Characteristics of each internal pressure mode.
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The high availability of technological systems, like defense, aerospace, automobile industries, and petrochemistry, is a central major objective of previous and latest developments in the technology of system design. Pipelines are the primary component of the systems of hydrocarbon transport in petrochemical industries. They are vital for human activities because they serve to transport water, natural gases, and oil from sources to all consumer sites. A novel analytic prognostic model was established in my earlier research work and applied to the case of pipelines subject to the effects of corrosion, to soil loading, and to internal pressure. These will initiate micro-cracks in the body of the tubes that can spread suddenly and can lead to failure. The increase of pipeline availability and the reduction of their global mission cost and performance necessitate to elaborate a suitable process of prognostic. Accordingly, a novel strategy based on degradation analytic laws was applied to diverse dynamic systems and was developed in my research work [1, 2, 3, 4, 5, 6]. Additionally, the remaining useful lifetime (
Moreover, prognostic is a process involving a prediction capacity. Using prognostic, we are able to evaluate the equipment remaining useful lifetime in terms of its future usage and its history of functioning. Predicting the remaining useful lifetime of industrial systems turns out to be presently a vital goal for industrialists knowing that the consequences of failure, which can occur suddenly, are usually very expensive. The traditional maintenance strategies [7, 8] founded on a static threshold of alarm are no more practical and efficient since they do not consider the instantaneous functioning state of a product. The establishment of a prognostic approach as an “intelligent” maintenance consists of the health follow-up, monitoring, and analysis, based on physical measurements utilizing sensors.
\nAlso, earlier expert studies of prognostic belong in general to three categories of technical approaches: the first category is the “experience-based prognostic” [9] which is based on measurements taken from a machine health monitoring, for example, those based on stochastic model, expert judgment, Bayesian approach, reliability analysis, Markovian process, optimization of preventive maintenance, etc. Their methodology of prognostic shows to be simple but inflexible toward changes in the environment and in the system behavior. The second category is the “estimation-based or trending prognostic” based on the statistics of vast measured data. We can cite as illustrations the work relying on the behavior of degradation expressed by abaci and utilizing a system expert description (process-mission-environment) [10]; the work relying on artificial intelligence, machine learning [11], neural network [12], and fuzzy logic [13]; and additionally the work based on dissipativity-based fuzzy integral sliding mode control of continuous time T-S fuzzy systems, SMC design for robust stabilization of nonlinear Markovian jump singular systems, sliding mode control of fuzzy singularly perturbed systems with application to electric circuits, the stabilization of quantized sampled-data neural network-based control systems, etc. Their methodologies are designated generally as not very precise, but they propose a powerful tool to the theory of prognostic. The third category is the “model-based prognostic” relying on the mathematical description of the process of degradation and its evolution level utilizing nondestructive inspection (NDI) monitoring. It is designated to be more precise and flexible than the two first categories. My earlier research illustrates a methodology of analytical prognostic relying on analytic laws of damage, such as the linear damage accumulation law of Palmgren-Miner and the fatigue crack propagation law of Paris-Erdogan. It fits in the third category of models. This approach is used whenever the law of damage of the studied system is analytically available. The advantage of this approach is consequently its precise and realistic features in evaluating the remaining useful lifetime of a system [14, 15, 16, 17].
\nAdditionally, pipes are petrochemical systems that transport natural gas and oil in huge quantities and over long distances. Their life prognostic is crucial in this industry because their availability has vital outcomes. Their major failures are due to soil settlements, seismic ground waves, deformations, buckling, internal and external corrosion, vibration and resonance, stress concentration in welding and fitting, and pressure fluctuation over long period. The failures due to fatigue by means of cracks propagation are noticed and measured by the tools of crack detection. Therefore, three case studies of pipelines were taken into consideration in my earlier publications [18, 19]: buried, unburied, and subsea (offshore pipelines). Each one of these situations necessitates different physical parameters like friction and soil pressure, atmospheric and water pressure, and corrosion. The buried pipes case will only be considered in the present chapter.
