using SOLUTION OF THE OPTIMIZATION PROBLEM ON THE CONTROL OVER OPERATION OF GAS PUMPING UNITS UNDER FUZZY

A network of gas pipelines that run through the territory of Ukraine ensures continuous supply of gas both to enterprises of the country and for export to the countries of Central and Eastern Europe. As present [1], the total length of gas pipelines reaches 34.8 thousand km. The estimated through-put capacity of gas transportation system when entering Ukraine is 288 billion m3 per year (800 million m3 per day). Design capacity of the gas transportation system (GTS) of Ukraine in the direction of countries of Western Europe and Turkey is 142 billion cubic meters per year while the actual volume of transit in 2014 was 60 billion cubic meters [2]. Thus, the Ukrainian GTS has excessive capacity, which makes it a relevant task to choose the number of parallel working units at each compressor station and their rotation frequency, based on the selected criteria for performance efficiency of the groups of parallel units.


Introduction
A network of gas pipelines that run through the territory of Ukraine ensures continuous supply of gas both to enterprises of the country and for export to the countries of Central and Eastern Europe. As present [1], the total length of gas pipelines reaches 34.8 thousand km. The estimated through-put capacity of gas transportation system when entering Ukraine is 288 billion m 3 per year (800 million m 3 per day).
Design capacity of the gas transportation system (GTS) of Ukraine in the direction of countries of Western Europe and Turkey is 142 billion cubic meters per year while the actual volume of transit in 2014 was 60 billion cubic meters [2].
Thus, the Ukrainian GTS has excessive capacity, which makes it a relevant task to choose the number of parallel working units at each compressor station and their rotation frequency, based on the selected criteria for performance efficiency of the groups of parallel units.

Literature review and problem statement
In papers [3][4][5][6][7][8][9], authors stated and solved problems of the optimal redistribution of flows between parallel working units. For the given cases, the authors apply different approaches to solving such problems. Article [3] sets the task to minimize consumption of fuel gas for the selected section of a gas pipeline, which has a certain number of compressor stations. The problem is set for the assigned pressures at the inlet and outlet of each element of a gas pipeline on finding such controlling actions so that a cost criterion acquires the minimum value. In order to solve the set problem, the authors employed a deterministic model disregarding an interaction between the system and the external environment, as well as neglecting a constraint for the controlling actions.
Paper [4] addresses the role of technical condition of GPU at periodic maintenance that affects the total cost of gas compression, however, the authors fail to take into account the impact of the environment on the technical condition of GPU, moreover, they chose a deterministic model.
Characteristics of productivity and energy efficiency of parallel working GPU are covered in [5], but the model that is employed in the paper does not consider significant errors of instruments for measuring performance of compressors.
The task of selecting the optimal mode for the operation of compressors is tackled in article [6], where it is solved by the redistribution of gas flows through GPU provided the effective overall efficiency of a group of parallelly connected units has the maximum possible magnitude. This work did not consider any other circuits for connecting GPU except for the parallel one.
In order to increase performance efficiency of a compressor station, author of paper [7] points to the need to stabilize productivity of parallel working compressors using

M . G o r b i y c h u k
PI-controllers. The author rightly indicates that the problem on stabilizing the performance of compressors is rather controversial, while achieving the lowest consumption of fuel gas requires a certain technique to distribute the load on working units, however, the model presented does not account for the fuzziness of technological parameters.
In article [8], authors developed an integrated algorithm of the optimal load on GPU for the optimal operation of CU. However, a decision on the minimization of costs is made by using a consistent computer-based brute-force method for the operational modes.
A slightly different approach to solving the task of optimal distribution of flows between the elements of a gas transportation system is proposed in paper [9]. In order to solve it, the authors apply methods of linear programming and the theory of networks taking into account constraints predetermined by the material balances of flows. Since the problem is determined, so the authors neglected such issues as the impact of the environment on the work of GTS, technical condition of separate units of the system, the effect of measurement errors on a decision-making process, etc.

