Modified Stochastic Petri Net-Based Modeling and Optimization of Emergency Rescue Processes during Coal Mine Accidents

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Introduction
Coal provides strong supports for the rapid development of the national economy as one of the main energy sources in China. However, frequent safety accidents during coal mining hindered the healthy development of coal mining enterprises. Government and enterprises have tried to change such a passive situation by improving safety management measures and introducing rescue equipment, but the accidents and casualties have not been fundamentally resolved [1][2][3]. The relevant accident statistics in 2020 show that 122 coal mines accidents occurred in China. Compared with 2019, the death toll caused by coal mine accidents slightly increased. For example, the "1·12" coal dust explosion in Shannxi Baiji Mining Company induced 21 deaths, and the "12·16" coal gas outburst accident in Guizhou Guanglong coal mine resulted in 16 deaths and 1 injury [4]. Therefore, the establishment of an early warning and rapid rescue framework for coal mine accidents is the key point of current works, which would improve the level of emergency rescue (ER) and reduce or even avoid the casualty caused by accidents [5,6].
To date, several studies have deeply investigated the ER measures of coal mine accidents [7,8]. The ER communication system can effectively improve the monitoring accuracy and the efficiency of searching dangerous situations [9]. It is found that the emergency stations covering all mining areas are set up underground to provide medical first aid and transportation of injured in the shortest time [10]. In addition, it is significant for rescue to improve the existing refuge chamber and strengthen the escape simulation training [11].
For major coal mine accidents, the core tasks of various ER processes are to construct the mechanism of ER response and strengthen the evaluation of emergency management. It is well known that effective evaluation and optimization of ER processes in coal mines can improve the efficiency of ER and reduce casualties [12,13]. A Petri net is an important tool to deal with discrete events, and it is widely used in various fields. Petri nets, as a visual tool for modeling and analysis of the system simulation, can accurately simulate the dynamic behavior of the system and analyze the model characteristics through the definition of assumptions. With the application in various complex systems, Petri nets have evolved into many advanced forms such as color networks, stochastic networks, and object networks. Stochastic Petri nets (SPN) based on time have more advantages in describing the dynamic behavior of a system network. Therefore, SPN is used to study the unreasonable problems in the ER process of the coal mines, which can improve the efficiency of ER and increase the survival probability of trapped miners [14][15][16][17].
In this paper, based on the "9·28" major water inrush accident, a coal mine ER model is constructed through SPN, and the effectiveness of the model is analyzed. The steady-state probability, the busy rate of the places, and the utilization rate of transitions are calculated and discussed by MC. The triangular fuzzy strategy is used to analyze the change value of the total time in the ER system after changing the firing rate of each transition and find out the key activities affecting the rescue time.

Related Work
2.1. Major ER Policies. In 1978, a national meeting about coal mine ER held by the Ministry of Coal Industry of China provided reliable policy supports for the development of coal mine ER. As time goes on, the coal mine ER management system is gradually improved, the ER team is expanded, and the capability of ER is additionally significantly improved. These improvements can be proved by the fact that 34% of 1363 trapped persons were rescued in 2009, and 72% of 306 trapped persons were rescued in 2014 [18]. Since there are still some gaps compared with developed countries, the Chinese government attaches great importance to improve the ER management system and has proposed to establish an emergency management department. It is believed that this measure will encourage the further improvement of ER level in coal mines.
2.2. Structure of the ER Team. The rescue team and the medical team are the two main components of the ER team. So far, China has built 482 coal mine rescue teams, which are distributed all over the country. The medical team is divided into the national and provincial rescue centers and mine enterprise rescue stations [19]. The rescue ability of the rescue team is very important, so it is necessary to strengthen the daily traditional physical training and skill training. The application of high-tech in the field of ER enhances the prac-tical ability of rescue teams and improves the efficiency of rescue work such as the development of a comprehensive simulation training system for ER, which achieves personnel positioning and environmental monitoring [20,21].

