Content

Introduction 1. Relevance of the topic 2. Aims and tasks of the research 3. Current status of the problem 3.1 Analysis and comparison of existing methods, the construction of an optimal route for laying a gas pipeline 3.2 Analysis and comparison of implemented software tools the construction of an optimal route for laying a gas pipeline 4. Statement of the mathematical model Conclusions References

Introduction

In connection with the development of the ubiquitous gasification of the country, as well as an increase in the pace of erection of low-rise, small-apartment and manor buildings in the territories of small settlements, questions arise of optimizing their gas supply systems. This factor determines the development and implementation of advanced engineering systems and equipment, which should provide the required sanitary and hygienic living conditions of the country's population and the necessary level of landscaping.

1. Relevance of the topic

Gas pipelines affect more subjects of infrastructure than other technical complexes and structures. The task of planning a route for laying pipeline networks includes the search for the optimal sequence of connecting territories of cities and other settlements, taking into account the restrictions determined by the demand indicator in each of them and the characteristics of the pipes. As a criterion for laying such a route, you can choose the minimum cost or maximize the profit of the operation of the pipeline [1].

Before starting the construction of the pipeline, the process of designing the future route is underway. At the stage of her selection, the foundation is laid for the profitability and reliability of the future transport route, since a set of tasks is solved that is associated with minimizing financial costs, taking into account the conditions of construction, limiting the construction time, reliability of the gas pipeline, and environmental protection.

From a huge number of factors affecting the position of the future route, designers need to single out the most important ones in order to take them into account when determining the so-called general direction of the route.

Thus, the choice of route is the first significant step in the design and construction of the gas pipeline, this step can have a significant impact on the deployment and operation of the gas pipeline as a whole, therefore, the optimization of this process can significantly affect the financial and material resources necessary to complete this task [2].

2. Aims and tasks of the research

The aim of the master's thesis is to research and develop a method for constructing the optimal route for laying a gas pipeline in a community.

To achieve this goal, you must perform the following tasks:

• analyze the current state of the problem of constructing an optimal route for laying a gas pipeline;
• analyze and compare existing algorithms and methods for finding the optimal path;
• to analyze and compare the implemented software for finding the best way;
• formulate a mathematical model;
• develop software, conduct testing;
• explore the effectiveness of the developed algorithm.

3. Current status of the problem

Pipeline construction, undoubtedly, requires long-term planning, including management decisions to determine the route, as well as effective protection of soils, water and people. A well-planned, reliably constructed, properly operated pipeline with proper maintenance and repair is a reliable and environmentally friendly means of transporting liquid and gaseous hazardous substances. Route planning and environmental studies are one of the most costly stages, therefore the success of the investment project as a whole depends on the effectiveness of their implementation. The task of planning a route for laying pipeline transport networks includes the need to find the optimal sequence of combining the territories of cities and other settlements according to the criterion of minimum costs (or, for example, maximizing profits), taking into account the restrictive conditions determined by the demand indicator in each of them and pipe characteristics [3].

Consider the well-known works and methods devoted to the construction of the optimal route for laying a gas pipeline in a settlement.

3.1 Analysis and comparison of existing methods, the construction of an optimal route for laying a gas pipeline

Local search (optimization)

In computer science, local search is a heuristic method for solving computationally difficult optimization problems. Local search is used for tasks that can be formulated as a search for a solution to the problem of maximizing the criterion of a number of possible points. Local search algorithms go from point to point in the space of feasible solutions, applying local changes until the optimal solution is found or a time limit is reached.

Local search algorithms are widely used in complex computational problems, including computer science (in particular, artificial intelligence), mathematics, operations research, engineering, and bioinformatics.

Heuristic Insertion Methods

The best solution for specific source data can be found by successively applying various heuristic methods, using for approximating the quality of approximation of the length of the obtained route. Consider the 4 most popular heuristic algorithms:

• Nearest Neighbor method
• Nearest Town method
• Most Cheap Inclusion method;
• Minimum Spanning Tree method.

In the closest neighbor method, the points of the plan are sequentially included in the route, and each next included point must be the closest to the last selected point among all the others that are not yet included in the route.

The method of the nearest city at each step of the algorithm builds a valid route for the current subset of points already included in the route, adding to it a new point from among those not yet included in the route for which there is a closest neighbor from the number of points already belonging to the route.

The method of the cheapest inclusion at each step of the algorithm conducts an acceptable route along the current subset of points already included in the route, adding a new point to it, the inclusion of which between some adjacent points leads to a minimum increase in the cost (length) of the route. However, any heuristic method is based on formally unsubstantiated considerations; therefore, it is impossible to prove that the heuristic algorithm for any initial data finds solutions close to optimal.

