} \cdot \mathbf{T} = (1 - \mathbf{P}) \cdot \mathbf{T} \tag{24}$$


$$\mathbf{M}\left(\boldsymbol{\gamma}(\mathbf{t})\right) = \boldsymbol{\lambda} \cdot \mathbf{P} \cdot \mathbf{T} = \boldsymbol{\mu} \cdot \mathbf{Q} \cdot \mathbf{T} \tag{25}$$

Reliability Centered Maintenance Optimization of Electric Distribution Systems 95

defining elements: *the type of wear* and intervention operations, *the renewal type and moment*.

a. The system is in operation, and has the probability function P0 ( <sup>0</sup> T ), known by the

b. the law of variation of system reliability is know, being expressd by he relationship (15); c. The system and its components will follow the same law of variation of reliability including during the periods of preventive maintenance ( T<sup>+</sup> ), presented in Figure 1; we

In the conditions of preventive maintenance actions for a period of [0, ∆T], with a number of renewals r performed during that period, we can express the system reliability function, on

( ) ()( ) ( ) <sup>1</sup>

Figure 5 illustrates the graph in the case of the OEL in study, with the parameters given in the relations (15) and (21) with he purpose of exemplifying the variation of the reliability function R(r,∆T), under the influence of PMA, carried out by varying the number of renewals r for [0,10,50,100], assuming a reliability of the OEL, at the beginning of the study

> ( ) () ( )( ) ( ) <sup>1</sup> R r, T P t exp r 1 T <sup>0</sup>

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> 0.1

Fig. 5. The influence of the renewals r on the reliability of the OEL 20 kV

R r, T exp r 1 T −β <sup>β</sup> Δ = −α + ⋅ Δ (28)

−β <sup>β</sup> Δ = ⋅ −α + ⋅ Δ (29)

Number of renewals

r0=0 r1=10 r2=50 r3=100

[days]

can shape the preventive renewal process for the following situations:

the relationship given by (Catuneanu & Popentiu, 1998), (Georgescu et al., 2010).

In the cases in which at the time t = <sup>0</sup> T presented in Figure 1:

and the nonreliability function *F(t),* through the relationship:

where: α ; β - Weibull parameters, the relationship (14); r – the number of preventive renewals during the study ∆T.

> 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Po=0.9922

relationship (21) ;

**4.1 Modeling on a time interval** 

period by P0 (t) = 0.9922.

#### **3.7 The availability function**

In the case of an exponential variation of *reliability function R(t)*, the following indices are specified, (Dub, 2008):


$$\mathbf{A}\left(\mathbf{t}\right) = \frac{\mu}{\lambda + \mu} + \frac{\lambda}{\lambda + \mu} \exp\left[-\left(\lambda + \mu\right)\mathbf{t}\right] \tag{26}$$


$$\bar{\mathbf{A}}(\mathbf{t}) = \frac{\lambda}{\lambda + \mu} \cdot \left[1 - \exp\left[-\left(\lambda + \mu\right)\cdot \mathbf{t}\right]\right] \tag{27}$$

For the OEL in study, based on the known indices λ and µ and on relations (26) and (27), Figure 4 presents a variation of the availability/unavailability function of the OEL.

Fig. 4. Availability and unavailability functions for OEL 20 kV

The proposed algorithm allows to establishing the parameters and reliability indices for each component of OEL: pillar, conductor, insulator, assuming that we have a specific database for each component.

Considering these parameters and indices as the intrinsic dimensions of the OEL at a time, we can move on to preventive maintenance planning, namely the establishment of preventive maintenance strategies for a future period.

#### **4. Modelling strategies by preventive maintenance with renewals**

In accordance with the information presented in section 2 and section 3, it follows that the prophylactic strategy related to a system with RCM represents the combination of two defining elements: *the type of wear* and intervention operations, *the renewal type and moment*. In the cases in which at the time t = <sup>0</sup> T presented in Figure 1:


#### **4.1 Modeling on a time interval**

94 Electrical Generation and Distribution Systems and Power Quality Disturbances


In the case of an exponential variation of *reliability function R(t)*, the following indices are


, probability to be in course of repairing at a time *t*, called *unavailability*

For the OEL in study, based on the known indices λ and µ and on relations (26) and (27),

The proposed algorithm allows to establishing the parameters and reliability indices for each component of OEL: pillar, conductor, insulator, assuming that we have a specific

Considering these parameters and indices as the intrinsic dimensions of the OEL at a time, we can move on to preventive maintenance planning, namely the establishment of

In accordance with the information presented in section 2 and section 3, it follows that the prophylactic strategy related to a system with RCM represents the combination of two

**4. Modelling strategies by preventive maintenance with renewals** 

Fig. 4. Availability and unavailability functions for OEL 20 kV

preventive maintenance strategies for a future period.

database for each component.

Figure 4 presents a variation of the availability/unavailability function of the OEL.

M t PT QT ( ) γ ( ) =λ⋅ ⋅ = μ ⋅ ⋅ (25)

A t( ) exp t ( ) μ λ = + − λ+ <sup>μ</sup> λ+μ λ+μ (26)

A t 1 exp t ( ) ( ) <sup>−</sup> <sup>λ</sup> = ⋅− − λ+ <sup>μ</sup> <sup>⋅</sup> λ+μ (27)

time interval (0,T)

**3.7 The availability function** 

specified, (Dub, 2008):


In the conditions of preventive maintenance actions for a period of [0, ∆T], with a number of renewals r performed during that period, we can express the system reliability function, on the relationship given by (Catuneanu & Popentiu, 1998), (Georgescu et al., 2010). and the nonreliability function *F(t),* through the relationship:

$$\mathcal{R}\left(\mathbf{r}, \Delta \mathbf{T}\right) = \exp\left(-\alpha \left(\mathbf{r} + \mathbf{1}\right)^{(1-\beta)} \cdot \Delta \mathbf{T}^{\beta}\right) \tag{28}$$

where: α ; β - Weibull parameters, the relationship (14);

r – the number of preventive renewals during the study ∆T.

