Математическая модель прогнозирования отказов автономного самоспасателя

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Introduction

Self-contained self-rescuers (SCSR) are devices used by miners to evacuate them-selves in the event of an emergency in a mine. There are two types of SCSRs: 30-minute and 60-minute, each of which is used once.

Compliance with norms and standards is a key factor in ensuring the safety of miners [1, 2]. According to the adopted norms, strict selection and quality testing of SCSRs is required. The tests are divided into laboratory tests – 50 % of the samples (minimum 6 units) are tested in an accredited laboratory according to the IS15803-2008 standard and practical tests, when the remaining SCSR samples undergo a performance test under conditions close to real ones, for example, at a rescue station. SCSR testing parameters and their limit values are given in Table 1.

Even if the above provisions are met, there is still a possibility that the device may fail during actual use or emergency use due to various reasons including human error.

Mathematical modeling Mathematical model selection

A large number of papers are devoted to mathematical modeling [3 – 12]. We use the grey Markov SCGM (1,1)c prediction model to study or predict the failures of SCSRs in emergency situations arise due to human error in the care and maintenance of SCSRs [13 – 15]. We have selected the grey Markov model to forecast or assess the errors due to following reasons:

1. Error is not regular and it cannot be generalized.
2. It is randomly scattered.
3. Information on raw data is limited.

 

Table 1

SCSR test parameters and limit values

Parameter	Acceptance Level
Rated Duration (as per label)	Not less than 90 % of the specified duration
Inhalation Oxygen Concentration (% by vol.)	Not less than 21*
Inhalation Carbon Dioxide Concentration (% by vol.)	Not more than 1.5**
Maximum inhalation temperature dry bulb	Not more than 55 °C
Inhalation breathing resistance	Not more than 10 mbar
Exhalation breathing resistance
* A short term deviation to a level of not less than 17 % for a period of not more than two minutes at the beginning of the test is permissible. ** Throughout the rated duration of the apparatus, carbon dioxide concentration of the inhaled air shall not exceed 3.0% (by volume).

 

4. Information is not valid for all the stake holders (for all the mining companies or mining players); hence we can say that information is neither perfect and nor certain.

Every mining company is not weak on budgets, hence it is very difficult to model such scenarios based on a probabilistic statistical model or mathematical statistics methods. The grey Markov SCGM (1,1)c is quite suitable for complex mathematical models.

Mathematical model development

Since human errors cannot be generalized, we can say that the error data is random. Taking into consideration of randomness of data, failure of SCSRs in time series x⁽⁰⁾ can be expressed as

$x^{\left(0\right)}=\left\{x^0\left(1\right),x^{\left(0\right)}\left(2\right),\mathrm{...,}{x}^{\left(0\right)}\left(n\right)\right\}$.                                                               (1)

First,  is integrated as follows, where

${\overline{x}}^{\left(1\right)}=\left\{{\overline{x}}^{\left(1\right)}\left(2\right),{\overline{x}}^{\left(1\right)}\left(3\right),\mathrm{...,}{\overline{x}}^{\left(1\right)}\left(n\right)\right\}$;                                                             (2)

${\overline{x}}^{\left(1\right)}\left(k\right)=\sum _{m=2}^k{\overline{x}}^{\left(1\right)}\left(m\right),k=\mathrm{2,}\mathrm{3,}\mathrm{...,}n$                                                         (3)

 x^-(0) is a close mean value generated sequence for x^-(0)

${\overline{x}}^{\left(1\right)}=\left\{{\overline{x}}^{\left(1\right)}\left(2\right),{\overline{x}}^{\left(1\right)}\left(3\right),\mathrm{...,}{\overline{x}}^{\left(1\right)}\left(n\right)\right\}$;                                                          (4)

where

${\overline{x}}^{\left(1\right)}\left(k+1\right)=\frac{x^{\left(0\right)}\left(k+1\right)+x^{\left(0\right)}\left(k\right)}{2}$.                                                         (5)

Random human errors in maintenance of SCSRs can lead to the failures of SCSRs; such failures of SCSRs are dynamic in nature and given that the integral sequence of failures of SCSRs time series is expressed as {x⁽⁰⁾(k)} that is associated with satisfaction trend of non-homogeneous index discrete-function expressed as

$f_r\left(k\right)=b{\text{e}}^{a\left(k-1\right)}-c$.

