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Identification Based Control For Wind Turbine
Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Proceedings of the 20th World Congress The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect IFAC PapersOnLine 50-1 (2017) 2272–2277
Identification Identification Based Based Control Control For For Wind Wind Turbine Turbine Identification Based Control For Wind Identification Based Control For Wind Turbine Turbine Albert G. Alexandrov* Vladimir N. Chestnov**
Albert G. Alexandrov* Vladimir N. Chestnov** Vadim A. Alexandrov*** Albert Vladimir N. Chestnov** Vadim A. Alexandrov*** Albert G. G. Alexandrov* Alexandrov* Vladimir N. Chestnov** Vadim A. Alexandrov*** Vadim A. Alexandrov*** *V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, *V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia (e-mail: [email protected]) *V.A. Trapeznikov Institute of Control of Academy of Moscow, Russia (e-mail:Sciences [email protected]) *V.A. Trapeznikov Institute of Control Sciences of Russian Russian Academy of Sciences, Sciences, ** V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Moscow, Russia (e-mail: [email protected]) ** V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Moscow, Russia (e-mail: [email protected]) Sciences, Moscow, Russia (e-mail: [email protected]) ** V.A. of Control Sciences Moscow, Institute Russia (e-mail: [email protected]) **Sciences, V.A. Trapeznikov Trapeznikov Institute of(e-mail:[email protected]) Control Sciences of of Russian Russian Academy Academy of of *** Adaplab LLC, Moscow, Russia Sciences, Moscow, Russia (e-mail: [email protected]) *** Adaplab LLC, Moscow, Russia (e-mail:[email protected]) Sciences, Moscow, Russia (e-mail: [email protected]) *** *** Adaplab Adaplab LLC, LLC, Moscow, Moscow, Russia Russia (e-mail:[email protected]) (e-mail:[email protected]) Abstract: The blade pitch angle control when the wind speed is above rated is considered in this Abstract: The blade pitch angle control when the wind speed is above rated is considered in this paper. Finite-frequency identification approach is used to obtain the linearized model. The identification Abstract: The blade pitch angle control when the wind speed is rated is considered in paper. Finite-frequency identification approach is used to obtain the linearized model. identification Abstract: The blade pitch angle control when the wind speed is above above rated isThe considered in this this procedure can be carried out in closed loop in spite of fluctuating wind speed. PI controller paper. Finite-frequency identification approach is used to obtain the linearized model. The identification procedure can be carried out in closed loop in spite of fluctuating wind speed. PI controller paper. Finite-frequency identification approach is used to obtain the linearized model. The identification coefficients determine limits for linearized model parameters for which the roots of characteristic procedure be out in closed loop in of speed. PI controller coefficientscan determine limits linearized model parameters for whichwind the roots of characteristic procedure can be carried carried out for in closed loop inIf spite spite of fluctuating fluctuating wind speed. PIare controller identified estimates of parameters outside equation of closed loop system are real-valued. coefficients determine limits for linearized model parameters for which the roots of characteristic equation of closed loop system are real-valued. If identified estimates of parameters are outside coefficients determine limits for linearized model parameters for which the roots of characteristic the limits maintained by previously designed PI controller then PI controller is tuned on the basis equation closed are real-valued. If identified estimates of parameters are outside the limits of maintained by system previously PI controller then PI controller is tuned on basis equation of closed loop loop system are designed real-valued. If the identified estimates of parameters arethe outside of obtained estimates. The simulation results show high efficiency of the proposed approach. the limits PI PI is on of estimates. by Thepreviously simulationdesigned results show the highthen efficiency of the proposed theobtained limits maintained maintained by previously designed PI controller controller then PI controller controller is tuned tuned approach. on the the basis basis Keywords: variable speed wind turbine, pitch control, PI control, identification, disturbance .approach. of obtained estimates. The simulation results show the high efficiency of the proposed © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. of obtainedvariable estimates. The simulation results show PI thecontrol, high efficiency of thedisturbance proposed .approach. Keywords: speed wind turbine, pitch control, identification, Keywords: Keywords: variable variable speed speed wind wind turbine, turbine, pitch pitch control, control, PI PI control, control, identification, identification, disturbance disturbance.. 1. INTRODUCTION 1. INTRODUCTION 1. Wind energy conversion to electric power is one of the 1. INTRODUCTION INTRODUCTION Wind energy conversion to electric power is one of the most important problems of electric modernpower powerisengineering. Wind energy conversion one of most important problemsto of electric modernpower powerisengineering. Wind energy conversion to one of the the The blade pitch angle control is considered in the paper. most important problems of modern power engineering. The blade pitch angle control is considered in the paper. most important problems of modern power engineering. One of the pitch main angle difficulties of is this control isin a significant The control considered the paper. One of the pitch main angle difficulties of is this control isin a significant The blade blade control considered theturbine paper. nonlinearity of the mathematical model of the One of the main difficulties of this control is a significant nonlinearity of the mathematical model of the turbine One of the main difficulties of this control is a significant rotor speed of toward the pitch angle (Burton, T.turbine et al. nonlinearity the model of rotor speed of toward the pitch angle (Burton, T.turbine et al. nonlinearity the mathematical mathematical model of the theare (2001)). Well-established PI and PID controllers used rotor speed toward the pitch angle (Burton, T. et al. (2001)). Well-established PI and PID controllers are used rotor speed toward thebypitch angle (Burton, T. et al. for wind turbine control pitch angle changing (Burton, (2001)). Well-established PI and PID controllers are used for wind turbine control by pitch angle changing (Burton, (2001)). Well-established PI and PID controllers are More used T. et al. (2001), Leithead, W.E. et al. (2000)). for wind by pitch changing (Burton, T. al. turbine (2001),control Leithead, W.E.angle et al. (2000)). More for et wind turbine control by H pitch angle changing (Burton, H2 and modern techniques ∞ are also practiced (Rocha, T. et al. (2001), Leithead, W.E. et al. (2000)). More H2 and H∞W.E. are also practiced (Rocha, modern techniques T. et al. (2001), Leithead, et al. (2000)). More R. et al. techniques (2005)). Recent papers utilised gain scheduling H and H are also practiced (Rocha, modern 2 ∞ R. et al. techniques (2005)). Recent papers utilised gain scheduling H and H are also practiced (Rocha, modern 2 ∞ (Bianchi, F.D. et al.Recent (2005)) , disturbance observer (Joo, Y. R. et al. papers utilised gain (Bianchi, F.D. et al.Recent (2005)) , disturbance observer (Joo, Y. R. et (2012)), al. (2005)). (2005)). papers utilised gain scheduling scheduling et al. Maximum Power Point Tracking algorithm (Bianchi, F.D. et al. (2005)) ,, disturbance observer (Joo, et al. (2012)), Maximum Power Point Tracking algorithm (Bianchi, F.D. et al. (2005)) disturbance observer (Joo, Y. Y. (Tilli, A. et al. (2011)), etc.Power Point Tracking algorithm et al. (2012)), Maximum (Tilli, A. et al. (2011)), etc.Power Point Tracking algorithm et al. (2012)), Maximum (Tilli, et (2011)), In all A. these it is assumed that all parameters (Tilli, et al. al.techniques (2011)), etc. etc. In all A. these techniques it is assumed that all parameters (or their limits) of the it wind turbine are defined. In this In all these techniques is that all parameters (or their limits) of the wind turbine are defined. In this In all these techniques it is assumed assumed that allobtained parameters paper it is assumed that controller has been for (or their of the turbine defined. In paper it islimits) assumed thatwind controller hasare been obtained for (or their limits) of the wind turbine are defined. In itthis this known limits of the wind turbine parameters. But is paper it is assumed controller been obtained known limits of thethat wind turbine has parameters. But itfor is paper it is assumed that controller has been obtained for considered that blade aerodynamics can change in time known limits of the wind turbine parameters. it is considered that blade aerodynamics can changeBut in time known limits of the wind turbine parameters. But it is and the wind turbine parameters go can out change of theseinlimits. considered that blade aerodynamics time and the wind turbine parameters go can out change of theseinlimits. considered that blade aerodynamics time The finite-frequency identification method (Alexandrov, and the wind parameters go out these The finite-frequency identification method and et the wind turbine turbine parameters go out of of (Alexandrov, theseislimits. limits. A.G. al.(2011), Alexandrov, A.G. (2005),(1994)) used The finite-frequency identification method (Alexandrov, A.G. et al.(2011), Alexandrov, A.G. (2005),(1994)) is used The finite-frequency identification method (Alexandrov, in the paper for obtaining linearized model parameters. A.G. et al.