Optimization of the Technical Parameters of Universal Freight Wagons

Abstract

The present study is devoted to the selection of a new criterion for the optimization of the specific volume/area of universal freight wagons. The currently used criterion—minimum of reduced costs—is practically not applied and the costs for different railway administrations and regions differ significantly. This leads to insufficient use of the effective volume/area of the wagon and to insufficient profitability when carrying out transport work. An analysis of the technical parameters of freight wagons was made, based on which the new criterion—loading tare coefficient—was proposed, depending only on the type of cargo and the wagon parameters. In the analysis, the classical methods for determining the technical parameters were used and an original method for finding the minimum of the new criterion was proposed. These methods were applied to optimize the parameters of a real wagon and the results show that it is expedient to reduce the volume of the wagon. With the optimized wagon, the same transport work can be carried out, as the number of trains for its execution is reduced. The conducted studies can be used by the designers of new freight wagons and by the companies—both the carriers or lessors of wagons.

1. Introduction

The production of new products like freight wagons is a complex, lengthy and expensive process. It usually includes the following main stages [1]: market research, formulation of technical specifications, development of a conceptual design, preparation of working documentation, theoretical strength testing, development of technological equipment, production of a prototype, testing of the structure, corrections of the design and technological documentation, commissioning and transition to regular production.

However, freight wagons are not products which are produced and consumed in large volumes and therefore there are mostly only a few producers and users at the national level. That makes the procedure described above almost not applicable. This is because most often the producers launch a vehicle into production based on a firm order from a beneficiary and a specifications sheet with the main characteristics of the vehicle. Thus, it is necessary to propose a methodology, which is primarily aimed at the beneficiaries, so that they can offer the most effective construction possible for them, considering their own policy for carrying out transport activities.

An important stage in creating a sustainable product is market research. Its purpose is to analyze the need to create a specific design that is efficient, competitive and guarantees a sufficiently long production period for the given series. This leads to a significant reduction in the company’s production costs and an increase in its profit. However, the main benefit is for the carrier/owner of the wagon, as it significantly reduces operating costs. This suggests that the main responsibility for proper research lies with the carrier/owner of the wagon.

Typically, the study includes determining the following: the purpose of the wagon, type of cargo to be transported, regulatory documents to which it must comply, technical parameters, environmental standards, etc. Most often, the indicators like purpose of the wagon, type of cargo, regulatory documents or environmental standards are trivially determined by enterprises owning their own wagon fleet or by companies carrying out transport activities or renting wagons. Therefore, the main problem in designing a new and efficient freight wagon remains the determination of its optimal technical parameters.

They have a clearly expressed technical nature, but in the specialized literature [2,3,4,5] they are called economic technical parameters (ETP), since they have a significant impact on the economic indicators like costs, price, production costs, profitability, etc. An important place among them is occupied by the parameters specific volume and specific area, which are fundamental for the optimization of all the other technical parameters and have a serious impact on the own weight of universal freight wagons, i.e., wagons for the transport of various types of cargo, characterized by a wide range of changes in their density.

In [2] it is proposed to use the indicator of “adjusted costs” as an optimization criterion, i.e., these are all costs (capital/investment and operational/working) described and analyzed in [6,7,8,9], attributed to a unit of transportation work. In principle, this is a correct business approach, but in practice it is inapplicable in modern world realities. The main reasons for this have been analyzed in several publications, some of which are [10,11,12,13,14,15]. In summary, they can be described as follows:

• Information on operating costs in the European Union (EU) is incomplete, difficult to access and completely non-functional [13]. As an example, it is stated that even in countries with published methodologies for determining infrastructure access charges, a carrier cannot calculate its costs accurately. The reason for this is the presence of numerous conditions depending on the type and prices of energy sources, the presence or absence of engineering facilities (tunnels, bridges, etc.), route traffic, track development and many others.
• There are huge differences in capital costs across countries and regions. For example, the cost of building a railway track varies from 1.5 to 70 million EUR/km [6]; the cost of fixed (station and ancillary) equipment ranges from EUR 2 to 100 million [6]; the cost of new rolling stock varies in a ratio of minimum to maximum price of 1:9 (own studies); the ratios for land, electrification, signaling systems, etc., are approximately similar.
• There are large differences in operating costs, including infrastructure access charges ranging from 0.5 to 11.5 EUR/train-km [13], cost of energy sources, cost of repair and maintenance of rolling stock, wages, etc.

The indicated data, analyses and reasons make the correct determination of the costs impossible and therefore, the “adjusted costs” parameter is not suitable for optimal selection of the specific volume/area of the freight wagons. This study aims to propose a new optimization criterion that meets the following requirements:

• It should depend solely on the design parameters of the wagon and the physical properties of the cargo;
• It should ensure maximum use of the permissible wagon gross weight on the given rail track;
• There must be sufficiently simple functional dependencies between it and the other parameters related to the optimization;
• There must be a minimum corresponding to the minimum of the “adjusted costs”. wages, etc.;
• It should be noted that some studies use different optimization criteria and modern engineering methods [16] and different lightweighting and advanced manufacturing/thermal management methods [17] to assure the optimization goals.

2. Analysis of Technical Optimization Parameters

The aim of the study requires a brief analysis of the technical parameters to establish the functional relationships between them and to study the influence of individual variables on the efficiency of the wagon. According to [2,3,4,5] they are divided into two large groups—absolute and relative.

