Design of Passenger Aerial Ropeway for Urban Environment
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Abstract
Aerial ropeway is an effective alternative to the conventional modes of land public transport in metropolitan areas and cities. Construction of passenger aerial ropeways in urban environment is a very costly enterprise in terms of engineering and economics, and requires significant financial resources. This article is aimed at the development of the design method of the passenger aerial ropeway, ensuring the reduction in its construction cost. For this purpose, the individual components of the construction cost are considered, and the approximate calculation dependencies are proposed. It is shown that the cost of the aerial ropeway is mainly influenced by the installation step, height of intermediate towers and carrying rope tension. The task of the conditional nonlinear optimization of the given parameters is formulated and solved in the research. This task ensures the minimum cost of the aerial ropeway. The optimization task is done by taking into account possible limitations on the ropeway laying in the severely urbanized environment (the terrain, urban infrastructure arrangement, altitude performance of the urban development, technical characteristics of the carrying rope, etc.). Implementing the solution findings of the given optimization task makes it possible to significantly reduce the construction cost of aerial ropeways in urban environment.
Keywords
Aerial passenger ropeway Urban environment Step of towers Height of towers Optimization CostList of Symbols
 \(\alpha_{Tnk}\)
The thermal elongation coefficient of the carrying rope
 [β]
The critical slope angle to the horizon of the carrying ropes
 \(\eta_{m}\)
The coefficient of the wind reduction on the mth carrying rope for the row of parallel ropes [28]
 \(\mu_{m} ,\mu_{wm}\)
The irregularity coefficients of distribution of weight and wind loads on the mth carrying rope from the passenger cabin
 \(\psi\)
The coefficient of the tower structure reinforcement when there is tension \(T_{k\,\hbox{max} } = R_{kn} /[n]_{k}\) having the maximum value under the permissible strength condition
 \(\psi_{f}\)
The coefficient of the permissible rope sag between the towers
 \(A_{cab}\)
The projected area of the cabin on the vertical plane
 C_{A}, C_{B}
The cost of the station buildings, A and B
 \(C_{eA} ,C_{eB}\)
The cost of the technological equipment mounted at A and B stations
 \(C_{ti} \,,\;C_{fi} ,\;C_{ei}\)
The unit cost of the metal structures, foundation and set of technological equipment for the ith intermediate tower
 \(C_{kt} ,C_{kn}\)
The cost of 1 running meter of the hauling and carrying ropes
 \(C_{sA(B)}\)
The cost of 1 m^{2} of the station building A(B)
 \(C_{wkn} ,C_{wcab}\)
The aerodynamic coefficients of the carrying ropeway and cabin
 \(d_{kn}\)
The maximum diameter of the carrying rope, determined from the condition of the aggregate strength
 \(d_{kt\;\hbox{max} } ,d_{kn\;\hbox{max} }\)
The maximum diameters of the hauling and carrying ropes
 \(d_{kn\;\hbox{min} }\)
The minimum diameters of the hauling and carrying ropes
 f
The maximum rope sag
 \(h_{cab}\)
The vertical dimension of the passenger cabin along with the suspension system
 \(H_{A(B)}\)
The height of the location of the station boarding site A (B)
 \(H_{tg}\)
The height of the intermediate tower
 \(H_{t\;\hbox{max} }\)
The limiting height of the intermediate height
 \([i_{t} ]\)
The permissible longitudinal slope of the arrangement of the carrying rope fastener assemblies to the neighboring intermediate towers
 I_{t}
The number of intermediate towers
 J_{d}
The number of exclusion zones
 \(k_{wkn} ,k_{wcab}\)
The coefficient of the wind pressure increase, depending on the height for the rope and cabin
 l_{k}
The length of the rope, taking into account its sagging in the span between adjacent intermediate towers
 \(L_{AB}\)
The distance between the boarding stations
 \(L_{cab}\)
The distance between the neighboring passenger cabins
 L_{t}
The distance between the neighboring towers
 \(L_{t\;\hbox{max} } ,L_{t\;\hbox{min} }\)
The maximum and minimum limiting distances between the intermediate towers
 \(n_{kn}\)
The number of carrying ropes
 \(n_{cab}\)
The number of passenger cabins, which are within one span
 \([n]_{k}\)
The minimum rope safety factor
 \(p_{cab}\)
The horizontal transverse statistic component of the wind pressure on the passenger cabin
 \(p_{kn}\)
The horizontal transverse statistic component of the wind pressure on the carrying ropes
 \(q_{kn}\)
The vertical uniformly distributed load of the dead weight of the rope with intensity
 \(q_{Rkn}\)
The resultant of the transverse distributed load on the carrying rope
 \(Q_{cab}\)
The vertical concentrated load of the weight of passenger cabins
 \(R_{kn}\)
The aggregate strength of the carrying rope