Clark-Wright Method

The Clark-Wright method was developed by scientists from Britain, G. Clark and J.V. Wright. The Clark-Wright method is one of the most commonly used heuristic methods used to find solutions to the transport problem. This method is one of iterative, approximate methods. Most often, this method is used to solve the transport problem using computers. In order to evaluate the operation of merging two routes, the term win is introduced. The measure by which the cost is reduced when combining two routes into one route is called a win. Scalability, flexibility and simplicity are the advantages of the Clark Wright method, and the error of the solution does not exceed 5-10%. Due to the greedy nature of the algorithm, the resulting solutions are inferior to the solutions obtained by a more complex algorithm. In problems with many constraints, after the first few iterations, the probability of route merges can be greatly reduced, as a result of which the ability to control the number of routes is lost [4].

3.2 Analysis and comparison of implemented software tools the construction of an optimal route for laying a gas pipeline

To date, the implemented software products for constructing an optimal gas pipeline route do not exist. However, this system was developed on the basis of the ant algorithm, which is implemented in the same software tools: Maxoptra, Zig-Zag, Ant logistics.

Maxoptra

Service for the management of urban delivery logistics. Automatic route planning taking into account time windows, traffic jams, cargo volumetric characteristics, transportation requirements, vehicle equipment, drivers and couriers work schedules. A mobile application for the driver and an online service for the dispatcher.

Zig-Zag

Service for building an optimal route. Automatic allocation of addresses by vehicles. Determining the optimal shipment order. Display on the map how the execution of applications takes place.

Ant logistics

The program of automation of transport logistics is a convenient tool for a logistician and an assistant manager in the management of an enterprise. Makes the route planning process convenient and efficient [5].

4. Statement of the mathematical model

The choice of the gas pipeline construction project is proposed to be carried out in 2 stages.

The first stage is finding the route least costly, the second stage is choosing the minimum cost gas pipeline project taking into account its characteristics.

To solve the problem of finding the route least costly, a graph was proposed, the nodes of which are gas control units (GRU), and the arcs between the nodes are the cost of laying a gas pipeline taking into account the local landscape, that is, the cost of construction work on the site С_ij (see Figure 1).

The engineer puts the estimated location of the GRU (nodes of the graph). To do this, he uses data that to some extent depend on the position of the future gas pipeline and the environmental conditions in which it may find itself:

• hydrogeological and climatic data of the route;
• the number of large and medium rivers, swamps, lakes, railways and roads, the intersection of which is possible during construction;
• the density of settlements and the presence of parallel roads and railways, river and sea routes;
• the presence of existing pipelines and other extended structures (power lines, cable trunk lines, etc.) passing along the proposed route;
• the quantity and quality of cultivated agricultural land (arable land, orchards, meadows), the intersection of which, taking into account restoration and alienation, increases the cost of the pipeline.

It is these factors in most cases that determine both the general direction and the detailed laying of the route on the ground.

Then, using the classical ant algorithm, the route with the minimum cost of work is calculated [6].

Let the ant be in node i , and node j – is one of the nodes available for transition: j∈S_i . We denote the weight of the edge connecting the nodes i and j , as w_ij , and the intensity of the pheromone on it as – как t_ij . Тhen the probability of the transition of the ant from i to j will be equal to:

where α and β – are adjustable parameters that determine the importance of the components (rib weight and pheromone level) when choosing a path. Obviously, for α=0, the algorithm turns into a classical greedy algorithm, and for β=0 it quickly converges to some suboptimal solution. The choice of the correct ratio of parameters is the subject of research, and in the general case is made on the basis of experience.

After the ant successfully passes the route, it leaves a trace on all the edges traveled, inversely proportional to the cost of the distance traveled:

where С – is the cost of laying pipes taking into account the landscape, and k – is an adjustable parameter. In addition, pheromone traces evaporate, that is, the pheromone intensity on all edges decreases at each iteration of the algorithm. Thus, at the end of each iteration, it is necessary to update the intensities:

Mathematically, this can be represented as:

Graphically, this can be represented as:

For safe and trouble-free operation of the gas supply, it must be designed and calculated. It is important to perfectly select pipes for highways of all types of pressure, ensuring a stable supply of gas to the devices. To make the selection of pipes, fittings and equipment as accurate as possible, in the second stage, hydraulic calculation is used to calculate the cost of the gas pipeline project.

Any hydraulic calculation performed is a determination of the parameters of the future gas pipeline. This procedure is mandatory, as well as one of the most important stages of preparation for construction. The correctness of the calculation determines whether the gas pipeline will function in optimal mode.

During each hydraulic calculation, the following is determined:

• the required diameter of the pipes, which will ensure efficient and stable transportation of the right amount of gas;
• whether pressure losses when moving the required volume of blue fuel in pipes of a given diameter will be acceptable.

Pressure losses occur due to the fact that in any gas pipeline there is hydraulic resistance. With an incorrect calculation, it can lead to the fact that consumers will not have enough gas for normal operation in all modes or at the moments of its maximum consumption.

The capacity of gas pipelines can be taken from the conditions of creating, with the maximum allowable loss of gas pressure, the most economical and reliable system in operation, ensuring the stability of hydraulic fracturing and gas control units (GRU), as well as the operation of consumer burners in acceptable gas pressure ranges. The calculated internal diameters of the gas pipelines are determined on the basis of the conditions for ensuring uninterrupted gas supply to all consumers during peak hours of gas consumption. Hydraulic calculation is performed according to the formulas below [7].