Thus the data of x⁽¹⁾(k) is fit to f_r(k)

According to the grey system cloud prediction method, the grey system SCGM (1,1)c prediction model can be expressed as

$\frac{d{x}^{\left(1\right)}\left(k\right)}{dk}=a{x}^{\left(1\right)}\left(k\right)+u,k\ge 2$.                                                               (6)

Its time response function can be expressed as follows

$x^{\left(1\right)}\left(k\right)=\left[x^{\left(1\right)}\left(1\right)+\frac{U}{a}\right]{\text{\hspace{0.17em}\hspace{0.17em}e}}^{ak}-\frac{U}{a}$,                                                            (7)

where

$a=$$\frac{\ln {\displaystyle \sum _{k=2}^n{x}^{\left(0\right)}\left(k-1\right)x^{\left(0\right)\left(k\right)}}}{{\displaystyle \sum _{k=2}^n{\left(x^{\left(0\right)}\left(k-1\right)\right)}^2}}$;                                                     (8)

$b=$$\frac{\left(n-1\right){\displaystyle \sum _{k=2}^n{\text{e}}^{a\left(k-1\right)}{\overline{x}}^{\left(1\right)}\left(k\right)}-\left({\displaystyle \sum _{k=2}^n{\text{e}}^{a\left(k-1\right)}}\right)\left({\displaystyle \sum _{k=2}^n{\overline{x}}^{\left(1\right)}\left(k\right)}\right)}{\left(n-1\right){\displaystyle \sum _{k=2}^n{\text{e}}^{2a\left(k-1\right)}}-{\left({\displaystyle \sum _{k=2}^n{\text{e}}^{a\left(k-1\right)}}\right)}^2}$;       (9)

$c=$$\frac{1}{n-1}$$\left[\left(\sum _{k=2}^n{\text{e}}^{a\left(k\right)}\right)b-\sum _{k=2}^n{\overline{x}}^{\left(1\right)}\left(k\right)\right]$.                                               (10)

If, ${\overline{x}}^{\left(1\right)}\left(1\right)=b-c$, $U=ac, {\overline{x}}^{\left(1\right)}\left(k\right)$ is reverted, the grey system SCGM (1,1)c prediction model in originality can be expressed as

${\widehat{x}}^{\left(0\right)}\left(k\right)=\frac{2b\left(1-{\text{e}}^{-a}\right)}{\text{1}+{\text{e}}^{-a}}{\text{\hspace{0.17em}e}}^{a\left(k-1\right)}$.                                                                    (11)

Then, the grey precision index Y(k), residual error e(k) and the relative error Dk can be calculated as follows:

$Y\left(k\right)=\frac{x^0\left(k\right)}{x^{\left(0\right)}\left(k\right)}$;                                                                                         (12)

$\epsilon \left(k\right)={\widehat{x}}^{\left(0\right)}\left(k\right)-x^0\left(k\right)$;                                                                                (13)

$\Delta k=\frac{\left|\epsilon \left(k\right)\right|}{x^0\left(k\right)}$.                                                                                           (14)

Equations (12) – (14) are the indicators of the grey fitting accuracy, which also reflect the degree of deviation in the predicted values to the original set of data.

