COMPREHENSIVE ASSESSMENT OF THE RELIABILITY OF A NON-REDUNDANT SYSTEM OF BIOGAS PLANTS BASED ON MARKOV PROCESS MODELS

ABSTRACT

This article describes ways to ensure efficient and uninterrupted operation of biogas plants that receive energy and biofertilizers in climatic conditions of Uzbekistan. Mathematical statistical methods, profitability assessment, functional integrity scheme and least reduction method, factor criticality assessment methodology were used in the processing of available data for feasibility study. The main result is a worldwide trend on this issue and a comparison of the results of different authors.

In order to verify our calculations, the results of analysis and monitoring of biogas plants, their failures and shutdowns were studied.

АННОТАЦИЯ

В данной статье описаны способы обеспечения эффективной и бесперебойной работы биогазовых установок, получающих энергию и биоудобрения в климатических условиях Узбекистана.

При обработке имеющихся данных для технико-экономического обоснования использовались методы математической статистики, оценки рентабельности, схемы функциональной целостности и метода наименьшего сокращения, методологии оценки факторной критичности. Основным результатом является мировая тенденция по данному вопросу и сравнение результатов разных авторов. Основным результатом является мировая тенденция по данному вопросу и сравнение результатов разных авторов.

Keywords: reliability indicators, forecasting techniques, biogas plant, faults and failures, diagnostics and fault finding

Ключевые слова: показатели надежности, методы прогнозирования, биогазовая установка, неисправности и отказы, диагностика и дефектация.

We describe a model that prevents unpleasant outages in the use of renewable energy-based biogas plants using the flow of events. In this case, from the starting point of the events, the object performs its work until the first stop, after the stop the object's ability to work is restored, the work continues until the next stop, and these cases are returned. Consider a system with i (i =1, …n) sets of discrete states and operating in a continuous time interval.

In this case , S ( t ) = i – the state of the system at time point t is equal to i ( 0 ) . ) = from state i in the time interval t S ( t + Д t ) = j (0 ) to the state t + Д in the time interval t , seeing the probability of the conditional transition of the system S ( t ) = i in the time interval S ( t + _Д t ) = j ( we analyze the transition rates to the state t + D t in the time interval t + D t. It is known that P _ij ( t + D t ) (0 transition probabilities do not depend on the behavior of the system until the time point t), we can see such a process in the example of a Markov process.

If P _ij ( t + D t )= P _ij ( D t ) = l _ij D t transition probabilities do not depend on t, the process taking place in the biogas extraction device is considered as a Markov process (one type) process.

For this process, the system S ( t )= i (0 We assume that the period in the state obeys an exponential distribution and that the rates of the transitions obey the following conditions:

,                                                                                ( 1)

= ,                                                                           ( 2)

where, - i (speed of keeping in state 0 .

(0 biogas systems in state i are represented by the following differential equations :

                                                     (3)

where, - i (speed of keeping in state 0

- initial conditions , .                                             (4)

This system of differential equations can be converted into a system of Kolmogorov equations. So, we can use different Markov process models to estimate the reliability indicators of spared and non-reserved systems (repairable and non-repairable links) of the power plant.

A non-reservable system can be in the following states at any time interval t : 0 - in an operational system; 1 will be in a non-functional and under-repair system.

We denote the probability of these cases as P ₀ ( t ) and P ₁ ( t ).

In this case, A (t) = P ₀ (t), U ( t ) = P ₁ ( t ). In the case of long-term use ( t → ∞), readiness A= R ₀ and persistence -U= R ₁ coefficients are considered to reach the specified values .

In our calculation books, we used the conditions for writing the following differential equations [6A; 5-b], where: we listed the states in which the devices can stop during their operation, and a mathematical model in the form of a state diagram in which states are represented by rectangles or circles and lines reflecting the transitions we made it. It is assumed that only one stop or one recovery is possible in one time frame.

Depending on the state scheme, we can construct a system of differential equations in the following order:

• the left parts of the equations contain the time derivatives of the respective states and are obtained by multiplying the transition intensity of each term on the right side of the equation by the corresponding probability of the above state on the axis associated with that state;
• the sign on the right of each multiplier depends on the direction of the pointer (if the pointer points to the position and vice versa);
• the number of equations is equal to the number of cases; the system of differential equations must be filled with the norm condition, in which the sum of the probabilities of all cases is equal to one.

First, the system of differential equations is solved by the Laplace formula or another method, which allows determining the required reliability indicators [1,6,7].

So, we present the following system of differential equations for a non-restorable joint:

                                                               ( 5)

Expressions for stationary readiness and unreadiness coefficients can be written based on the following rule depending on the state scheme [ 107; 194-207-b ]:

stationary probability P _{k of finding the system in the k-state, it is} necessary to travel the shortest path in the direction of indicators from each extreme state to the k-state and multiply all transition intensities corresponding to the previous indicators.

Thus, all paths from all extreme states to each state of the system are explored. In rare cases, it is recommended to pass some paths in the system several times. In this case, it is necessary to take into account the transition speeds of these parts. The probability that the system is in state _k can be determined as follows:

,                                                                          ( 6)

where, D _k , D _j - the intensity product of transitions from all limit states to states k and j with the shortest path depending on the direction of the indicators; m+1 is the number of system states.

In determining the stationary probability, this algorithm can be used for various types of reserves, especially in the case of reduced conditions, as well as in cases where the number of repair groups r is m >r> 1. Taking into account the above, when moving from state 1 to state 0, the transition intensity is equal to m, and when moving from state 0 to state 1, it is equal to 1-l. So,

P ₀ ( t ) = ; P ₁ ( t ) = .                                                   ( 7)

Given that λ = 1/ T ₀ , m =1/ T _{V ,} for the law of exponential distribution of time , then:

                                                   ( 8)

The analysis of the formula 3.11 shows that the availability coefficient is the fraction of time the system is operational, and the persistence coefficient is the fraction of time spent on repair.

We can assume a linear Markov process for a non-reserved renewable system of biogas plants (Fig. 1), in which we can draw a diagram of the current states of the system and transitions from one state to another.

Figure 1. Schematic of non-reserved system states: here, state numbers 0 and 1; On m and l-indicators – transition speeds

where, l-stop frequency; m- recovery frequency

where the constants λ mean the speed of system shutdowns, µ means the speed of recovery.

Based on the Matlab/Simulink program, we solve the following system and create its visual model (Fig. 3.6), where λ = 0.0001(1\s); µ =1(1\s). We can see the model results for different parameters λ and µ in Figure 2 and Table 1.

Table 1.

Reliability assessment for a non-redundant renewable system

Here: A is ready, U is not ready

Figure 2 Visual solution model for a non-reserved non-recovery system

From the visual model shown in Figure 3, we can see that when the recovery frequency in the range of 0.0001 -1 exceeds the stop frequency in the range of 0.0001 -1, the device is found to have continuous operation, otherwise, stoppage occurs. We can see this through the graph. (Figure 3)

Figure 3 Visual solution model for a non-reserved non-recovery system 3 Model graph for a non-reserved non-regenerative system at the stop frequency l- based on Markov processes

3-4 show the dimensions of the model for different values of parameters λ and µ

Figure 4. Model graph for non-reserved non-renewable system with m-recovery frequency based on Markov processes

Reducing the downtime of the biogas plant (increasing the quality of the elements used, applying the best design solutions, improving the system of protection against external influences, encouraging the operator's work, etc.) increases the readiness coefficient and reduces the downtime coefficient.

Backing up systems and joints in biogas plants leads to increased reliable performance.

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