The aim and objectives of the study
The aim of present work is to solve a problem on the optimal control over operation of parallel working gas pumping units under a condition that the total consumption of fuel gas is minimal, while satisfying the constraints for technological parameters and taking into account a significant error in the measurement of pressure drop on confuser.
To accomplish the aim, the following tasks have been set: -to determine a membership function for the productivity of a compressor; -to construct mathematical models and to formalize a problem on the optimal control over the process of natural gas compression; -to determine rotation frequency of the rotors of centrifugal compressors of natural gas.

Formalization of a problem on the optimal control over the process of natural gas compression
The total costs of operating compressors at a compressor station with a gas-turbine drive can be calculated from the following formula: where ( ) J n is the cost of work m of parallel operating units relative to the unit of time; c g is the cost of volumetric unit of the fuel gas used to drive a gas-turbine unit; G(n i ) is the fuel gas consumption, relative to normal conditions, which is used by the i-th GPU; n i is the frequency of the rotor of the i-th compressor.
In line with the technological regime, it is necessary to limit gas temperature at the outlet from compressor T out and the temperature of combustion products at the outlet of a low-pressure turbine -T v . For the case of a pump-free work, it is necessary to limit lower frequency of the compressor's rotor [10]: are the lower and upper limit for the compressor's rotor frequency. The Temperature of exhaust gases ( ) i T ν of the i-th GPU, gas temperature at the outlet from the i-th compressor ( ) i out T are the functions of such technological factors as the rotor's frequency n i , temperature T in and gas pressure P in at the inlet to compressor, a degree of pressure increase ε, and depend on the parameters of the environment -atmospheric pressure P c and temperature T c When constraints (2)-(4) are satisfied, a requirement must be fulfilled to ensure the preset performance efficiency Q of the compressor station where Q i is the performance efficiency of the i-th compressor. Productivity of the i-th compressor is a function of such parameters as n i , P in , T in , ε, P c and T c ( ) ( ) , , , , , .
Dependences (5), (6) and (8) are given in paper [11] as empirical models obtained based on the results of observation of work of GPU. Approximation of the results of observation was carried out by a polynomial of degree r where M is the number of terms of the polynomial; a i are the coefficients of the polynomial; s ij are the degree of arguments, which must satisfy the constraint; k is the number of independent variables The number of terms M in polynomial (10) is determined from formula [12] ( )! .
Since dependences (5), (6), and (8) are the functions of technological parameters P in , T in , ε and environmental parameters P c and T c , which are known, then, by substituting them in the derived regression equations that take the form of polynomial (9), we shall obtain dependences ( ) These dependences will be the functions of only one variable n i . Constraints (2) and (3) assign the upper bound of temperature values ( ) As a result, we obtain the equations whose positive roots will be magnitudes ( ) Thus, we shall solve the following problem on the optimal control over the work of GPU: Using a shop automation, performance efficiency of the compressor was determined by measuring pressure drop i P ∆ at the inlet confuser with subsequent calculation using the following formula [13]: A is the coefficient of volumetric productivity of confuser of the i-th compressor; i ρ is the gas density at the inlet to a compressor.
Calculation of volumetric productivity Q i of the centrifugal compressor (CC) using formula (15) produces an error of tens of percent [14].
In order to obtain empirical models, by employing results of observations over the work of GPU, we applied a method of synthesis of empirical models based on genetic algorithms [15]. The essence of the method is that it is required to choose a polynomial of degree r, in which part of the coefficients is assigned with a value of zero while the rest of the coefficients are different from zero.
We shall construct an ordered structure of length M, where the i-th place is taken by unity or zero depending on whether parameter , (9) differs from zero, or is zero.
The problem of synthesis of an empirical model will be stated as follows: it is required to choose by evolutionary selection such a chromosome from the initial population of chromosomes, which provides the best value for the adaptation function (a minimum value of the selection criterion).
In order to choose a model from a set of all possible models, the set of experimental values is divided into two partstraining and testing. The training part of the total set is used to calculate parameters of the model, while the testing one is employed in order to select the best model out of a given set of models. Such a selection is implemented using the criteria of regularity, or shift [11,15].
We shall use a method of synthesis of empirical models based on genetic algorithms. We shall obtain a model of dependence (8) in the form of a polynomial (9). Substituting numerical values of magnitudes n i , P in , T in , ε, P c and T c , which are measured by means of the shop automation, we shall obtain a polynomial in the form: Given the fact that measurement of the productivity of the i-th compressor yields a significant error, then there is every reason to consider coefficients of model (16) fuzzy numbers. We shall assume that a membership function ( ) ( ) i k a µ of the fuzzy magnitude ( ) i k a takes a triangular shape (Fig. 1). In Fig. 1, the following denotations are adopted   z z z z z z z z z z z A triangular membership function is inconvenient for practical use. This is explained by its piecewise-linear form, where there are no derivatives at points A, B, and C (Fig. 1).
We shall approximate a piecewise-linear membership function (17) by exponential function Parameter 2 , σ which is the parameter of fuzziness concentration, will be chosen such that the membership function (18) passes through point 1 B with coordinates ( ) 1 ;1 2 A B z (Fig. 1). Let us find abscissa .
A z To do this, we shall consider triangles ABD and AB 1 A 1 . We obtain from the condition of similarity of these triangles  In this case, one should bear in mind that when one performs operations of addition and when multiplying a Gauss number by a clear magnitude, we shall again obtain the Gauss number [8], which means that