 \(R_{kn} (d_{kn\;\hbox{max} } )\)
The aggregate strength of the maximum diameter of the chosen construction
 S_{k}
The horizontal longitudinal tension force of the rope
 \(S_{fA(B)} \,,\;h_{fA(B)}\)
The area and height of the building floor A (B)
 t
The temperature
 \(t_{\hbox{max} }\)
The maximum ambient temperature
 T_{k}
The axial tension forces of the carrying rope on the tower
 \(\bar{u}_{dj} ,\Delta u_{dj}\)
The coordinate of the center and halfwidth of the jth exclusion zone
 \(v_{\hbox{min} }\)
The minimum permissible height of proximity of passenger cabins to the ground
 \(w_{0}\)
The regulatory wind pressure
1 Introduction
Aerial ropeways are widely used as continuous transport for organization of passenger and cargo carriage in many countries worldwide [1, 2]. Passenger aerial ropeways are mostly used for rapid and convenient movement of people to the sports, tourist, ecological and healthimproving facilities within nature areas with difficult terrain which are difficult to access [3, 4]. Cargo ropeways are used in many sectors of economics for transportation of different cargos. The industries where one can see the use of cargo ropeways are mining, coal, chemical, metallurgical, power, timber and agricultural [5, 6, 7, 8]. According to the data of the comparative technoeconomic study [2, 9, 10, 11], aerial ropeways are more economically and environmentally beneficial than land transport (road, conveyor and rail), especially in cases when the terrain, high density of the housing and industrial development and various urban planning restrictions impede the development of the ground traffic.
The theory of passenger and cargo aerial ropeways was actively developed in the middle of the 20th century. At the same time, the studies in the area were conducted in England, Austria, Germany, Italy, Russia and other countries [1, 2, 5].
As it is shown in the related literatures [2, 6, 9, 11, 12], aerial ropeways are an effective alternative to the conventional modes of land public transport in metropolitan areas and cities. Aerial ropeways may be labeled as highspeed urban transport. The average speed of the passenger cabins may be from 18 to 40 km/h [9, 10, 13]. This value is higher than an average travel speed of the conventional land transport in straitened urban conditions. In addition, the problem of transport accessibility is becoming increasingly important when evaluating projects for the modernization of transport infrastructure in major urban centers [14]. This indicator is an apparent advantage of aerial ropeways as well.
A detailed review of exploitation of aerial ropeways in different cities is given in [2, 13, 15]. In the urbanized environment, aerial ropeways have begun to play an active role in the last 10~15 years [16]. Therefore, due to the lack of theoretical studies and scientific publications on the topic, at present a lot of specific questions related to designing, calculation and modeling of operational processes in aerial ropeways should be considered specifically for the urbanized environment. Among the early publications on the issue is the research presented in [17]. Questions related to the productivity, cost and possibility of applying of cablepropelled systems in the urban environment are considered in the article. A number of papers addressed the issues of the effect of climatic factors (wind and air temperature difference) on the dynamics of passenger cabins and the cable system of aerial ropeways [18, 19]; calculation of strength and tension of carrying ropes [20, 21, 22]; safety of passenger transportation [2, 3].
The research problem of passenger aerial ropeways not only includes the technical aspect. For example, [4] deals with the questions of social and economic impact of aerial ropeway construction on the development of adjacent areas; [23] addresses the issues of acquiring rights to the air space in urban environments for ropeways. To date, the economic aspects of the construction of ropeways have not been sufficiently discussed in previously published articles. However, it is these aspects that determine the prospects and economic feasibility of modernizing the urban transport infrastructure on the basis of passenger aerial ropeways.
2 Statement of the Research Task
It is clear that the arrangement of the intermediate towers of the ropeway is the task of technical and economic optimization. The aim of optimization is to provide the minimum cost of the erection of boarding stations, intermediate towers, procurement of hauling and carrying cables and the set of technological equipment installed on the tower [24]. Statement and solving of the optimization problem make it possible to substantially reduce the costs of construction of passenger aerial ropeways in the urbanized environment [2, 25].
3 Mathematical Model of Passenger Aerial Ropeway Line