Figure 5 illustrates the graph in the case of the OEL in study, with the parameters given in the relations (15) and (21) with he purpose of exemplifying the variation of the reliability function R(r,∆T), under the influence of PMA, carried out by varying the number of renewals r for [0,10,50,100], assuming a reliability of the OEL, at the beginning of the study period by P0 (t) = 0.9922.

$$\mathbf{R}\left(\mathbf{r},\Delta\mathbf{T}\right) = \mathbf{P}\_0\left(\mathbf{t}\right) \cdot \exp\left(-\alpha \left(\mathbf{r}+\mathbf{1}\right)^{(1-\beta)} \cdot \Delta\mathbf{T}^{\beta}\right) \tag{29}$$

Fig. 5. The influence of the renewals r on the reliability of the OEL 20 kV

Reliability Centered Maintenance Optimization of Electric Distribution Systems 97

For the purpose of studying the PM of the system components, we consider the OEL with




Through the OEL components, the structural model of the OEL reliability is the *series model*, such as the one presented in Figure 7. The reliability parameters and indicators of each

In the line of these assumptions we can model the probability of operating with renewals, on each component over an interval ∆T, knowing the operating probability for each

In this case, the model of reliability of the OEL with renewals on components will take the

( ) () ( )( )<sup>i</sup> <sup>i</sup> 3 3 <sup>1</sup> i 0i i

Let us study the variation of the OEL reliability parameters and of its components for a period of time ∆T = 350 days, assuming that the reliability parameters of the components and the number of preventive maintenance actions on each component are known,

If we have the number of renewals ri|i=1,2,3, each in four variants, the results are presented

*i* Element R0i α <sup>i</sup> ∗ 10-4 βi ri[buc] 1 Pillars 0.99888 2.66883 1.29 0/5/10/20 2 Conductor 0.99750 9.56379 1.21 0/10/20/25 3 Insulators 0.99610 15.45002 1.33 0/10/15/30

\*\* The data are estimates, because there is no information on the component maintenance

R r, T R T Pi exp r 1 T −β <sup>β</sup>

−β <sup>β</sup> Δ = ⋅ −α + ⋅ Δ = (33)

Δ = Δ = ⋅ −α + ⋅ Δ ∏ ∏ (34)

() ()( ) ( ) <sup>i</sup> <sup>i</sup> <sup>1</sup> R r, T Pi exp r 1 T i 1,2,3 <sup>i</sup> <sup>0</sup> i i

basic components: pillars, conductors, insulators, with the following assumptions:

**4.3 Modeling on the components of the OEL** 

reliability indicators for each component;

form (6);

component are known.

form:

according to Table 2.

each component, for a certain period of operation;

Fig. 7. Equivalent scheme of reliability on components of the OEL

where: ri - the number of renewals on the component i = 1, 2, 3.

i 1 i 1

= =

component in the early of period Pi0, under the form:

in Table 3. and their variation is presented in Figure 8.

Table 2. Reliability parameters of OEL components

From the graphic of Figure 5, we note an increase reliability of the OEL at the end of the period ∆T, under the influence of renewals r0, r1, r2, r3.

#### **4.2 Modeling over n time intervals**

Let us study the variation probability for a system operating with PM, over a period of time Tn*,* composed of *n* time intervals ∆Ti. A number ri of renewals is performed over each interval ∆Ti for i=1,n, while the reliability of system at the beginning of the interval ∆T1 is known Po|t=0.

$$\mathbf{R}\left(\mathbf{T}\_{\mathrm{n}}\right) = \mathbf{P}\_{\mathrm{o}}\left(\mathbf{t}\right)\prod\_{\mathrm{i}=1}^{n}\mathbf{R}\_{\mathrm{i}}\left(\mathbf{r}\_{\mathrm{i}}\,\Delta\mathbf{T}\_{\mathrm{i}}\right)\tag{30}$$

$$\text{where:}\\
\text{where:}\\
\qquad \qquad \qquad \mathcal{R}\_i \left(\mathbf{r}\_i, \Delta \mathbf{T}\_i\right) = \exp\left[-\alpha \left(\mathbf{r}\_i + \mathbf{1}\right)^{(1-\beta)} \cdot \Delta \mathbf{T}\_i^{\beta}\right] \\
\quad \left| \quad \mathbf{i} = \mathbf{1}, \mathbf{n} \right.\tag{31}$$

$$\text{corr.} \tag{3.1} \\ \qquad \qquad \qquad \mathbf{R}\left(\mathbf{T}\_n\right) = \mathbf{P}\_o\left(\mathbf{t}\right) \prod\_{l=1}^n \exp\left[-\alpha \left(\mathbf{r}\_i + 1\right)^{(1-\beta)} \cdot \Delta \mathbf{T}\_i^{\beta}\right] \tag{32}$$

In the case of OEL in study, the variation of reliability was studied for:




With the parameters given in relations (15) and (21), we obtain the graph of variation presented in Figure 6:

Fig. 6. Variation of reliability over time periods Δ = T i 1,2,3

From the graph of Figure 6, we note an increased reliability of OEL at the end of the three periods ∆Ti | i = 1, 2, 3, under the influence of renewals.

#### **4.3 Modeling on the components of the OEL**

96 Electrical Generation and Distribution Systems and Power Quality Disturbances

From the graphic of Figure 5, we note an increase reliability of the OEL at the end of the

Let us study the variation probability for a system operating with PM, over a period of time Tn*,* composed of *n* time intervals ∆Ti. A number ri of renewals is performed over each interval ∆Ti for i=1,n, while the reliability of system at the beginning of the interval ∆T1 is

> ( ) () ( ) <sup>n</sup> n o ii i i 1 R T P t R r, T =

R r , T exp r 1 T i 1,n ii i <sup>i</sup> <sup>i</sup> −β <sup>β</sup> Δ = −α + ⋅ Δ =

n 0 i i

R T P t exp r 1 T −β <sup>β</sup>

With the parameters given in relations (15) and (21), we obtain the graph of variation

From the graph of Figure 6, we note an increased reliability of OEL at the end of the three

i 1

=

In the case of OEL in study, the variation of reliability was studied for: - T T = Δ <sup>i</sup> , with equal periods of time of ∆Ti = 600 days |i=1,2,3; - for each period i having a number rij | j = 1, 2, 3 renewals, or: - ∆T1 | r1 = 0,5,20,50 ; ∆T2 | r2 = 0,3,15,30 ; ∆T3 | r3 = 0,1,10,20 ;

= Δ ∏ (30)

(31)

= − α+ ⋅ <sup>Δ</sup> ∏ (32)

period ∆T, under the influence of renewals r0, r1, r2, r3.

where: ( ) ()( ) <sup>1</sup>

or: ( ) () ( )( ) <sup>n</sup> <sup>1</sup>

Fig. 6. Variation of reliability over time periods Δ = T i 1,2,3

periods ∆Ti | i = 1, 2, 3, under the influence of renewals.

**4.2 Modeling over n time intervals** 

known Po|t=0.

presented in Figure 6:

For the purpose of studying the PM of the system components, we consider the OEL with basic components: pillars, conductors, insulators, with the following assumptions:


Through the OEL components, the structural model of the OEL reliability is the *series model*, such as the one presented in Figure 7. The reliability parameters and indicators of each component are known.

$$\begin{array}{|c|c|c|} \hline \text{1. Pillars} \\ \hline \text{P1. 0: } \text{a}; \beta 1 \\ \hline \end{array} \begin{array}{|c|c|} \hline \text{2. Conductors} \\ \hline \text{P2. 0: } \text{a}; \beta 2 \\ \hline \end{array} \begin{array}{|c|c|} \hline \text{3. Insalators} \\ \hline \text{P3. 0: } \text{a}; \beta 3 \\ \hline \end{array}$$

Fig. 7. Equivalent scheme of reliability on components of the OEL

In the line of these assumptions we can model the probability of operating with renewals, on each component over an interval ∆T, knowing the operating probability for each component in the early of period Pi0, under the form:

$$\mathbf{R}\_{\mathbf{i}}\left(\mathbf{r}\_{\prime}\Delta\mathbf{T}\right) = \mathbf{P}\mathbf{i}\_{0} \cdot \exp\left(-\alpha\_{\mathrm{i}}\left(\mathbf{r}\_{\mathrm{i}} + \mathbf{1}\right)^{(1-\beta\_{\mathrm{i}})} \cdot \Delta\mathbf{T}^{\beta\_{\mathrm{i}}}\right) \quad \left|\mathbf{i} = \mathbf{1}\right.\tag{33}$$

where: ri - the number of renewals on the component i = 1, 2, 3.