Considering the values of
Parameters ( ) i q m and ( ) i q σ of the membership function (20) will be found by using the rules of arithmetic operations on fuzzy numbers of the (L-R)-type in the Gauss basis [16,17].
Following the structure of model (16), in order to determine parameters ( ) It is obvious that , , .
We obtain  In the case when a fuzzy number is multiplied by a clea number, the membership function remains Gaussian [16].
Let a fuzzy number z be multiplied by clear number . For function (20), we shall set a γ-cut. As a result, we receive equation Let us introduce the following denotation: and take into account the value of ( ) , i q m which is computed from formula (24). Then Q n a n K a n Thus, we have obtained the following problem on the optimal control over a group of parallel working compressor units: By solving the optimization problem, we obtained the following results: n 1 =3,200 rpm; n 2 =3,550 rpm.

Discussion of results of examining the optimization of the process of natural gas compression
Existing methods for solving optimization problems on the control over the process of natural gas compression employ deterministic models. However, the inaccuracy of measuring technological parameters allows us to consider performance of the compressor as a fuzzy magnitude with a triangular shape of the membership that makes it possible to build an adequate mathematical model of the process of natural gas compression.
A triangular shape of the membership function is inconvenient when used to solve the problem on the optimal control over the process of natural gas compression. That is why, in the present work, we proposed approximating it with a Gaussian membership function. This allowed us to state the optimization problem in terms of fuzzy magnitudes and, based on it, to develop effective methods for its solution with consideration of constraints for controlling actions. An introduction of fuzziness to the optimization problem changed a structure of the constraint, which determines a balance of gas flows through parallel-connected compressors. As a result, there appears an additive component, which is a kind of "payment" for the fuzziness in measuring productivity of compressors. The presence of such a component transforms a determined problem of discrete integral linear programming into a nonlinear discrete programming problem, which significantly complicates the process of solving such an optimization problem. It is natural that the introduction of fuzziness to the optimizing problem somewhat impairs efficiency of the process of control and such a deterioration can vary within 3-5 % compared with a problem in the determined statement, which does not take into account the specificity of measuring compressors' productivity given the existing methods and technical means. Since technological parameters of the gas pumping unit change over time, there is a need to recalculate the model's parameters in order to solve the optimization problem on the process of natural gas compression. Determination of frequency of recalculation of the model's parameters is an unsolved scientific problem at present and it is a promising task for further research.

Conclusions
Performance efficiency of centrifugal compressors are considered as fuzzy magnitudes with triangular membership functions, approximated by exponential functions, with their parameters calculated, which allowed us to formalize the problem on the optimal control over the process of natural gas compression taking into account the fuzziness in the productivity of compressors and constraints for controlling actions. We developed algorithmic provision and software in the MATLAB environment and solved the problem on the optimal control, which made it possible to determine rotation frequency of the rotors of centrifugal compressors of natural gas, at which total costs of the fuel gas for the natural gas compression are minimal.