the cost of the buildings of boarding stations A and B;

the cost of the technological equipment mounted inside A and B stations;

the cost of the metal structures of the intermediate towers and the foundation under them;

the cost of the technological equipment mounted on the towers;

the cost of the hauling and carrying ropes.

dead weight of the 1 running meter of the rope length

aggregate strength of the rope (breaking tension)

cost of 1 running meter of the rope length

vertical uniformly distributed load of the dead weight of the rope with intensity \(q_{kn}\);

vertical concentrated load of the weight of passenger cabins \(Q_{cab}\);

horizontal longitudinal tension force of the rope S_{k};

horizontal transverse statistic component of the wind pressure on the passenger cabin \(p_{cab}\);

horizontal transverse statistic component of the wind pressure on the carrying ropes \(p_{kn}\).
The carrying ropes also undergo dynamic stresses from the swaying of passenger cabins during their movement and swaying of the ropes themselves under the wind pressure. To take into account these dynamic effects, the dynamic coefficient \(\psi_{d} > 1\) is used.

shape I (the section of the maximum sagging is within the span, between the intermediate towers, shown in Fig. 3);

shape II (the section of the maximum sagging is outside the span);

shape III (the section of the maximum sagging coincides with one of the towers).
When \(K_{f} \notin (  1;\; + 1)\), shape I of the sagging is implemented; when \(K_{f} \in (  1;\; + 1)\), shape II is implemented; when \(K_{f} = \pm 1\), shape III is implemented.
Taking into account the required values of the minimum diameter of the carrying rope for every \(I_{t} + 1\) bay of the aerial ropeway line, completely its minimum diameter is selected to be equal to the maximum diameter determined from Eqs. (7) and (8).
In this equation, the distances \(a_{i + 1} (t)\) and \(b_{i + 1} (t)\) are the functions of the sought quantity \(S_{k} (t)\) according to Eqs. (5) and (6).
The geometrical line of the carrying rope sagging in the (i + 1)th span and maximum rope sag \(f_{i + 1}\) at unconditioned temperature t will be determined by Eqs. (3) and (4) or (9) when substituting the adjusted values \(S_{k} (t)\) and \(a_{i + 1} (t)\) for the corresponding members in these equations.
4 Problem Statement of Optimization of Aerial Ropeway Line

number of the intermediate towers I_{t};

point data of the intermediate ith tower arrangement along the aerial ropeway line \(u_{i} \;(i \in [1\,;\;I_{t} ])\);

heights of the intermediate towers \(H_{tgi} \;(i \in [1\,;\;I_{t} ])\) and heights of the boarding stations \(H_{A} = H_{tgi = 0}\) and \(H_{B} = H_{{tgi = I_{t} + 1}}\);

tension forces of the carrying ropes S_{k}.
The number of the vector components is \(N = 2I_{t} + 3\). When the distances (specific to aerial ropeways) between the neighboring boarding stations are L_{AB} from 3 to 5 km, the number of variables in the optimization task will be N, whose values is within 20 to 100.
The algorithm of the optimal design of the aerial ropeway line includes multiple sequential minimization of the objective function (10) when the characteristic I_{t} takes various values. The absolute minimum will determine the characteristics of the optimal option for the projected line.
When determining the minimum of the objective function (10), the following limitations in the form of inequations should be implemented:

