Belt conveyor systems play a key role in ensuring the continuous transportation of bulk materials in mining enterprises, both in open and underground conditions. One of the urgent tasks is to increase energy efficiency and material loading factor, which allows to reduce operating costs and meet modern environmental requirements. In this paper, the behavior of the transport system with a stochastic nature of the input material flow is investigated. A mathematical model describing the movement of material along the conveyor route is developed, taking into account the statistical characteristics of the input flow, modeled as a stationary random process. A review of modern approaches to transport system control is given, including belt speed regulation, flow control from accumulation bins, and the use of reversible conveyors for route optimization. Particular attention is paid to the influence of the randomness of the input flow on the system performance parameters. The obtained results allow us to identify dependencies between the stochastic characteristics of the input flow and key parameters of the system, which contributes to the development of more efficient and adaptive conveyor transport control strategies.
2 Wroclaw Universi
Belt conveyors play a key role in ensuring continuous transportation of bulk materials and are an integral part of technological processes in the mining industry [1, 2]. Their widespread use is due to their ability to effectively move large volumes of rock and minerals over significant distances in open pits and underground mines [3, 4]. In modern operating conditions, increased demands are placed on the reliability, durability, and energy efficiency of such systems [5]. A significant share of operating costs is electricity consumption, which is especially relevant against the current trend of rising energy prices and tightening environmental standards [6]. In addition, in modern production environments, there is increasing variability and instability in input material flows, which requires that stochastic factors be taken into account when designing and operating conveyor systems [7]. Ignoring these factors can lead to significant deviations in belt loading and a decrease in the overall efficiency of the transport process. One of the urgent problems is to increase the material loading factor of the transport conveyor, which contributes to a more rational use of power and resources [8 ,9]. In this regard, this study aims to model the movement of material along a transport conveyor with a stochastic nature of the incoming flow, to identify patterns affecting the load factor and develop approaches to optimizing the operation of the transport system.
An increase in the material loading coefficient of a belt conveyor can be achieved by eliminating the unevenness of the material flow along the conveying line [10]. For this purpose, various approaches to the control of the bulk material feeding and conveying system are considered in the literature [11].
The first approach is to control the input flow of material from the accumulation bunker [12]. At a fixed belt speed, a flow is formed that ensures an optimal value of the loading factor [13, 14]. This method allows achieving a uniform distribution of material along the length of the conveyor and reducing power losses [15, 16].
The second approach is related to the regulation of the belt speed [17, 18]. In these papers, the linear density of the material on the belt is achieved by proportionally changing the speed depending on the size of the input flow [19, 20]. This approach allows adaptive maintenance of target load with variable input material flow.
The third direction is the application of energy management methodology [21, 22], which involves operating the conveyor system during hours with the minimum cost of electricity. This allows for a reduction in total energy costs while maintaining the required productivity.
Another important area of research is the optimization of material transportation routes [23]. The use of reversible conveyors makes it possible to flexibly redirect flows, taking into account the current load and energy conditions, increasing the efficiency of the entire system. A characteristic feature of most of the listed studies is the assumption of determinism of the input material flow, which simplifies modelling but does not always reflect real conditions. At the same time, some works present experimental data indicating the stochastic nature of the input material flow [24, 25]. The analysis of statistical characteristics and modelling of a random input flow are considered [26, 27], in particular, in the papers [29, 30], where the influence of fluctuations and irregularities in input material flow values on the operating parameters of the transport system is shown. This study examines the movement of material under stochastic input flow and analyses the influence of its statistical characteristics on the flow parameters of the system.
Despite the wide range of models of material movement along a transport conveyor, existing models are focused on taking into account the stochastic nature of real input flows of conveyors or on taking into account the dynamics of flow parameters of a transport conveyor. Paper [9] proposes a belt speed control model that adapts to changes in material flow, but their deterministic assumptions limit applicability under conditions of stochastic input material flow. The study [28] focuses on modeling stochastic input material flow without interrelating with the dynamic characteristics of the conveyor. The assumption of unlimited size of the input bins implies the absence of input material spillage. The presence of spillage leads to the fact that for model [9] and model [28] the dynamics of the material flow along the conveyor can be significantly changed. Neural network-based models such as [20] require extensive training data, which assumes stable operating conditions of the conveyor system.