(2011), Alexandrov, A.G. (2005),(1994)) is in for obtaining linearized model parameters. A.G.the et paper al.(2011), Alexandrov, A.G. is used used Advantages of this method are(2005),(1994)) abilities to identify in the paper for obtaining linearized model parameters. Advantages of this method are abilities to identify in the paper for obtaining linearized model parameters. linearized model parameters onare running turbine, in real Advantages of this method abilities to identify linearized model parameters onare running turbine, in real Advantages of loop this method abilities to external identify time, in closed system, in spite of unknown linearized model parameters on running turbine, in time, in closed loop system, in spite of unknown external linearized model parameters runningcan turbine, in real real disturbances. Coefficients ofinon controller be changed time, in closed loop system, spite of unknown external disturbances. Coefficients of controller can be changed time, in closed loop system, in spite of unknown external based on identification results. disturbances. Coefficients of based on identification results. disturbances. Coefficients of controller controller can can be be changed changed based on identification results. The case when the wind speed is above rated as operating based on identification results. The case when the wind speed is above rated as operating region of when pitch the control isspeed considered in rated the paper. In case The case wind is as region of when pitch the control considered in rated the paper. In case The case windisup speed is above above as operating operating of low wind with speed to rated the pitch angle should region of control in the case of low wind with speedis to rated the angle In should region of pitch pitch control isupconsidered considered in pitch the paper. paper. In case of low wind with speed up to rated the pitch angle should of low wind with speed up to rated the pitch angle should
be at minimum value for maximum power. This value may be at minimum value for maximum power. This value may slightly differ from zeromaximum due to power. the design of wind be value for This value may slightly differ from zeromaximum due to power. the design of wind be at at minimum minimum value for This25-30 value may turbine. In casefrom of very strong wind more than m/s slightly differ zero due to the design of wind turbine. In casefrom of very strong wind more than 25-30 m/s slightly differ zero due to the design of wind the turbine is stopped for safety. Therefore the pitch turbine. In case of very strong wind more than 25-30 m/s the turbine is stopped for safety. Therefore the pitch turbine. In case of very strong wind more than 25-30 m/s controller operates in a limited rangeTherefore of wind speed. the turbine is stopped for safety. the pitch controller operates in a limited rangeTherefore of wind speed. the turbine is stopped for safety. the pitch controller controller operates operates in in aa limited limited range range of of wind wind speed. speed. 2. PROBLEM STATEMENT 2. PROBLEM STATEMENT STATEMENT Control system 2. of PROBLEM wind turbine is considered. It consists STATEMENT Control system 2. of PROBLEM wind turbine is considered. It consists of the control object and the controller. The controller is Control system of wind turbine considered. It consists of the control object and the controller. The controller is Controlpitch system of wind turbine is is The considered. Itobject consists blade angle PIand controller. control is of the control object the controller. The controller is blade pitch angle PI controller. The control object is of the control object and the controller. The controller considered inangle conditions when theThe wind speed is above blade pitch PI controller. control object is considered inangle conditions when theThe wind speed object is above blade pitch PI controller. control is rated and the pitch controller restricts wind turbine rotor considered in conditions when the wind speed is above rated and the pitch controller restricts wind turbine rotor considered in conditions when the wind speed is above speed while the torque controller tracks the turbine rated power. rated and the pitch controller restricts wind rotor speed while torque controller tracks the turbine rated power. rated and thethe pitch controller restricts wind rotor In this case the control object is described by equation speed while the torque controller tracks the rated power. In thiswhile case the the torque controlcontroller object is tracks described by equation speed the rated power. (Tilli, A. et al.the (2011); Joo,object Y. et al.is(2012)) In this control described (Tilli, et al.the (2011); Joo,object Y. et al.is(2012)) In thisA.case case control described by by equation equation (Tilli, A. et al. (2011); Joo, Y. et al. (2012)) 𝜌𝜌𝜌𝜌 2 3Joo, Y. et al. (2012)) 1 𝑃𝑃𝑅𝑅 (1) (Tilli, A. et al. (2011); 𝜔𝜔̇ = 𝜌𝜌𝜌𝜌 𝑅𝑅2 𝑉𝑉 3 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) − 1 𝑃𝑃𝑅𝑅 , (1) (𝜔𝜔, 𝜔𝜔̇ = 2𝐽𝐽𝐽𝐽 𝑅𝑅 𝑉𝑉 𝐶𝐶 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) − , 𝐽𝐽 𝜔𝜔 𝜌𝜌𝜌𝜌 1 𝑃𝑃 (1) 𝑅𝑅 2 3 𝑝𝑝 (𝜔𝜔, 𝜌𝜌𝜌𝜌 1 𝑃𝑃 2𝐽𝐽𝐽𝐽 𝐽𝐽 𝜔𝜔 (1) 𝑅𝑅 𝜔𝜔̇ = = 𝑅𝑅 𝛽𝛽, 𝑉𝑉, 𝑉𝑉, 𝑐𝑐) 𝑐𝑐) − − 𝐽𝐽 𝜔𝜔 ,, 𝑅𝑅2 𝑉𝑉 𝑉𝑉 3 𝐶𝐶 𝐶𝐶𝑝𝑝𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝜔𝜔̇ 2𝐽𝐽𝐽𝐽 2𝐽𝐽𝐽𝐽 𝐽𝐽 𝜔𝜔 where 𝜔𝜔 is the measured turbine rotor speed, 𝛽𝛽 is the where 𝜔𝜔 is the measured turbine rotor speed, 𝛽𝛽 is the controlled blade pitch angle, 𝑉𝑉 is rotor the wind speed, is where 𝜔𝜔 is is the measured measured turbine speed, 𝛽𝛽 is is𝜌𝜌 the controlled blade pitch angle, 𝑉𝑉 is rotor the wind speed, 𝜌𝜌the is where 𝜔𝜔 the turbine speed, 𝛽𝛽 R is the blade rotor radius, 𝐽𝐽 is the the air density, controlled blade pitch angle, 𝑉𝑉 is is the the wind speed, speed, 𝜌𝜌the is R is the angle, blade rotor radius, 𝐽𝐽 is 𝜌𝜌 the air density, controlled blade pitch 𝑉𝑉 wind is is rated power, 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽,is 𝑉𝑉, 𝑐𝑐)the is moment of inertia, 𝑃𝑃𝑅𝑅 blade R is the rotor radius, 𝐽𝐽 the air density, is rated 𝑉𝑉, 𝑐𝑐)the is moment of inertia, 𝑃𝑃𝑅𝑅 blade R is the rotorpower, radius,𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝐽𝐽 𝛽𝛽,is the density, the air power coefficient as nonlinear function depended on rated power, 𝐶𝐶 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) is moment inertia, 𝑃𝑃 𝑅𝑅as is 𝑝𝑝 (𝜔𝜔, the powerof coefficient nonlinear function depended on is rated power, 𝐶𝐶 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) isa moment of inertia, 𝑃𝑃 𝑅𝑅 𝑝𝑝 turbine specifics withasparameters vector depended 𝑐𝑐. This ison the power coefficient nonlinear function turbine specifics withasparameters vector depended 𝑐𝑐. This isona the power coefficient nonlinear function simplified model with without detailed vector aerodynamics anda turbine specifics parameters 𝑐𝑐. simplified model with without detailed vector aerodynamics anda turbine specifics parameters 𝑐𝑐. This This is is mechanics and without generator and grid models. simplified model without detailed aerodynamics and mechanics without generator and grid models. simplified and model without detailed aerodynamics and mechanics and without generator and grid models. The pitch angle 𝛽𝛽 is the control value from PI controller mechanics and without generator and grid models. The pitch angle 𝛽𝛽 is the control value from PI controller The is control (𝜔𝜔 − 𝜔𝜔 ) +value 𝑧𝑧, from The pitch pitch angle angle𝛽𝛽 𝛽𝛽 𝛽𝛽= 𝑘𝑘 is the the control value from PI PI controller controller (2) 𝛽𝛽 = 𝑘𝑘𝑝𝑝𝑝𝑝 (𝜔𝜔 − 𝜔𝜔𝑅𝑅𝑅𝑅 ) + 𝑧𝑧, (2) 𝛽𝛽 (2) 𝛽𝛽 = = 𝑘𝑘 𝑘𝑘𝑝𝑝𝑝𝑝 (𝜔𝜔 (𝜔𝜔 − − 𝜔𝜔 𝜔𝜔𝑅𝑅𝑅𝑅 )) + + 𝑧𝑧, 𝑧𝑧, (2)
Copyright © 2017 IFAC 2308 Copyright © 2017 IFAC 2308 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 2308Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 responsibility IFAC 2308Control. Peer review©under of International Federation of Automatic 10.1016/j.ifacol.2017.08.196
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277
𝑧𝑧̇ = 𝑘𝑘𝑖𝑖 (𝜔𝜔 − 𝜔𝜔𝑅𝑅 ),
(3)
where 𝜔𝜔𝑅𝑅 is the determined desired turbine rotor speed, 𝑧𝑧 is the controller integrator output, 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 are positive PI coefficients. Reasonable to assume that controlled blade pitch angle 𝛽𝛽 is limited from 0 to 90 degrees. Although in practice these boundaries may be another.
Finite-frequency identification method (Alexandrov, A.G. (2005),(1994)) that works in spite of unknown external disturbances will be solving this problem. The limits of parameters of the linearized model (10) 𝑎𝑎 and 𝑏𝑏 should be determined by coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 of PI controller (2),(3) such that the roots of the characteristic equation of closed loop system are real-valued: 𝐴𝐴 = [𝑎𝑎, 𝑎𝑎], 𝐵𝐵 = [𝑏𝑏, 𝑏𝑏]
Dinamics of pitch actuator, gear and torque controller are not considered for simplicity. 𝜌𝜌𝜌𝜌 2 3 1 𝑃𝑃𝑅𝑅 𝑅𝑅 𝑉𝑉 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, V, 𝑐𝑐) − 2𝐽𝐽𝐽𝐽 𝐽𝐽 𝜔𝜔
(4)
Problem: if identified estimates 𝑎𝑎̂, 𝑏𝑏̂ are not within these limits then coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 should be tuned.
then the equation (1) takes the form 𝜔𝜔̇ = 𝜑𝜑(𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐).
(5)
The Taylor series for function (4) around the point 𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , where 𝛽𝛽𝑠𝑠𝑠𝑠 provides speed 𝜔𝜔𝑆𝑆𝑆𝑆 close to 𝜔𝜔𝑅𝑅 at a wind speed 𝑉𝑉𝑠𝑠𝑠𝑠 is: 𝜑𝜑(𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) = 𝜑𝜑(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) + 𝑎𝑎(𝜔𝜔 − 𝜔𝜔𝑅𝑅 ) + 𝑏𝑏(𝛽𝛽 − 𝛽𝛽𝑠𝑠𝑠𝑠 ) + 𝑑𝑑(𝑉𝑉 − 𝑉𝑉𝑠𝑠𝑠𝑠 ) + 𝜓𝜓(𝜔𝜔, 𝑅𝑅, 𝑉𝑉, 𝑐𝑐)
where 𝑎𝑎 = 𝑎𝑎(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) =
∂𝜑𝜑 | , ∂𝜔𝜔 𝜔𝜔𝑅𝑅 ,𝛽𝛽𝑠𝑠𝑠𝑠 ,𝑉𝑉𝑠𝑠𝑠𝑠
𝑏𝑏 = 𝑏𝑏(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) =
∂𝜑𝜑
| , ∂𝛽𝛽 𝜔𝜔𝑅𝑅 ,𝛽𝛽𝑠𝑠𝑠𝑠 ,𝑉𝑉𝑠𝑠𝑠𝑠
(11)
where 𝑎𝑎, 𝑏𝑏 are low limits and 𝑎𝑎, 𝑏𝑏 are high limits of the parameters.