when the rope sagging takes shape I

when the rope sagging takes shape II and III

when the rope sagging takes shape I

when the rope sagging takes shape II or III

when the rope sagging takes shape I

when the rope sagging takes shape II or III
To determine the minimum of the objective function (10) taking into account the acceptable limitations, it is necessary to use one of the direct methods of the conditional optimization [28], based on the immediate calculation of the objective function value \(O(\{ x\} )\).
5 Solutions Findings of the Optimization Task and Their Analysis
The task of the optimal design of aerial passenger ropeway for the conditions of a highly urbanized environment and high irregularity of the earth’s surface is a very complex mathematical task. Indeed, with the distances between neighboring stations for boarding passengers, which are characteristic of ropeway lines, the number of variable parameters in the optimization problem can reach up to 100 unknown values. The number of these unknown quantities determines the dimension of the optimization problem. At the same time, it is also necessary to take into account 11 types of various structural, strength and operational constraints, which should be imposed on variables during the solution of the optimization problem. These constraints are expressed using 99 mathematical dependencies in the form of inequalities. Thus, the practical implementation of the proposed mathematical model and the problem of minimizing the objective function (Eq. 1) are possible only through the use of numerical mathematical methods and computer equipment.
To this end, the authors developed a computer program “RopewayOptimization.” The original text of the program was protected by the Patent Office of the Russian Federation [29]. As a mathematical method of optimization, the method of the HookeJeeves type was used [27]. The need to take into account a large number of constraints in the form of inequalities complicates the form of the domain of possible solutions of the optimization problem in the Ndimensional space of variables. Also, the presence of a large number of unknown variables leads to the fact that the objective function \(\left. {O(\{ x\} )} \right_{{I_{t} = const}}\) (Eq. 10) has several local minima within the domain of possible solutions. When optimizing for test and real conditions of designing ropeways, the authors recorded from 3 to 7 local minima of the objective function. Obviously, only one of these local minima can be considered the best solution to the optimization problem (global minimum). However, some of the local minima have objective function values that are quite close in magnitude to the value of the objective function in the global minimum. Therefore, when developing a real project of a ropeway, it is also advisable to consider such local minima. Perhaps an additional consideration of any other considerations (for example, political, legal or property reasons) will lead to the need to use the solution of the optimization problem not at the global minimum point of the objective function, but at a point of the near local minimum. Thus, it is important to correctly set the initial optimization point (i.e., the initial combination of the values of all N variables). The starting point must satisfy 1) all 99 constraints and 2) ensure that the global minimum of the objective function is found. In the RopewayOptimization program, the optimization start point was set using the repeated iteration algorithm for possible combinations of two variables—the height of intermediate towers H_{t} (it is assumed the same for all supports) and the horizontal longitudinal tension force of the rope S_{k}. Parameters H_{t} and S_{k} change with step \(\Delta H_{t}\) and \(\Delta S_{k}\) within the intervals of their possible change (\(H_{t} \le H_{t\;\hbox{max} }\) and \(S_{k} < R_{kn} (d_{kn\;\hbox{max} } )/[n]_{k}\)). The experience of carrying out test calculations showed that the recommended values of these steps are \(\Delta H_{t}\) ~ 10 m, \(\Delta S_{k}\) ~ 40 kN. If a possible combination of parameters H_{t} and S_{k} satisfies the requirements for choosing the starting point, then the optimization problem is solved and the local minimum point of the objective function \(\left. {O(\{ x\} )} \right_{{I_{t} = const}}\) (Eq. 10) is determined. As a result, one point is selected from the set of points of local minima thus obtained, which has the lowest value of the objective function. This point is the global minimum point and, accordingly, the solution to the optimization problem.
The efficiency of the computer program “RopewayOptimization” was tested by solving several test tasks. The test tasks had a different dimension (the number of variables in the optimization problem varied within 10 to 400), various quantitative parameters of the shape of the earth’s surface and the location of exclusion zones. Regular and irregular forms of the earth’s surface along the ropeway line were considered. The regular shape was modeled by rectilinear (horizontal and oblique with a slope of up to 60 degrees) and sinusoidal functions. To simulate the irregular shape of the earth’s surface, real topographic maps of the large cities of RostovonDon and Bryansk (Russian Federation) were used. At the same time, the real maps of these cities were used to specify the location and size of the exclusion areas, and thus, the actual location of the street infrastructure objects was taken into account. On the basis of the calculated data, proposals were formulated for the development of promising projects for the construction of passenger aerial ropeways in these cities [2, 11].
The further calculation results of the aerial ropeway cost are expressed in conventional units (C.U.). As a conventional unit, the Russian rouble was used when setting the cost indicators. When conducting cost optimization in other countries, the summands in Eq. (1) should be expressed in the national currency. At the same time, the mathematical model and calculation dependencies of the optimization task do not change.
The graphs in Fig. 5 a show the influence of the terrain diversity on the aerial ropeway line. (The terrain diversity grows with increasing the number of halfwaves n.) When the terrain is slightly uneven (n < 3), the line cost is minimum and virtually the same for the different number of properties. However, when the terrain is considerably uneven (n > 4), an increasing number has a positive impact on the cost due to the possibility of installing the towers in the elevated areas. This leads to reducing the tower height and reducing their total cost despite the large number.
6 Conclusion
The proposed mathematical model and optimization problem should be used at the initial stage of designing passenger aerial ropeways to analyze the influence of a substantial number of cost factors, as well as construction and geometric factors on the optimal placement, height and number of intermediate towers and the tension force of the supporting ropes.
The optimal method put forward in this article would be helpful in determining the most costeffective construction option in aerial ropeway in the given conditions, taking into account the terrain, urban infrastructure arrangement, altitude performance of the urban development, technical characteristics of the carrying rope, etc.
Implementing the solution findings of the given optimization task makes it possible to significantly reduce the construction cost of aerial ropeways in the urban environment.
 1.
consideration of important physical processes that may affect the reliability, safety and cost of ropeways (in particular, dynamic processes during the movement of passenger cabins, increased negative environmental effects, possible deviations from normal operation, etc.);
 2.
the use of multicriteria objective functions to determine the best options for aerial ropeways, not only taking into account the cost of construction, but also taking into account capacity, operational safety, maintenance costs, etc.
Notes
Open Access
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