4.3 Analysis and extreme chaotic and random conditions

Although the analytic linear laws of prognostic are very well-known and deterministic in [14, 16], there are general influences and aspects that can be chaotic and stochastic (like humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.). Moreover, various variables in the expressions (5) and (6) of degradation which are considered as deterministic can also have a random aspect, such as the magnitude of applied pressure (due to the different conditions of pressure profile) and the length of the initial crack (potentially existing from the process of manufacturing). All those stochastic factors, embodied in the model by their mean values, influence the buried pipeline system and make its function of

degradation diverge from its computed trajectory modeled by these deterministic laws. An updated follow-up of the degradation performance and behavior with cycle number or time, which is subject to non-chaotic and chaotic influences, is made possible by Prð Þ tk =ψ<sup>j</sup> due to its definition that evaluates the jumps in D. In fact, chaos modifies and affects all the environment and system parameters included in the degradation equations (Eqs. (5) and (6)). Consequently, chaos total effect on the pipelines contributes to shape the degradation curve D and is materialized by and counted in the pipeline system failure probability Prð Þ tk =ψ<sup>j</sup>

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

Actually, Prð Þ tk =ψ<sup>j</sup> quantifies the resultant of all the nonrandom (deterministic) and random (nondeterministic) parameters and aspects which are contained in the equation of D, which affect the system and which lead to the consequent final curve of degradation. Consequently, an accentuated influence of chaos on the pipeline can lead to a smaller (or bigger) jump in the trajectory of degradation and therefore to a smaller (or bigger) failure probability Prð Þ tk =ψj. If, for example, due to extreme deterministic causes and random factors, D jumps directly from D<sup>0</sup> ≈0 to 1 then RUL goes straight from t<sup>C</sup> to 0 and consequently Prð Þ tk =ψ<sup>j</sup> jumps instantly

Prð Þ tk <sup>=</sup>ψ<sup>j</sup> <sup>¼</sup> D tð Þ� <sup>k</sup> D tð Þ¼ <sup>k</sup>�<sup>1</sup> D tð Þ� <sup>C</sup> <sup>D</sup>ð Þ <sup>0</sup> <sup>≈</sup> <sup>1</sup> � <sup>0</sup> <sup>¼</sup> <sup>X</sup><sup>t</sup>¼tC

In the extreme ideal case, if the pipeline system never deteriorates (no stresses or pressure) and with zero random causes and chaotic factors, then the resultant of all the nondeterministic and deterministic influences is null (like in the pipeline isolated and idle state). Accordingly, the system remains indefinitely at D<sup>0</sup> ≈ 0 and RUL stays equal to tC. So consequently, the jump in D is constantly zero. Hence, the

Prð Þ tk =ψ<sup>j</sup> ¼ D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ � ¼ ½ �¼ D<sup>0</sup> � D<sup>0</sup> 0

Figure 6 illustrates the real probability of failure Pr(t) in terms of the random degradation step CDF of the pipeline as a function of the cycle time t of pressure for

Figure 9 illustrates the real probability of failure Pr(t) in terms of the random

4.4 The flowchart of the complex probability analytic linear prognostic model

The following flowchart summarizes all the procedures of the proposed complex

where D tð Þ¼ <sup>0</sup> D tð Þ¼ <sup>1</sup> … ¼ D tð Þ¼ <sup>k</sup>�<sup>1</sup> D tð Þ¼ <sup>k</sup> D tð Þ¼ <sup>k</sup>þ<sup>1</sup> … ¼ D<sup>0</sup> ¼

Figure 7 illustrates the real probability of failure Pr(t) in terms of the random degradation of the pipeline as a function of the cycle time t of pressure for

Figure 8 illustrates the real probability of failure Pr(t) and the random degradation D(t) of the pipeline in terms of the number of cycle time t of pressure

degradation D(t) of the pipeline and the random RUL(t) of the pipeline as a

function of the cycle time t (in years) of pressure for mode 1.

where t jumps directly from 0 to tC.