2.1. Absolute Technical Parameters

The absolute technical parameters include the following (Figure 1): total length of the wagon (with buffers)—2L_w; length of the wagon body (without buffers)—2L; pivot distance—2l (distance between the axles for two-axle wagons or between the pivots—for bogie wagons); length of the end part—n_e (distance from the pivot (for bogie wagons) or from the wheelbase (for two-axle wagons) to the end of the wagon); base of the bogie (distance between the axles)—2l_b; width of the wagon—2B; distance between the buffers—2b_b; track gauge—2s; height of the wagon body—H; height of the buffers from the rail head—h_b; total height of the wagon from the rail head—h. Other parameters not shown in Figure 1 are area of the body—F; volume of the body—V; own weight of the wagon (tare weight)—T; load capacity (maximum permissible mass of the payload)—P; the gross weight of the wagon—Q; the number of axles—n, etc.

Absolute parameters enable the user or the owner of the wagon to assess whether a given type of vehicle can transport one or a group of different loads, i.e., whether the dimensions, area, volume and load capacity allow the placement of the loads in the designed wagon.

2.2. Relative Technical Parameters

Relative parameters represent a ratio between two or more absolute parameters. They allow us to evaluate the efficiency of a wagon when comparing it with other wagons of the same type. The more important ones are discussed below.

2.2.1. Axle Load

Axle load p₀ (in t/axle) is a parameter that defines the maximum permissible mass for loading on one axle. It is determined by Equation (1):

$$p_0= {\displaystyle \frac{Q}{n}}={\displaystyle \frac{T+P}{n}}$$

(1)

where Q is the gross weight of the wagon, T tare weight, P load capacity and n number of axles.

The axle load is an important parameter for both the wagon and the railway track. That is why it is standardized by the UIC (International Union of Railways) and the European Union [18,19,20]. It has a significant impact on the wagon, since it and Equation (1) determine the gross weight Q. The carriers strive to use the maximum permissible values of the axle load depending on the parameters of the railway infrastructure. It is important to note that in some cases it is appropriate for the newly ordered wagon to have a lower axle load.

The examples are specialized wagons for the transport of low-density cargo, in which the volume, area, length, width, etc., do not allow reaching the maximum permissible value of the axle load on the railway track.

The axle load is also an important characteristic of the track. Depending on the type of track, the following track categories have been standardized within the framework of the UIC and TSI (Technical Specification for Interoperability [20]): category A—16 t/axle; B—18 t/axle; C—20 t/axle; D—22.5 t/axle; E—25 t/axle; F—27.5 t/axle and G—30 t/axle. At present, all main sections of the railway tracks included in the European railway traffic organization are equipped with rails for an axle load of 22.5 t/axle.

2.2.2. Load per Linear Meter

Load per linear meter q (in t/m) is determined by Equation (2):

$$q= {\displaystyle \frac{Q}{2L_w}}={\displaystyle \frac{T+P}{2L_w}}$$

(2)

where 2L_w is the total length of the wagon with buffers.

Physically, the parameter expresses the gross weight of the wagon per unit of the total length of the vehicle. It is important for the railway track and for the railway bridges, whose maximum load is determined by it. In this sense, the railway track designations (A, B, C, D, E, F and G) are further differentiated by adding numbers after the corresponding letter. They correspond to the following linear loads [20]: 1 for 5 t/m, 2 for 6.4 t/m, 3 for 7.2 t/m, 4 for 8.0 t/m, 5 for 8.8 t/m and 6 for 10 t/m.

2.2.3. Tare Coefficients

Tare coefficients are one of the most important parameters of wagon structures. In general, they express directly or in a more complex way the relationship between the tare weight T and the load capacity P of the vehicles k = f(T/P). Taking this into account, it follows that the smaller the corresponding tare coefficient, the more efficient a given wagon design is. The reason is that small values of the coefficient are obtained at low tare weight and high load capacity of the wagon. The correctness of the above statement can be proven, considering that the gross weight of the wagon Q is expressed as the sum of the tare weight T and the load capacity P and determined by Equation (3):

$$Q= T+P$$

(3)

On the other hand, this absolute parameter is a limited quantity, the maximum value of which is determined by the permissible axle load p₀ and the number of axles n as given in Equation (4).

$$Q=p_0·n$$

(4)

Using Equations (3) and (4) and solving the resulting expression for P, it follows that the load capacity is a function of the tare weight, the axle load and the number of axles as determined by Equation (5).

$$P=p_0·n-T$$

(5)

Equation (5) shows that for pre-selected values of p₀ and n for a given wagon structure, the load capacity depends only on the tare weight of the wagon. Moreover, the smaller the tare weight, the greater the load capacity. The arguments prove that reducing the wagon’s tare weight leads to a significant increase in the productivity of railway transportation. This is achieved not only at the expense of the reduced amount of metal material inserted into the wagon structure, but mainly at the expense of reduced operating costs [6,7,8,9,21]. For example, if a lighter wagon structure is used, the planned/agreed work (t/km) can be performed with a smaller number of wagons and therefore with a smaller number of locomotives and train crews. Under these conditions, energy costs, the number of repairs, the number of spare parts used, infrastructure fees, personal costs etc., are reduced [22,23,24].

