UNLOADED OIL-SUBMERSIBLE CABLE REDUCED DYNAMIC MODEL ORDER OPTIMIZATION BASED ON THE FREQUENCY RESPONSE APPROXIMATION

Abstract

The relevance. An oil-submersible cable transfers electrical energy from an energy source to submersible electric motors of oil production plants. Resonant peaks localized in a narrower frequency band characterize the frequency response of an unloaded cable, which allows us to correctly optimize the order of the reduced cable line model based on the minimum error criterion. The principal methods of mathematical description of the oil-submersible cable model are systems of ordinary differential equations and systems of partial differential equations. The first method of mathematical description of the model of an oil-submersible cable comprising one elementary four-pole with a longitudinal active-inductive and transverse active-capacitive components allows calculating the power balance and the energy transfer at the first harmonic, considering losses on the ohmic (active) resistance, as well as losses in cable insulation. Using a mathematical model comprising one elementary four-pole is unacceptable for more complex tasks, such as diagnostics of the cable damage site, identification and evaluation of the parameters of the submersible engine based on ground measurements, control of the transient process of the submersible engine during start-up, and others. The second method of mathematical description of the oil-submersible cable model is based on «telegraphic» equations as partial differential equations. The main advantage of such a mathematical model is the increased accuracy of the analysis of specific modes inherent in long electric lines, such as forward and reverse waves, resonant phenomena that occur during the mutual exchange of energy between the electric and magnetic components of the electromagnetic field. The disadvantages include the technical difficulties of combining a mathematical model of a cable based on a partial differential systems of differential equations and a mathematical model of a submersible engine based on a system of differential equations in the normal Cauchy form. Also, it is extremely difficult to implement models of long electric lines based on «telegraph» equations in practice in a microcontroller for real-time systems built based on digital signal processes. From the point of view of applying dynamic models in real-time digital systems, the transition from a mathematical model of a cable with distributed parameters to a mathematical model of a cable with concentrated parameters, i. e. to four-pole cables, is relevant. With such a transition, it is unacceptable to use a mathematical model represented by one link, since unacceptable inconsistencies arise between the work of a proper object and the mathematical model of this object. This does not allow us to reliably display the processes occurring in the cable when powered by a frequency converter. Using a model with an almost infinite number of four-poles is not advisable, since the calculation of such a model will take a large amount of processor time, which is not acceptable in real-time systems and negates the advantages of switching from a mathematical model with distributed parameters. Considering the above, determining the optimal order of the reduced dynamic model of an unloaded oil-submersible cable based on the approximation of its frequency response is an urgent, scientific and practically significant task. The main aim of the research is to develop a method for determining the minimum required and sufficient number of links of a reduced mathematical model of an oil submersible cable with lumped parameters for its operation with an accuracy in transient and steady-state operating modes. Methods: system of differential equations in Cauchy normal form, system of partial differential equations, nonlinear algebraic equations, Cramer's method, numerical methods, theory of four-poles, theory of long lines with distributed parameters, state space method, methods of frequency analysis, optimization methods. Results. The authors have got a reduced dynamic model, comprising the minimum required and sufficient number of two-port networks, which makes it possible with an accuracy to analyze transient and steady-state electromagnetic processes along the length of the cable line and at various forms of voltage at the input. The resulting reduced model applies for identification of cable and submersible motor parameters; constructing observers of variable states of a submersible electric motor; diagnostics of the integrity of electrical circuits; determination of stress distribution along the length of the cable line. The resulting reduced dynamic model allows as well evaluating the entire range of dynamic modes of operation, in contrast to the prototype model. The developed reduced dynamic model of a submersible cable, described in ordinary differential equations, is a convenient notation for the mathematical description of the cable as a subsystem of the electrical complex of an electric centrifugal pump installation.

Keywords

• long cable
• cascade connection of four-port networks
• reduced model of a cable line
• long line with distributed parameters
• frequency response
• oil submersible cable
• electrical submersible pump