Introduction of ER Equipment and Communication
Technology. The rescue teams are required to equip with a lot of rescue equipment to deal with different types of coal mine accidents. For instance, the large drainage equipment and mining submersible pumps are used to deal with flood disaster, the rapid sealing and spraying machine are used to deal with fire accidents, the positive pressure blower and exhaust fan are used to deal with gas and coal dust explosion accidents, and the light disaster relief drill is used to deal with roof accidents. Besides, life detectors, support vehicles of multifunctional rescue equipment, and detection robots have also been developed. The use of advanced equipment has greatly enhanced the ability of ER and improved the survival possibility of trapped miners [22][23][24].
With the development of Internet technology, many new communication technologies have been applied to mine ER such as the video communication between the underground and ground and between the accident scene and ER command departments at all levels [25,26].

Research Methodology
At first, SPN was used to describe and analyze the causal relationship between computer system events. After more than a century of development and improvement, SPN has been extensively used in other fields such as finite state machines, communication protocol, synchronous control, production system, and business process. SPN has the characteristics of graphic visualization and mathematical logic, which can describe highly discrete complex systems [27][28][29][30][31].
3.1. Definition of SPN. SPN can simulate and analyze discrete events of the system during an uncertain time. SPN is a seven-tuple directed graph including place, transition, arc, and firing rate [32].

Geofluids
Events t i , t j ∈ T accept the weight on mark M, resulting in a concurrency relationship, the satisfied relations are as follows: as shown in Figure 2. Events t i , t j ∈ T have a conflicted relationship on mark M [33]. That is,t i ≠ t j ∧ M½t j i∧¬M½ft i , t j gi, as shown in Figure 3. Figure 4. fp 1 , p 2 , p 3 , p 4 , p 5 g refers to a set of classical places. ft 2 , t 2 , t 3 g refers to a transition set containing time. For example, t 1 represents the time taken from p 1 to p 2 and p 3 . After t 1 is fired, p 2 generates two tokens and p 3 generates one token. The number marked on the directed arc indicates the weight. If there is no number, the weight is 1.

Model Construction of SPN. The SPN model is shown in
The analysis process of the SPN model is as follows: firstly, give place p 1 a token to fire transition t 1 , so that place p 2 generates two tokens and place p 3 generates one token. Then, it is judged whether the number of tokens in places p 2 and p 3 meets the condition of firing transition t 2 , and if so, place p 4 can be reached. Finally, we judge the condition of firing transition t 3 according to the previous method. If the condition is met, it will reach the termination place p 5 , and the process will end.
The implementation steps of using the SPN model to analyze the performance of the ER process for coal mine accidents are as follows [34][35][36]: (1) The systematic SPN model is established to verify the validity of the model. Here, T-invariant method is chosen to realize this process. A linear algebraic matrix equation is constructed to determine the performance of the SPN model where C refers to the incidence matrix, O refers to the postmatrix, I is the prematrix, and Y is a row vector, representing T-invariant (2) According to the initial mark M 0 and firing rate λ, the isomorphic Markov chain (MC) is obtained [37] (3) The MC is studied. Based on the steady-state probability, the system performance is analyzed and evaluated. The steady-state probability in MC is regarded as a row vector X = ðx 1 , x 2 , ⋯, x n Þ, and the linear equations XQ = 0 are listed where Q represents the state transition matrix The linear equations are calculated to obtain the steadystate probability.
For the nondiagonal element q ij , For the diagonal element q ij , where k refers to the number of states connected to M i . ∀p ∈ P, ∀i ∈ N, P½MðpÞ = i is the probability that each p with i marks. The busy probability of place indicator can be calculated using Equation (8). Figure 1: Sequential relationship of an SPN model.

Geofluids
The utilization of a transition indicator refers to the sum of all steady-state probability of firing transition t and is calculated as follows: The token rate of transition indicator represents the average token number of transition flowing into the place p of transition t per unit time. The calculation formula is as follows: Suppose ∀p ∈ P, the average token number N of the system refers to the average value of marks contained by p i in any reachable marking graph. The marked average value of the set p i ∈ P of a place is the sum of the average value of each place.
The average delay time t of the system is the ratio of the average token number N and the token rate Rðt, pÞ of transition.
3.4. The Optimization Strategy Based on Triangular Fuzzy (TF). The TF is a widely used function of fuzzy mathematics, which is mainly suitable for the situation of insufficient field data and low precision of monitoring instruments. The variables are fuzzified to form a trust interval with the upper and lower limits [38,39]. Assume that the fuzzy variable is where e refers to the value of the variable. e − and e + represent the lower and upper limits of fuzzy variables, respectively.
The corresponding triangular membership function is where η E is the membership degree of the fuzzy variable and x is a variable. The triangular membership function is calculated by using X cut set to obtain a fuzzy interval.
Optimization steps of the triangular fuzzy are as follows: (1) The firing rateλ = ðλ 1 , λ 2 ,⋯,λ n Þ in the SPN model is fuzzified according to different proportions [40]. The fuzzy intervals ½λ − i , λ i and ½λ i , λ + i of firing rate λ i are constructed (2) a is valued in the interval ½0, 1, which meets the rule of increasing step by step. For example, a = 0, 0:1, 0:2, ⋯, 1. Substituting different a values into formula (15) to obtain E ðaÞ i . Then, substituting λ ′ = ðλ 1 , λ 2 , The total time of system change under unit transition rate is (4) When δ i is larger than most δ values, it shows that the change of transition has great influences on the average delay time of the system. Therefore, the optimized SPN model can be obtained only by simulating δ i with a large numerical value