In general, the reduction in the wagon’s tare weight can be achieved by improving the methods for strength calculation of wagon structures and their testing, by improving the manufacturing technology, using new lighter materials [25], etc. Freight wagons are characterized by three tare coefficients: technical, loading and operational.

The technical tare coefficient is a dimensionless quantity and expresses the relationship between the tare weight T and the load capacity P, as given in Equation (6):

$$k_T= {\displaystyle \frac{T}{P}}={\displaystyle \frac{T}{p_0-T}}$$

(6)

It largely reflects the design and technological level of the manufacturing plant, as it depends practically only on the quantity of materials used to build the wagon. At the same time, the coefficient also serves as an indicator of the potential capabilities of the vehicle (which are rarely used in operation) for transporting various types of cargo.

The loading tare coefficient is the ratio between the tare weight T and the actual load capacity P·λ of the wagon, as given in Equation (7):

$$k_P= {\displaystyle \frac{T}{P·\lambda }}={\displaystyle \frac{k_T}{\lambda }}={\displaystyle \frac{T}{\left(p_0·n-T\right)·\lambda }}$$

(7)

where λ is the coefficient of load capacity utilization when transporting various types of cargo. In the general case λ ≤ 1, which indicates that only in rare cases (for some specialized wagons) is it possible to realize full load capacity utilization. The coefficient of load capacity utilization is determined by Kogan’s equations [2,3,4,5], shown in Equation (8):

$$\begin{gathered} {\lambda }_v= {\displaystyle \frac{\sum a_u+\sum a_n}{\sum a_u+\frac{1}{v_y}\sum a_n{·v}_{yT}}} \\ {\lambda }_f= {\displaystyle \frac{\sum a_u+\sum a_n}{\sum a_u+\frac{1}{f_y}\sum a_n{·f}_{yT}}} \end{gathered}$$

(8)

where λ_v is the coefficient of load capacity utilization for specific volume, λ_f is the coefficient of load capacity utilization for specific area, Σa_u is the relative share in the freight turnover of the cargo goods for which the wagon’s load capacity is used, Σa_n is the relative share in the freight turnover of the cargo goods for which the wagon’s load capacity is not used, v_y and f_y are specific volume and specific area of the wagon and v_yT and f_yT are the specific volume and specific area of the respective loads.

Issues related to specific volume, specific area and Kogan’s equations will be discussed in detail below.

The operational tare coefficient is given in Equation (9):

$$k_O= {\displaystyle \frac{T\left(1+{\alpha }_e\right)}{P_{ad}}}$$

(9)

where α_e is the empty mileage coefficient for the given type of wagon, equal to the ratio of the empty mileage to the loaded mileage, and P_ad is the average dynamic load, obtained from the ratio of the transport work (in ton–kilometers) to the total distance traveled (in kilometers).

The analysis of Equations (6), (7) and (9) shows that the efficiency of the freight wagon is most fully characterized by the operational tare coefficient, since it considers not only the real or actual, but also the average dynamic load capacity, i.e., it considers the mileage in the loaded state.

Nevertheless, the other two tare coefficients also provide essential information about the efficiency of the wagon structure. It should also be noted that the load tare coefficient is very important; below, its use as a criterion is proposed for the optimization of all economic technical parameters (ETPs).

2.2.4. Specific Volume and Specific Area

The specific volume and specific area of the wagon are determined by the ratio of the geometric volume V or the geometric area F of the wagon body to its load capacity P, as given in Equation (10):

$$v_y= {\displaystyle \frac{V}{P}}, f_y= {\displaystyle \frac{F}{P}}$$

(10)

In addition to the parameter geometric volume V of the body, the concept of cargo volume V_T is used in practice, which is the volume of the cargo in the wagon. The relationship between them is given by the volume utilization coefficient φ as determined in Equation (11):

$$\phi = {\displaystyle \frac{V_T}{V}}$$

(11)

The maximum values of φ depend on the type of wagon and the type of cargo. They can be obtained using the data in

Table 1. Maximum values of the volume utilization coefficient.

Wagon Type	Cargo Type	φmax
Open and flat wagons	bulk	1 + (3/8)·tg αi ≤ 1.35
	non-bulk	1.35
Covered wagons	all	0.9
Tank wagons	all	0.99

[5], where α_i denotes the angle of repose of bulk cargo.

This angle is determined empirically and is equal to the angle between the horizontal base and the descent of the cone obtained when the cargo is freely poured into the wagon structure, as shown in Figure 2 [2,3,4,5].

From Equations (10) and (11) follows Equation (12):

$$v_y= {\displaystyle \frac{V_T}{P·\phi }}, f_y= {\displaystyle \frac{V_T}{P·H_T·\phi }}$$

(12)

where H_T is the height of the cargo.

Specific volume (area) is an important indicator of the efficiency of freight wagons. For example, if it is necessary to transport a cargo with density γ, volume V_T and volume utilization factor φ in a wagon with specific volume (area) v_y (f_y), then optimal conditions will be created if Equation (13) is satisfied:

$$v_y= {\displaystyle \frac{1}{\gamma ·\phi }}, f_y= {\displaystyle \frac{V_T}{\gamma ·H_T·\phi }}$$

(13)

This will ensure maximum use of both the load capacity and the volume of the wagon.