Amended residual SCGM (1,1) c model to predict failures in SCSRs

The data for statistics of the failures of SCSRs due to non-maintenance could be larger and as a human’s behaviour is highly uncertain, therefore, regularity cab be weak in nature or may not be very strong. Hence, it is incorrect to predict accuracy based only on SCGM(1,1)c in order to predict SCSR failures due to human errors in SCSR maintenance. In order to improve the accuracy of the prediction rate and meet closely to the actual situation, the model is corrected to improve the accuracy. The amended principle and steps are as follows:

1. The first time residuals data sequence has the following predicted value and the actual value, respectively:

${\epsilon }^{\left(0\right)}\left(k\right)={\widehat{x}}^{\left(0\right)}\left(k\right)-x^0\left(k\right),k=\mathrm{1,}\mathrm{2,}\mathrm{...,}n;$                                                            (15)

${\epsilon }^{\left(0\right)}\left(k\right)=\left\{{\epsilon }^0\left(1\right),{\epsilon }^0\left(2\right),\mathrm{...,}{\epsilon }^0\left(n\right)\right\}$.                                                                     (16)

2. Processing residuals correction sequence is as follows, if

$M={\left(1+{\text{e}}^{-a}\right)}^{-1}{\left(1-{\text{e}}^{-a}\right)}^b$,

then the SCGM (1,1) prediction can be expressed as

${\widehat{x}}^{\left(0\right)}\left(k\right)=2{\text{e}}^{a\left(k-1\right)}M$.

If ${\epsilon }^{\left(0\right)}\left(k\right)\ge \mathrm{0,}k=\mathrm{1,}\mathrm{2,}\mathrm{...,}n,$ the SCGM (1,1)c model can be expressed as

${\widehat{\epsilon }}^{\left(0\right)}\left(k\right)=2{\text{e}}^{a_1\left(k-1\right)}{M}_1$

and a₁, b₁ and M₁ can be obtained in the same manner as a, b and M as earlier used method.

Hence, the first time residual corrected SCGM (1,1)c model can be expressed as

${\widehat{x}}_{{\epsilon }_1}{\epsilon }^{\left(0\right)}\left(k\right)=2\left({\text{e}}^{a\left(k-1\right)}M-{\text{e}}^{a_1\left(k-1\right)}{M}_1\right)$.                                                           (17)

If the first time residuals corrected SCGM (1,1)c model can be expressed as

${\widehat{x}}_{{\epsilon }_1}^{\left(0\right)}\left(k\right)=2\left({\text{e}}^{a\left(k-1\right)}M+{\text{e}}^{a_1\left(k-1\right)}{M}_1\right)$.                                                               (18)

If the first time the residuals amendment could not meet the accuracy of the required forecast, a second residual correction shall be carried out and so on until the required accuracy requirements are meet.

The residual correction generic model can be expressed as

${\widehat{x}}_{{\epsilon }_2}{\epsilon }^{\left(0\right)}\left(k\right)={\widehat{x}}_{{\epsilon }_1}{\epsilon }^{\left(0\right)}\left(k\right)-2{\text{e}}^{a_2\left(k-1\right)}{M}_2,$                                                           (19)

where ${\epsilon }_1{\epsilon }^{\left(0\right)}\left(k\right)\ge \mathrm{0,}k=\mathrm{1,}\mathrm{2,}\mathrm{...,}n,$  and 

${\epsilon }_1^{\left(0\right)}\left(k\right)\le \mathrm{0,}k=\mathrm{1,}\mathrm{2,}\mathrm{...,}n.$                                                                            (20)

where ${\epsilon }_1^{\left(0\right)}\left(k\right)\le \mathrm{0,}k=\mathrm{1,}\mathrm{2,}\mathrm{...,}n.$

Amended Markov SCGM (1,1)c model for prediction of failures in SCSRs

The curve that we obtained as a result of forecasting SCGM(1,1)c is, in fact, an exponential curve, and the result of forecasting is a smooth curve. Since, human errors are the main part of the failures of SCSRs, the SCGM (1,1)c model solely applied cannot meet the required accuracy for the forecast. Markov’s theories have no aftereffect, that is to say, the future of the system is only related to the current state and there is no effect from the past state. Meanwhile, the Markov model is adopted to predict state trends through the transfers of probability which it can adapt to the randomness and variability of the state. Application of Markov theory to make corrections in the SCGM (1,1)c prediction model of failure of SCSRs due to non-maintenance as a human error can better solve the variability and randomness in maintenance of SCSRs caused by human errors to improve the prediction accuracy.