DOI: http://dx.doi.org/10.5772/intechopen.90157

failure probability remains ideally 0:

0:020408 ≈0, for k ¼ 0, 1, 2, 3, … ∞.

probability prognostic model:

from 0 to 1:

mode 1.

mode 1.

81

for mode 1.

.

t¼0

ProbðÞ¼ t 1

Figure 8. Degradation and Pr.

Figure 9. Pr, D, and RUL.

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

degradation diverge from its computed trajectory modeled by these deterministic laws. An updated follow-up of the degradation performance and behavior with cycle number or time, which is subject to non-chaotic and chaotic influences, is made possible by Prð Þ tk =ψ<sup>j</sup> due to its definition that evaluates the jumps in D. In fact, chaos modifies and affects all the environment and system parameters included in the degradation equations (Eqs. (5) and (6)). Consequently, chaos total effect on the pipelines contributes to shape the degradation curve D and is materialized by and counted in the pipeline system failure probability Prð Þ tk =ψ<sup>j</sup> . Actually, Prð Þ tk =ψ<sup>j</sup> quantifies the resultant of all the nonrandom (deterministic) and random (nondeterministic) parameters and aspects which are contained in the equation of D, which affect the system and which lead to the consequent final curve of degradation. Consequently, an accentuated influence of chaos on the pipeline can lead to a smaller (or bigger) jump in the trajectory of degradation and therefore to a smaller (or bigger) failure probability Prð Þ tk =ψ<sup>j</sup> . If, for example, due to extreme deterministic causes and random factors, D jumps directly from D<sup>0</sup> ≈0 to 1 then RUL goes straight from t<sup>C</sup> to 0 and consequently Prð Þ tk =ψ<sup>j</sup> jumps instantly from 0 to 1:

$$P\_r(\mathbf{t}\_k) / \mu\_j = D(\mathbf{t}\_k) - D(\mathbf{t}\_{k-1}) = D(\mathbf{t}\_C) - D(\mathbf{0}) \approx \mathbf{1} - \mathbf{0} = \sum\_{t=0}^{t=t\_C} P\_{mb}(t) = \mathbf{1}$$

where t jumps directly from 0 to tC.

In the extreme ideal case, if the pipeline system never deteriorates (no stresses or pressure) and with zero random causes and chaotic factors, then the resultant of all the nondeterministic and deterministic influences is null (like in the pipeline isolated and idle state). Accordingly, the system remains indefinitely at D<sup>0</sup> ≈ 0 and RUL stays equal to tC. So consequently, the jump in D is constantly zero. Hence, the failure probability remains ideally 0:

$$P\_r(\mathfrak{t}\_k)/\mathfrak{\mu}\_j = [D(\mathfrak{t}\_k) - D(\mathfrak{t}\_{k-1})] = [D\_0 - D\_0] = \mathbf{0}$$

where D tð Þ¼ <sup>0</sup> D tð Þ¼ <sup>1</sup> … ¼ D tð Þ¼ <sup>k</sup>�<sup>1</sup> D tð Þ¼ <sup>k</sup> D tð Þ¼ <sup>k</sup>þ<sup>1</sup> … ¼ D<sup>0</sup> ¼ 0:020408 ≈0, for k ¼ 0, 1, 2, 3, … ∞.

Figure 6 illustrates the real probability of failure Pr(t) in terms of the random degradation step CDF of the pipeline as a function of the cycle time t of pressure for mode 1.

Figure 7 illustrates the real probability of failure Pr(t) in terms of the random degradation of the pipeline as a function of the cycle time t of pressure for mode 1.

Figure 8 illustrates the real probability of failure Pr(t) and the random degradation D(t) of the pipeline in terms of the number of cycle time t of pressure for mode 1.

Figure 9 illustrates the real probability of failure Pr(t) in terms of the random degradation D(t) of the pipeline and the random RUL(t) of the pipeline as a function of the cycle time t (in years) of pressure for mode 1.