Equation (13) allows us to obtain, with high accuracy, the optimal values of the specific volume (area) of all specialized wagons, which are intended for the transportation of one or a group of goods with similar physical properties. In the case of universal wagons, which are used to transport various goods with a wide range of densities γ, the problem of the correct selection of v_y and f_y is significantly more complicated. The specific case under consideration can be analyzed as follows: It is known that a newly designed wagon will transport m types of goods, each of which has a density γ_i. Using the proposed data from Table 1, the maximum values of the volume utilization coefficients φ_maxi can be determined. Using Equations (13) and (11), it is possible to determine the optimal values of v_yi and V_i for the transportation of the i-th cargo, for which full use of both the load capacity P and the body volume V would be realized. In this case, the Inequalities (14), (15) and (16) are satisfied:

$${\gamma }_{max}= {\gamma }_1>{\gamma }_2>{\gamma }_3>\cdots >{\gamma }_m= {\gamma }_{min}$$

(14)

$$v_{ymin}=v_{y1}<v_{y2}<v_{y3}<\cdots <v_{ym}=v_{ymax}$$

(15)

$$V_{min}=V_1<V_2<V_3<\cdots <V_m=V_{max}$$

(16)

If the designed wagon is built with a volume V_max, then only for the lightest cargo m with density γ_m = γ_min will full use of the volume V_max and the load capacity P_max be obtained. For all other cargo types, the load capacity P_max will be fully used, but the geometric volume will not be used, as given in Equation (17):

$$V_{Ti}<V_{max}, P_i= P_{max}$$

(17)

valid for all i = 1, …, m − 1, where V_Ti is the volume occupied by the i-th load.

Therefore, for all cargo types, except the lightest, there will be free and unused space in the wagon. This means that the structure is designed irrationally, with an excess amount of metal material being used, i.e., the tare weight T of the wagon is unreasonably high.

The other limit case leads to similar results. Let us assume that the wagon is designed in accordance with the parameters of the heaviest cargo type γ_max = γ₁. From Inequalities (15) and (16) it follows that v_y = v_min and V = V_min. The analysis of this solution shows that when operating such a wagon for all cargo types, except for the heaviest cargo, there will be full use of volume V = V_min and incomplete use of the load capacity P_max, as given in Equation (18):

$$V_{Ti}=V_{min}, P_i< P_{max}$$

(18)

valid for all i = 2, 3, …, m. In this case, the limitation will be in volume, while the maximum load capacity will not be used.

Such a wagon design is also extremely irrational, since the non-use of the load capacity of the wagon leads to insufficient force loading of the supporting elements of the wagon body and therefore to incomplete use of the strength of the materials used.

The conclusion in this limiting case is that the own mass of the wagon is unreasonably high. Solving the problem using the arithmetic mean values of the products γ_i·φ_i also does not lead to good results. Therefore, to find the appropriate values of the specific volume (area), it is necessary to solve an optimization problem.

3. Basic Principles for Optimization of the Specific Volume (Area) of Freight Wagons

3.1. Optimization Criterion

From the research conducted on the topic, it became clear that in the existing methods for determining the specific volume (area) of universal freight wagons, the parameter reduced costs is used as an optimization criterion. The inexpediency of its use was emphasized and the requirements for the application of a new criterion were formulated. The parameter loading tare coefficient from Equation (7) most fully meets these requirements. To prove this statement, it is necessary to represent the own mass T as a linear function of absolute volume V or area F, as shown in Equation (19):

$$\begin{gathered} T= T_p+t_y·V \\ T= T_p+t_y^{\prime }·F \end{gathered}$$

(19)

and Kogan’s Equation (8) can be transformed into the form shown in Equation (20):

$$\begin{gathered} {\lambda }_v= {\displaystyle \frac{B}{\sum _{i=1}^k{a}_i+\frac{1}{v_y}\sum _{i=k+1}^m{a}_i{·v}_{yi}}} \\ {\lambda }_f= {\displaystyle \frac{B}{\sum _{i=1}^k{a}_i+\frac{1}{f_y}\sum _{i=k+1}^m{a}_i{·f}_{yi}}} \end{gathered}$$

(20)

Then the technical tare coefficient k_T and the loading tare coefficient k_p can be determined from Equations (21) and (22), respectively:

$$k_T= {\displaystyle \frac{T}{P}}=A·v_y, k_T= {\displaystyle \frac{T}{P}}=A{·f}_y$$

(21)

$$\begin{gathered} k_p={\displaystyle \frac{A·v_y+C}{B}}·\left({\sum }_{i=1}^k{a}_i+{\displaystyle \frac{1}{v_y}}{\sum }_{i=k+1}^m{a}_i·v_{yi}\right) \\ {k}_p={\displaystyle \frac{A^{\prime }·f_y+C}{B}}·\left({\sum }_{i=1}^k{a}_i+{\displaystyle \frac{1}{f_y}}{\sum }_{i=k+1}^m{a}_i·f_{yi}\right) \end{gathered}$$

(22)

In Equations (19)–(22), the following notations are used:

• t_y and t′_y are relative variable components of the tare weight per unit volume or area, respectively, determined by Equation (23):

$$t_y={\displaystyle \frac{T-T_p}{V}}, t_y^{\prime }={\displaystyle \frac{T-T_p}{F}}$$

(23)