Case Study
4.1. Case Introduction. The "9·28" major water inrush accident in the Shanxi Fenxi Zhengsheng Coal Company of China is used as a case to validate the proposed model. In this case, the water inrush occurred in the east flank return airflow roadway to the coal mine dispatching room. Though the leader of underground miners executed emergency evacuations immediately, many miners were still trapped for the far distance between the working face and the shaft. According to the guidance of higher authorities, coal mine leaders executed ER plan, set up an ER headquarter, and launched rescue operations. The ER process for this accident is shown in Figure 5.

Performance Analysis of Water Inrush ER Process
Based on SPN Model. Figure 6 shows the SPN model of the ER process. The SPN model combines all the ER activities and points out the relationship among these activities, as shown in Figure 6. In Figure 6, a place represents the event state and a transition represents the event activity during the ER process. The process includes 25 places and 20 transitions, and their respective definitions are shown in Table 1. T3  T4  T5   P8   P9  T7   P10   P11   P12   P13   T8   T9   T10   T11   P14   P15   P16   P17   T13   T12   P19   P18   T14  P20  T15  P21   T16   P22   T18   T17   P23  P24 T19 T20 P25 T2 T6 Figure 6: The SPN of ER process during coal mine accident.  Figure 5: The ER process of the coal mine.

Results and Discussion
The time parameters that obeyed a certain probability distribution are added in the SPN, and the time of random variables is associated with the transitions. Through the accident investigation, the time consumed in each link of the accident is collected to obtain the average time t of each transition. According to the firing rate formula, the average firing rate of 20 transitions is obtained, and the results are as shown: λ = 1/t = ð1/10, 1/5:1/5, 1/20, 1/150, 1/20/1/60, 1/  Figure 7.
According to formulas (4), (8), and (9), the steady-state probability, the busy rate of the places, and the utilization rate of the transitions are calculated, respectively, as shown in Tables 2-4. In Table 3, the busy rates of places of P9 and P14~P19 are higher, which indicates that the state information of these places cannot be effectively processed and results in the accumulation of information.
The onsite command team (place P9) is composed of coal mine leaders, experts, technicians, rescue team, and medical team, which formulates and adjusts the ER plan according to the onsite situation, and maintains communication with relevant superior departments. Since the experts of research  6 Geofluids institutes and universities were in different cities, they cannot arrive at the accident site in time, which caused the waste of the ER time. Therefore, the emergency center should improve work efficiency and optimize deployment time.
After the accident, the busy rate of the places of the P14 and P16 (coal enterprise) is very high. Through deep analysis, it is found that the number of emergency training is less and the rescue work is not proficient enough for employees in the communication department and the safety supervision department. The mine leaders should pay attention to the daily emergency training and strengthen the emergency response awareness of the miners. S0 S1 S2 S3 S5 S7 S8 S9 S6 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21    The medical rescue team (place P15) includes medical rescue stations of the mining enterprise and medical personnel assisted by other areas. Because the two teams had not trained together before, the cooperation of rescue teams was not good enough, which caused the redundancy of place P15.
The construction team (place P17) mainly includes the rescue team from the provincial mine rescue base and the mine enterprise rescue team. Because many rescue machines are required to be transported, it is very likely that vehicles will be congested and cannot arrive quickly due to the influence of traffic conditions. Places P18 and P19 are in a synchronized state. Due to the uncertainty of the accident location and the influence of the complex underground environment on the equipment such as life detectors, it is difficult to find the location of the trapped miners for a short time. In the early stage of the rescue, the rescue team can only use drainage equipment to drain the floodwater in the roadway. Therefore, we should increase the number of rescue equipment and use more advanced rescue equipment to deal with ER of coal mine accidents.
It can be seen from Table 4 that the utilization rates of transitions T12 and T13 are much higher than that of the remaining transitions, which indicates that the activities they represented spend more time.
T12 indicates the draining in the flooded roadway. During this process, it is difficult to determine the exact position of the trapped miners, so the ER team can only gradually rescue along the rescue roadway based on the known information and experience. In the flooded roadway, life detectors and other equipment are used to detect the vital signs nearby. This work is difficult because it not only needs to clean up the accumulated water and gravel in the roadway but also needs to stabilize the roadway to ensure its safety, which leads to the total time of T12 is very long.
T13 indicates the exploring information underground. In this process, because the communication staff need to follow the construction team to carry out underground detection and feedback the real-time information to the ER headquarters immediately, which also takes a lot of time.