In this case, the model of reliability of the OEL with renewals on components will take the form:

$$\mathbf{R}\left(\mathbf{r}, \Delta \mathbf{T}\right) = \prod\_{i=1}^{3} \mathbf{R}\_{i}\left(\Delta \mathbf{T}\right) = \prod\_{i=1}^{3} \mathbf{P} \mathbf{i}\_{0} \cdot \exp\left[-\alpha\_{i}\left(\mathbf{r}\_{i} + \mathbf{1}\right)^{\left(1-\beta\_{i}\right)} \cdot \Delta \mathbf{T}^{\beta\_{i}}\right] \tag{34}$$

Let us study the variation of the OEL reliability parameters and of its components for a period of time ∆T = 350 days, assuming that the reliability parameters of the components and the number of preventive maintenance actions on each component are known, according to Table 2.

If we have the number of renewals ri|i=1,2,3, each in four variants, the results are presented in Table 3. and their variation is presented in Figure 8.


Table 2. Reliability parameters of OEL components

\*\* The data are estimates, because there is no information on the component maintenance

Reliability Centered Maintenance Optimization of Electric Distribution Systems 99

Or: TC r C r C r C r [cost / year] () () () () =+ + F PM INT (35)

We express these costs depending on the parameters of reliability and on the number of planned interventions *r* during the period ΔT of by PMA for an OEL belonging to the EDS,

The costs due to unplanned interruptions, due to system failure, can be expressed as

where: N*F*, average number of failures for the reference interval ∆T from the relation (25),

or: NR T F SS = ⋅λ ⋅Δ (37)

λS - the failure rate of a EDS, considered constant during the study ∆T, OEL being half way

C r P exp r 1 T T c F 0 S F

The cost of preventive maintenance activities performed on a system or system element is determined on the basis of existing statistical information available at each EDS operating unit. This cost can be appreciated as a function directly proportional to the number of

The penalties suported by the *electricity supplier*, due unplanned interruptions, for duration



interventions, i.e. renewals *r* on the system or on the system element, of the form:

−β <sup>β</sup> = ⋅ −α + ⋅ Δ from the relationship (28) (38)

−β <sup>β</sup> = ⋅ −α + ⋅ Δ ⋅ λ ⋅ Δ ⋅ (39)

C r rc PM ( ) = ⋅ PM (40)

Cr Nc F FF ( ) = ⋅ (36)



through the following components.

follows:

**5.2 Costs with unplanned interuptions** 

where: ( )( ) ( ) <sup>1</sup>

*α ; β* - Weibull parameters, the relationship (14);

with: cPM - average cost of a preventive renewal.

average power on OEL by POEL, and

through its life cycle, the relationship (18); cF - the average cost to fix a failure.

**5.3 Renewal costs** 

**5.4 Penalty costs** 

ΔT, can be by two types:

connected to the OEL.

R P exp r 1 T S 0

or: ( ) ( )( ) ( ) <sup>1</sup>


Table 3. Renewal influence on OEL components reliability

From the study and comparison of the values presented in Table 3, we note an increase of reliability for each component and for the whole OEL, when the number of renewals is increased. Figure 8., graphically presents the influence of renewals on the components and on the OEL.

Fig. 8. The influence of renewals on the components and on the OEL

#### **5. The economic modeling of preventive maintenance with renewals**

The second component of the actions of the PM, after *strategy modeling*, is the *cost management* involved in the prophylactic strategy associated to the OEL systems with maintenance. In turn, the action of management and control of the maintenance plan of the electricity provider and in particular the implications of these actions on the financial relationship between the *electricity supplier* and *electricity consumers*, impose realizing an *economic model of PM*. Such a model should summarize all the costs and effects generated by the actions of the preventive and corrective maintenance, in all their aspects: planning, execution, management, etc., including the effect of inflation through the update method (Sarchiz et al., 2009), (IEC 60300-3-11, 2009).

To optimize the effect of RCM through the action and through the effect of the renewal processes for a period ∆T, belonging to the interval ( T<sup>+</sup> ), presented in Figure 1, we propose an economic model, based on the state parameters of the OEL and the number of *r* renewals.

#### **5.1 Total costs**

The economic model of the PMA, from the perspective of the total cost (TC), has three basic components, (Anders et al., 2007):


Or: TC r C r C r C r [cost / year] () () () () =+ + F PM INT (35)

We express these costs depending on the parameters of reliability and on the number of planned interventions *r* during the period ΔT of by PMA for an OEL belonging to the EDS, through the following components.

#### **5.2 Costs with unplanned interuptions**

The costs due to unplanned interruptions, due to system failure, can be expressed as follows:

$$\mathbf{C}\_{\rm F} \left( \mathbf{r} \right) = \mathbf{N}\_{\rm F} \cdot \mathbf{c}\_{\rm F} \tag{36}$$

where: N*F*, average number of failures for the reference interval ∆T from the relation (25),

98 Electrical Generation and Distribution Systems and Power Quality Disturbances

ri = 5/10/10 i =1,2,3

R1 0.9988 0.8762 0.9230 0.9344 0.9447 R2 0.9975 0.7300 0.8264 0.8465 0.8527 R3 0.9961 0.4055 0.6632 0.6954 0.7464 ROEL 0.9922 0.2593 0.5058 0.5501 0.6012

From the study and comparison of the values presented in Table 3, we note an increase of reliability for each component and for the whole OEL, when the number of renewals is increased. Figure 8., graphically presents the influence of renewals on the components and

The second component of the actions of the PM, after *strategy modeling*, is the *cost management* involved in the prophylactic strategy associated to the OEL systems with maintenance. In turn, the action of management and control of the maintenance plan of the electricity provider and in particular the implications of these actions on the financial relationship between the *electricity supplier* and *electricity consumers*, impose realizing an *economic model of PM*. Such a model should summarize all the costs and effects generated by the actions of the preventive and corrective maintenance, in all their aspects: planning, execution, management, etc., including the effect of inflation through the update method

To optimize the effect of RCM through the action and through the effect of the renewal processes for a period ∆T, belonging to the interval ( T<sup>+</sup> ), presented in Figure 1, we propose an economic model, based on the state parameters of the OEL and the number of *r* renewals.