• T_p is the constant component of the tare weight, including the weight of the drawing gear equipment, doors, hatches, running gear, elements of the braking equipment and other elements whose weight does not depend on the linear dimensions of the wagon;
• v_y and f_y are the current values of the specific volume and specific area of the wagon;
• a_i is the relative share in the cargo turnover of the i-th cargo as determined in Equation (26);
• v_yi and f_yi are the specific volume and specific area of the i-th cargo;
• $B=\sum _{i=1}^m{a}_i$ is a constant, which, in general, can be equal to or less than 1 (B < 1 if the data on the transported cargo is incomplete);
• k is the nearest number from the row indices in Equation (15) for which the load capacity P is used (k = 1, …, m);
• A, A′ and C are coefficients, constant for a given type of wagon, which are determined by Equation (24):

  $$A={\displaystyle \frac{n·p·t_y}{n·p_0-T_p}}, A^{\prime }={\displaystyle \frac{n·p_0·t_y^{\prime }}{n·p_0-T_p}}, C={\displaystyle \frac{T_p}{Fn·p_0-T_p}}$$

  (24)

  The graphical variation in λ, k_T and k_p is shown in Figure 3. For completeness of the figure, the graph of the change in the adjusted cost parameter R is also given. The nature of the change in the functions R = R(v_y) and k_p = k_p(v_y) is similar. The difference between the two curves is that the theoretical curve R = R(v_y) is defined as a continuous function, while the curve k_p = k_p(v_y) is considered as a partially continuous function.

The analysis of Equations (20)–(22), as well as Figure 3, allows us to draw the following important conclusions:

• The load capacity utilization coefficient λ (see Equation (20)) increases non-linearly with increasing specific volume of the wagon body. It reaches its maximum λ_max = 1 when the specific volume (area) of the wagon body corresponds to the lightest cargo type, i.e., when Equation (25) is satisfied:

  $$v_y=v_{ym}={\displaystyle \frac{1}{{\phi }_m·{\gamma }_m}}$$

  (25)

  In this case, all expected cargo types can be loaded into the wagon, but to a height corresponding to the maximum load capacity P_max = n·p₀ − T of the wagon. From this, two main conclusions follow:

  -

  The parameters specific volume v_y, and therefore geometric volume V should not be increased above v_ymax, as this leads to an increase in the tare weight T of the wagon and a decrease in its load capacity P;

  -

  At v_y = v_ym = v_ymax, the geometric volume of the wagon will be fully used only when transporting the lightest cargo. For all other cargoes, there will be an unusable volume, leading to the transportation of excess tare weight T and a decrease in the load capacity of the vehicle P.

2.

The technical tare coefficient k_T increases linearly with increasing specific volume (area) of the wagon body as given in Equation (21). Therefore, using the value of v_y > v_ymax increases the tare weight of the wagon T and reduces its load capacity P, which makes the vehicle inefficient.

3.

The loading tare coefficient k_p is changed as follows:

• According to hyperbolic law in the interval 0 < v_y < v_y1 (zone I in Figure 3), there is a constant decrease with increasing specific volume (area). The reason for this is that in this interval none of the cargoes will use the wagon’s load capacity and, therefore, the sum $\sum _{i=1}^k{a}_{i }$ in the denominator of Equation (20) will be equal to zero;
• In the interval v_y1 ≤ v_y ≤ v_ym (zone II in Figure 3), the function k_p = k_p(v_y) is partially continuous, with the inflection points corresponding to the values of the argument v_y = v_y1, v_y2…, v_ym. The reason for this is that the sum $\sum _{i=1}^k{a}_{i }$ of Equation (20) increases with increasing v_y, and the sum $\sum _{i=k+1}^m{a}_i·v_{yi}$ decreases. The minimum of the function is located exactly in this interval;
• In the interval v_ym < v_y < ∞ (zone III in Figure 3), the loading tare coefficient k_p equals the technical one (k_T where λ = 1) and increases linearly with increasing specific volume of the wagon body.

The conclusions drawn and the analysis of Equation (22) show that the loading tare coefficient k_p satisfies the conditions for the optimization criterion of the specific volume (area) of the wagon body, due to the following: k_p depends only on the design parameters of the wagon and the physical properties of the cargo; Equation (22) is simple enough; the maximum use of the gross weight of the wagon is ensured by the product n·p₀ in Equation (24); the function k_p = k_p(v_y) or k_p = k_p(f_y) has a minimum that corresponds to the minimum of the reduced costs. The latter statement is proven in detail in [26]. The main argument is that when designing a new wagon with a minimum value of the loading tare coefficient k_p, the net weight of the trains increases while maintaining their gross weight. This leads to a decrease in the number of wagons and therefore the locomotives, necessary for the transportation of certain quantities of cargo, i.e., the costs of purchasing new wagons and locomotives decrease; the volume of repair work decreases, which enables the reduction of the number of workers engaged in this type of activity; the costs of access to the railway infrastructure decrease; the throughput capacity of the railway tracks and stations increases, etc. The above directly affects the two main components of the costs of providing the freight transportation service—capital and operational—and leads to an increase in the profitability of the carrier company.

3.2. Specific Volume (Area) Optimization Methodology

From what is stated in Section 3.1 above, it follows that the optimal specific volume (area) of the wagon body corresponds to the minimum of the partially continuous functions k_p = k_p(v_y) or k_p = k_p(f_y), which is in the interval v_y1 ≤ v_y ≤ v_ym or f_y1 ≤ f_y ≤ f_ym.