Let the right part of the equation (1) be denoted as 𝜑𝜑(𝜔𝜔, 𝛽𝛽, 𝑣𝑣, 𝑐𝑐) =
2273
(6)
(7)
3. OPEN LOOP IDENTIFICATION The system (1)-(3) with disabled PI controller is considered: 𝑘𝑘𝑝𝑝 = 𝑘𝑘𝑖𝑖 = 0.
(12)
𝛽𝛽 = 𝛽𝛽𝑠𝑠𝑠𝑠 + 𝑞𝑞 sin 𝛾𝛾𝛾𝛾
(13)
In this case pitch angle 𝛽𝛽 is formed as a sum of 𝛽𝛽𝑠𝑠𝑠𝑠 defined by operating point and test sine wave: where 𝛾𝛾 is test frequency, 𝑞𝑞 is amplitude of the test sine wave. The test sine wave parameters based on the finite-frequency identification method (Alexandrov, A.G. (2005)). Then equation (10) has the form
𝑑𝑑 = 𝑑𝑑(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) =
∂𝜑𝜑
| , ∂𝑉𝑉 𝜔𝜔𝑅𝑅 ,𝛽𝛽𝑠𝑠𝑠𝑠 ,𝑉𝑉𝑠𝑠𝑠𝑠
𝑦𝑦̇ = −𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 sin 𝛾𝛾𝛾𝛾 + 𝑑𝑑𝑉𝑉𝑑𝑑 .
(8)
(14)
𝜓𝜓(𝜔𝜔, 𝑅𝑅, 𝑉𝑉, 𝑐𝑐) contains deviations in the second degree or higher.
A particular solution of this equation where the turbine rotor speed 𝜔𝜔 oscillates about 𝜔𝜔𝑅𝑅 is
In (6) 𝑎𝑎 < 0 for the condition of the stability of the equation (1) therefore hereafter – 𝑎𝑎 where 𝑎𝑎 > 0 will be used. Value b in (7) is always negative because rotor speed 𝜔𝜔 decreases when the 𝛽𝛽 increases. Hereafter – 𝑏𝑏 where 𝑏𝑏 > 0 will be used.
(15)
Let 𝑦𝑦 = 𝜔𝜔 − 𝜔𝜔𝑅𝑅 ,
𝑢𝑢 = 𝛽𝛽 − 𝛽𝛽𝑠𝑠𝑠𝑠 ,
𝑉𝑉𝑑𝑑 = 𝑉𝑉 − 𝑉𝑉𝑠𝑠𝑠𝑠
(9)
𝑎𝑎 > 0, 𝑏𝑏 > 0
where 𝜈𝜈 and 𝜇𝜇 are frequency parameters of object on the frequency 𝛾𝛾 ; 𝑦𝑦𝑑𝑑 (𝑡𝑡) is the output component depending on the wind speed fluctuations. 𝑦𝑦̇ (𝑡𝑡) obtaining from (15) and 𝑦𝑦(𝑡𝑡) from (15) are pluged to (14). Then coefficients at sin𝛾𝛾𝛾𝛾 and cos𝛾𝛾𝛾𝛾 are equated. Thereby parameters 𝑎𝑎 and 𝑏𝑏 are expressed through frequency parameters 𝜈𝜈 and 𝜇𝜇 as 𝜈𝜈
Then linearized equation of wind turbine is 𝑦𝑦̇ = −𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 + 𝑑𝑑𝑉𝑉𝑑𝑑 ,
𝑦𝑦(𝑡𝑡) = 𝑞𝑞𝑞𝑞 sin 𝛾𝛾𝛾𝛾 + 𝑞𝑞𝑞𝑞 cos 𝛾𝛾𝛾𝛾 + 𝑦𝑦𝑑𝑑 ,
𝑎𝑎 = − 𝛾𝛾,
𝑏𝑏 = 𝜇𝜇𝜇𝜇 − 𝑎𝑎𝑎𝑎
𝜇𝜇
(10)
Problem of identification: to find out estimates 𝑎𝑎̂, 𝑏𝑏̂ of parameters of the linearized model (10).
and conversely
The need of identification is caused by the deformation of blades during continuous working of wind turbine which leads to undefined change of the power coefficient 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) in equation (1).
Estimates of frequency parameters 𝜈𝜈̂ and 𝜇𝜇̂ calculated as Fourier filter outputs on the frequency
𝜈𝜈 = −
Identification is complicated by unknown variations of the wind speed 𝑉𝑉𝑑𝑑 relatively constant 𝑉𝑉𝑠𝑠𝑠𝑠 . 2309
𝑏𝑏𝑏𝑏
𝛾𝛾2 +𝑎𝑎2
𝜈𝜈̂ = 𝜈𝜈(𝜏𝜏) = 𝜇𝜇̂ = 𝜇𝜇(𝜏𝜏) =
,
𝜇𝜇 =
𝑏𝑏𝑏𝑏
𝛾𝛾2 +𝑎𝑎2
2 𝑡𝑡𝐹𝐹 +𝜏𝜏 𝑦𝑦(𝑡𝑡) sin 𝛾𝛾𝛾𝛾 𝑑𝑑𝑑𝑑 ∫ 𝑞𝑞𝑞𝑞 𝑡𝑡𝐹𝐹
2 𝑡𝑡𝐹𝐹 +𝜏𝜏 𝑦𝑦(𝑡𝑡) cos 𝛾𝛾𝛾𝛾 𝑑𝑑𝑑𝑑 ∫ 𝑞𝑞𝑞𝑞 𝑡𝑡𝐹𝐹
(16)
(17) are (18) (19)
Proceedings of the 20th IFAC World Congress 2274 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277 Toulouse, France, July 9-14, 2017
where 𝜏𝜏 is filtering time, 𝑡𝑡𝐹𝐹 is filtering start time. Fourier filter outputs converge for 𝜏𝜏 → ∞ to the true values if the wind speed does not contain harmonic oscillations with a frequency of 𝛾𝛾 (Alexandrov, A.G. (2005),(1994)). Implementation of frequency parameters 𝜈𝜈̂ and 𝜇𝜇̂ real-time calculation is acquainted (Alexandrov, A.G. et al.(2008))
Estimates of 𝜈𝜈𝑦𝑦 , 𝜇𝜇𝑦𝑦 , 𝜈𝜈𝑢𝑢 , 𝜇𝜇𝑢𝑢 values are obtained by Fourier filter (18),(19). Hence,
Hence,
Step 2. To find out estimates of frequency parameters 𝜈𝜈̂𝑦𝑦 , 𝜇𝜇̂ 𝑦𝑦 , 𝜈𝜈̂𝑢𝑢 , 𝜇𝜇̂ 𝑢𝑢 as Fourier filter outputs (18), (19) and to calculate frequency parameters of object by (27),(28).
Algorithm 3.1 The open loop identification of a wind turbine consists of operations: Step 1. To form the blades motion in accordance with the expression (13). Step 2. To find out estimates of frequency parameters as Fourier filter outputs (18), (19).
Algorithm 4.1. The closed loop identification of a wind turbine consists of operations: Step 1. To form the blades motion in accordance with the expression (20).
Step 3. To calculate estimates of the object parameters by formulae (16). In a real system it is needed to check that the torque controller should not react to our test sine wave.
Step 3. To calculate estimates of the object parameters by formulae (16) using estimates of frequency parameters.