In contrast, the model proposed in this paper integrates a canonical decomposition of stochastic input, considers probability-based spillage estimation, and introduces dimensionless inter-conveyor dynamics. This enables a more accurate prediction of operational behavior under variability, supporting adaptive control strategies that are grounded in rigorous mathematical representation.
In the conditions of the modern mining industry, an important problem is to increase the efficiency of transport systems based on belt conveyors. A significant role in this is played by the nature of the input flow of material, which in real conditions is subject to stochastic fluctuations due to uneven extraction and supply from accumulating bunkers. This study is aimed at constructing a model of a conveyor transport system taking into account the stochastic nature of the input material flow. The purpose of the modelling is to analyze the influence of statistical characteristics of the input flow on the flow parameters of the transport system. The following assumptions were used in the modelling: a) the flow of material coming from the place of material extraction to the input of each of the conveyors is stationary and has the same statistical characteristics; b) the transport scheme considered in [23] is used as the basic configuration for the analysis; c) In the qualitative analysis of the movement of material along the transportation route, a zero approximation was used for the input flow of material; d) the speed of the belt for each conveyor is constant.
The model of a conveyor with a constant belt speed, studied in detail in [15], can be represented by
a partial differential equation of the following form:
χ0(t, S)+ χ1(t, S) = (S ) (t), χ1(t, S) = a(t) χ0(t, S) , χ0( 0,S) = H (S ) (S ), (
To describe the flow of parameters, dimensionless parameters are introduced:
= t , = S , g( ) = a(t) Td , ( ) = Sd (S ), H (Sd ) = H (S ), (
Td Sd Sd 0 ( , ) = χ0 (t,S) , ( ) = χ0max (S ) , 1( , ) = χ1(t,S) Td = 0 ( , )g( ), χ0max χ0max Sd ( ) = (t) Td ,
χ0max Sd
where Td is the characteristic time for the material to transport the length of the route; χ0max is
the maximum permissible linear density of the material for the conveyor. Considering the
dimensionless parameters (
The equation (
( − ( ))
0 ( , ) = H ( ) − H − 0 g(z)dz g( − ( )) + H − 0 g(z)dz) − 0 g(z)dz),
G( ) = g( z)dz, ( ) = − G−1(G( ) − ),
0
(
g0 1( ,1) = 1 − H (1 − g0 ) ( − 1/ g0 ) + g0H (1 − g0 ) (1 − g0 )), 1( ) = 1/ g0.
G−1(G( ) − 1) = − 1/ g0,
(
0 ( ,1) = ( −g10/ g0 ) , 1( ,1) = ( − 1/ g0 ), g0 1. (
To model a transport system consisting of m-separate conveyors, dimensionless parameters are introduced: = t , m = Sm , gm ( ) = am (t) Td , ( ) = Sd (S ), H (Sd ) = H (S ), m = 1..M ,
Td Sd Sd
0m ( , ) = χ0m (t,S ) , m ( ) = m (S) , 1m ( , ) = 0m ( , )gm ( ), m ( ) = m (t) Td .