#### 4.4 The flowchart of the complex probability analytic linear prognostic model

The following flowchart summarizes all the procedures of the proposed complex probability prognostic model:

4.3 Analysis and extreme chaotic and random conditions

Fault Detection, Diagnosis and Prognosis

Figure 9. Pr, D, and RUL.

80

Figure 8.

Degradation and Pr.

Although the analytic linear laws of prognostic are very well-known and deterministic in [14, 16], there are general influences and aspects that can be chaotic and stochastic (like humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.). Moreover, various variables in the expressions (5) and (6) of degradation which are considered as deterministic can also have a random aspect, such as the magnitude of applied pressure (due to the different conditions of pressure profile) and the length of the initial crack (potentially existing from the process of manufacturing). All those stochastic factors, embodied in the model by their mean values, influence the buried pipeline system and make its function of

The degree of our knowledge

DOI: http://dx.doi.org/10.5772/intechopen.90157

¼ 1–2Prð Þþ tk 2Pr

¼ �2Prð Þþ tk 2Pr

¼ 2Prð Þ� tk 2Pr

Pc tð Þ<sup>k</sup>

prognostic parameters and CPP.

¼ 1

5. The simulation of the new paradigm

The magnitude of the chaotic factor MChf:

2 ð Þ tk

2 ð Þ tk

2

Chf tð Þ¼ <sup>k</sup> 2iPrð Þ tk Pmð Þ¼� tk 2Prð Þ tk Pmð Þ tk =i ¼ �2Prð Þ tk ½ � 1 � Prð Þ tk

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

Chf is null when Pr(Nk) = Pr(0) = 0 (point J) and when Pr(tk) = Pr(tC)=1

MChf tð Þ¼ <sup>k</sup> ∣Chf tð Þ<sup>k</sup> ∣ ¼ �2iPrð Þ tk Pmð Þ¼ tk 2Prð Þ tk Pmð Þ tk =i ¼ 2Prð Þ tk ½ � 1 � Prð Þ tk

MChf is null when Pr(tk) = Pr(0) = 0 (point J) and when Pr(tk) = Pr(tC) = 1 (point L)

At any instant tk 0≤ ∀tk ≤ tC, the probability expressed in the complex set C is

<sup>2</sup> <sup>¼</sup> ½ � Prð Þþ tk Pmð Þ tk <sup>=</sup><sup>i</sup> <sup>2</sup> <sup>¼</sup> j j Z tð Þ<sup>k</sup>

¼ DOK tð Þ� <sup>k</sup> Chf tð Þ<sup>k</sup>

¼ DOK tð Þþ <sup>k</sup> MChf tð Þ<sup>k</sup>

then, Pc(tk) = Pr(tk) + Pm(tk)/i = Pr(tk) + [1 � Pr(tk)] = 1 always.

5.1 The parameter simulation in the pipeline prognostic for mode 1

Therefore, the prognostic of RUL(tk) and D(tk) of the pipeline in the set C is forever certain. The buried pipeline system is considered thereafter under three modes of pressure in order to simulate the cumulative distribution function D(tk) = F(tk) and hence in order to visualize, to quantify, as well as to draw all the

We will simulate in this section the original model of prognostic for the three internal pressure modes. We note that we have used the 64-Bit MATLAB version 2019 software to evaluate and find all the numerical values of the paradigm func-

<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>iPrð Þ tk Pmð Þ¼ tk <sup>1</sup>–2Prð Þ tk Pmð Þ tk <sup>=</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup>–2Prð Þ tk ½ � <sup>1</sup>–Prð Þ tk

ð Þ tk (22)

<sup>2</sup> � <sup>2</sup>iPrð Þ tk Pmð Þ tk

(21)

(23)

(24)

DOK tð Þ¼ <sup>k</sup> j j Z tð Þ<sup>k</sup>

The chaotic factor

(point L) (Figures 5a and 5b).

(Figures 5a and 5b).

the following:

tions analysis.

83

See Figures 10–12.