\nComputing probabilities is the main work of classical probability theory. Adding new dimensions to the stochastic experiments will lead to a deterministic expression of probability theory. This is the original idea at the foundations of this work. Actually, the theory of probability is a nondeterministic system in its essence; that means that the event outcomes are due to the chance and randomness. The addition of novel imaginary dimensions to the chaotic experiment occurring in the set \n
Furthermore, although the analytic linear prognostic laws are deterministic and very well-known in [14, 16], there are chaotic and stochastic influences and aspects (such as humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.) that 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 what I called the system failure probability due to its definition that estimates the jumps in the function of degradation
Additionally, my objective in this present work is to connect the complex probability paradigm to the buried pipeline system analytic prognostic in the case of linear damage accumulation which is subject to fatigue. In fact, the system failure probability derived from prognostic will be applied to and included in the complex probability paradigm. This will lead to the original and novel model of prognostic illustrated in this chapter. Thus, by determining the new prognostic model parameters, it becomes possible to evaluate the degree of our knowledge, the magnitude of the chaotic factor, the complex probability, the
Accordingly, the advantages and the purpose of the current chapter are to:
Extend classical probability theory to the set of complex numbers and therefore to link the theory of probability to the field of complex variables and analysis. This job was started and elaborated in my previous 12 papers.
Do an updated follow-up of the degradation
Apply the new axioms of probability and paradigm to system prognostic; thus, I will extend the prognostic concepts to the set of complex probabilities \n
Show that all stochastic phenomena can be expressed deterministically in the set of complex probabilities \n
Measure and compute both the degree of our knowledge and the chaotic factor of the system remaining useful lifetime and its degradation.
Draw and illustrate the graphs of the parameters and functions of the original paradigm corresponding to a buried pipeline prognostic.
Show that the classical concepts of random remaining useful lifetime and degradation possess a probability permanently equal to one in the complex set; hence, no randomness, no chaos, no uncertainty, no ignorance, no disorder, and no unpredictability exist in:\n
Show that by adding new and supplementary dimensions to any stochastic phenomenon, whether it is a pipeline system or any other random experiment, it becomes possible to do prognostic in a deterministic way in the set \n
Pave the way to implement this novel model to other areas in stochastic processes and to the field of prognostics in science and engineering. These will be the topics of my future research works.
Concerning some applications of the original elaborated paradigm and as a future work, it can be applied to a wide set of dynamic systems like vehicle suspension systems and offshore and buried petrochemical pipelines which are subject to fatigue and in the cases of nonlinear and linear damage accumulation. Furthermore, compared with existing literature, the main contribution of the present research work is to apply the novel paradigm of complex probability to the concepts of random remaining useful lifetime and degradation of a buried pipeline system hence to the case of analytic prognostic in the case of linear damage accumulation subject to fatigue. The following figure shows the main purposes of the complex probability paradigm (
The diagram of the main purposes of the complex probability paradigm and research work.
To conclude and to summarize, in the real probability universe \n
The EKA or the CPP diagram.
In this section a comprehensive summary of a part of my previously published PhD thesis [16] and of the formerly published IFAC conference paper [14] will be done, and the results that this current chapter needs will be just cited.
\nThe objective of my earlier research study, which will be enhanced in the present chapter and will be linked to
Petrochemical pipelines are systems that are used to transport natural gas and oil between sites. We believe that pipeline tubes are a major element in petrochemical industries. As a matter of fact, the prognostic of their life is essential in this industry since their availability has decisive and critical consequences on the cost of exploitation. Fatigue, which is due to internal pressure-depression variation along time, is the major failure cause of these systems. These pipelines are typically devised for ultimate limit states (resistance). Additionally, due to soil aggression influences, buried pipelines are subject to corrosion. Pipelines are designed as cylindrical tubes of thickness
A target failure probability of about 10−5 for pipelines is suggested by the DNV 2000 rules. Their major failure causes are soil settlements, seismic ground waves, deformations, buckling, stress concentration in welding and fitting, internal and external corrosion, pressure fluctuation over a long period, and vibration and resonance. Moreover, crack detection tools detect the crack propagation caused by failures due to fatigue.