χ0max χ0max χ0max Sd
(
As characteristic values Sd , Td , χ0 max of these parameters of one of the conveyors
(characteristic length of the conveyor; characteristic time of transportation of material along the
conveyor) of the transport system are used. Another option for choosing values Sd , Td is the
maximum length of material transportation along the route of the transport system and the
characteristic time of material transportation along this route. With this choice of parameters Sd ,
Td average conveyor length can be estimated as max ( m ) ~ 1/ M . With a large number of
conveyors in the transport system (M 1) , the value max ( m ) becomes much less than 1, which
affects the scale of the values of the flow parameters of the transport system. Taking these
considerations into account, as characteristic values Sd , Td , χ0 max , let's select the characteristic
values of one of the basic conveyors of the transport system Sd = Sb , Td = Tb , χ0 max = χ0 max b ,
m = b . When choosing such a basic conveyor, preferences may be given to the conveyor with the
maximum material loading coefficient. Since the choice of Sd , Td is arbitrary, then at a constant
base conveyor belt speed am (t) = ab , the values of Sd , Td are determined from the condition
1 = TSbb ab. (
If the maximum permissible value of the linear density has the same value for each conveyor of the transport system
χ0 max m = χ0 max b , (
The dimensionless function m ( ) is the ratio of the value of the input flow of material entering the input of the m − th conveyor to the maximum allowable value of the material flow for the base conveyor 1b max = χ0max ab for a dimensionless moment in time : m (t) = m (t) Td = m ( ). χ0max ab χ0max Sd (15)
For a constant belt speed of each conveyor, the dimensionless quantity gm is considered as the ratio of the belt speed am for m − th conveyor to the belt speed ab for the base conveyor: aamb = am TSbb ab TSbb = am TSbb = gm. (16)
Thus, the introduced dimensionless parameters gm , 0m ( , ) , m ( ) have a strictly defined physical meaning and are convenient dimensionless parameters for modeling the movement of material in a transport system consisting of a large number of conveyors. The dimensionless parameter 1m ( , ) is expressed through the introduced dimensionless parameters 0m ( , ) , gm ( ) . The physical meaning of the dimensionless parameter 1m ( , ) is defined as the ratio of the material flow χ1m (t,S ) at a certain point in time and point of the transportation route to the maximum permissible flow χ1max b = χ0max ab in the base conveyor:
χχ10mma(xt,aSb) = χ0χm0(mta,xSa)bam = χ0χm0 m(ta,xS) aamb = 0m ( , )gm ( ) =1m ( , ). (17)
Taking into account the introduced parameters and their physical meaning, dimensionless
parameters 1m ( , ) , gm ( ) , m ( ) for modeling the transport system are used, taking into
account the constraint (
gm ( ) The input and output flow of material is related by the ratio 1m ( ,d m ) = 1 − H (d m − gm ) m − gdmm + gmH (d m − gm ) (d m − gm )), d m = SSdbm , (19) where Sd m the length of m − th the conveyor. For the steady-state operating mode of the transport system, which is of both theoretical and practical interest, the output flow of material is determined by the following relationship
1m ( , d m ) = m ( − d m / gm ), d m − gm 0. (20)
For two successively located (m − 1) − th and m − th conveyors without an accumulation bunker between them, the output flow of the (m − 1) − th conveyor is equal to the input flow of the m − th conveyor 1(m-1)( , d (m−1)) = 1m ( ,0) = m ( ). (21)
For certainty, the parameter m ( ) will denote only the input flow entering the transport system. For the flow of input material m − th conveyor, which is equal to the output flow of the (m − 1) − th conveyor, the notation 1m ( ,0) is used, hereby emphasizing that the value of the material flow for the m − th conveyor is the result of the operation of the (m − 1) − th conveyor. Expressions (20), (21) allow us to calculate the output flow of material from the transport system with known values of the input flow of material of the transport system.
The flow of material entering the input of the transport system can be represented on the considered interval [0; pr ] in the form of a canonical decomposition [29]
зк 1, if i = , m ( ) = m0 ( ) + i=0 mi i ( ) , M mi = 0 , 0 i ( ) ( )d = i , i = 0, if i , (22) where mi are uncorrelated centered random variables with mathematical expectation equal to zero; i ( ) are non-random coordinate orthogonal functions of the expansion of a random process m ( ) on the interval [0; pr ] . The non-random function m0 ( ) is the mathematical expectation mm ( ) = M ( ) of the stochastic flow of material m ( ). Centered random variables mi are the coefficients of the expansion of the stochastic flow of material m ( ) into coordinate functions i ( ) . The canonical decomposition of a stochastic material flow m ( ) is in general, an infinite series that can be bounded with a given degree of accuracy by a finite sum of terms [29]. The statistical characteristics of the stochastic input flow of material m ( ), entering the transport system are defined as follows
mm ( ) = M m ( ) = M m0 ( ) + mi i ( ) = M m0 ( )+ M mi i ( ) = m0 ( ), i=0 i=0 m2 ( ) = M m2 ( )= M mii ( )m ( ) = i2 ( )M m2i = i2 ( ) m2i, i=0 =0 i=0 i=0
km ( ) = M m ( ) m ( + ) = M mii ( )m ( + ) =
i=0 =0 = i ( ) i ( + )M m2i = i ( ) i ( + ) , i =0 i =0 (23) (24) (25) where mm ( ) , m ( ) , km ( ) are the mathematical expectation, the mean square deviation and the correlation function of the stochastic input flow for the m − th conveyor, respectively. Using expressions (23),(24),(25) for calculating the characteristics of the stochastic input flow of material m ( ) , entering the transport system, as well as coefficients (20), (21), it is possible to calculate the statistical characteristics of the input flow of material for the subsequent conveyor of the transport system.