The economic model of the PMA, from the perspective of the total cost (TC), has three basic

Fig. 8. The influence of renewals on the components and on the OEL

(Sarchiz et al., 2009), (IEC 60300-3-11, 2009).

components, (Anders et al., 2007):

**5.1 Total costs** 

**5. The economic modeling of preventive maintenance with renewals** 

ri = 10/20/15 i =1,2,3

ri = 20/25/30 i =1,2,3

Ri0

on the OEL.

ri = 0 / 0 / 0 i =1,2,3

Table 3. Renewal influence on OEL components reliability

$$\mathbf{N}\_{\text{F}} = \mathbf{R}\_{\text{S}} \cdot \mathbf{\hat{\lambda}}\_{\text{S}} \cdot \boldsymbol{\Delta T} \tag{37}$$

−β <sup>β</sup> = ⋅ −α + ⋅ Δ from the relationship (28) (38)

*α ; β* - Weibull parameters, the relationship (14);

R P exp r 1 T S 0

where: ( )( ) ( ) <sup>1</sup>

λS - the failure rate of a EDS, considered constant during the study ∆T, OEL being half way through its life cycle, the relationship (18);

cF - the average cost to fix a failure.

$$\mathbf{C} \cdot \mathbf{r} = \mathbf{C}\_{\mathrm{F}} \cdot \exp\left(-\alpha \left(\mathbf{r} + \mathbf{1}\right)^{(1-\theta)} \cdot \Delta \mathbf{T}^{\theta}\right) \cdot \mathbf{\hat{\lambda}}\_{\mathrm{S}} \cdot \Delta \mathbf{T} \cdot \mathbf{c}\_{\mathrm{F}} \tag{39}$$

#### **5.3 Renewal costs**

The cost of preventive maintenance activities performed on a system or system element is determined on the basis of existing statistical information available at each EDS operating unit. This cost can be appreciated as a function directly proportional to the number of interventions, i.e. renewals *r* on the system or on the system element, of the form:

$$\mathbf{C}\_{\rm PM}(\mathbf{r}) = \mathbf{r} \cdot \mathbf{c}\_{\rm PM} \tag{40}$$

with: cPM - average cost of a preventive renewal.

#### **5.4 Penalty costs**

The penalties suported by the *electricity supplier*, due unplanned interruptions, for duration ΔT, can be by two types:


$$\mathbf{C\_{INT}}\left(\mathbf{r}\right) = \text{PEN}\_{\mathbb{S}} + \text{PEN}\_{\mathbb{U}} \tag{41}$$

$$\text{with:}\tag{4}\tag{5}$$

$$\text{PEN}\_{\text{S}} = \text{P}\_{\text{OEL}} \cdot \text{c}\_{\text{w}} \cdot \text{T}\_{\text{F}}\tag{4}$$

Reliability Centered Maintenance Optimization of Electric Distribution Systems 101

The strategies of optimizing PM of OEL, depending on the optimal number of preventive renewals r over a given period (0, T), can be approached from the perspective of the consequences it has on the relationship between electricity supplier and electricity consumer, based on two different criteria, which from the standpoint of the electricity

a. *The economic criterion:* through the total cost involved in providing a safe supply of

min TC T,r { ( )} (47)

( ) min R t,r R S S ≥ or ( ) max N t,r N F F ≤ (48)

max R T,r or min 1 R T,r { S S ( )} { − ( )} (49)

( ) max TC T,r TC ≤ (50)

**6.1 Study assumptions** 

supplier are (Sarchiz, 1993, 2005), (Georgescu, 2009):

reliability on interval (0, T).

duration, *T*, a year, on these assumptions:

Mahdavi 2009), (Teresa Lam & Yeh 1993):

constant time between two renewals for:

studied interval.

electricity to consumers. This optimization model is:

in presence of technical constraints imposed on safety criteria:

in presence of economic constraints due to maintenance actions:

where: - min R , the minimum reliability imposed on the study <sup>S</sup> interval, and - max N , the maximum number of failures permitted on the study <sup>F</sup> interval.

b. *The tehnical criterion,* aims to maximize safety in operation, respectively of system

where, max TC , the maximum cost allocated to the exploitation of the OEL on the

The study of the strategies used to optimize the RCM based on models (47) and (49), can be performed depending on the degree of safety imposed to ensure electricity and/or depending on the degree of assurance of financial resources during the PM during (0,T). We will exemplify the application of the two models to the OEL 20KV in study, for the


Simple/minimal preventive maintenance and/or; Maximal preventive replacement; Sequential and/or continuous inspection strategies. - we admit a periodic distribution of preventive renewals on the interval (0, T), with a

OEL t T /r Δ = or

OEL component i i Δ= = t T /r i 1,2,3

$$\text{PEN}\_{\text{U}} = \sum\_{\mathbf{k}=1}^{K} \mathbf{P}\_{\mathbf{k}} \cdot \mathbf{c}\_{\mathbf{k}} \cdot \mathbf{T}\_{\text{F}} \tag{43}$$

where:

POEL - average power on OEL; Pk - average power at the k consumer; cW - electricity cost; ck - production cost unrealized for a kilowatt hour of electricity undelivered, at the k consumer; TF - total average duration of OEL nonoperation during the reference period ∆T, relation (24)*;* K - number of consumers connected to the OEL in the study.

$$\begin{array}{cc} \text{with:} & \mathbf{T}\_{\mathbf{f}} = \left(\mathbf{1} - \mathbf{R}\_{\mathbf{s}}\right) \cdot \Delta \mathbf{T} \end{array} \tag{44}$$

By replacing the relations established in (41), it follows:

$$\mathbf{C}\_{\rm INI}\left(\mathbf{r}\right) = \left(\mathbf{P}\_{\rm OEL} \cdot \mathbf{c}\_{\rm w} + \sum\_{\mathbf{k}=1,\mathbf{K}} \mathbf{P}\_{\rm k} \cdot \mathbf{c}\_{\rm k}\right) \cdot \left(\mathbf{1} - \mathbf{P}\_{\rm o} \cdot \exp\left(-\alpha \left(\mathbf{r} + \mathbf{1}\right)^{(1-\beta)} \cdot \Delta \mathbf{T}^{\beta}\right)\right) \cdot \Delta \mathbf{T} \tag{45}$$

By replacing the relations (39), (40) and (45) in relation (35), we obtain the expression of the total costs over a period of time ∆T, depending on the safety operating parameters of OEL and depending on the renewal actions of the PM.

$$\begin{split} \mathbf{T} \mathbf{C}\_{(\mathbf{r})} &= \mathbf{P}\_{\mathbf{0}} \cdot \exp\left(-\boldsymbol{\alpha} \left(\mathbf{r} + \mathbf{1}\right)^{(1-\beta)} \cdot \boldsymbol{\Delta} \mathbf{T}^{\beta}\right) \cdot \mathbf{\boldsymbol{\lambda}}\_{\mathrm{S}} \cdot \boldsymbol{\Delta} \mathbf{T} \cdot \mathbf{c}\_{\mathbf{r}} + \mathbf{r} \cdot \mathbf{c}\_{\mathbf{r}\mathbf{M}} + \\ &+ \left(\mathbf{P}\_{\mathrm{OEL}} \cdot \mathbf{c}\_{\mathbf{w}} + \sum\_{\mathbf{k}=1,\mathbf{K}} \mathbf{P}\_{\mathbf{k}} \cdot \mathbf{c}\_{\mathbf{k}}\right) \cdot \left(\mathbf{1} - \mathbf{P}\_{\mathrm{o}} \cdot \exp\left(-\boldsymbol{\alpha} \left(\mathbf{r} + \mathbf{1}\right)^{(1-\beta)} \cdot \boldsymbol{\Delta} \mathbf{T}^{\beta}\right)\right) \cdot \boldsymbol{\Delta} \mathbf{T} \end{split} \tag{46}$$

With the following specifications on the relashionsip (46)


#### **6. The optimization of RCM strategies**

The literature in this field approaches a wide range of classifications, according to different criteria and parameters, used in the design and optimization of RCM strategies (Hilbert, 2008), (Anders et al., 2007). Further on, we will give examples of RCM strategies for an OEL belonging to EDS, for the following cases and mathematical models.