#### 4.5 The evaluation of the new paradigm parameters

We can infer from what has been elaborated previously the following:

$$\text{The real probability is } P\_r(t\_k) = \boldsymbol{\nu}\_j \times [D(t\_k) - D(t\_{k-1})], \text{ for pressure modes } j = 1, 2, 3 \tag{17}$$

The imaginary probability is Pmð Þ¼ tk <sup>i</sup> � ½ �¼ <sup>1</sup>–Prð Þ tk <sup>i</sup> � <sup>1</sup>–ψ<sup>j</sup> � D tð Þ<sup>k</sup> –D tð Þ <sup>k</sup>�<sup>1</sup> ½ � n o (18)

The complementary probability is Pmð Þ tk =i ¼ 1–Prð Þ¼ tk 1–ψ<sup>j</sup> � D tð Þ<sup>k</sup> –D tð Þ <sup>k</sup>�<sup>1</sup> ½ � (19)

The complex probability vector is Z tð Þ¼ <sup>k</sup> Prð Þþ tk Pmð Þ¼ tk Prð Þþ tk i � ½ � 1–Prð Þ tk (20)

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

The degree of our knowledge

$$\begin{split} \text{DOK}(\mathbf{t}\_k) &= |\mathbf{Z}(\mathbf{t}\_k)|^2 = \mathbf{1} + \mathbf{2}\mathbf{i}P\_r(\mathbf{t}\_k)P\_m(\mathbf{t}\_k) = \mathbf{1} - \mathbf{2}P\_r(\mathbf{t}\_k)P\_m(\mathbf{t}\_k)/i = \mathbf{1} - \mathbf{2}P\_r(\mathbf{t}\_k)[\mathbf{1} - P\_r(\mathbf{t}\_k)] \\ &= \mathbf{1} - \mathbf{2}P\_r(\mathbf{t}\_k) + \mathbf{2}P\_r^{-2}(\mathbf{t}\_k) \end{split}$$

(21)

The chaotic factor

$$\begin{split} \text{Cdf}(\mathbf{t}\_k) &= 2\mathbf{i}\mathbf{P}\_r(\mathbf{t}\_k)\mathbf{P}\_m(\mathbf{t}\_k) = -\mathfrak{D}\_r(\mathbf{t}\_k)\mathbf{P}\_m(\mathbf{t}\_k)/i = -\mathfrak{D}\_r(\mathbf{t}\_k)[\mathbf{1} - \mathbf{P}\_r(\mathbf{t}\_k)] \\ &= -\mathfrak{D}\_r(\mathbf{t}\_k) + \mathfrak{D}\_r^{-2}(\mathbf{t}\_k) \end{split} \tag{22}$$

Chf is null when Pr(Nk) = Pr(0) = 0 (point J) and when Pr(tk) = Pr(tC)=1 (point L) (Figures 5a and 5b).

The magnitude of the chaotic factor MChf:

$$\begin{split} \text{MCdf}(\mathbf{t}\_k) &= |\mathbf{C}\mathbf{h}f(\mathbf{t}\_k)| = -2\mathbf{i}P\_r(\mathbf{t}\_k)P\_m(\mathbf{t}\_k) = 2P\_r(\mathbf{t}\_k)P\_m(\mathbf{t}\_k)/\mathbf{i} = 2P\_r(\mathbf{t}\_k)[\mathbf{1} - P\_r(\mathbf{t}\_k)] \\ &= 2P\_r(\mathbf{t}\_k) - 2P\_r^{-2}(\mathbf{t}\_k) \end{split} \tag{23}$$

MChf is null when Pr(tk) = Pr(0) = 0 (point J) and when Pr(tk) = Pr(tC) = 1 (point L) (Figures 5a and 5b).