Estimated pressure losses in high and medium pressure pipelines are accepted within the pressure category adopted for the gas pipeline. The calculated total loss of gas pressure in low pressure gas pipelines (from the gas supply source to the most distant device) is accepted no more than 180 daPa, including 120 daPa in distribution pipelines, 60 daPa in inlet pipelines and internal gas pipelines. The values ​​of the calculated loss of gas pressure during the design of gas pipelines of all pressures for industrial, agricultural and domestic enterprises and public service organizations are taken depending on the gas pressure at the connection point, taking into account the technical characteristics of the gas equipment accepted for installation, safety automation devices and process control automatics thermal units.

The pressure drop in the gas network section can be determined:

- for medium and high pressure networks according to the formula

where

P_н – absolute pressure at the beginning of the gas pipeline, MPa;

P_к – absolute pressure at the end of the pipeline, MPa;

P₀ =0,101325 MPa;

λ – coefficient of hydraulic friction;

l – design length of a gas pipeline of constant diameter, m;

d – gas pipeline inner diameter, cm;

ρ₀ – gas density under normal conditions, kg/m;

Q₀ – gas consumption, m/h, under normal conditions;

- for low pressure networks according to the formula

where

P_н – pressure at the beginning of the gas pipeline, Pa;

P_к – pressure at the end of the pipeline, Pa;

λ ,l ,d ,ρ₀ ,Q₀ – the notation is the same as in the formula (5).

The coefficient of hydraulic friction λ is determined depending on the mode of gas movement in the gas pipeline, characterized by the Reynolds number,

where ν – coefficient of kinematic viscosity of the gas, м²/с, under normal conditions;

Q₀ ,d – the designations are the same as in formula (5), and the hydraulic smoothness of the inner wall of the gas pipeline, determined by condition (8),

where Re – Reynolds number;

n – equivalent absolute roughness of the inner surface of the pipe wall, which is assumed to be equal to – 0,01 cm for new steel, – 0,1 cm for used steel, – 0,0007 cm for polyethylene, regardless of the operating time;

d – the notation is the same as in formula (5).

Depending on the value of Re, the coefficient of hydraulic friction λ determined by:

- for laminar gas flow 2000

- for critical regime of gas movement Re =2000-4000

- at Re > 4000 - depending on the fulfillment of condition (8);

- for a hydraulically smooth wall (inequality (8) is true):

- at 4000 < Re < 100,000 according to the formula

- at Re >100000

- for rough walls (inequality (8) is unfair) at Re > 4000

where n – the designation is the same as in formula (8);

d – the notation is the same as in formula (5).

Using the hydraulic calculation formulas, we obtain the calculated pressure losses in gas pipelines for pipes of different diameters. Pipes are made of different materials (steel or plastic), which affects their cost, which also depends on the diameter of the pipe [8].

As a result, we obtain a decision-making problem with two criteria:

• С₁ – minimum total cost of pipes of various types and diameters;
• С₂ – minimum total pressure loss in the gas pipeline.

Any decision-making task requires determining a list of alternatives, and obtaining a utility matrix that evaluates alternatives for each criterion. As alternatives, options are proposed for using pipes of different types and diameters for laying a pipeline on a selected route. In this case, pipes of different types and different diameters can be used on each individual section of the route. For example, if a polymer pipe with a diameter of 5 cm is laid in the area between 1 and 2 of the gas regulator, the losses will be 12.26 Pa. and the cost is 10,000 rubles., and if you take a steel pipe with the same dimeter, then the loss will be 10.19 Pa. but the cost will be 20,000 rubles. But in some areas you can lay a steel pipe, but not in others, or you can lay a polymer or steel pipe, but the losses and cost will be different. Thus, we obtain a certain number of alternatives and fill out the utility matrix, the elements of which are estimates of alternatives according to two selected criteria. The obtained problem can be solved by any of the multicriteria methods [9].

The following is an example of a utility matrix. The first column describes alternatives, for example, for an optimal route consisting of three sections, such alternative solutions for laying a gas pipeline are possible:

• А1 – T1D5T1D6T1D8;
• A2 – T1D8T1D7T2D5; etc.

Here Ti – is the type of pipe used for laying in the corresponding section, D – is the pipe diameter in cm. Two types of pipes are used: polymer (T1) and steel (T2)

Different methods can be used to solve this problem: the additive convolution method, the multiplicative convolution method, the assignment method, the target programming method, the main criterion method, the ranking method, etc [10].

Conclusions

In the framework of this article, the problem of finding the optimal gas supply route is formulated, a two-stage solution to the problem is proposed. For each stage, a mathematical model is developed. At the first stage, it is proposed to use the classical ant algorithm to determine the route that is minimal in cost of laying the gas pipeline taking into account the landscape. At the second stage, hydraulic calculation formulas were used to formalize the multicriteria task of deciding on the choice of the gas pipeline project. According to the results of the article, it is necessary to investigate methods for solving the problem obtained in the second stage.

References