State divided criterion for the prediction of failures in SCSRs

Failures in SCSRs due to non-maintenance could change annually and are dynamic non-stationary random process. Therefore, the indicators of precision fitting the prediction shall also be variable and random as the boundary and connotation of the different annual states can change. Hence, an adaptive state divided criterion needs to be obtained, and the criterion should be consistent with the basic timing trend of the Failures of SCSRs due to non-maintenance.

Hence, k was divided into m states, and each state can be expressed as

$E_i\in \left[{\epsilon }_{1i},{\epsilon }_{2i}\right],i=\mathrm{1,}\mathrm{2,}\mathrm{...,}m,$                                                                     (21)

where ${\epsilon }_{1i}=Y\left(k\right)+A_i; {\epsilon }_{2i}=Y\left(k\right)+B_i,$

E_i is expressed as 𝑖 state, ε₁_i and ε₂_i are, respectively, expressed as the upper and lower bounds of the 𝑖 state, A_i and B_i are constants determined according to prediction data.

Since 𝑘 is a time function, ε₁_i and ε₂_i will be changed with time, so the state possesses variability.

When we say that the state is divided, the numbers of different intervals can be reasonably divided according to the actual situation. If our raw data will be less, the interval division shall be less to increase the number of transfers between the various states, and thus the transfer law shall be more objectively reflected between states.

Conversely, if raw data will be more, the interval division shall be less to excavate more information from a large number of data to improve the prediction accuracy. It will be suitable to adapt the clustering classification method to determine class number and classification intervals due to fewer data and the uncertain status of failures of SCSRs due to non-maintenance as a human error.

Developing state transition rate matrix for the prediction of failures in SCSRs

Let the original number of samples is expressed as s M_ij(k) from the state E_i transiting to the state E_j by k step, and the number of occurrences of the state E_i is expressed as

M_i, so state transition probability can be expressed as

${\overline{P}}_{ij}\left(k\right)=\frac{M_{ij}\left(k\right)}{M_i},i,j=\mathrm{1,}\mathrm{2,}\mathrm{...,}\overline{M}$,                                                                 (22)

M ´ M state transition probability matrix can be obtained as

$P\left(k\right)=$$\left[\begin{array}{cccc}{P}_{11}\left(k\right)& P_{12}\left(k\right)& \cdots & P_{13}\left(k\right)\\ P_{21}\left(k\right)& P_{22}\left(k\right)& \cdots & P_{23}\left(k\right)\\ ⋮& ⋮& ⋰& ⋮\\ P_{m1}\left(k\right)& P_{m2}\left(k\right)& \cdots & P_{m3}\left(k\right)\end{array}\right]$.                                              (23)

Determining the predictive value for the prediction of failures in SCSRs

The state transition probability matrix k reflects all statistical regularities of state transition, and the future system state steering can be predicted by investigating the matrix. In the actual analysis of the process, one step transition probability matrix is generally only examined. Given that predicted moment object is in the state E_k, investigating the k row of (1) can get the following below results;

1. If max P_ij = P_kl, the next time system should most likely shift from the state E_k to the state E_l.
2. If there are two or more probability values identical or similar to k row in the matrix P(1), the future state steering will be difficult to determine; it needs to consider probability transition matrix P(2) or P(n) (n ≥ 3).

The system's future state is determined by investigating the state transition probability matrix, and the grey change interval of relative prediction value in the future moments will also be determined; it can be expressed as [ε₁_i, ε₂_i]. The predicted value of the future moment can be expressed as the interval median as Y(k)Y'(k)

$Y^{\text{'}}\left(k\right)=\frac{x^0\left(k\right)}{x^{\left(0\right)}\left(k\right)}=\frac{1}{2}\left({\epsilon }_{1i}+{\epsilon }_{2i}\right)=Y\left(k\right)\frac{1}{2}\left(A_i+B_i\right)$.                                             (24)

Error analysis

In order to examine the precision and accuracy of the proposed model, ground tests are required to determine the error between the forecast value and actual value. In this study, we use the relative percentage error (RPE) analysis to assess the model precision and accuracy.