3.2.1. Finding the Minimum of a Non-Linear Partially Continuous Function

Finding the minimum of the loading tare coefficient k_p will be illustrated only for the variant k_p = k_p(v_y). The reason is that the two functions k_p = k_p(v_y) or k_p = k_p(f_y) are derivatives of each other and differ (see Equation (13)) only in the presence of the parameter H_T—height of the cargo in the wagon body.

Partially continuous functions of mixed hyperbolic and parabolic type, such as functions from Equation (22), consist of multiple sections whose boundary values coincide—as shown in Figure 3. For each of the intervals v_yk1 ≤ v_y ≤ v_yk+1, the function is continuous and can have one of the forms shown in Figure 4a–c.

The fact that for each of the intervals v_yk ≤ v_y ≤ v_yk+1 the function k_p = k_p(v_y) is continuous allows us to find the local minimum in this interval. For this purpose, we use the classical method from mathematics: we find the first derivative of the function in the domain v_yk ≤ v_y ≤ v_yk+1, set it equal to zero and solve for the variable v_y. The resulting value of v_ykopt can be as follows:

• v_yk opt > v_yk+1 (Figure 4a)—since the function is decreasing in the entire interval [v_yk, v_yk+1], the function is decreasing too. Therefore, the local minimum of k_p will be obtained at the value of the argument v_y = v_yk+1;
• v_yk < v_yk opt < v_yk+1 (Figure 4b)—the local minimum of k_p is in the interval [v_yk, v_yk+1] and corresponds to the value of the argument vy = v_yk opt;
• v_yk opt < v_yk (Figure 4c)—in this case, the function is increasing in the entire interval [v_yk, v_yk+1], from which it follows that the local minimum of k_p is obtained at the value of the argument vy = v_yk.

In practice, the partially continuous function k_p consists of multiple intervals [v_yk, v_yk+1]. Therefore, to find the absolute minimum, it is necessary to determine all local minima and compare them with each other. The smallest of them corresponds to the absolute minimum of the function.

3.2.2. Algorithm for Optimization of Specific Volume (Area)

Due to the identical nature of the dependencies k_p = k_p(v_y) and k_p = k_p(f_y), only the algorithm for optimization of the specific volume of the wagon will be considered. It includes the following stages:

• For optimization purposes, the following initial data are required for each load:—number of cargo types—m, density—γi, angle of repose—α_i, mass of the cargo—M_i, average transportation distance—L_i. The mass of the cargo—M_i and average transportation distance—L_i can be replaced by the parameter average transportation work S_i = M_i·L_i.
• The relative share in the total cargo turnover for each of the transported cargo types is determined using Equation (26):

$$a_i={\displaystyle \frac{M_i·L_i}{\sum _{i=1}^m{M}_i·L_i}}={\displaystyle \frac{S_i}{\sum _{i=1}^m{S}_i}}$$

(26)

3.

The maximum values of the wagon volume utilization coefficients φ_imax for each load are determined using the data from Table 1.

4.

The specific volumes v_yi required for the full use of the wagon capacity and volume for each cargo type are calculated using Equation (27):

$$v_{yi}={\displaystyle \frac{1}{{\gamma }_i·{\phi }_{imax}}}$$

(27)

5.

The ascending order of specific volumes is formed using Equation (28):

$$v_{y1}< v_{y2}<v_{y3}<\cdots <v_{ym}$$

(28)

6.

The absolute minimum of the loading tare coefficient k_p is determined in the following sequence:

• The optimal values of the specific volume v_ykopt are calculated for each of the intervals v_yk < v_y < v_yk+1 from the row obtained in step 5 using Equation (29):

  $$v_{ykopt}=\sqrt{{\displaystyle \frac{C·\sum _{i=k+1}^m{a}_i·v_{yi}}{A·\sum _{i=1}^k{a}_i}}}$$

  (29)

  Equation (29) is obtained after finding the first derivative of the function k_p = k_p(v_y) in the interval [v_yk, v_yk+1], equating the obtained result to zero and solving for v_y:
• For the intervals for which the condition in Equation (30) is met

  $$v_{yk}< v_{ykopt}<v_{yk+1}$$

  (30)
• The local minima k_{pk min} are calculated using Equation (31) after replacing v_y with v_ykopt from Equation (29) in Equation (22):

  $$k_{pk min}={\displaystyle \frac{1}{B}}{\left(\sqrt{A·{\sum }_{i=k+1}^m{a}_i·v_{yi}}+\sqrt{C·{\sum }_{i=1}^k{a}_i}\right)}^2$$

  (31)
• For the intervals for which condition (30) is not met, the boundary values k′_pk are determined at v_y = v_yk using Equation (32):

  $$k_{pk}^{\prime }={\displaystyle \frac{A·v_{yk}+C}{B}}·\left({\displaystyle \frac{1}{v_{yk}}}·{\sum }_{i=k+1}^m{a}_i·v_{yi}{\sum }_{i=1}^k{a}_i\right)$$

  (32)
• The loading tare coefficient k_p is calculated for the boundary value between zone II and III (Figure 3) at v_y = v_ym using Equation (33):

  $$k_{pk}^{″}=A·v_{ym}+C$$

  (33)
• The local minima from Equations (31)–(33) are compared, with the absolute minimum k_{p min} corresponding to the smallest of them using Equation (34):

  $$k_{p min}=\min \left(k_{pk min},{ k}_{pk}^{\prime }, k_{pk}^{″}\right)$$

  (34)

7.