In the absence of accumulation bunker between conveyors in the transport system, material spillage occurs [12] due to non-fulfilment of the restriction (27). Material spillage will occur if the value of the material flow entering the conveyor input exceeds a certain critical value 1m ( ,0) 1m cr ( ) = gm ( ) . This critical value 1m cr ( ) determines the minimum conveyor belt speed at which the amount of spilt material per unit of time is limited to specified values. The probability that the material flow value will exceed the critical value is x P(1m ( ,0) 1m cr ( )) = 1.0 − Fm ( ,1m cr ( )) , Fm ( , x) = fm ( , )d , (26) 0 where the function Fm ( , x) = P(X ( ) x) is a one-dimensional distribution law of a random process X ( ) = 1m ( ,0) , characterizing the value of the input flow of material 1m ( ,0) at the moment of time . The function Fm ( , x) depends on two arguments: firstly, on the value , for which the cross-section is taken; secondly, on the critical value x = 1m cr ( ) , when exceeded by the value of the stochastic material flow X ( ) = 1m ( ,0) , material spillage occurs past the conveyor belt. The statistical characteristics of the flow of material fall 1m ( ) = max (1m ( ,0) −1m cr ( ), 0) , spilling past the conveyor, are as follows: M fall 1m ( ) = 1m cr ( ) 0 fm ( , x)dx + 1m cr ( ) (x −1m cr ( )) fm ( , x)dx = (x −1m cr ( )) fm ( , x)dx, (27) M f2all 1m ( ) =
Expressions (27), (28) make it possible to calculate the critical value 1m cr ( ) for the stochastic flow of material 1m ( ,0) . For a conveyor with a constant belt speed, the critical value 1m cr 0 is a constant value for m − th conveyor. As one of the methods for calculating the critical value 1m cr 0 the condition can be adopted under which the average flow of material that spills at the conveyor inlet is limited by the maximum permissible value fall 1m 0 (27). Then the constant speed of the conveyor belt is calculated from condition (18): 1m cr 0 1. (29)
gm0
Equations (28), (29) together with equations (20), (21), (22) form a model of a transport system with a stochastic input flow of material and a constant conveyor belt speed.
When modelling the transport system, a typical material transportation route is considered, similar
copper [23]. The test transport system is represented by 7 conveyors, in which conveyor 3(
For the input flow of material [31] 2 = M ( (t) − m )2 = 1.367 kg/s, incoming at the input of a conveyor of length Sd = 7.0 m at a belt speed of a = 1.0 m/s and a planned material loading factor of the belt equal to 0.795, expressions for calculating the dimensionless parameters are obtained: ( ) = (t) = (t) (t) , = t = t = t = t . (30) χ0max ab 8.5111.0 / 0.795 10.7 Td Sd / a 7.0 /1.0 7.0
with statistical characteristics m = M (t) = 8.911 kg/s, histogram of the distribution of values of the input material flow.
The dimensionless input material flow with statistical characteristics given in Table 1 is the
ratio of the input material flow to the maximum allowable material flow at the belt speed
characteristic of the conveyor system ab . The characteristic time Td is selected from condition
(
a) N = 0;
b) N = 1; c) N = 2;
d) N = 4; e) N = 32; f) N = 64;
The dimensionless realization of the approximated material flow a ( ) for different numbers of terms of the approximating expression (31) is shown in Figure 4. The statistical characteristics of the input material flow ( ) are reflected in the graphs presented in Figure 5. The correlation function k ( ) decreases exponentially with increasing correlation time , and asymptotically tends to zero with a characteristic correlation time cor , much less than the characteristic time of material transportation along the conveyor Td , cor 0.01 Td , Figure 5a. The presented time dependence between the values of the stochastic material flow in different sections of the random process is typical for stationary processes. The relationship between material flow values separated by a time interval weakens with increasing correlation time . The density distribution of the dimensionless input material flow values (Figure 3b) corresponds to a distribution law close to the one-dimensional normal distribution law, which is confirmed by the Q-Q plot of the material flow values ( ) (Figure 5b).