Relative percentage error analysis

Relative percentage error compares the real and predicted values at specific time k. The RPE is defined as equation (14). The total model precision can be defined by average relative percentage error (ARPE) as below

$\text{ARPE}=\frac{1}{n}\sum _{k=1}^n\text{RPE, where k = 2, 3, 4}$.                                                  (25)

Analysis

A field study shall be able to determine the accuracy of the model. Since the field data is very difficult to get and study cannot be carried out, this mathematical model is accurate and it will determine the desired results.

We simulate a data point of one of the cluster of mines with 2500 employees going underground every day to check the accuracy of the SCGM (1,1) c mathematical model. This cluster of mines has a total of 2500 units of SCSRs, but they are not maintained to proper safety standards or there is a human negligence in maintaining those SCSRs. On forceful annual check or during the escape exercise annually, the rate of failure of SCSRs is illustrated below. The number of failed SCSRS in one of the mines from 2018 to 2022 are shown in Table 2.

 

Table 2

Values of failed SCSRs in one of the mines from 2018 to 2022

No.	Year	Annual failure of SCSRs
1	2018	54
2	2019	50
3	2020	52
4	2021	55
5	2022	47

Analysis based on SCGM (1,1)с prediction model

Based on the above data and calculations done on MATLAB, the values for a and b were 0.0072 and 7.359e + 03, respectively, then the grey prediction model was established. The obtained data from SCGM (1,1) c analysis of data with grey precision index are shown in Table 3.

 

Table 3

SCGM (1,1)c analysis of data with grey precision index

No.	Year	Real value	Prediction value	Relative error	Grey precision index, %
1	2018	54	52.93	0.01	102
2	2019	50	52.55	0.05	95
3	2020	52	52.18	0.003	99
4	2021	55	51.81	0.05	106
5	2022	47	51.44	0.09	91

 

Analysis of results based on SCGM (1,1)c prediction model

The analysis of results based on SCGM (1,1)c prediction model suggests that if the negligence on the maintenance of SCSRs due to human error happens, it results into the SCSRs failures. The grey precision index suggests that it can predict close to accurate results for the SCSRs failure or the said conditions.

This analysis implies that on an average 50 people are at risk of carrying failed SCSRs, which might not work or might not work in full capacity during the actual escape. Accidents in the mines cannot be assessed in advance and people working underground if they are carrying failed SCSRs; they are risking their lives and it can result into big fatal accident.

Conclusion

Mines are vulnerable to accidents and people’s safety becomes one of the priority assignments in case of mine accidents. With the help of an appropriate SCSR, a person can easily escape in various accident conditions such as influx of toxic gases or increase in explosive gases or when the mine environment has become suffocating. Given the life of the apparatus which is 10 years and maintaining such apparatus manually for such long period and following all the norms of the regulatory authority is a daunting task although it is in the benefit of the people.

Therefore, the time has come to think beyond to use a SCSR which is smart and which need not to be monitored all the time which can diagnose the faults by itself and can raise alarm in case of any fault. By using such SCSRs, miners can always be updated of the health of the SCSRs and it will give ultimate confidence to their preparedness of self-escape.

Об авторах

Валерий Григорьевич Матвейкин

ФГБОУ ВО «ТГТУ»

Автор, ответственный за переписку.
Email: terehova.aa@mail.tstu.ru

доктор технических наук, профессор, заведующий кафедрой «Информационные процессы и управление»

Россия, Тамбов

Правин Кумар Правин

ФГБОУ ВО «ТГТУ»

Email: terehova.aa@mail.tstu.ru

аспирант кафедры «Информационные процессы и управление»

Россия, Тамбов

Борис Сергеевич Дмитриевский

ФГБОУ ВО «ТГТУ»

Email: terehova.aa@mail.tstu.ru

доктор технических наук, профессор кафедры «Информационные процессы и управление»

Россия, Тамбов

Анастасия Андреевна Терехова

ФГБОУ ВО «ТГТУ»

Email: terehova.aa@mail.tstu.ru

аспирант кафедры «Информационные процессы и управление»

Россия, Тамбов

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