Optimization Analysis of ER Process.
In this process, the SPN model of the emergency rescue process was analyzed, the busy places and high utilization of transitions were identified, and improvement measures were suggested. We changed some transition's time and calculated the corresponding average delay time of the whole ER system.
The method of changing the firing rate of each transition is adopted to optimize the total delay time of the system. When the speed range of the firing rate of each transition is constant, the different decreases of the total delay time have different impacts on the whole ER system. A lot of computations are needed to substitute the firing rate of each transition into the SPN model, so a triangular fuzzy strategy is proposed to optimize the SPN, which greatly simplifies the calculation.
Firstly, 5% fuzzy degree is set for each transition to get the fuzzy interval of each transition according to formula (15). Then, the average delay time of the system corresponding to the transition is calculated in the SPN model. Finally, according to formula (16), the change value δ of the total time of the ER system is calculated when each transition unit changes, and the results are shown in Table 5. Table 5 shows that when the fuzzy degrees of transitions are same, the δ values corresponding to transitions are quite different. The value of δ 12 corresponding to λ 12 is 1265 times of δ 8 corresponding to λ 8 . Therefore, changing the value of λ 12 can reduce the total system time more than changing the value of λ 8 . In addition, the values of λ 13 ,λ 19 ,λ 20 , and λ 5 have the same effect on reducing the total time, so only five parameters need to be changed to optimize the SPN model.
Firstly, it is assumed that only the value of λ 12 is changed and the remaining values are constant. The results are shown in Figure 8. Figure 8 shows that when the value of λ 12 changes from 0 to 0.003, the probability of PðM 34 Þ = ðP15, P16, P17, P19Þ is greatly decreased. The probability of PðM 31 Þ = ðP14, P15, P 16, P17Þ is slightly increased when the value of λ 12 changes from 0 to 0.0009 but decrease gradually when the value of λ 12 changes from 0.0009 to 0.003.
With the advancing speed of the rescue roadway in the mining area increases, the communication department will find the location of trapped miners underground faster in the rescue process and promptly report the rescue progress to the ER headquarters of coal mine. The command team masters the latest rescue situation in order to quickly adjust the plan and deploy rescue measures. Therefore, it is necessary to carry out a reasonable allocation of emergency resources and special drainage equipment so that the medical rescue team, the safety supervision department, the construction team, and the communication department can quickly put into the ER work. Meanwhile, the safety supervision department, communication department, and medical rescue team from the mine enterprise must strengthen the daily management and emergency training. The emergency center should request nearby hospitals to allocate medical teams for 8 Geofluids support, but it is necessary to ensure that these rescue teams are well trained. Secondly, it is assumed that only the value of λ 13 is changed and the remaining values are constant. The results are shown in Figure 9. Figure 9 shows that when the value of λ 13 changes from 0 to 0.003, the probability of PðM 35 Þ = ðP14, P18Þ is greatly decreased. The probability of P ðM 31 Þ = ðP14, P15, P16, P17 Þ is slightly increased when the value of λ 13 changes from 0 to 0.0009, but gradually decreased when the value of λ 13 changes from 0.0009 to 0.003.
The probability value and variation range of P ðM 31 Þ in Figure 10 is basically consistent with that in Figure 9. This result indicates that if the location information of trapped miners can be grasped as soon as possible, the rescue work of the communications department, rescue teams and medical teams, safety supervision department, and construction team will proceed successfully. With the increase of λ 13 , the probability value of P ðM 35 Þ gradually decreases from 0.86151 to 0.06308. It means that the communications department works more efficiently, and the rescue roadway advances faster and strives for golden rescue time for trapped miners. Therefore, more detection equipment is needed to expand the range of underground information exploration, to grasp the location information of trapped miners and feedback information to the headquarters faster.
Thirdly, it is assumed that only the value of λ 19 is changed and the remaining values are constant. The results are shown in Figure 10. Figure 10 shows that when the value of λ 19 changes from 0 to 0.01, the probability of P ðM 40 Þ = ðP23Þ is greatly decreased. The rescue team encountered obstacles in the process of roadway advancement. To solve this problem, the command team immediately adjusted the plan to increase emergency rescue efforts. The most effective way is to mobilize large rescue equipment to remove obstacles in the roadway, which will shorten the rescue time. Therefore, we need to ensure the supply of large-scale rescue equipment to deal with some serious problems in water inrush accidents. The "expanding ER" should also be added to the emergency plan to improve the efficiency of ER.
Fourthly, it is assumed that only the value of λ 20 is changed and the remaining values are constant. The results are shown in Figure 11. Figure 11 shows that when the value of λ 20 changes from 0 to 0.01, the probability of P ðM 41 Þ = ðP24Þ is greatly decreased. It can be explained that when the rescue speed of the trapped miners is accelerated, the distance between the rescuer and the trapped person is decreased. In this way, if there are some miners in poor health, the rescuer can transport the drugs to the trapped miners by drilling, so that they can get emergency treatment and prevent the deterioration of illness. For this reason, the medical team should prepare all kinds of first-aid medicines in advance to deal with the possible illness of the miners.
Finally, it is assumed that only the value of λ 5 is changed and the remaining values are constant. The results are shown in Figure 12.    Figure 12 shows that when the value of λ 5 changes from 0 to 0.01, the probability of P ðM 7 Þ = ðP5, P6, P7, P9Þ is greatly decreased. It can be explained that if the speeds of the experts, rescue teams, medical teams, and technicians arrive at the accident site are larger, the on-site command team can be set up more quickly. However, because most of the experts are distributed in universities and research institutes, it takes the longest time for them to arrive at the coal mine, which has a great influence on the establishment of the onsite command team. Therefore, the government should provide an emergency green channel for experts to ensure that they can arrive at the accident site faster.

