title: Ð¡Ð¸Ð½ÑÐµÐ· Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð³Ð¸Ð´ÑÐ¾ÑÑÑÐ±Ð¸Ð½Ñ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Synthesis of a mathematical model of a hydraulic turbine
Creators: name: ÐÐ»Ð¸Ð¼ÐµÐ½ÑÐ¾Ð², ÐÐ½Ð´ÑÐµÐ¹
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2024
Subjects: Ð³Ð¸Ð´ÑÐ¾ÑÑÑÐ±Ð¸Ð½Ð°
Ð¼Ð¾Ð´ÐµÐ»Ñ
Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐ°
ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ðµ
ÑÐ½ÐµÑÐ³Ð¾ÑÐ¸ÑÑÐµÐ¼Ð°
hydraulic turbine
model
dynamics
control
power system

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐ¸ Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÑÐ°Ð´Ð¸Ð°Ð»ÑÐ½Ð¾-Ð¾ÑÐµÐ²Ð¾Ð¹ Ð³Ð¸Ð´ÑÐ¾ÑÑÑÐ±Ð¸Ð½Ñ, Ð° ÑÐ°ÐºÐ¶Ðµ ÑÐ¸Ð½ÑÐµÐ·Ñ ÑÐ¿ÑÐ°Ð²Ð»ÑÑÑÐµÐ³Ð¾ Ð²Ð¾Ð·Ð´ÐµÐ¹ÑÑÐ²Ð¸Ñ Ð´Ð»Ñ ÑÑÐ°Ð±Ð¸Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½ÑÑ Ð´Ð²Ð¸Ð¶ÐµÐ½Ð¸Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ. ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: 1) ÐÐ·ÑÑÐµÐ½Ð¸Ðµ Ð¿ÑÐµÐ´Ð¼ÐµÑÐ½Ð¾Ð¹ Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ð¸ ÑÐ¸Ð½ÑÐµÐ· Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð³Ð¸Ð´ÑÐ¾ÑÑÑÐ±Ð¸Ð½Ñ. 2) ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐ¸ ÑÐ¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð½Ð¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ. 3) Ð¡Ð¸Ð½ÑÐµÐ· ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð´Ð»Ñ ÑÑÐ°Ð±Ð¸Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½ÑÑ Ð´Ð²Ð¸Ð¶ÐµÐ½Ð¸Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ. 4) ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐ¸ ÑÐ¸ÑÑÐµÐ¼Ñ Ð¿Ð¾Ð´ Ð²Ð¾Ð·Ð´ÐµÐ¹ÑÑÐ²Ð¸ÐµÐ¼ ÑÐ¸Ð½ÑÐµÐ·Ð¸ÑÐ¾Ð²Ð°Ð½Ð½Ð¾Ð³Ð¾ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ. Ð ÑÐ°Ð±Ð¾ÑÐµ Ð±ÑÐ» Ð¿ÑÐ¾Ð¸Ð·Ð²ÐµÐ´ÐµÐ½ Ð¾Ð±Ð·Ð¾Ñ Ð¸ ÐºÐ»Ð°ÑÑÐ¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¸ Ð²Ð¸Ð´Ð¾Ð² Ð³Ð¸Ð´ÑÐ¾ÑÑÑÐ±Ð¸Ð½. ÐÐ°ÑÐµÐ¼ Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÐ°Ð´Ð¸Ð°Ð»ÑÐ½Ð¾-Ð¾ÑÐµÐ²Ð¾Ð¹ Ð³Ð¸Ð´ÑÐ¾ÑÑÑÐ±Ð¸Ð½Ñ Ð±ÑÐ»Ð° ÑÐ¸Ð½ÑÐµÐ·Ð¸ÑÐ¾Ð²Ð°Ð½Ð° Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ Ð² Ð¼Ð°ÑÑÐ¸ÑÐ½Ð¾Ð¼ Ð²Ð¸Ð´Ðµ. ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð»Ð¸ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÑ Ð¸ ÑÐ²Ð¾Ð¹ÑÑÐ²Ð° Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸, Ð° ÑÐ°ÐºÐ¶Ðµ ÐµÐµ Ð´Ð¸Ð½Ð°Ð¼Ð¸Ðº Ñ ÑÑÐµÑÐ¾Ð¼ Ð²Ð¾Ð·Ð´ÐµÐ¹ÑÑÐ²Ð¸Ñ Ð½Ð°Ð³ÑÑÐ·ÐºÐ¸ Ð½Ð° Ð³Ð¸Ð´ÑÐ¾ÑÑÑÐ±Ð¸Ð½Ñ, Ð° ÑÐ°ÐºÐ¶Ðµ Ð±ÐµÐ· ÑÑÐµÑÐ° Ð½Ð°Ð³ÑÑÐ·ÐºÐ¸. ÐÐ°ÑÐµÐ¼ Ð±ÑÐ»Ð° ÑÐ´ÐµÐ»Ð°Ð½Ð° ÑÐ¾ÑÐ¼Ð°Ð»ÑÐ½Ð°Ñ Ð¿Ð¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐ° Ð·Ð°Ð´Ð°ÑÐ¸ ÑÐ¸Ð½ÑÐµÐ·Ð° ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð´Ð»Ñ ÑÑÐ°Ð±Ð¸Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½ÑÑ Ð´Ð²Ð¸Ð¶ÐµÐ½Ð¸Ð¹. ÐÐ°Ð´Ð°ÑÐ° ÑÐ¸Ð½ÑÐµÐ·Ð° ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ ÑÐµÑÐ°Ð»Ð°ÑÑ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½Ð¾-Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÐ½ÑÐ¼ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Ð² Ð²Ð¸Ð´Ðµ Ð²ÑÐ¿ÑÐºÐ»Ð¾Ð¹ Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ ÐºÐ¾Ð¼Ð±Ð¸Ð½Ð°ÑÐ¸Ð¸ ÑÐºÑÑÑÐµÐ¼Ð°Ð»ÑÐ½ÑÑ ÑÐµÑÐµÐ½Ð¸Ð¹. ÐÑÐ°Ð½Ð¸ÑÐ½ÑÐµ ÑÐµÑÐµÐ½Ð¸Ñ Ð²ÑÑÐ¸ÑÐ»ÑÐ»Ð¸ÑÑ Ð¿ÑÐ¸ Ð¿Ð¾Ð¼Ð¾ÑÐ¸ Ð¿ÑÐ¾ÐµÐºÑÐ¾ÑÐ° Ð½Ð° Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½Ð¾Ðµ Ð´Ð¾Ð¿Ð¾Ð»Ð½ÐµÐ½Ð¸Ðµ Ðº Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¼Ñ Ð¿Ð¾Ð´Ð¿ÑÐ¾ÑÑÑÐ°Ð½ÑÑÐ²Ñ Ð¸ Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½Ð¾Ð³Ð¾ Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÐ° Ð½Ð° Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ðµ Ð¼Ð½Ð¾Ð³Ð¾Ð¾Ð±ÑÐ°Ð·Ð¸Ðµ. ÐÐ° Ð¾ÑÐ½Ð¾Ð²Ðµ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½Ð¾Ð³Ð¾ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ ÑÐ°ÐºÐ¶Ðµ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²ÑÐ²Ð°Ð»Ð°ÑÑ Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»ÑÐ½Ð°Ñ ÑÐ¾ÑÑÐ°Ð²Ð»ÑÑÑÐ°Ñ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð¸Ñ ÐÐ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ.
description: This work is devoted to the study of the dynamics of the radial-axial hydraulic turbine model, as well as the synthesis of a control action to stabilize the program movements of the system. The research set the following goals: 1) Study of the subject area and synthesis of a mathematical model of a hydraulic turbine. 2) Investigation of the dynamics of the simulated system. 3) Control synthesis to stabilize the program movements of the system. 4) Study of the dynamics of the system under the influence of synthesized control. The paper reviewed and classified the types of hydraulic turbines. Then, a mathematical model in matrix form was synthesized on the basis of a radial-axial hydraulic turbine. The parameters and properties of the mathematical model were studied, as well as its dynamics, taking into account the impact of the load on the hydraulic turbine, as well as without taking into account the load. Then a formal statement of the problem of control synthesis was made to stabilize program movements. The problem of control synthesis was solved by the projection-operator method in the form of a convex linear combination of extreme solutions. The boundary solutions were calculated using a projector on an orthogonal complement to a linear subspace and an orthogonal projection operator on a linear manifold. On the basis of projection control, an integral control component was also implemented to obtain PI control.