Conveyor Model with Input and Output Accumulating Bunker

In this article, a model of a conveyor-type transport system with an input and output bunker is developed. The transport conveyor is presented in the form of a dynamic distributed system. It is shown that the material flow is proportional to the linear density of material distribution along the transport route. The coefficient of proportionality is the speed of the belt. When constructing the model, the assumption of the absence of oscillatory processes associated with the tension of the conveyor belt is introduced, which corresponds to the case when the function determining the speed of the belt is only a function of time. A solution is given, that determines the state of the flow parameters of the conveyor section for a given point of the transport route at an arbitrary point in time. It is shown that the state of the flow parameters for an arbitrary place in the transport route is determined by the state of the flow parameters at the input of the conveyor section, considering the transport delay. An expression is written that allows to calculate the amount of transport delay. The relationship of the transport delay value with the algorithm for controlling the conveyor belt speed is demonstrated. A system of equations for the model of a conveyor-type transport system with an input and output bunker is obtained. The behavior of the model for several characteristic cases of the functioning of the transport system is analyzed. The constructed model of the control object can be used to design highly efficient control systems for the flow parameters of the transport system with an input and an output bunker.


INTRODUCTION
A characteristic feature of Industry 4.0 is fully automated production. The conveyor belt is an integral part of an intelligent industrial automation system. Smart modern factories create transport systems that can improve the efficiency of production process control. The conveyor is the link between the production modules. The modern conveyor includes various intelligent control components and sensors to increase the efficiency of using [1]. Industrial Internet of Things opens a direct path to the creation of fully automated industries. The industrial Internet of things opens a direct path to the creation of fully automated industries, in which conveyor systems play an important role. But that is not enough. For optimal control of the flow parameters of the transport system, it is necessary to build more and more efficient models that allow us to describe the transport system as an object of intelligent control. Of particular importance within the framework of the Industry 4.0 concept is the conveyor transport in the mining industry [2]. There are special reasons for this. A conveyor belt is a universal means of transporting bulk materials due to the low unit cost of the material [3,4]. For normative mode the average share of the cost of transporting a unit mass of material is 20% of the total cost of coal mining [5] and depends on the length of the transport route. For long conveyors, the cost of transportation increases in proportion to the length of the conveyor. Energy costs are especially noticeable for long multi-sectional transport systems and branched transport systems, the characteristics of some of them are given in Table 1. The length of the transport route exceeds 100 km [6] and continues to increase. To increase reliability, the transport route is divided into sections. The technology for transporting material provides a section length of up to 20 km.
One of the problems with using a conveyor is that often the conveyor is loaded below the nominal value. In this case, the cost of transporting the material can increase several times. Throughput control of the conveyor transport system is reduced to controlling the intensity of the flow of the material at the input of the section [14,15] or to controlling the speed of the conveyor belt [16,17]. Dividing the transport route into sections allows providing in the transport system a different speed of conveyor belt for each section. In the general case, dividing the transport system into sections allows to use your own flow control algorithm for a separate section. The presence of an accumulating bunker at the inlet and outlet of the conveyor section makes it possible to stop the movement of the belt for maintenance work. In this case, the incoming material from the previous section is accumulated in the bunker. There is no feed to the conveyor belt. To ensure the continuous operation of the transport system, the material is fed from the output bunker of the stopped section to the input of the next section, the belt of which is in motion. When designing a control system for flow parameters of a conveyor line, limitations should be taken into account. The specific linear density of the material, which is located on the conveyor belt, is limited by the maximum permissible value [19]. Absence of consideration of this factor leads to disruption of continuous operation of the conveyor line as a result of the conveyor belt rupture, Fig. 1. The total power of the section engines used to move the material is a limitation on the total rock mass within the conveyor section [19]. A significant limitation that affects the choice of algorithm for controlling the flow parameters of the transport system is the limitation on the maximum capacity of accumulating bunkers for storing material before and after the conveyor section [15,16]. Conveyor line accident [18] The belt conveyor is a distributed system. The material entering the input of the conveyor for long transport systems (Table I) is in the process of transportation for several hours. To ensure a time-specified value of the output flow of the material, which is required to ensure a continuous production process at the processing plant, effective algorithms are needed to control the state of the material in input and output bunkers. An increased interest in the problem of designing transport system control systems [20,21], which has an input and output bunker, is associated with a tendency to increase the material transportation route and the throughput of the transport system [22][23][24]. To solve this problem, it is necessary to develop new, more efficient models of the transport system and improve existing models. The relevance of this problem and the growing interest in its solution determined the purpose of this article.