4. CLOSED LOOP IDENTIFICATION Let PI coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 be available. Then pitch angle 𝛽𝛽 is formed as 𝛽𝛽 = 𝑘𝑘𝑝𝑝 (𝜔𝜔 − 𝜔𝜔𝑅𝑅 ) + 𝑧𝑧 + 𝑞𝑞 sin 𝛾𝛾𝛾𝛾 𝑧𝑧̇ = 𝑘𝑘𝑖𝑖 ⋅ (𝜔𝜔 − 𝜔𝜔𝑅𝑅 )
(21) 𝑧𝑧̇ = 𝑘𝑘𝑖𝑖 𝑦𝑦
and object input is (24)
where 𝜈𝜈𝑦𝑦 and 𝜇𝜇𝑦𝑦 are output frequency parameters which estimates can be calculated according to (18),(19), 𝜈𝜈𝑢𝑢 and 𝜇𝜇𝑢𝑢 are input frequency parameters which estimates can be calculated by formulae analogous as (18),(19) for 𝑢𝑢(𝑡𝑡) rather than 𝑦𝑦(𝑡𝑡). Plugging (23),(24) into (10) similar as for (16),(17) 𝜈𝜈𝑦𝑦 and 𝜇𝜇𝑦𝑦 are expressed as 𝜈𝜈𝑦𝑦 = −𝑏𝑏
𝛾𝛾2 +𝑎𝑎2
, 𝜇𝜇𝑦𝑦 = 𝑏𝑏
−𝜇𝜇𝑢𝑢𝑎𝑎+𝛾𝛾𝜈𝜈𝑢𝑢 𝛾𝛾2 +𝑎𝑎2
𝑢𝑢 = 𝑘𝑘𝑝𝑝 y + z,
𝜇𝜇𝑦𝑦 = 𝜇𝜇𝑢𝑢 𝜈𝜈 + 𝜈𝜈𝑢𝑢 𝜇𝜇
𝑦𝑦̈ + (𝑎𝑎 + 𝑏𝑏 𝑘𝑘𝑝𝑝 )𝑦𝑦̇ + 𝑏𝑏 𝑘𝑘𝑖𝑖 𝑦𝑦 = 0
𝑠𝑠 2 + (𝑎𝑎 + 𝑏𝑏𝑘𝑘𝑝𝑝 )𝑠𝑠 + 𝑏𝑏𝑘𝑘𝑖𝑖 = 0
𝜇𝜇 =
𝜈𝜈𝑦𝑦 𝜈𝜈𝑢𝑢 + 𝜇𝜇𝑦𝑦 𝜇𝜇𝑢𝑢 𝜈𝜈𝑢𝑢2 + 𝜇𝜇𝑢𝑢2
𝜇𝜇𝑦𝑦 𝜈𝜈𝑢𝑢 − 𝜈𝜈𝑦𝑦 𝜇𝜇𝑢𝑢 𝜈𝜈𝑢𝑢2 + 𝜇𝜇𝑢𝑢2
(30)
2
(𝑎𝑎 + 𝑏𝑏𝑘𝑘𝑝𝑝 ) − 4𝑏𝑏𝑘𝑘𝑖𝑖 > 0
(32)
𝑎𝑎2 + 2𝑎𝑎𝑎𝑎𝑘𝑘𝑝𝑝 + 𝑏𝑏 2 𝑘𝑘𝑝𝑝2 − 4𝑏𝑏𝑘𝑘𝑖𝑖 > 0
(33)
or
Sufficient conditions for validity of this inequation are
(25)
(26)
(27)
(31)
All parameters of this equation (𝑎𝑎, 𝑏𝑏, 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 ) are positive, so the roots of this characteristic equation have negative real parts. Conditions should be found so that these roots are real-valued to ensure the quality of transients. It is seen that this condition has the form
𝑎𝑎 >
2𝑘𝑘𝑖𝑖 𝑘𝑘𝑝𝑝
(34)
or
Frequency parameters of object 𝜈𝜈 and 𝜇𝜇 can be expressed from (26): 𝜈𝜈 =
(29)
Differential equation obtaining after differentiating of equation (10) for 𝑉𝑉𝑑𝑑 = 0 and excluding variable u is
or using (17) as 𝜈𝜈𝑦𝑦 = 𝜈𝜈𝑢𝑢 𝜈𝜈 − 𝜇𝜇𝑢𝑢 𝜇𝜇,
ż = 𝑘𝑘𝑖𝑖 y
Then characteristic equation has the form
𝑦𝑦̅(𝑡𝑡) = 𝑞𝑞𝜈𝜈𝑦𝑦 sin 𝛾𝛾𝛾𝛾 + 𝑞𝑞𝜇𝜇𝑦𝑦 cos 𝛾𝛾𝛾𝛾 + 𝑦𝑦̅𝑑𝑑 (23)
𝜈𝜈𝑢𝑢 𝑎𝑎−γ𝜇𝜇𝑢𝑢
PI controller formulae (2),(3) are written as
(22)
Output of the system (10), (22) is
𝑢𝑢(𝑡𝑡) = 𝑞𝑞𝜈𝜈𝑢𝑢 sin 𝛾𝛾𝛾𝛾 + 𝑞𝑞𝜇𝜇𝑢𝑢 cos 𝛾𝛾𝛾𝛾
Let estimates 𝑎𝑎̂ and 𝑏𝑏̂ be found. They will be used as values of the object parameters in the linearized equation (10).
(20)
or using (9) as 𝑢𝑢 = 𝑘𝑘𝑝𝑝 𝑦𝑦 + 𝑧𝑧 + 𝑞𝑞 sin 𝛾𝛾𝛾𝛾 ,
5. PI COEFFICIENTS
𝑏𝑏 > 4
𝑘𝑘𝑖𝑖
𝑘𝑘𝑝𝑝 2
(35)
Then low limits for parameters interval (11) for available coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 can be evaluated as
(28)
𝑎𝑎 = 2310
2𝑘𝑘𝑖𝑖 𝑘𝑘𝑝𝑝
(36)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277
𝑘𝑘𝑖𝑖
𝑏𝑏 = 4
𝑘𝑘𝑝𝑝
6.1 Open loop identification simulation
(37)
2
2275
𝑟𝑟𝑟𝑟𝑟𝑟
High limits for parameters interval (11) are not evaluated because they can not violate the inequation (33) when low limits do not violate it.
Algorithm 5.1. PI coefficients correction.
𝑚𝑚
Value of rotor speed 𝜔𝜔𝑅𝑅 = 6.7 for 𝑉𝑉𝑠𝑠𝑠𝑠 = 17 is 𝑠𝑠 𝑐𝑐 achieved by 𝛽𝛽𝑠𝑠𝑠𝑠 = 20.6245 deg . The test signal for identification is formed as (13) where 𝑞𝑞 = 0.1 and 𝛾𝛾 = 0.1. Figure 1 shows pitch angle values as the test signal and rotor speed values as the system response.
Step 1. To identify estimates 𝑎𝑎̂, 𝑏𝑏̂ by Algorithm 3.1 or Algorithm 4.1. Step 2. To check inequations (34),(35). If both of them are violated then inequation (33) is checked as condition of necessity and sufficiency. Step 3. If inequation (33) is violated then coefficietn 𝑘𝑘𝑝𝑝 is increased as 𝑘𝑘𝑝𝑝 >
2𝑘𝑘𝑖𝑖 𝑎𝑎̂
𝑘𝑘𝑝𝑝 2 >
4𝑘𝑘𝑖𝑖 𝑏𝑏̂
(38)
or (39)
It should be considered that 𝑘𝑘𝑝𝑝 is constrained by unmodelled dynamics of blades, gear etc. If because of that 𝑘𝑘𝑝𝑝 can not be increased then 𝑘𝑘𝑖𝑖 should be decreased for validity of inequation (34) or (35). The procedure for starting the algorithm 5.1 is not discussed here. It may be started manually during maintenance or auto intermittently or as an event handling. 6. SIMULATION The wind turbine is simulated by nonlinear model (1) where the power coefficient is approximated (Tilli, A. et al. (2011)) by 𝑐𝑐5 1 𝜔𝜔𝜔𝜔 , 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) = 𝑐𝑐1 (𝑐𝑐2 − 𝑐𝑐3 𝛽𝛽 − 𝑐𝑐4 ) 𝑒𝑒 − 𝐿𝐿 − 𝑐𝑐6 𝐿𝐿 𝑉𝑉
1 𝑉𝑉 0,035 = − . 𝐿𝐿 𝑅𝑅𝑅𝑅 + 0.08𝛽𝛽𝛽𝛽 1 + 𝛽𝛽 3
Fig.1. Object input/output during open loop identification Algorithm 3.1 is used to obtain estimates of the object linearized model parameters from the data of this test: 𝑎𝑎̂ = 0.04984, 𝑏𝑏̂ = 0.06414
(43)
Simulation of nonlinear model (1),(40),(41) with parameters (42) and linearized model (10) with parameters (43) are carried out for step of 𝛽𝛽𝑠𝑠𝑠𝑠 from 20.6245 deg to 20.7 deg. Figure 2 shows that outputs of these models are closely adjacent and estimates (43) are verified.
(40) (41)
with the following parameters values (Tilli, A. et al. (2011)): 𝐽𝐽 = 40000kg𝑚𝑚2 , 𝜌𝜌 = 1.225 𝑃𝑃𝑅𝑅 = 200𝑘𝑘𝑘𝑘, 𝜔𝜔𝑅𝑅 = 6.7 с1 = 0.5176, с2 = 116,
𝑘𝑘𝑘𝑘
𝑚𝑚3
𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠
, 𝑅𝑅 = 13𝑚𝑚,
(42)
𝑚𝑚
, 𝑉𝑉𝑠𝑠𝑠𝑠 = 17 ,
Fig 2. Step response of nonlinear and linearized models
𝑐𝑐
𝑐𝑐3 = 0.4, 𝑐𝑐4 = 5,
𝑐𝑐5 = 21, 𝑐𝑐6 = 0.0068
6.2 Open loop identification for fluctuating wind speed simulation Simulation of object (1),(40),(41) for test signal (13) with parameters 𝛽𝛽𝑠𝑠𝑠𝑠 = 20.6245 deg , 𝑞𝑞 = 0.1, 𝛾𝛾 = 0.1 when 2311
Proceedings of the 20th IFAC World Congress 2276 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277 Toulouse, France, July 9-14, 2017
wind speed is 𝑉𝑉𝑠𝑠𝑠𝑠 added by random value is shown on Figure 3.
Algorithm 4.1 return estimates of the object linearized model parameters: 𝑎𝑎̂ = 0.04948, 𝑏𝑏̂ = 0.06405
(45)
It is practically the same values that were obtained by open loop identification.
6.4 Closed loop identification for fluctuating wind speed simulation Closed loop identification when wind speed is 𝑉𝑉𝑠𝑠𝑠𝑠 added by random value is shown on Figure 5.
Fig.3. Object input/output during open loop identification for fluctuating wind speed The shown wind speed is used for a simulation but identification procedure operates with the pitch angle (input) and rotor speed (output) data only. Algorithm 3.1 returns estimates of the object linearized model parameters: 𝑎𝑎̂ = 0.05499, 𝑏𝑏̂ = 0.06398
(44)
They do not differ significantly from estimates (43). Fig.5. Object input/output during open loop identification for fluctuating wind speed
6.3 Closed loop identification simulation The object (1) ,(40),(41) with parameters (42) with PI controller (2),(3) is simulated. Let PI coefficietns 𝑘𝑘𝑝𝑝 = 100 and 𝑘𝑘𝑖𝑖 = 10.
The pitch angle 𝛽𝛽 is formed by (20),(21). Object input and output are shown on Figure 4.
Estimates of the object linearized model parameters from the data of this test are obtained by Algorithm 4.1: 𝑎𝑎̂ = 0.04917, 𝑏𝑏̂ = 0.05793
(46)
They do not differ significantly from previous estimates. Obtained estimates obey inequations (34) and (35).