\nAn important part of the main pipes is exposed to external cracking, which is a dangerous setback for the industry of pipes, for example, in the USA, Canada, and Russia. External crack identification is accomplished using diverse nondestructive evaluation (NDE) methods. If cracks were detected during inspection, we should evaluate their influence on the remaining useful lifetime of the pipeline in order to select the action of maintenance that should be applied: do nothing/repair/replace. We judge the integrity of pipes by assuming that some defects after in-line inspection (ILI) can be still undetected; detected, but not measured; detected and measured.
\nMoreover, the objective in my publications was to assess the evolution of the lifetime of a system at each instant. Consequently, and for this purpose, the trajectories of degradation had been utilized in terms of the time of operation or cycles’ number. Hence, we deduce the
The stress intensity factor was introduced to calculate the correlation between the crack growth rate,
where
\n\n\n
\n\n
\n\n
\n\n
\n
To do the prognostic of a degrading element, my approach was to evaluate and to predict the end of life of the element by modeling and tracking the function of degradation. My model of damage, whose progress is up to the macro-crack initiation point, is illustrated in \nFigure 3\n by the damage linear rule of Palmgren-Miner.
\nPalmgren-Miner’s linear rule of damage.
As a matter of fact, this law [7] is used to calculate the cumulative damage
The initial detectable crack
Or in terms of the pressure cycle time
To simplify the study, it is suitable to adopt a measurement of damage denoted by
Therefore, my general prognostic analytic linear model function, which is a recursive relation for the sequence of
where
Or in terms of the pressure cycle time
Consequently, the previous recursive relation leads to a sequence of
We will consider three different levels of internal pressure to take into account the diverse states of pressure conditions which are low, middle, and high. Moreover, as the stress load is a function of the cycles
We will consider in our current work a pipeline transporting natural gas of radius
The soil specific weight is
The internal pressure
Triangular variation of internal pressure.
We will consider three maximal levels of
For the purposes of simulations, in \nTable 1\n, the values of pressure
Pressure mode | \n\n | \nModel | \n
---|---|---|
High (mode 1) | \n8 | \nTriangular | \n
Middle (mode 2) | \n5 | \nTriangular | \n
Low (mode 3) | \n3 | \nTriangular | \n
Characteristics of each internal pressure mode.
A huge amount of pressure simulations of the order of hundreds of millions are required to estimate the real system lifetime; hence, we have used an approximated model of lifetime simulation of the order of 10,000,000 iterations. Accordingly, we have considered for this purpose a high-capacity computer system: a workstation computer with parallel microprocessors, a 64-Bit operating system, a 64 GB RAM, as well as a 64-Bit MATLAB version 2019 software.
\nThe evaluation of the remaining useful lifetime of the system is the major objective in a prognostic study. Since the
where
Consequently, my prognostic model computes the
For mode 3,
For mode 2,
For mode 1,
Two parameters which are
In this section, the novel complex probability paradigm will be presented after applying it to prognostic.
\nIt is very well-known that in systems engineering, the remaining useful lifetime and the degradation prediction is profoundly linked to many aspects (like humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.) that usually have a stochastic and chaotic behavior which reduces the degree of our certain system knowledge [32, 33, 34, 35]. Consequently, the lifetime of the system becomes a random variable and is computed by the arbitrary time
Let us consider a system degradation trajectory
CPP. (a) The prognostic of degradation and (b) The prognostic of RUL.