At any instant tk 0≤ ∀tk ≤ tC, the probability expressed in the complex set C is the following:

$$\begin{aligned} \left[Pr(t\_k)\right]^2 &= \left[P\_r(t\_k) + P\_m(t\_k)/i\right]^2 = \left|Z(t\_k)\right|^2 - 2iP\_r(t\_k)P\_m(t\_k) \\ &= DOK(t\_k) - Cly'(t\_k) \\ &= DOK(t\_k) + MClf(t\_k) \\ &= 1 \end{aligned} \tag{24}$$

then, Pc(tk) = Pr(tk) + Pm(tk)/i = Pr(tk) + [1 � Pr(tk)] = 1 always.

Therefore, the prognostic of RUL(tk) and D(tk) of the pipeline in the set C is forever certain. The buried pipeline system is considered thereafter under three modes of pressure in order to simulate the cumulative distribution function D(tk) = F(tk) and hence in order to visualize, to quantify, as well as to draw all the prognostic parameters and CPP.

#### 5. The simulation of the new paradigm

We will simulate in this section the original model of prognostic for the three internal pressure modes. We note that we have used the 64-Bit MATLAB version 2019 software to evaluate and find all the numerical values of the paradigm functions analysis.

#### 5.1 The parameter simulation in the pipeline prognostic for mode 1

See Figures 10–12.

4.5 The evaluation of the new paradigm parameters

Fault Detection, Diagnosis and Prognosis

82

We can infer from what has been elaborated previously the following:

The real probability is Prð Þ¼ tk ψ<sup>j</sup> � D tð Þ<sup>k</sup> –D tð Þ <sup>k</sup>�<sup>1</sup> ½ �, for pressure modes j ¼ 1, 2, 3

The imaginary probability is Pmð Þ¼ tk i � ½ �¼ 1–Prð Þ tk i � 1–ψ<sup>j</sup> � D tð Þ<sup>k</sup> –D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

The complementary probability is Pmð Þ tk =i ¼ 1–Prð Þ¼ tk 1–ψ<sup>j</sup> � D tð Þ<sup>k</sup> –D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

The complex probability vector is Z tð Þ¼ <sup>k</sup> Prð Þþ tk Pmð Þ¼ tk Prð Þþ tk i � ½ � 1–Prð Þ tk

(17)

(18)

(19)

(20)

n o

Figure 10. Pipeline degradation (a) and RUL (b) under linear damage law for high-pressure mode of excitation (mode 1).

Figure 13.

Figure 14.

85

DOK and Chf in terms of t and of each other for mode 1.

DOI: http://dx.doi.org/10.5772/intechopen.90157

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

Pr and Pm/i in terms of t and of each other for mode 1.

#### Figure 11.

Degradation and CPP parameters with Chf (a) and with MChf (b) for mode 1.

#### Figure 12.

Degradation, rescaled RUL, and CPP parameters with Chf (a) and with MChf (b) for mode 1.

5.1.1 The complex probability cubes for mode 1

See Figures 13–15.

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

Figure 13. DOK and Chf in terms of t and of each other for mode 1.

Figure 14. Pr and Pm/i in terms of t and of each other for mode 1.

5.1.1 The complex probability cubes for mode 1

Degradation, rescaled RUL, and CPP parameters with Chf (a) and with MChf (b) for mode 1.

Degradation and CPP parameters with Chf (a) and with MChf (b) for mode 1.

Pipeline degradation (a) and RUL (b) under linear damage law for high-pressure mode of excitation (mode 1).

See Figures 13–15.

Figure 12.

84

Figure 10.

Fault Detection, Diagnosis and Prognosis

Figure 11.

5.2.1 The complex probability cubes for mode 2

DOI: http://dx.doi.org/10.5772/intechopen.90157

Degradation, rescaled RUL, and CPP parameters with Chf (a) and with MChf (b) for mode 2.

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

5.3 The parameter simulation in the pipeline prognostic for mode 3

See Figures 19–21.

Figure 18.

See Figures 22–24.

Figure 19.

87

See Figures 25–27.

5.3.1 The complex probability cubes for mode 3

DOK and Chf in terms of t and of each other for mode 2.

Figure 15. The complex probability vector Z in terms of t for mode 1.