7. CONCLUSIONS Approach to first order linearized model identification based on sine wave test signal is considered in the paper. The identification procedure can be carried out in closed loop in spite of fluctuating wind speed. Results of first order linearized model identification can be the basis for PI controller tuning. Coefficients of PI controller are chosen from the condition that the roots of the characteristic equation of closed loop system of linearized model with PI controller are real-valued for wide limits of linearized model parameters. The simulation results show the high efficiency of the proposed approach. Fig.4. Object input/output during closed loop identification
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ACKNOWLEDGMENT The authors wish to thank Dr. Kitch Wilson (Applied Theoretic Systems, Inc.) for valuable comments on the blade pitch angle control.
REFERENCES Alexandrov, A.G., Palenov, M.V. (2011). Self-tuning PID-I controller. Preprints of the 18th IFAC World Congress.
Milano, Italy. 28 Aug. - 2 Sept.2011, pp.3635-3640 Alexandrov, A.G., Khomutov D., Shchegoleva K. (2008). Dynamic System Identifier "DPI-3" : Experimental Investigations. Proceedings of the 27-th IASTED
International Conference on Modelling, Identification and Control, Innsbruck, Austria. Alexandrov, A.G. (2005). Finite-frequency identification: self-tuning of test signal. Proceeding of 16th World
Congress of IFAC, Preprints, p.p. 295–301. Alexandrov, A.G. (1994). Finite-frequency method of identification. Proceeding of 10th IFAC Symposium on
System Identification, Preprints, vol. 2, p.p. 523–527. Burton, T., Sharpe, D., Jenkins, N., Bossanyi, E. (2001). Wind energy handbook. Wiley & Sons, Ltd Baffins Lane, Chichester, West Sussex, England. Bianchi, F.D., Mantz, R.J., Christiansen, C.F. (2005). Gain scheduling control of variable-speed wind energy conversion systems using quasi-LPV models. Control
Engineering practice, Vol. 13, pp. 247-255. Joo, Y., Back, J. (2012). Power Regulation of Variable Speed Wind Turbines using Pitch Control based on Journal of Electrical Disturbance Observer.
Engineering & Technology Vol. 7, No. 2, pp. 273-280. Leithead, W.E., Connor, P. (2000). Control of variable speed wind turbines: design task. International Journal of Control, IEEE, Vol. 73, n. 13, pp. 1189-1212. Rocha, R., Filho, L.S.M., Bortolus, M.V. (2005). Optimal Multivariable Control for Wind Energy Conversion System - A comparison between H2 and H∞ controllers. IEEE Conference on Decision and Control, December 2005, pp. 7906-7911. Tilli, A., Conficoni, C. (2011). Speed Control for Medium PowerWind Turbines: An Integrated Approach oriented to MPPT. Preprints of the 18th IFAC World
Congress Milano (Italy) August 28 - September 2 2011, pp. 545-550.
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Identification Identification Based Based Control Control For For Wind Wind Turbine Turbine Identification Based Control For Wind Identification Based Control For Wind Turbine Turbine Albert G. Alexandrov* Vladimir N. Chestnov**
Albert G. Alexandrov* Vladimir N. Chestnov** Vadim A. Alexandrov*** Albert Vladimir N. Chestnov** Vadim A. Alexandrov*** Albert G. G. Alexandrov* Alexandrov* Vladimir N. Chestnov** Vadim A. Alexandrov*** Vadim A. Alexandrov*** *V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, *V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia (e-mail: [email protected]) *V.A. Trapeznikov Institute of Control of Academy of Moscow, Russia (e-mail:Sciences [email protected]) *V.A. Trapeznikov Institute of Control Sciences of Russian Russian Academy of Sciences, Sciences, ** V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Moscow, Russia (e-mail: [email protected]) ** V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Moscow, Russia (e-mail: [email protected]) Sciences, Moscow, Russia (e-mail: [email protected]) ** V.A. of Control Sciences Moscow, Institute Russia (e-mail: [email protected]) **Sciences, V.A. Trapeznikov Trapeznikov Institute of(e-mail:[email protected]) Control Sciences of of Russian Russian Academy Academy of of *** Adaplab LLC, Moscow, Russia Sciences, Moscow, Russia (e-mail: [email protected]) *** Adaplab LLC, Moscow, Russia (e-mail:[email protected]) Sciences, Moscow, Russia (e-mail: [email protected]) *** *** Adaplab Adaplab LLC, LLC, Moscow, Moscow, Russia Russia (e-mail:[email protected]) (e-mail:[email protected]) Abstract: The blade pitch angle control when the wind speed is above rated is considered in this Abstract: The blade pitch angle control when the wind speed is above rated is considered in this paper. Finite-frequency identification approach is used to obtain the linearized model. The identification Abstract: The blade pitch angle control when the wind speed is rated is considered in paper. Finite-frequency identification approach is used to obtain the linearized model. identification Abstract: The blade pitch angle control when the wind speed is above above rated isThe considered in this this procedure can be carried out in closed loop in spite of fluctuating wind speed. PI controller paper. Finite-frequency identification approach is used to obtain the linearized model. The identification procedure can be carried out in closed loop in spite of fluctuating wind speed. PI controller paper. Finite-frequency identification approach is used to obtain the linearized model. The identification coefficients determine limits for linearized model parameters for which the roots of characteristic procedure be out in closed loop in of speed. PI controller coefficientscan determine limits linearized model parameters for whichwind the roots of characteristic procedure can be carried carried out for in closed loop inIf spite spite of fluctuating fluctuating wind speed. PIare controller identified estimates of parameters outside equation of closed loop system are real-valued. coefficients determine limits for linearized model parameters for which the roots of characteristic equation of closed loop system are real-valued. If identified estimates of parameters are outside coefficients determine limits for linearized model parameters for which the roots of characteristic the limits maintained by previously designed PI controller then PI controller is tuned on the basis equation closed are real-valued. If identified estimates of parameters are outside the limits of maintained by system previously PI controller then PI controller is tuned on basis equation of closed loop loop system are designed real-valued. If the identified estimates of parameters arethe outside of obtained estimates. The simulation results show high efficiency of the proposed approach. the limits PI PI is on of estimates. by Thepreviously simulationdesigned results show the highthen efficiency of the proposed theobtained limits maintained maintained by previously designed PI controller controller then PI controller controller is tuned tuned approach. on the the basis basis Keywords: variable speed wind turbine, pitch control, PI control, identification, disturbance .approach. of obtained estimates. The simulation results show the high efficiency of the proposed © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. of obtainedvariable estimates. The simulation results show PI thecontrol, high efficiency of thedisturbance proposed .approach. Keywords: speed wind turbine, pitch control, identification, Keywords: Keywords: variable variable speed speed wind wind turbine, turbine, pitch pitch control, control, PI PI control, control, identification, identification, disturbance disturbance.. 1. INTRODUCTION 1. INTRODUCTION 1. Wind energy conversion to electric power is one of the 1. INTRODUCTION INTRODUCTION Wind energy conversion to electric power is one of the most important problems of electric modernpower powerisengineering. Wind energy conversion one of most important problemsto of electric modernpower powerisengineering. Wind energy conversion to one of the the The blade pitch angle control is considered in the paper. most important problems of modern power engineering. The blade pitch angle control is considered in the paper. most important problems of modern power engineering. One of the pitch main angle difficulties of is this control isin a significant The control considered the paper. One of the pitch main angle difficulties of is this control isin a significant The blade blade control considered theturbine paper. nonlinearity of the mathematical model of the One of the main difficulties of this control is a significant nonlinearity of the mathematical model of the turbine One of the main difficulties of this control is a significant rotor speed of toward the pitch angle (Burton, T.turbine et al. nonlinearity the model of rotor speed of toward the pitch angle (Burton, T.turbine et al. nonlinearity the mathematical mathematical model of the theare (2001)). Well-established PI and PID controllers used rotor speed toward the pitch angle (Burton, T. et al. (2001)). Well-established PI and PID controllers are used rotor speed toward thebypitch angle (Burton, T. et al. for wind turbine control pitch angle changing (Burton, (2001)). Well-established PI and PID controllers are used for wind turbine control by pitch angle changing (Burton, (2001)). Well-established PI and PID controllers are More used T. et al. (2001), Leithead, W.E. et al. (2000)). for wind by pitch changing (Burton, T. al. turbine (2001),control Leithead, W.E.angle et al. (2000)). More for et wind turbine control by H pitch angle changing (Burton, H2 and modern techniques ∞ are also practiced (Rocha, T. et al. (2001), Leithead, W.E. et al. (2000)). More H2 and H∞W.E. are also practiced (Rocha, modern techniques T. et al. (2001), Leithead, et al. (2000)). More R. et al. techniques (2005)). Recent papers utilised gain scheduling H and H are also practiced (Rocha, modern 2 ∞ R. et al. techniques (2005)). Recent papers utilised gain scheduling H and H are also practiced (Rocha, modern 2 ∞ (Bianchi, F.D. et al.Recent (2005)) , disturbance observer (Joo, Y. R. et al. papers utilised gain (Bianchi, F.D. et al.Recent (2005)) , disturbance observer (Joo, Y. R. et (2012)), al. (2005)). (2005)). papers utilised gain scheduling scheduling et al. Maximum Power Point Tracking algorithm (Bianchi, F.D. et al. (2005)) ,, disturbance observer (Joo, et al. (2012)), Maximum