#### **6.1 Study assumptions**

100 Electrical Generation and Distribution Systems and Power Quality Disturbances

with: PEN P c T S OEL w F = ⋅⋅ (42)

K U kk F k 1 PEN P c T =

POEL - average power on OEL; Pk - average power at the k consumer; cW - electricity cost; ck - production cost unrealized for a kilowatt hour of electricity undelivered, at the k consumer; TF - total average duration of OEL nonoperation during the reference period ∆T,

with: T 1R T F S = − ⋅Δ ( ) (44)

C r P c P c 1 P exp r 1 T T −β <sup>β</sup>

By replacing the relations (39), (40) and (45) in relation (35), we obtain the expression of the total costs over a period of time ∆T, depending on the safety operating parameters of OEL

= ⋅ + ⋅ ⋅ − ⋅ −α + ⋅ Δ ⋅ Δ

P c P c 1 P exp r 1 T T

+ ⋅ + ⋅ ⋅ − ⋅ −α + ⋅ Δ ⋅ Δ

1. The term (43) is included into the optimization calculations for situations where an OEL provides electricity for consumer, which can calculate the costs of production according to the electric energy supplied, which allows the calculation of the

2. To certain categories of electricity consumers, the higher production losses occur depending on the number of interruptions NF during ΔT and less influence on time of

3. In a system with multiple components, each having a specific number of renewals *ri*, the

The literature in this field approaches a wide range of classifications, according to different criteria and parameters, used in the design and optimization of RCM strategies (Hilbert, 2008), (Anders et al., 2007). Further on, we will give examples of RCM strategies for an OEL belonging to EDS, for the following cases and mathematical

( ) ( )( )

relation (24)*;* K - number of consumers connected to the OEL in the study.

By replacing the relations established in (41), it follows:

and depending on the renewal actions of the PM.

coefficient *ck*;

interruption TF;

models.

( ) ( )( )

With the following specifications on the relashionsip (46)

INT OEL w k k 0 k 1,K

=

( )

OEL w k k 0 k 1,K

=

total costs are the sum of costs on each system element.

**6. The optimization of RCM strategies** 

1 r 0 S F PM

−β β

= ⋅ −α + ⋅ Δ ⋅ λ ⋅ Δ ⋅ + ⋅ +

TC P exp r 1 T T c r c

where:

C r PEN PEN INT ( ) = +<sup>S</sup> <sup>U</sup> (41)

( ( )) <sup>1</sup>

( )( ) ( ( ))

1

−β β

(45)

(46)

= ⋅⋅ (43)

The strategies of optimizing PM of OEL, depending on the optimal number of preventive renewals r over a given period (0, T), can be approached from the perspective of the consequences it has on the relationship between electricity supplier and electricity consumer, based on two different criteria, which from the standpoint of the electricity supplier are (Sarchiz, 1993, 2005), (Georgescu, 2009):

a. *The economic criterion:* through the total cost involved in providing a safe supply of electricity to consumers. This optimization model is:

$$\min\left[\text{TC}(\text{T}, \text{r})\right] \tag{47}$$

in presence of technical constraints imposed on safety criteria:

$$\mathbf{R\_{S}\left(t,\mathbf{r}\right)} \succeq \mathbf{R\_{S}^{\min}} \qquad \text{or} \qquad \mathbf{N\_{F}\left(t,\mathbf{r}\right)} \leq \mathbf{N\_{F}^{\max}} \tag{48}$$

where: - min R , the minimum reliability imposed on the study <sup>S</sup> interval, and


b. *The tehnical criterion,* aims to maximize safety in operation, respectively of system reliability on interval (0, T).

$$\max\left[\mathcal{R}\_{\rm s}(\mathcal{T},\mathbf{r})\right] \text{ or } \min\left[1-\mathcal{R}\_{\rm s}(\mathcal{T},\mathbf{r})\right] \tag{49}$$

in presence of economic constraints due to maintenance actions:

$$\text{TC} \left( \text{T}, \text{r} \right) \le \text{TC}^{\text{max}} \tag{50}$$

where, max TC , the maximum cost allocated to the exploitation of the OEL on the studied interval.

The study of the strategies used to optimize the RCM based on models (47) and (49), can be performed depending on the degree of safety imposed to ensure electricity and/or depending on the degree of assurance of financial resources during the PM during (0,T).

We will exemplify the application of the two models to the OEL 20KV in study, for the duration, *T*, a year, on these assumptions:


Simple/minimal preventive maintenance and/or;

Maximal preventive replacement;

Sequential and/or continuous inspection strategies.


$$\text{OEL } \Delta \text{t} = \text{T } / \text{r } \text{ or }$$

OEL component i i Δ= = t T /r i 1,2,3

Reliability Centered Maintenance Optimization of Electric Distribution Systems 103

costs with reliability min RS imposed to the OEL or with the optimum number of renewals,

0 10 20 30 40 50 60

Fig. 9. Variation of the optimum number of renewals with reliability imposed to the OEL

( ) ( )( ) ( )

where : TC x P exp x 1 T T c x c

By running the application program, for different maximum values imposed to the numbers of unplanned interruptions on the OEL, we obtain the optimum number of renewals that are

OEL w 0

or: ( )( ) ( ) <sup>1</sup> max max P exp x 1 T T N ; x 0; x x <sup>0</sup> s F

with xmax – the maximum imposed number of renewals.

( )( ) ( ( ))

−β <sup>β</sup> ⋅ −α + ⋅ λ ⋅ ≤ − ≤ ≤ (56)

P c 1 P exp x 1 T T

+ ⋅ ⋅ − ⋅ −α + ⋅

−β β

= ⋅ −α + ⋅ λ ⋅ ⋅ + ⋅ +

1

( ) max max Nx N ;0xx F f ≤ ≤≤ (55)

−β β

(54)

1 0 S F PM

We determine the optimum number of renewals with minimum total cost, in conditions in which the number of failures (unplanned interruptions) NF does not exceed a maximum number imposed per year, max N . In this hypothesis, the st <sup>F</sup> ructure of the mathematical

**6.3 The model: Minimum costs and number of interruptions imposed** 

1. Vector of decision variable: x ≡ r the number of renewals

2. Objective function: min TC x { ( )}

r-renewals [pieces]

i.e.: ( ) min TC f R = S or ( ) opt TC f r = .