The optimal value of the specific volume v_yopt corresponds to the absolute minimum of the loading tare coefficient k_{p min}.

Similarly, the optimal value of the specific area of the wagon body f_{y opt} can be obtained. For this purpose, it is necessary to determine the height of the cargo, use the coefficient A′ from Equation (24) instead of A and replace the notation v with f in Equations (28)–(32).

Based on the above algorithm, the calculation program for optimization of the specific volume and specific area of freight wagons has been developed at the Department of Railway Engineering at the Technical University of Sofia. This optimization procedure is proposed to domestic wagon manufacturers when developing new freight wagons.

3.3. Methodology for Determining the Parameters Volume, Tare Weight and Load Capacity of the Optimized Wagon

After finding the optimal value of the specific volume/area of the newly designed universal freight wagon, it is possible to determine with sufficient accuracy the parameters geometric volume V, tare weight T and load capacity P.

3.3.1. Determining the Geometric Volume of the New Freight Wagon

The geometric volume of the newly designed freight wagon is determined using Equation (35):

$$V_{new}=\zeta ·V_{prot}$$

(35)

where ζ is a correction coefficient and V_prot is the volume of the prototype wagon. The correction coefficient ζ is determined by the ratio between the optimal specific volume v_{y opt} of the new wagon and the specific volume of the prototype wagon v_{y prot} using Equation (36):

$$\zeta =1-{\displaystyle \frac{v_{y opt}}{v_{y prot}}}$$

(36)

where v_{y prot} is calculated using Equation (10) after substituting the real values V_prot and P_prot of the prototype wagon.

3.3.2. Determining the Tare Weight and Load Capacity of the New Freight Wagon

The tare weight of the new freight wagon is determined using Equation (37), considering Equation (23):

$$T_{new }= V_{new}· t_y+ T_p$$

(37)

The load capacity of the new wagon is determined using Equation (38), considering Equation (5):

$$P_{new}=n· p_o- T_{new}$$

(38)

The final values of V_new, T_new and P_new are obtained by iteration until the correction coefficient ζ from Equation (36) becomes approximately equal to zero. It should be noted that the iteration process is rapidly convergent and ends already on the second or third iteration.

From a theoretical point of view and to obtain the most accurate solution possible, iteration is appropriate. The reason for this is that when determining the optimal value of the specific volume v_yopt, a change occurs in the coefficients A and C, as well as in the relative variable components of the tare weight per unit volume or area t_y. For this purpose, the values of the wagon obtained in iteration 1 are substituted as the values of the prototype and the data for iteration 2 are obtained.

From a practical point of view, optimization through subsequent iterations is not worth carrying out because the values obtained in the first iteration are very close to the values calculated in the subsequent ones. They differ in the order of 1–2 percent and are within the engineering error.

The results obtained using the above method for the tare weight and load capacity of the newly designed freight wagon T_new and P_new will certainly undergo some corrections in the design process. The reason for this is that their obtainment is based on the appropriate selection of an existing prototype, the design solutions adopted by the designers, the theoretical study of the strength of the supporting structure, the materials used in the design and other features. Nevertheless, the method allows us to create a new freight wagon structure that meets the requirements for the best use of the volume and load capacity of the newly designed freight wagon.

4. Results and Discussion

The proposed method was used to optimize the technical parameters of a real open wagon series Eamnos, used by the national carrier Bulgarian State Railways—Freight Transport Ltd. Sofia, Bulgaria (BDZ-TP). Its main dimensions and parameters are given in Figure 5 and in

Table 2. Main technical parameters of open wagon series Eamnos.

Parameter	Unit	Value
Number of axles	-	4
Axle load	t/axle	20
Maximum load capacity	t	61
Tare weight	t	19
Pivot distance	m	6.5
Wagon floor area	m2	30
Geometric volume of the wagon	m3	56

.

Data on transported cargo from the year 2022 were obtained from the BDZ-TP website [28] and the National Statistical Institute [29]. They are summarized and presented in

Table 3. Data on transported cargo and calculated parameters.

Cargo No.	Unit	1	2	3	4	5	6	7	Total
Cargo type	-	Solid mineral and plant-based fuels	Ferrous metal ores	Scrap	Limestone	Inert materials	Copper ore	Other	-
Density γi	t/m3	0.8	1.6	2	1.4	1.5	1.8	-	-
Angle of repose αi	Degree °	20	22	0	25	25	22	-	-
Cargo weight Mi	103 t	320.24	732.36	440.06	314.6	734.06	452	103.45	3096.77
Average transport work Si = Mi·Li	106 t·km	108	221.42	133.06	100.04	233.4	174.28	193.45	1163.65
Volume utilization coefficient φmax	-	1.136	1.151	1	1.174	1.1748	1.151	-	-
Relative share in cargo turnover ai	-	0.093	0.190	0.114	0.086	0.201	0.150	0.166	1
Specific volume for the full use of the wagon capacity and volume vyi	m3/t	1.100	0.543	0.5	0.608	0.567	0.482

. The correct approach to effectively apply the proposed methodology from Section 3 is that the beneficiary should make a long-term (10–15 years) analysis of the transported cargo and forecast the trends for their change. The reason for this recommendation is that the life cycle of freight wagons is between 30 and 40 years. The last three rows of the table contain the results of the calculated values of the wagon volume utilization coefficients φ_imax, the relative share in the total freight turnover for each of the transported goods a_i, and the specific volumes v_yi, necessary for full utilization of the wagon’s load capacity and volume for each cargo. In Table 3, it is striking that six groups of cargoes have been analyzed. Groups 1, 2 and 5 include many types of cargoes and it is characteristic of each group that they have the same or very close values of the density parameter. The last group, the seventh (column 7), includes all types of cargoes that are transported with the given type of wagons and have an insignificant relative share in the transport activity of the carrier or for which statistical data are missing. The smaller the relative share of these cargoes in the transport work, the more effective the structure will be created.