Power Point Tracking algorithm (Bianchi, F.D. et al. (2005)) disturbance observer (Joo, Y. Y. (Tilli, A. et al. (2011)), etc.Power Point Tracking algorithm et al. (2012)), Maximum (Tilli, A. et al. (2011)), etc.Power Point Tracking algorithm et al. (2012)), Maximum (Tilli, et (2011)), In all A. these it is assumed that all parameters (Tilli, et al. al.techniques (2011)), etc. etc. In all A. these techniques it is assumed that all parameters (or their limits) of the it wind turbine are defined. In this In all these techniques is that all parameters (or their limits) of the wind turbine are defined. In this In all these techniques it is assumed assumed that allobtained parameters paper it is assumed that controller has been for (or their of the turbine defined. In paper it islimits) assumed thatwind controller hasare been obtained for (or their limits) of the wind turbine are defined. In itthis this known limits of the wind turbine parameters. But is paper it is assumed controller been obtained known limits of thethat wind turbine has parameters. But itfor is paper it is assumed that controller has been obtained for considered that blade aerodynamics can change in time known limits of the wind turbine parameters. it is considered that blade aerodynamics can changeBut in time known limits of the wind turbine parameters. But it is and the wind turbine parameters go can out change of theseinlimits. considered that blade aerodynamics time and the wind turbine parameters go can out change of theseinlimits. considered that blade aerodynamics time The finite-frequency identification method (Alexandrov, and the wind parameters go out these The finite-frequency identification method and et the wind turbine turbine parameters go out of of (Alexandrov, theseislimits. limits. A.G. al.(2011), Alexandrov, A.G. (2005),(1994)) used The finite-frequency identification method (Alexandrov, A.G. et al.(2011), Alexandrov, A.G. (2005),(1994)) is used The finite-frequency identification method (Alexandrov, in the paper for obtaining linearized model parameters. A.G. et al.(2011), Alexandrov, A.G. (2005),(1994)) is in for obtaining linearized model parameters. A.G.the et paper al.(2011), Alexandrov, A.G. is used used Advantages of this method are(2005),(1994)) abilities to identify in the paper for obtaining linearized model parameters. Advantages of this method are abilities to identify in the paper for obtaining linearized model parameters. linearized model parameters onare running turbine, in real Advantages of this method abilities to identify linearized model parameters onare running turbine, in real Advantages of loop this method abilities to external identify time, in closed system, in spite of unknown linearized model parameters on running turbine, in time, in closed loop system, in spite of unknown external linearized model parameters runningcan turbine, in real real disturbances. Coefficients ofinon controller be changed time, in closed loop system, spite of unknown external disturbances. Coefficients of controller can be changed time, in closed loop system, in spite of unknown external based on identification results. disturbances. Coefficients of based on identification results. disturbances. Coefficients of controller controller can can be be changed changed based on identification results. The case when the wind speed is above rated as operating based on identification results. The case when the wind speed is above rated as operating region of when pitch the control isspeed considered in rated the paper. In case The case wind is as region of when pitch the control considered in rated the paper. In case The case windisup speed is above above as operating operating of low wind with speed to rated the pitch angle should region of control in the case of low wind with speedis to rated the angle In should region of pitch pitch control isupconsidered considered in pitch the paper. paper. In case of low wind with speed up to rated the pitch angle should of low wind with speed up to rated the pitch angle should
be at minimum value for maximum power. This value may be at minimum value for maximum power. This value may slightly differ from zeromaximum due to power. the design of wind be value for This value may slightly differ from zeromaximum due to power. the design of wind be at at minimum minimum value for This25-30 value may turbine. In casefrom of very strong wind more than m/s slightly differ zero due to the design of wind turbine. In casefrom of very strong wind more than 25-30 m/s slightly differ zero due to the design of wind the turbine is stopped for safety. Therefore the pitch turbine. In case of very strong wind more than 25-30 m/s the turbine is stopped for safety. Therefore the pitch turbine. In case of very strong wind more than 25-30 m/s controller operates in a limited rangeTherefore of wind speed. the turbine is stopped for safety. the pitch controller operates in a limited rangeTherefore of wind speed. the turbine is stopped for safety. the pitch controller controller operates operates in in aa limited limited range range of of wind wind speed. speed. 2. PROBLEM STATEMENT 2. PROBLEM STATEMENT STATEMENT Control system 2. of PROBLEM wind turbine is considered. It consists STATEMENT Control system 2. of PROBLEM wind turbine is considered. It consists of the control object and the controller. The controller is Control system of wind turbine considered. It consists of the control object and the controller. The controller is Controlpitch system of wind turbine is is The considered. Itobject consists blade angle PIand controller. control is of the control object the controller. The controller is blade pitch angle PI controller. The control object is of the control object and the controller. The controller considered inangle conditions when theThe wind speed is above blade pitch PI controller. control object is considered inangle conditions when theThe wind speed object is above blade pitch PI controller. control is rated and the pitch controller restricts wind turbine rotor considered in conditions when the wind speed is above rated and the pitch controller restricts wind turbine rotor considered in conditions when the wind speed is above speed while the torque controller tracks the turbine rated power. rated and the pitch controller restricts wind rotor speed while torque controller tracks the turbine rated power. rated and thethe pitch controller restricts wind rotor In this case the control object is described by equation speed while the torque controller tracks the rated power. In thiswhile case the the torque controlcontroller object is tracks described by equation speed the rated power. (Tilli, A. et al.the (2011); Joo,object Y. et al.is(2012)) In this control described (Tilli, et al.the (2011); Joo,object Y. et al.is(2012)) In thisA.case case control described by by equation equation (Tilli, A. et al. (2011); Joo, Y. et al. (2012)) 𝜌𝜌𝜌𝜌 2 3Joo, Y. et al. (2012)) 1 𝑃𝑃𝑅𝑅 (1) (Tilli, A. et al. (2011); 𝜔𝜔̇ = 𝜌𝜌𝜌𝜌 𝑅𝑅2 𝑉𝑉 3 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) − 1 𝑃𝑃𝑅𝑅 , (1) (𝜔𝜔, 𝜔𝜔̇ = 2𝐽𝐽𝐽𝐽 𝑅𝑅 𝑉𝑉 𝐶𝐶 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) − , 𝐽𝐽 𝜔𝜔 𝜌𝜌𝜌𝜌 1 𝑃𝑃 (1) 𝑅𝑅 2 3 𝑝𝑝 (𝜔𝜔, 𝜌𝜌𝜌𝜌 1 𝑃𝑃 2𝐽𝐽𝐽𝐽 𝐽𝐽 𝜔𝜔 (1) 𝑅𝑅 𝜔𝜔̇ = = 𝑅𝑅 𝛽𝛽, 𝑉𝑉, 𝑉𝑉, 𝑐𝑐) 𝑐𝑐) − − 𝐽𝐽 𝜔𝜔 ,, 𝑅𝑅2 𝑉𝑉 𝑉𝑉 3 𝐶𝐶 𝐶𝐶𝑝𝑝𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝜔𝜔̇ 2𝐽𝐽𝐽𝐽 2𝐽𝐽𝐽𝐽 𝐽𝐽 𝜔𝜔 where 𝜔𝜔 is the measured turbine rotor speed, 𝛽𝛽 is the where 𝜔𝜔 is the measured turbine rotor speed, 𝛽𝛽 is the controlled blade pitch angle, 𝑉𝑉 is rotor the wind speed, is where 𝜔𝜔 is is the measured measured turbine speed, 𝛽𝛽 is is𝜌𝜌 the controlled blade pitch angle, 𝑉𝑉 is rotor the wind speed, 𝜌𝜌the is where 𝜔𝜔 the turbine speed, 𝛽𝛽 R is the blade rotor radius, 𝐽𝐽 is the the air density, controlled blade pitch angle, 𝑉𝑉 is is the the wind speed, speed, 𝜌𝜌the is R is the angle, blade rotor radius, 𝐽𝐽 is 𝜌𝜌 the air density, controlled blade pitch 𝑉𝑉 wind is is rated power, 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽,is 𝑉𝑉, 𝑐𝑐)the is moment of inertia, 𝑃𝑃𝑅𝑅 blade R is the rotor radius, 𝐽𝐽 the air density, is rated 𝑉𝑉, 𝑐𝑐)the is moment of inertia, 𝑃𝑃𝑅𝑅 blade R is the rotorpower, radius,𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝐽𝐽 𝛽𝛽,is the density, the air power coefficient as nonlinear function depended on rated power, 𝐶𝐶 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) is moment inertia, 𝑃𝑃 𝑅𝑅as is 𝑝𝑝 (𝜔𝜔, the powerof coefficient nonlinear function depended on is rated power, 𝐶𝐶 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) isa moment of inertia, 𝑃𝑃 𝑅𝑅 𝑝𝑝 turbine specifics withasparameters vector depended 𝑐𝑐. This ison the power coefficient nonlinear function turbine specifics withasparameters vector depended 𝑐𝑐. This isona the power coefficient nonlinear function simplified model with without detailed vector aerodynamics anda turbine specifics parameters 𝑐𝑐. simplified model with without detailed vector aerodynamics anda turbine specifics parameters 𝑐𝑐. This This is is mechanics and without generator and grid models. simplified model without detailed aerodynamics and mechanics without generator and grid models. simplified and model without detailed aerodynamics and mechanics and without generator and grid models. The pitch angle 𝛽𝛽 is the control value from PI controller mechanics and without generator and grid models. The pitch angle 𝛽𝛽 is the control value from PI controller The is control (𝜔𝜔 − 𝜔𝜔 ) +value 𝑧𝑧, from The pitch pitch angle angle𝛽𝛽 𝛽𝛽 𝛽𝛽= 𝑘𝑘 is the the control value from PI PI controller controller (2) 𝛽𝛽 = 𝑘𝑘𝑝𝑝𝑝𝑝 (𝜔𝜔 − 𝜔𝜔𝑅𝑅𝑅𝑅 ) + 𝑧𝑧, (2) 𝛽𝛽 (2) 𝛽𝛽 = = 𝑘𝑘 𝑘𝑘𝑝𝑝𝑝𝑝 (𝜔𝜔 (𝜔𝜔 − − 𝜔𝜔 𝜔𝜔𝑅𝑅𝑅𝑅 )) + + 𝑧𝑧, 𝑧𝑧, (2)
Copyright © 2017 IFAC 2308 Copyright © 2017 IFAC 2308 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 2308Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 responsibility IFAC 2308Control. Peer review©under of International Federation of Automatic 10.1016/j.ifacol.2017.08.196
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277
𝑧𝑧̇ = 𝑘𝑘𝑖𝑖 (𝜔𝜔 − 𝜔𝜔𝑅𝑅 ),
(3)
where 𝜔𝜔𝑅𝑅 is the determined desired turbine rotor speed, 𝑧𝑧 is the controller integrator output, 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 are positive PI coefficients. Reasonable to assume that controlled blade pitch angle 𝛽𝛽 is limited from 0 to 90 degrees. Although in practice these boundaries may be another.