As a matter of fact, at
If
If 0 <
Furthermore, by applying here the
Additionally, we can express two complementary phenomena or events \n
Therefore, let the probability \n
where the classical and usual cumulative distribution function (
Since \n
In addition, two particular instants can be defined:
\n\n
\n
Consequently, we can state the boundary conditions as follows:
\nFor
For
We note also that since
The novel model of prognostic basic assumption will be presented now [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. We assume first the cumulative probability distribution function
We mention here that we are working with discrete random functions that depend on the discrete random time
This basic assumption is reasonable because:
Both
Both are non-decreasing functions.
Both functions are without measure units:
Afterward, we suppose that, at the instant \n
= \n
Pr, degradation, and the
Pr as a function of degradation D(t).
where \n
Hence, since
\n
\n\n
Thus, initially we have
\nMoreover,
\nwhere \n
This result is reasonable since \n
Therefore, we can deduce that
\nWe can understand that
We can notice the following:
\nand
If \n
if \n
This, since the degradation is very flat near 0 and starts increasing with
Degradation and Pr.
Furthermore, we have:
\n\n
\n\n
This implies that (\nFigure 9\n)
\nPr, D, and RUL.
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 \n
where
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 \n
where \n
\n\nFigure 6\n illustrates the real probability of failure
\n\nFigure 7\n illustrates the real probability of failure
\n\nFigure 8\n illustrates the real probability of failure
\n\nFigure 9\n illustrates the real probability of failure
The following flowchart summarizes all the procedures of the proposed complex probability prognostic model:
\n\n\n
We can infer from what has been elaborated previously the following:
\nThe degree of our knowledge
\nThe chaotic factor
\n\n
The magnitude of the chaotic factor
\n
At any instant
then,
Therefore, the prognostic of
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.
\nSee \nFigures 10\n–\n12\n.
\nPipeline degradation (a) and
Degradation and
Degradation, rescaled
See \nFigures 13\n–\n15\n.
\nDOK and Chf in terms of t and of each other for mode 1.
Pr and Pm/i in terms of t and of each other for mode 1.
The complex probability vector Z in terms of t for mode 1.
See \nFigures 16\n–\n18\n.
\nPipeline degradation (a) and
Degradation and
Degradation, rescaled
See \nFigures 19\n–\n21\n.
\nDOK and Chf in terms of t and of each other for mode 2.
Pr and Pm/i in terms of t and of each other for mode 2.
The complex probability vector Z in terms of t for mode 2.
See \nFigures 22\n–\n24\n.
\nPipeline degradation (a) and RUL (b) under linear damage law for low-pressure mode of excitation (mode 3).
Degradation and CPP parameters with Chf (a) and with MChf (b) for mode 3.
Degradation, rescaled RUL, and CPP parameters with Chf (a) and with MChf (b) for mode 3.
See \nFigures 25\n–\n27\n.
\nDOK and Chf in terms of t and of each other for mode 3.
Pr and Pm/i in terms of t and of each other for mode 3.
The complex probability vector Z in terms of t for mode 3.
We will present in this section the original general prognostic equations, we will interpret all the achieved simulations and the obtained data, and we will do a final analysis. Also, we will illustrate the results and a detailed discussion of the all the previous simulations and figures and of the following corresponding tables.
\nFirstly, we have linked prognostic characterized by the degradation
Subsequently, we do an updated follow-up of the performance of the random degradation in terms of time or cycle number, which is subject to non-chaotic and chaotic influences, by using the quantity \n
Referring to classical probability theory, this makes \n
In addition, in the simulations, a constant and very small increments in
Therefore, \n
Hence, if \n
This implies that \n
If \n
for any mode
Moreover, since \n
In the case of general prognostic, if we possess the
And the recursive relation
\nwith \n
It is crucial to indicate here that the
Furthermore, by applying
We can conclude from all the above that since
Furthermore, in the new model, we have
\nNote that since
Then, we get always \n
This implies that \n
and \n
Hence, we reach a new and general prognostic equation for
for any mode
Moreover, from Eqs. (25), (26), (27), and (28) and for any mode
where \n
In the ideal case, if all the factors are 100% deterministic, then we have in \n
Furthermore, at each instant
The new prognostic model parameters for any pipeline internal pressure mode.