0.91

model to be optimized is:

3. Constraints of the model:

0.92

0.93

0.94

Reliability of EL

0.95

0.96

0.97


$$\mathbf{c\_{F}} = 1000 \text{ m.u.} ; \quad \mathbf{c\_{PM}} = 17500 \text{ m.u.} ; \quad \mathbf{c\_{W}} = 530 \text{ m.u.} / \text{MWh.} ;$$

Note: The values of costs used in the program are not real, that is why the obtained results are only *demonstrative theoretical results*.


The solution of the mathematical optimization models (47) or (49) in relation with *r variable optimization*, impose the use of nonlinear optimization techniques in relation with *criterion functions* and the *restrictions of the models*. In order to solve the RCM strategy models presented below, we used the software package MATLab 7.0\Optimization Toobox\procedure *fmincon*.

The design of preventive strategies for renewal by type (47) or (49) can be done based on different criteria, which will be listed below within each RCM model optimization strategies.

#### **6.2 The model: The minimum costs and imposed reliability**

We will determine the optimum number of renewals, in the situation of minimum total costs, in such away that OEL relaibility does not drop below the imposed value min R . S

The *fmincon* procedure imposes the following structure of mathematical models to be optimized:


$$\begin{aligned} \text{where:} \quad \mathbf{TC}(\mathbf{x}) &= \mathbf{P}\_0 \cdot \exp\left(-\alpha \left(\mathbf{x} + \mathbf{1}\right)^{(1-\beta)} \mathbf{T}^{\beta}\right) \cdot \boldsymbol{\lambda}\_{\mathbf{s}} \cdot \mathbf{T} \cdot \mathbf{c}\_{\mathbf{F}} + \mathbf{x} \cdot \mathbf{c}\_{\mathbf{PM}} + \\ &+ \mathbf{P}\_{\text{OEL}} \cdot \mathbf{c}\_{\mathbf{w}} \cdot \left(1 - \mathbf{P}\_0 \cdot \exp\left(-\alpha \cdot \left(\mathbf{x} + \mathbf{1}\right)^{(1-\beta)} \mathbf{T}^{\beta}\right)\right) \cdot \mathbf{T} \end{aligned} \tag{51}$$

3. Constraints of the model:

$$\mathcal{R}\_{\mathbb{S}}\left(\mathbf{x}\right) \ge \mathcal{R}\_{\mathbb{S}}^{\min}; \quad 0 \le \mathbf{x} \le \mathbf{x}^{\max} \tag{52}$$

$$\mathbf{0} \text{ or } \mathbf{:} \qquad \mathbf{P}\_0 \cdot \exp\left(-\mathbf{a}\left(\mathbf{x} + \mathbf{1}\right)^{(1-\beta)} \mathbf{T}^{\beta}\right) \ge \mathbf{R}\_{\mathbb{S}}^{\min}; \ -\mathbf{x} \le \mathbf{0}; \ \mathbf{x} \le \mathbf{x}^{\max} \tag{53}$$

with xmax – the maximum imposed number of renewals.

By running the application program for different values imposed to the minimum reliability min R , we obtain the optimum number of renewals S *ropt,* for PM of OEL during a year, in conditions of the minimum total cost TC (T, r), presented in Figure 9, i.e. the graph ( ) opt min <sup>S</sup> r fR = . From the graph of variation we remark that in order to ensure reliability of 0.95, it is required to perform a number of 16 preventive renewals per year and for a reliability of 0.96, it is impose a number of 41 renewals. Also, we can extract the variation costs with reliability min RS imposed to the OEL or with the optimum number of renewals, i.e.: ( ) min TC f R = S or ( ) opt TC f r = .

Fig. 9. Variation of the optimum number of renewals with reliability imposed to the OEL

#### **6.3 The model: Minimum costs and number of interruptions imposed**

We determine the optimum number of renewals with minimum total cost, in conditions in which the number of failures (unplanned interruptions) NF does not exceed a maximum number imposed per year, max N . In this hypothesis, the st <sup>F</sup> ructure of the mathematical model to be optimized is:


$$\begin{split} \text{where}: \quad \mathrm{TC}(\mathbf{x}) &= \mathrm{P}\_{0} \cdot \exp\left(-\alpha \left(\mathbf{x} + \mathbf{1}\right)^{(1-\beta)} \mathbf{T}^{\beta}\right) \cdot \lambda\_{\mathrm{S}} \cdot \mathbf{T} \cdot \mathbf{c}\_{\mathrm{F}} + \mathbf{x} \cdot \mathbf{c}\_{\mathrm{PM}} + \\ &+ \mathbf{P}\_{\mathrm{OEE}} \cdot \mathbf{c}\_{\mathrm{w}} \cdot \left(\mathbf{1} - \mathbf{P}\_{0} \cdot \exp\left(-\alpha \left(\mathbf{x} + \mathbf{1}\right)^{(1-\beta)} \mathbf{T}^{\beta}\right)\right) \cdot \mathbf{T} \end{split} \tag{54}$$

3. Constraints of the model:

$$\mathbf{N}\_{\mathbf{F}}\left(\mathbf{x}\right) \le \mathbf{N}\_{\mathbf{f}}^{\text{max}}; \quad 0 \le \mathbf{x} \le \mathbf{x}^{\text{max}}\tag{55}$$

102 Electrical Generation and Distribution Systems and Power Quality Disturbances


cF = 1000 m.u. ; cPM = 17500 m.u.; cW = 530 m.u./MWh. Note: The values of costs used in the program are not real, that is why the obtained


The solution of the mathematical optimization models (47) or (49) in relation with *r variable optimization*, impose the use of nonlinear optimization techniques in relation with *criterion functions* and the *restrictions of the models*. In order to solve the RCM strategy models presented below, we used the software package MATLab 7.0\Optimization

The design of preventive strategies for renewal by type (47) or (49) can be done based on different criteria, which will be listed below within each RCM model optimization

We will determine the optimum number of renewals, in the situation of minimum total

( )

1 0 S F PM

( )( )

( ) min max Rx R ;0xx S S ≥ ≤≤ (52)

−β <sup>β</sup> ⋅ −α + ≥ − ≤ ≤ (53)

1

−β β

(51)

( ( ))

P c 1 P exp x 1 T T

+ ⋅ ⋅ − ⋅ −α ⋅ + ⋅

−β β

= ⋅ −α + ⋅ λ ⋅ ⋅ + ⋅ +

costs, in such away that OEL relaibility does not drop below the imposed value min R . S The *fmincon* procedure imposes the following structure of mathematical models to be

( ) ( )( )

OEL w 0

( )( ) ( ) <sup>1</sup> min max or : P exp x 1 T R ; x 0; x x 0 S

By running the application program for different values imposed to the minimum reliability min R , we obtain the optimum number of renewals S *ropt,* for PM of OEL during a year, in conditions of the minimum total cost TC (T, r), presented in Figure 9, i.e. the graph

<sup>S</sup> r fR = . From the graph of variation we remark that in order to ensure reliability of 0.95, it is required to perform a number of 16 preventive renewals per year and for a reliability of 0.96, it is impose a number of 41 renewals. Also, we can extract the variation

where : TC x P exp x 1 T T c x c



results are only *demonstrative theoretical results*.