Finite-frequency identification method (Alexandrov, A.G. (2005),(1994)) that works in spite of unknown external disturbances will be solving this problem. The limits of parameters of the linearized model (10) 𝑎𝑎 and 𝑏𝑏 should be determined by coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 of PI controller (2),(3) such that the roots of the characteristic equation of closed loop system are real-valued: 𝐴𝐴 = [𝑎𝑎, 𝑎𝑎], 𝐵𝐵 = [𝑏𝑏, 𝑏𝑏]
Dinamics of pitch actuator, gear and torque controller are not considered for simplicity. 𝜌𝜌𝜌𝜌 2 3 1 𝑃𝑃𝑅𝑅 𝑅𝑅 𝑉𝑉 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, V, 𝑐𝑐) − 2𝐽𝐽𝐽𝐽 𝐽𝐽 𝜔𝜔
(4)
Problem: if identified estimates 𝑎𝑎̂, 𝑏𝑏̂ are not within these limits then coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 should be tuned.
then the equation (1) takes the form 𝜔𝜔̇ = 𝜑𝜑(𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐).
(5)
The Taylor series for function (4) around the point 𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , where 𝛽𝛽𝑠𝑠𝑠𝑠 provides speed 𝜔𝜔𝑆𝑆𝑆𝑆 close to 𝜔𝜔𝑅𝑅 at a wind speed 𝑉𝑉𝑠𝑠𝑠𝑠 is: 𝜑𝜑(𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) = 𝜑𝜑(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) + 𝑎𝑎(𝜔𝜔 − 𝜔𝜔𝑅𝑅 ) + 𝑏𝑏(𝛽𝛽 − 𝛽𝛽𝑠𝑠𝑠𝑠 ) + 𝑑𝑑(𝑉𝑉 − 𝑉𝑉𝑠𝑠𝑠𝑠 ) + 𝜓𝜓(𝜔𝜔, 𝑅𝑅, 𝑉𝑉, 𝑐𝑐)
where 𝑎𝑎 = 𝑎𝑎(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) =
∂𝜑𝜑 | , ∂𝜔𝜔 𝜔𝜔𝑅𝑅 ,𝛽𝛽𝑠𝑠𝑠𝑠 ,𝑉𝑉𝑠𝑠𝑠𝑠
𝑏𝑏 = 𝑏𝑏(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) =
∂𝜑𝜑
| , ∂𝛽𝛽 𝜔𝜔𝑅𝑅 ,𝛽𝛽𝑠𝑠𝑠𝑠 ,𝑉𝑉𝑠𝑠𝑠𝑠
(11)
where 𝑎𝑎, 𝑏𝑏 are low limits and 𝑎𝑎, 𝑏𝑏 are high limits of the parameters.
Let the right part of the equation (1) be denoted as 𝜑𝜑(𝜔𝜔, 𝛽𝛽, 𝑣𝑣, 𝑐𝑐) =
2273
(6)
(7)
3. OPEN LOOP IDENTIFICATION The system (1)-(3) with disabled PI controller is considered: 𝑘𝑘𝑝𝑝 = 𝑘𝑘𝑖𝑖 = 0.
(12)
𝛽𝛽 = 𝛽𝛽𝑠𝑠𝑠𝑠 + 𝑞𝑞 sin 𝛾𝛾𝛾𝛾
(13)
In this case pitch angle 𝛽𝛽 is formed as a sum of 𝛽𝛽𝑠𝑠𝑠𝑠 defined by operating point and test sine wave: where 𝛾𝛾 is test frequency, 𝑞𝑞 is amplitude of the test sine wave. The test sine wave parameters based on the finite-frequency identification method (Alexandrov, A.G. (2005)). Then equation (10) has the form
𝑑𝑑 = 𝑑𝑑(𝜔𝜔𝑅𝑅 , 𝛽𝛽𝑠𝑠𝑠𝑠 , 𝑉𝑉𝑠𝑠𝑠𝑠 , 𝑐𝑐) =
∂𝜑𝜑
| , ∂𝑉𝑉 𝜔𝜔𝑅𝑅 ,𝛽𝛽𝑠𝑠𝑠𝑠 ,𝑉𝑉𝑠𝑠𝑠𝑠
𝑦𝑦̇ = −𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 sin 𝛾𝛾𝛾𝛾 + 𝑑𝑑𝑉𝑉𝑑𝑑 .
(8)
(14)
𝜓𝜓(𝜔𝜔, 𝑅𝑅, 𝑉𝑉, 𝑐𝑐) contains deviations in the second degree or higher.
A particular solution of this equation where the turbine rotor speed 𝜔𝜔 oscillates about 𝜔𝜔𝑅𝑅 is
In (6) 𝑎𝑎 < 0 for the condition of the stability of the equation (1) therefore hereafter – 𝑎𝑎 where 𝑎𝑎 > 0 will be used. Value b in (7) is always negative because rotor speed 𝜔𝜔 decreases when the 𝛽𝛽 increases. Hereafter – 𝑏𝑏 where 𝑏𝑏 > 0 will be used.
(15)
Let 𝑦𝑦 = 𝜔𝜔 − 𝜔𝜔𝑅𝑅 ,
𝑢𝑢 = 𝛽𝛽 − 𝛽𝛽𝑠𝑠𝑠𝑠 ,
𝑉𝑉𝑑𝑑 = 𝑉𝑉 − 𝑉𝑉𝑠𝑠𝑠𝑠
(9)
𝑎𝑎 > 0, 𝑏𝑏 > 0
where 𝜈𝜈 and 𝜇𝜇 are frequency parameters of object on the frequency 𝛾𝛾 ; 𝑦𝑦𝑑𝑑 (𝑡𝑡) is the output component depending on the wind speed fluctuations. 𝑦𝑦̇ (𝑡𝑡) obtaining from (15) and 𝑦𝑦(𝑡𝑡) from (15) are pluged to (14). Then coefficients at sin𝛾𝛾𝛾𝛾 and cos𝛾𝛾𝛾𝛾 are equated. Thereby parameters 𝑎𝑎 and 𝑏𝑏 are expressed through frequency parameters 𝜈𝜈 and 𝜇𝜇 as 𝜈𝜈
Then linearized equation of wind turbine is 𝑦𝑦̇ = −𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 + 𝑑𝑑𝑉𝑉𝑑𝑑 ,
𝑦𝑦(𝑡𝑡) = 𝑞𝑞𝑞𝑞 sin 𝛾𝛾𝛾𝛾 + 𝑞𝑞𝑞𝑞 cos 𝛾𝛾𝛾𝛾 + 𝑦𝑦𝑑𝑑 ,
𝑎𝑎 = − 𝛾𝛾,
𝑏𝑏 = 𝜇𝜇𝜇𝜇 − 𝑎𝑎𝑎𝑎
𝜇𝜇
(10)
Problem of identification: to find out estimates 𝑎𝑎̂, 𝑏𝑏̂ of parameters of the linearized model (10).
and conversely
The need of identification is caused by the deformation of blades during continuous working of wind turbine which leads to undefined change of the power coefficient 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) in equation (1).
Estimates of frequency parameters 𝜈𝜈̂ and 𝜇𝜇̂ calculated as Fourier filter outputs on the frequency
𝜈𝜈 = −
Identification is complicated by unknown variations of the wind speed 𝑉𝑉𝑑𝑑 relatively constant 𝑉𝑉𝑠𝑠𝑠𝑠 . 2309
𝑏𝑏𝑏𝑏
𝛾𝛾2 +𝑎𝑎2
𝜈𝜈̂ = 𝜈𝜈(𝜏𝜏) = 𝜇𝜇̂ = 𝜇𝜇(𝜏𝜏) =
,
𝜇𝜇 =
𝑏𝑏𝑏𝑏
𝛾𝛾2 +𝑎𝑎2
2 𝑡𝑡𝐹𝐹 +𝜏𝜏 𝑦𝑦(𝑡𝑡) sin 𝛾𝛾𝛾𝛾 𝑑𝑑𝑑𝑑 ∫ 𝑞𝑞𝑞𝑞 𝑡𝑡𝐹𝐹
2 𝑡𝑡𝐹𝐹 +𝜏𝜏 𝑦𝑦(𝑡𝑡) cos 𝛾𝛾𝛾𝛾 𝑑𝑑𝑑𝑑 ∫ 𝑞𝑞𝑞𝑞 𝑡𝑡𝐹𝐹
(16)
(17) are (18) (19)
Proceedings of the 20th IFAC World Congress 2274 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277 Toulouse, France, July 9-14, 2017
where 𝜏𝜏 is filtering time, 𝑡𝑡𝐹𝐹 is filtering start time. Fourier filter outputs converge for 𝜏𝜏 → ∞ to the true values if the wind speed does not contain harmonic oscillations with a frequency of 𝛾𝛾 (Alexandrov, A.G. (2005),(1994)). Implementation of frequency parameters 𝜈𝜈̂ and 𝜇𝜇̂ real-time calculation is acquainted (Alexandrov, A.G. et al.(2008))
Estimates of 𝜈𝜈𝑦𝑦 , 𝜇𝜇𝑦𝑦 , 𝜈𝜈𝑢𝑢 , 𝜇𝜇𝑢𝑢 values are obtained by Fourier filter (18),(19). Hence,
Hence,
Step 2. To find out estimates of frequency parameters 𝜈𝜈̂𝑦𝑦 , 𝜇𝜇̂ 𝑦𝑦 , 𝜈𝜈̂𝑢𝑢 , 𝜇𝜇̂ 𝑢𝑢 as Fourier filter outputs (18), (19) and to calculate frequency parameters of object by (27),(28).