Finally, we say that we have applied for pressure modes 2 and 3 the same analysis, logic, and methodology that we have used for pressure mode 1 regarding the remaining useful lifetime, the degradation, as well as all the
The new prognostic model and the relative pipeline pressure mode comparisons for \n
The new prognostic model and the relative pipeline pressure mode comparisons for \n
The high availability of technological systems, like defense, aerospace, automobile industries, and petrochemistry, is a central major objective of previous and latest developments in the technology of system design where it is very well-known that expensive failure may in general happen unexpectedly. A novel model of analytic prognostic was established in my earlier work and publications as a counterpart of existent classical strategies of maintenance in order to take into account the evolving environment and product state and in order to make them more efficient. We have applied this model to systems of petrochemical pipes that are exposed to fatigue failure under cyclic repetitive triangular pressure. It is known that the effects of fatigue will initiate micro-cracks that can spread rapidly and hence will lead to failure. This model is founded on existing laws of damage in fracture mechanics which are the law of Palmgren-Miner of linear damage accumulation and the law of Paris-Erdogan of crack propagation. This prognostic model estimates the system
In the present research work, the novel extended Kolmogorov paradigm of eight axioms (
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 \n
Additionally, as it was verified and shown in the novel model, when the degradation index is 0 or 1 and correspondingly the
As a prospective and future work and challenges, and concerning some applications to practical engineering, it is planned to more elaborate the original created prognostic paradigm and to implement it to a varied set of nondeterministic and dynamic systems like vehicle suspension systems and offshore and buried petrochemical pipes which are under the influence of fatigue and in the cases of nonlinear and linear damage accumulation. Furthermore, we will apply also
No potential conflict of interest was reported by the author.
\n\n\nR\n\n\n | The set of real probabilities of events |
\n\n\nM\n\n\n | The set of imaginary probabilities of events |
\n\n\nC\n\n\n | The set of complex probabilities of events |
\ni\n | The imaginary number where \n\n\ni\n2\n\n=\n−\n1\n\n and \n\ni\n=\n\n\n−\n1\n\n\n\n\n |
\nEKA\n | extended Kolmogorov axioms |
\nCPP\n | Complex probability paradigm |
\nProb\n\n | any event probability |
\nPr\n\n | system failure probability, probability in the real set \n\nR\n\n\n |
\nPm\n\n | system survival probability in \n\nM\n\n, probability in the imaginary set \n\nM\n\n corresponding to the real probability in \n\nR\n\n\n |
\nPm/i\n | system survival probability in \n\nR\n\n\n |
\nPc\n | probability in the complex set \n\nC\n\n, probability of an event in \n\nR\n\n with its associated event in \n\nM\n\n\n |
\nZ\n | the sum of Pr\n and Pm\n, complex probability number and vector |
\nDOK\n | = |Z|2, the square of the norm of Z, degree of our knowledge of the random event and experiment |
\nChf\n | chaotic factor |
\nMChf\n | magnitude of the chaotic factor |
\nt\n | pressure cycle time |
\ntC\n\n | pressure cycle time till system failure |
\nPj\n\n | pipelines internal triangular pressure |
\nfj\n(t) | probability density function for each pressure mode j\n |
\nF(t) | cumulative probability distribution function |
\n\n\n\nψ\nj\n\n\n\n | simulation magnifying factors for each pressure mode j\n |
\n\n\n1\n/\n\nψ\nj\n\n\n\n | the normalizing constant of Pr\n(t) for each pressure mode j\n |
\nD\n | degradation indicator of a system |
\nRUL\n | remaining useful lifetime of a system |
\nProb\n[RUL(t)] | probability of RUL after a pressure cycle time t.\n |