**6.2 The model: The minimum costs and imposed reliability** 

1. Vector of decision variable: x ≡ r the number of renewals

2. Objective function: min TC x { ( )}

with xmax – the maximum imposed number of renewals.

units [m.u.];

electric line.

strategies.

optimized:

( ) opt min

Toobox\procedure *fmincon*.

3. Constraints of the model:

 or: ( )( ) ( ) <sup>1</sup> max max P exp x 1 T T N ; x 0; x x <sup>0</sup> s F −β <sup>β</sup> ⋅ −α + ⋅ λ ⋅ ≤ − ≤ ≤ (56)

with xmax – the maximum imposed number of renewals.

By running the application program, for different maximum values imposed to the numbers of unplanned interruptions on the OEL, we obtain the optimum number of renewals that are

Reliability Centered Maintenance Optimization of Electric Distribution Systems 105

0.93 0.935 0.94 0.945 0.95 0.955 0.96 0.965 0.97 <sup>2</sup>

Reliability of EL

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> <sup>2</sup>

Renewals [pieces]

We determine the optimum number of renewals on the three components of the OEL (pillars, conductors, and insulators), in the assumption of minimum total costs and on

In this case, the structure of the mathematical model to be optimized will include the total

1. In this case, the vector of variables to be optimized will have three components, corresponding to the number of renewals on the three components of the electric line:

X x x x with x r i 1,2,3 = ≡ <sup>123</sup> i i =

<sup>10</sup> x 106

Fig. 12. Variation of the number of renewals with TCmax imposed

costs TC(T, ri) corresponding to those three components.

1-pillars; 2-conductors; 3-insulators.

**6.5 The model on components: The minimun costs and reliability imposed** 

condition that the reliability of the OEL does not fall below an imposed value min R . S

[ ]<sup>T</sup>

Total Cost

Fig. 11. Variation of the reliability of the OEL with TCmax imposed

Total Cost

<sup>10</sup> x 106

required to apply to the OEL during a year, in conditions of minimum total cost TC(T,r), presented in Figure 10, i.e. ropt = *f( NF ).* 

Fig. 10. Variation of the optimum number of renewals with the number of unplanned interruption

#### **6.4 The model: Maximum safety and imposed costs**

We determine the optimum number of renewals, to maximize the reliability of the OEL in conditions of total costs TC(T,r) does not exceed a maximum value imposed max TC . In this hypothesis, the structure of mathematical model to be optimized is:

1. Vector of decision variable: x ≡ r the number of renewals

2. Objective function: min F x 1 R x { () () = − <sup>S</sup> }

$$\text{where:}\tag{7} = 1 - \mathbf{P}\_0 \cdot \exp\left(-\alpha \left(\mathbf{x} + \mathbf{1}\right)^{(1-\beta)} \mathbf{T}^{\beta}\right)\tag{8}$$

3. Constraints of the model:

$$\text{TC}(\mathbf{x}) \le \text{TC}^{\text{max}}; \quad 0 \le \mathbf{x} \le \mathbf{x}^{\text{max}} \tag{58}$$

$$\begin{split} \text{for } \mathbf{c}: \qquad & \mathbf{P}\_{\mathbf{0}} \cdot \exp\left(-\alpha \left(\mathbf{x} + \mathbf{1}\right)^{(1-\emptyset)} \mathbf{T}^{\emptyset}\right) \cdot \lambda\_{\mathbf{s}} \cdot \mathbf{T} \cdot \mathbf{c}\_{\mathbf{F}} + \mathbf{x} \cdot \mathbf{c}\_{\mathbf{PM}} + \\ & + \mathbf{P}\_{\text{OtL}} \cdot \mathbf{c}\_{\mathbf{w}} \cdot \left(1 - \mathbf{P}\_{\text{0}} \cdot \exp\left(-\alpha \left(\mathbf{x} + \mathbf{1}\right)^{(1-\emptyset)} \mathbf{T}^{\emptyset}\right)\right) \cdot \mathbf{T} \leq \mathbf{T} \mathbf{C}^{\text{max}}; \quad -\mathbf{x} \leq \mathbf{0}; \ \mathbf{x} \leq \mathbf{x}^{\text{max}}; \end{split} \tag{59}$$

with, xmax – the maximum imposed number of renewals.

By running the application program, for different values imposed for max TC , we obtain the maximum reliability of the OEL, presented in Figure 11 and the optimum number of renewals required to achieve that reliability, presented in Figure 12.

required to apply to the OEL during a year, in conditions of minimum total cost TC(T,r),

20 25 30 35 40 45 50 55

Renewals [pieces]

We determine the optimum number of renewals, to maximize the reliability of the OEL in

F x 1 P exp x 1 T <sup>0</sup>

( )( )

+ ⋅ ⋅ − ⋅ −α + ⋅ ≤ − ≤ ≤

By running the application program, for different values imposed for max TC , we obtain the maximum reliability of the OEL, presented in Figure 11 and the optimum number of

P c 1 P exp x 1 T T TC ; x 0; x x ;

−β β

( ( ))

( ) <sup>1</sup>

−β <sup>β</sup> = − ⋅ −α + (57)

( ) max max TC x TC ; 0 x x ≤ ≤≤ (58)

1 max max

(59)

Fig. 10. Variation of the optimum number of renewals with the number of unplanned

conditions of total costs TC(T,r) does not exceed a maximum value imposed max TC .

In this hypothesis, the structure of mathematical model to be optimized is: 1. Vector of decision variable: x ≡ r the number of renewals

presented in Figure 10, i.e. ropt = *f( NF ).* 

2.6

3. Constraints of the model:

**6.4 The model: Maximum safety and imposed costs** 

2. Objective function: min F x 1 R x { () () = − <sup>S</sup> }

where: ( ) ( )( )

( )( ) ( )

or : P exp x 1 T T c x c

OEL w 0

with, xmax – the maximum imposed number of renewals.

1 0 s F PM

renewals required to achieve that reliability, presented in Figure 12.

−β β

⋅ −α + ⋅ λ ⋅ ⋅ + ⋅ +

interruption

2.7

2.8

2.9

3

3.1

Failures

3.2

3.3

3.4

3.5

Fig. 11. Variation of the reliability of the OEL with TCmax imposed

Fig. 12. Variation of the number of renewals with TCmax imposed

#### **6.5 The model on components: The minimun costs and reliability imposed**

We determine the optimum number of renewals on the three components of the OEL (pillars, conductors, and insulators), in the assumption of minimum total costs and on condition that the reliability of the OEL does not fall below an imposed value min R . S

In this case, the structure of the mathematical model to be optimized will include the total costs TC(T, ri) corresponding to those three components.