II. BACKGROUNG
The main problems that arise when designing conveyor-type transport systems for mining enterprises are that the productivity of a coal seam varies over time due to a large number of technological and geological factors. In this regard, the throughput of the transport system should be designed for both short-term fluctuations flow parameters and long-term ones, due to depletion or saturation of the rock formation with the required material. In [15], a model of the transport system of a mining enterprise is presented. For each section of the conveyor containing input and output accumulating bunkers, equations of levels are written (system dynamics model). The proposed model was used to build policies to control the rock mass in the bunker. Recommendations for the designed values of the maximum capacity of bunkers of the conveyor section are presented.
The most widely used for developing models of conveyor systems are FEM models (finite element method) [3,[25][26][27], FDM models (finite difference method) [22], DEM models (discrete element method) [5]. These models, belonging to the class of numerical models, allowed researchers to take into account the distribution of material along the route of the transport, and, accordingly, the transport delay of the transport system.
As an alternative to numerical models for the estimation calculation, aggregated models of the conveyor system are proposed [28], which consider the values of the flow parameters averaged over a typical period of time. A separate class is models using artificial intelligence: the equations of multiple regressions [29,30] and the equations of the neural network [27,31,32]. To build models require a large amount of test data, which is difficult to obtain for non-stationary modes of functioning of the transport conveyor system. It limits the use of this class of models.
Among the articles discussed above, models of a transport system with accumulating bunkers are considered in the papers [15,33]. An analytical model of the conveyor section was proposed in [20,24]. In this article, we consider a model of a conveyor section with input and output bunker. The model is built to design a control system for the conveyor section, in which the flow of material from the accumulating bunker and the speed of the conveyor belt can act as control parameters.

III. CONVEYOR SECTION MODEL WITH ONE INPUT BUNKER
For the building a model of the section with two bunkers, we use the model of the conveyor section with one input bunker [7]: is the linear density of the material; λ were studied in [15].

IV. CONVEYOR SECTION MODEL WITH INPUT AND OUTPUT BUNKER
We supplement the system of equations (1) -(5) with an equation simulating the operation of the bunker at the output from the conveyor section, we obtain a system of equations that simulates the behavior of the flow parameters of the conveyor section with two accumulating bunker (input and output): λ is the intensity of the flow of material from the output bunker, which is determined by the need for production.

V. DIMENSIONLESS CONVEYOR SECTION MODEL WITH INPUT AND OUTPUT BUNKER
We use dimensionless parameters [7]: i max , present the system of equations (6) in a dimensionless form: where [ ] 0max χ is maximum allowable linear density of the material. The solution of equation (8) by the method of characteristics is presented in [20] and has the form: A schematic diagram of a conveyor line, which is described by equations (8) - (10), is shown in Fig. 2. The equation (12) gives the duration of the transition period tr τ , when the linear density of the material at the exit from the transport system is determined through the distribution ) (ξ ψ at the initial time. Considering (11), (12) we will present a system of equations in the form: The system of equations (13) is a dimensionless model of the section with input and output accumulating bunkers (Fig. 2).

VI. RESULTS
For a constant belt speed 0 ) ( g g = τ the values of the flow parameters at the output from the conveyor can be expressed in terms of the values of the flow parameters at the input to the conveyor line and the initial distribution ( ) ξ ψ is described by equations (14).
In the case tr τ τ < the system of equations (14) is divided into separate independent equations.  (14) Each bunker has his control, and the bunker controls are independent. In the case tr τ τ ≥ the conveyor section model is described , which provides a stationary state in which the rock mass in the bunker remains constant, has the form: In conclusion of the analysis, we consider the model of the transport system with 0 ) ( g g = τ and Material from the input bunker enters the output bunker after a period 0 1 . The duration of the transition period is 0 / 1 g tr = τ . The system of equations (13) in this case is simplified: in tr Control the intensity of the output flow ) ( 2 τ γ , which provides a stationary state has the view: The transport system does not exclude the presence of a situation in which overflow of the input bunker is possible.    The control switching point can also be selected in end states. These states correspond to an empty bunker and a fully filled bunker (Fig. 5). In this case, control can be selected from an admissible set of solutions  It should be noted that the switching point can be selected for an arbitrary value of the amount of material in the bunker 1 n ( ) τ . The transition trajectory can also be represented by a monotonously decreasing (increasing) curve, which will provide a transition from one admissible phase trajectory to another admissible phase trajectory.

VII. CONCLUSIONS
The basic result of the article is that a model of the conveyor section with two accumulating bunkers was built.
The scientific novelty of the results is that for the first time an analytical model of the conveyor with two bunkers is presented, which takes into account the time-varying transport delay. Using the developed model of the control object, optimal control of the flow parameters of the conveyor-type transport system with an input and an output bunker can be synthesized. The dependence of the flow parameters at the output from the conveyor section on the initial distribution of material along the route is shown. The duration of the transition mode for the conveyor section is determined. The next step in the development of the issue discussed in the article is the design of a control system for the conveyor section with input and output bunkers, which would allow taking into account restrictions on the phase coordinates and transport delay in phase coordinates.