a) correlation function k ( ) ;
The fact that a normal distribution law can approximate the distribution of material flow values ( ) makes it possible to fairly easily calculate the critical value 1m cr ( ) , above which the probability of no material spillage is no more than a given value, for example, 0.05. For the standard normal distribution at (z) = 0.95 , z~1.65 the critical value is given by 11= cr = M ( ) + z = 0.7956 + 1.6449 0.1276 = 1.0055 1. (32)
The numerical solution for the dimensionless material flow ( ) is presented in Figure 6 and gives a close result 11= cr = 1.0088 , which confirms the assumption about the distribution law of the values of the dimensionless material flow ( ) . For the steady-state operating mode of the transport system, the statistical characteristics of the material flow of each conveyor of the transport system can be expressed through the known statistical characteristics of the input material flow for conveyors 1 and 2 (Table 2): a) at the input to the conveyor m 1 = m 2 , 1 = 2 ; ) at the output from the conveyor m 10 = m 20 , 10 = 20 taking into account the effect of material spillage.
For a qualitative analysis of the features of material movement in the transport system, this paper examines the material flow presented by the approximation of Figure 4-a. For a more indepth analysis of the transport system in subsequent papers, the type of approximating expression will be determined by the required accuracy specified when setting the problem. In the current paper, the approximation used is sufficient to demonstrate the general patterns of movement of the material in the transport system. The length Sd 1 of the first conveyor is taken as a characteristic value. Each of the conveyors of the transport system has a limitation on the belt speed am am max , which is specified by the condition of no spillage of the material (29). For a constant value of material flow, the output material flow for the m − th conveyor can be calculated according to expression (20). The calculation results are presented in Table 3. The transport delay m = d m / gm for each conveyor is determined by the ratio of the conveyor length d m to the conveyor belt speed gm . Dynamics of material flow in a transport system with time-varying material flow values at the input and output of the m − th conveyor, Figure 7a,b. Thus, the conducted numerical analysis confirms the significant influence of the stochastic nature of the input flow on the flow parameters of the transport system. The developed model makes it possible to obtain dependencies between the input parameters and output flow parameters of the transport system, to justify the value of the material loading coefficient of each of the conveyors. The obtained results are the basis for improving the control systems of the flow parameters of the transport conveyor. The mathematical model developed in this study is a further step in the significant improvement of the model [10] and serves as a basis for adaptive control strategies, allowing real-time estimation of material flow parameters and critical threshold values for each conveyor segment. Since the stochastic input flow is represented by a canonical decomposition, the model allows continuous estimation of the probability of the expected input flow exceeding the permissible limit at an arbitrary time step, which allows making predictive control decisions and preventing unplanned conveyor stops due to material spillage.
This study examines the impact of stochastic input flow on belt conveyor performance, which is particularly relevant under variable load conditions typical of the mining industry. Analysis of existing scientific and technical developments shows that uneven loading significantly affects energy consumption, equipment wear and tear, and overall efficiency of the conveyor system. Modeling the movement of material along the transport line made it possible to identify key patterns of flow distribution and determine the parameters that have the greatest impact on the load factor and energy efficiency. The use of approaches that take into account the random nature of cargo receipt opens up opportunities for more precise adjustment of control systems, as well as the development of adaptive algorithms for regulating the speed and power of drives. The results obtained can be used to optimize the operation of existing transport systems, as well as when designing new lines, especially in conditions of unstable production and limited energy resources. Promising areas for further research include the development of flow parameter control systems for a transport system with a stochastic input flow of material. Of particular interest will be transport systems with material transportation routes that have a variable conveyor structure (reversible conveyor-type transport systems). Numerical simulation showed that the average value of the dimensionless input material flow is m = 0.7956 with a standard deviation of = 0.1276 . The critical flow value, above which material spillage occurs, is 11= cr = 1.0088 , which indicates the need to increase the belt speed by approximately 26% to prevent losses during peak loads. In the conditions of the considered transport system with a reversible conveyor, maintaining a constant belt speed gm = 1.0 ensures stable operation with a load factor close to 0.8. Numerical estimates can be used as a basis for qualitative analysis in the design of adaptive control systems and for the verification of boundary conditions in real time.