1. In this case, the vector of variables to be optimized will have three components, corresponding to the number of renewals on the three components of the electric line: 1-pillars; 2-conductors; 3-insulators.

$$\mathbf{X} = \begin{bmatrix} \mathbf{x}\_1 \mathbf{x}\_2 \mathbf{x}\_3 \end{bmatrix}^\mathrm{\!\!\!
\quad \text{with} \quad \mathbf{x}\_i \equiv \mathbf{r}\_i \text{ \!\!\!
\/} \mathbf{i} = \mathbf{1} \text{ \!\!\!\/} \mathbf{2} \text{ \!\!\!\/}$$

Reliability Centered Maintenance Optimization of Electric Distribution Systems 107

In conclusion, through the models of RCM optimization strategies that have been developed, we can obtain technical and economic information on the analysis, policy and

Based on the algorithm for calculating and optimizing PM, presented in the previous sections, we propose an integrated model of software for the implementation and exploitation of the RCM. This model is specifically designed for OEL belonging to EDS. To achieve the proposed information system we used the Matlab environment, because it offers the possibility to write and add programmes to the original files, allowing the development of the application characteristic to the EDS domain. The choice of Matlab was decisively influenced by the facilities offered by this environment in terms of achieving interactive user interfaces in the form of windows and menus, with the toolbox GUI (Graphical User Interface) and statistical processing with the Statistics Toolbox (Dulau et al.,

planning maintenance actions for a period of time. This information pertains to:


For interactive control of various representations, we do the following steps:



The user interface that opens as a result of startup operations identifies two types of overhead electric lines, of 20 kV and 110 kV, and allows choosing the overhead electric line that will be examined by pressing the corresponding button, as in Figure 14 (Dulau et al.,

**7. RCM - maintenance management integrated software** 




2007), (Dulau et al., 2010).



**7.1 The application startup** 

Fig. 14. User interface

2010).

2. Objective function: min TC X { ( )} where:

$$\begin{aligned} \text{TC}\left(\mathbf{X}\right) &= \sum\_{i=1}^{3} \text{TC}\_{i}\left(\mathbf{X}\right) \text{ where } : \\ \text{TC}\_{i}\left(\mathbf{X}\right) &= \text{P}\_{0i} \exp\left(-\alpha\_{i}\left(\mathbf{x}\_{i} + \mathbf{1}\right)^{(1-\beta i)}\mathbf{T}^{\beta i}\right) \cdot \lambda\_{i} \cdot \mathbf{T} \cdot \mathbf{c}\_{\text{Fi}} + \mathbf{x}\_{i} \cdot \mathbf{c}\_{\text{PM}} + \\ &+ \mathbf{P}\_{\text{OEL}} \cdot \mathbf{c}\_{\text{w}} \cdot \left(1 - \mathbf{P}\_{0i} \exp\left(-\alpha\_{i}\left(\mathbf{x}\_{i} + \mathbf{1}\right)^{(1-\beta i)}\mathbf{T}^{\beta i}\right)\right) \cdot \mathbf{T} \end{aligned} \tag{60}$$

3. Constraints of the model:

$$\mathbf{R}\_{\rm S} \left( \mathbf{x} \right) \ge \mathbf{R}\_{\rm S}^{\rm min}; \quad 0 \le \mathbf{x}\_{\rm l} \le \mathbf{x}\_{\rm l}^{\rm max} \tag{61}$$

$$\text{cor}: \prod\_{\mathsf{i}=1,2,3} \mathsf{P}\_{\mathsf{i}0} \exp \left[ -\mathsf{a}\_{\mathsf{i}} \left( \mathsf{x}\_{\mathsf{i}} + \mathbf{1} \right)^{(\mathsf{i}-\mathsf{j}\mathsf{i})} \mathsf{T}^{\mathsf{j}\mathsf{i}} \right] \geq \mathsf{R}\_{\mathsf{S}}^{\min}; \ -\mathsf{x}\_{\mathsf{i}} \leq \mathsf{0}; \ \mathsf{x}\_{\mathsf{i}} \leq \mathsf{x}\_{\mathsf{i}}^{\max}; \tag{62}$$

with: max x – the maximum imposed number of renewals on the i *i* component for |i=1, 2, 3; ii i α ; ; β λ - the reliability parametres of components;

Fi PMi c ;c - the cost with the corrective and preventive maintenance on components. where:

The reliability parameters of the components have the estimated values presented in the Table 2.

The maintenance costs of the components can be assessed as a percentage of the costs related to the maintenance on the electric line, in the absence of a database with the costs on the components.

By running the application program, for different values imposed to the reliability of the OEL, we obtain the optimum number of renewals ri optim | i=1,2,3; on each component of the OEL, to ensure the minimum reliability imposed min R , in the conditions of the minimum S total cost TC, presented in Figure 13.

Fig. 13. Variation of the optimum number of renewals with the reliability imposed to the OEL

In conclusion, through the models of RCM optimization strategies that have been developed, we can obtain technical and economic information on the analysis, policy and planning maintenance actions for a period of time. This information pertains to:


106 Electrical Generation and Distribution Systems and Power Quality Disturbances

( )

TC X P exp x 1 T T c x c

OEL w 0i i i

( )( )

1 i i

( ) min max Rx R ;0x x S S ii ≥ ≤≤ (61)

PILLARS CONDUCTORS INSULATORS

optim | i=1,2,3; on each component of the

−β β

(60)

( ( ))

( )( ) 1 i i min max

−α + ≥ − ≤ ≤ ∏ (62)

1 i i i 0i i i i Fi i PMi

−β β

= −α + ⋅ λ ⋅ ⋅ + ⋅ +

P c 1 P exp x 1 T T

+ ⋅ ⋅ − −α + ⋅

0i i i S i ii

or : P exp x 1 T R ; x 0; x x ; −β <sup>β</sup>

with: max x – the maximum imposed number of renewals on the i *i* component for |i=1, 2, 3;

The reliability parameters of the components have the estimated values presented in the

The maintenance costs of the components can be assessed as a percentage of the costs related to the maintenance on the electric line, in the absence of a database with the costs on

By running the application program, for different values imposed to the reliability of the

OEL, to ensure the minimum reliability imposed min R , in the conditions of the minimum S

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> 0.96

Fig. 13. Variation of the optimum number of renewals with the reliability imposed to the OEL

Optimal number of renewals

Fi PMi c ;c - the cost with the corrective and preventive maintenance on components.

2. Objective function: min TC X { ( )}

() ()

3

i 1

=

=

( ) ( )( )

i

TC X TC X where :

where:

where:

Table 2.

the components.

3. Constraints of the model:

i 1,2,3

ii i α ; ; β λ - the reliability parametres of components;

OEL, we obtain the optimum number of renewals ri

0.962 0.964 0.966 0.968 0.97 0.972 0.974 0.976 0.978 0.98

Reliability of EL

total cost TC, presented in Figure 13.

=