Geofluids
Above all, in the process of ER, coal mining enterprises should be equipped with large rescue equipment to deal with emergencies. When multicooperation is carried out for ER, it is necessary to optimize resource allocation, coordinate internal and external relations, enhance the awareness of cooperation, and organize daily emergency training, which will be better to carry out ER operations.

Conclusions and Further Work
6.1. Conclusions. In this paper, a new method is proposed to model and simulate the coal mine ER process based on modified SPN. To validate the correctness of this method, the SPN method is applied to the major water inrush accident in Shanxi Fenxi Zhengsheng Coal Company.
(1) The theoretical method of SPN is adopted to establish the ER process of coal mine and found that busy rates of places P9 and P14~P19 are higher and the utilization rates of translations T12 and T13 are higher. In addition, the problems existing in these places and transitions are analyzed (2) The triangular fuzzy strategy was introduced into the SPN. The change value of the total time of the ER system is analyzed when each transition changes, which illustrates that the key activities affecting the rescue time (3) In order to better carry out the ER process, it is suggested that larger rescue equipment should be added, team training should be organized regularly, cooperation should be strengthened, and green channels should be provided for experts

Further Work
(1) The ER of the coal mine is a very complex process, which can divide many links in more detail, such as organizational structure and emergency supplies. Through this division, a more complete ER process and SPN model can be constructed (2) When the SPN model is analyzed, it is assumed that the SPN model obeys an exponential distribution. However, this assumption may have some differences with the actual conditions. Therefore, more in-depth researches should be conducted to help us choose a more appropriate probability distribution (3) Visualization software should be developed for coal mines, which can show the ER process more vividly and give more supports on the ER work using information technology

Data Availability
All data, models, or code generated or used during the study are available from the corresponding author by request.

Conflicts of Interest
The authors declare no conflicts of interest.