In accordance with the algorithm from Section 3.2.2, the absolute minimum of the loading tare coefficient k_p (0.256) and the corresponding optimal value of the specific volume v_{y opt} (0.6079) have been determined. The data from the calculations are presented graphically in Figure 6.

Using the methodology from Section 3.3, the optimal values of the volume V_new, the tare weight T_new and the load capacity of the new wagon P_new have been determined. The data from the calculations are given in

Table 4. Calculated values for optimized wagon.

Parameter	Unit	Values for Prototype Wagon	Values for Optimized Wagon Iteration 1	Values for Optimized Wagon Iteration 2	Values for Optimized Wagon Iteration 3
Geometric volume of the wagon V	m3	56	37.086	38.000	37.957
Maximum load capacity P	t	61	62.520	62.427	62.432
Tare weight T	t	19	17.480	17.573	17.568

.

The analysis of the data from Table 4 shows that already in first iteration lower value for tare weight and higher value for load capacity are achieved compared to third iteration. For the first iteration, the corresponding optimal value of the specific volume v_{y opt} is 0.59, which differs from optimal value from Figure 6 (0.6079). This is a reason why the following of two iterations is needed to determine the final optimal value of v_{y opt}. Data in Table 4 shows also that the values in iteration 1 and iteration 3 differ by 2.3% for volume, 0.5% for tare weight and 0.1% for load capacity. This is within the engineering error and in practice it is not worth performing. The optimized design of the new wagon has a lower tare weight T by 1.432 t, which increases the maximum load capacity P by the same amount. The reduction in the tare weight is at the expense of the height of the wagon (reduced by 0.6 m) while maintaining its width and length. This approach was used due to the existing regulatory restrictions on the pivot distance parameter (6.5 m is the minimum pivot distance for freight wagons with bogies).

The financial benefits for the carrier and the shippers can be determined trivially by considering the total weight of the transported cargo (see Table 3) and the optimized wagon load capacity. For the transportation of total 3,096,770 tons of cargo with a non-optimized wagon (61 tons load capacity), approximately 50,766 wagons are needed, and for the same cargo, approximately 49,602 optimized wagons (62.432 tons load capacity) are needed. From this, it follows that the carrier will need 1164 fewer wagons on an annual basis or approximately 3 fewer traveling wagons daily to perform the same work. Provided that an average statistical train consists of 20 wagons, the carrier will reduce the number of trains in motion per year by 55. Converting these data sets into real money would not be correct, as access fees to the railway infrastructure of EU countries vary widely [13] from 0.5 to 11.5 EUR/train km.

5. Conclusions

In this study, a comprehensive review of the parameters of freight wagons and their functional relationships has been made. The dependence between them and their influence on the profitability of transport companies has been studied. It has been found that the criterion proposed in the literature for optimization of the specific volume and specific area (adjusted costs) is inapplicable in modern economic conditions.

New requirements have been formulated for the criterion and it has been proposed to use the loading tare coefficient indicator as such. This criterion depends solely on the parameters of the wagon and the physical properties of the transported cargo, ensures maximum use of the permissible gross weight of the wagon by rail and has a direct impact on the costs of providing the transport service.

A methodology and algorithm for optimization of the specific volume/area have been developed. They are based on the proposed method for finding the minimum of a partially continuous function.

A methodology for determining the geometric (useful) volume of the optimized wagon, its tare weight and maximum load capacity has been developed. For this purpose, the iteration method was used. It was found that the process is highly convergent and that the exact solution is obtained already on the second or third iteration.

The proposed new methods have been applied to optimize the parameters of a real open wagon series Eamnos from the fleet of the national carrier of Bulgaria—BDZ-TP Ltd.—using official data on the type of cargo and the work performed during transportation. As a result of the study, it was found that it is appropriate to reduce the volume of the wagon by 18 m³, which leads to a decrease in its tare weight by 1.342 t and an increase in the load capacity by the same amount. The optimized wagon can perform the same transportation work, while the number of trains for its implementation is reduced by 55 on an annual basis.

The proposed methodologies and algorithms can be used to optimize the technical parameters of all categories of universal freight wagons—flat, open and covered—which will lead to a significant reduction in the costs of performing the transportation work. For this purpose, the railway operator/client needs to determine the regions in which it will operate; to collect the necessary statistical data and to study the trend of changes in the expected quantities of cargo (type, mass, mileage, density, etc.); and to use a real prototype wagon with parameters close to the new design. This will lead to a significant reduction in the costs of performing the transportation work. The above data are strictly specific, depending on the commercial policy of each carrier. This requires an individual approach when determining the technical parameters of each universal wagon.

The research creates prerequisites for developing new methods related to the optimization of the linear dimensions of freight wagons—width, length, height, pivot distance, length of the end part and others.