Algorithm 3.1 The open loop identification of a wind turbine consists of operations: Step 1. To form the blades motion in accordance with the expression (13). Step 2. To find out estimates of frequency parameters as Fourier filter outputs (18), (19).
Algorithm 4.1. The closed loop identification of a wind turbine consists of operations: Step 1. To form the blades motion in accordance with the expression (20).
Step 3. To calculate estimates of the object parameters by formulae (16). In a real system it is needed to check that the torque controller should not react to our test sine wave.
Step 3. To calculate estimates of the object parameters by formulae (16) using estimates of frequency parameters.
4. CLOSED LOOP IDENTIFICATION Let PI coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 be available. Then pitch angle 𝛽𝛽 is formed as 𝛽𝛽 = 𝑘𝑘𝑝𝑝 (𝜔𝜔 − 𝜔𝜔𝑅𝑅 ) + 𝑧𝑧 + 𝑞𝑞 sin 𝛾𝛾𝛾𝛾 𝑧𝑧̇ = 𝑘𝑘𝑖𝑖 ⋅ (𝜔𝜔 − 𝜔𝜔𝑅𝑅 )
(21) 𝑧𝑧̇ = 𝑘𝑘𝑖𝑖 𝑦𝑦
and object input is (24)
where 𝜈𝜈𝑦𝑦 and 𝜇𝜇𝑦𝑦 are output frequency parameters which estimates can be calculated according to (18),(19), 𝜈𝜈𝑢𝑢 and 𝜇𝜇𝑢𝑢 are input frequency parameters which estimates can be calculated by formulae analogous as (18),(19) for 𝑢𝑢(𝑡𝑡) rather than 𝑦𝑦(𝑡𝑡). Plugging (23),(24) into (10) similar as for (16),(17) 𝜈𝜈𝑦𝑦 and 𝜇𝜇𝑦𝑦 are expressed as 𝜈𝜈𝑦𝑦 = −𝑏𝑏
𝛾𝛾2 +𝑎𝑎2
, 𝜇𝜇𝑦𝑦 = 𝑏𝑏
−𝜇𝜇𝑢𝑢𝑎𝑎+𝛾𝛾𝜈𝜈𝑢𝑢 𝛾𝛾2 +𝑎𝑎2
𝑢𝑢 = 𝑘𝑘𝑝𝑝 y + z,
𝜇𝜇𝑦𝑦 = 𝜇𝜇𝑢𝑢 𝜈𝜈 + 𝜈𝜈𝑢𝑢 𝜇𝜇
𝑦𝑦̈ + (𝑎𝑎 + 𝑏𝑏 𝑘𝑘𝑝𝑝 )𝑦𝑦̇ + 𝑏𝑏 𝑘𝑘𝑖𝑖 𝑦𝑦 = 0
𝑠𝑠 2 + (𝑎𝑎 + 𝑏𝑏𝑘𝑘𝑝𝑝 )𝑠𝑠 + 𝑏𝑏𝑘𝑘𝑖𝑖 = 0
𝜇𝜇 =
𝜈𝜈𝑦𝑦 𝜈𝜈𝑢𝑢 + 𝜇𝜇𝑦𝑦 𝜇𝜇𝑢𝑢 𝜈𝜈𝑢𝑢2 + 𝜇𝜇𝑢𝑢2
𝜇𝜇𝑦𝑦 𝜈𝜈𝑢𝑢 − 𝜈𝜈𝑦𝑦 𝜇𝜇𝑢𝑢 𝜈𝜈𝑢𝑢2 + 𝜇𝜇𝑢𝑢2
(30)
2
(𝑎𝑎 + 𝑏𝑏𝑘𝑘𝑝𝑝 ) − 4𝑏𝑏𝑘𝑘𝑖𝑖 > 0
(32)
𝑎𝑎2 + 2𝑎𝑎𝑎𝑎𝑘𝑘𝑝𝑝 + 𝑏𝑏 2 𝑘𝑘𝑝𝑝2 − 4𝑏𝑏𝑘𝑘𝑖𝑖 > 0
(33)
or
Sufficient conditions for validity of this inequation are
(25)
(26)
(27)
(31)
All parameters of this equation (𝑎𝑎, 𝑏𝑏, 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 ) are positive, so the roots of this characteristic equation have negative real parts. Conditions should be found so that these roots are real-valued to ensure the quality of transients. It is seen that this condition has the form
𝑎𝑎 >
2𝑘𝑘𝑖𝑖 𝑘𝑘𝑝𝑝
(34)
or
Frequency parameters of object 𝜈𝜈 and 𝜇𝜇 can be expressed from (26): 𝜈𝜈 =
(29)
Differential equation obtaining after differentiating of equation (10) for 𝑉𝑉𝑑𝑑 = 0 and excluding variable u is
or using (17) as 𝜈𝜈𝑦𝑦 = 𝜈𝜈𝑢𝑢 𝜈𝜈 − 𝜇𝜇𝑢𝑢 𝜇𝜇,
ż = 𝑘𝑘𝑖𝑖 y
Then characteristic equation has the form
𝑦𝑦̅(𝑡𝑡) = 𝑞𝑞𝜈𝜈𝑦𝑦 sin 𝛾𝛾𝛾𝛾 + 𝑞𝑞𝜇𝜇𝑦𝑦 cos 𝛾𝛾𝛾𝛾 + 𝑦𝑦̅𝑑𝑑 (23)
𝜈𝜈𝑢𝑢 𝑎𝑎−γ𝜇𝜇𝑢𝑢
PI controller formulae (2),(3) are written as
(22)
Output of the system (10), (22) is
𝑢𝑢(𝑡𝑡) = 𝑞𝑞𝜈𝜈𝑢𝑢 sin 𝛾𝛾𝛾𝛾 + 𝑞𝑞𝜇𝜇𝑢𝑢 cos 𝛾𝛾𝛾𝛾
Let estimates 𝑎𝑎̂ and 𝑏𝑏̂ be found. They will be used as values of the object parameters in the linearized equation (10).
(20)
or using (9) as 𝑢𝑢 = 𝑘𝑘𝑝𝑝 𝑦𝑦 + 𝑧𝑧 + 𝑞𝑞 sin 𝛾𝛾𝛾𝛾 ,
5. PI COEFFICIENTS
𝑏𝑏 > 4
𝑘𝑘𝑖𝑖
𝑘𝑘𝑝𝑝 2
(35)
Then low limits for parameters interval (11) for available coefficients 𝑘𝑘𝑝𝑝 and 𝑘𝑘𝑖𝑖 can be evaluated as
(28)
𝑎𝑎 = 2310
2𝑘𝑘𝑖𝑖 𝑘𝑘𝑝𝑝
(36)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277
𝑘𝑘𝑖𝑖
𝑏𝑏 = 4
𝑘𝑘𝑝𝑝
6.1 Open loop identification simulation
(37)
2
2275
𝑟𝑟𝑟𝑟𝑟𝑟
High limits for parameters interval (11) are not evaluated because they can not violate the inequation (33) when low limits do not violate it.
Algorithm 5.1. PI coefficients correction.
𝑚𝑚
Value of rotor speed 𝜔𝜔𝑅𝑅 = 6.7 for 𝑉𝑉𝑠𝑠𝑠𝑠 = 17 is 𝑠𝑠 𝑐𝑐 achieved by 𝛽𝛽𝑠𝑠𝑠𝑠 = 20.6245 deg . The test signal for identification is formed as (13) where 𝑞𝑞 = 0.1 and 𝛾𝛾 = 0.1. Figure 1 shows pitch angle values as the test signal and rotor speed values as the system response.
Step 1. To identify estimates 𝑎𝑎̂, 𝑏𝑏̂ by Algorithm 3.1 or Algorithm 4.1. Step 2. To check inequations (34),(35). If both of them are violated then inequation (33) is checked as condition of necessity and sufficiency. Step 3. If inequation (33) is violated then coefficietn 𝑘𝑘𝑝𝑝 is increased as 𝑘𝑘𝑝𝑝 >
2𝑘𝑘𝑖𝑖 𝑎𝑎̂
𝑘𝑘𝑝𝑝 2 >
4𝑘𝑘𝑖𝑖 𝑏𝑏̂
(38)
or (39)
It should be considered that 𝑘𝑘𝑝𝑝 is constrained by unmodelled dynamics of blades, gear etc. If because of that 𝑘𝑘𝑝𝑝 can not be increased then 𝑘𝑘𝑖𝑖 should be decreased for validity of inequation (34) or (35). The procedure for starting the algorithm 5.1 is not discussed here. It may be started manually during maintenance or auto intermittently or as an event handling. 6. SIMULATION The wind turbine is simulated by nonlinear model (1) where the power coefficient is approximated (Tilli, A. et al. (2011)) by 𝑐𝑐5 1 𝜔𝜔𝜔𝜔 , 𝐶𝐶𝑝𝑝 (𝜔𝜔, 𝛽𝛽, 𝑉𝑉, 𝑐𝑐) = 𝑐𝑐1 (𝑐𝑐2 − 𝑐𝑐3 𝛽𝛽 − 𝑐𝑐4 ) 𝑒𝑒 − 𝐿𝐿 − 𝑐𝑐6 𝐿𝐿 𝑉𝑉
1 𝑉𝑉 0,035 = − . 𝐿𝐿 𝑅𝑅𝑅𝑅 + 0.08𝛽𝛽𝛽𝛽 1 + 𝛽𝛽 3
Fig.1. Object input/output during open loop identification Algorithm 3.1 is used to obtain estimates of the object linearized model parameters from the data of this test: 𝑎𝑎̂ = 0.04984, 𝑏𝑏̂ = 0.06414
(43)
Simulation of nonlinear model (1),(40),(41) with parameters (42) and linearized model (10) with parameters (43) are carried out for step of 𝛽𝛽𝑠𝑠𝑠𝑠 from 20.6245 deg to 20.7 deg. Figure 2 shows that outputs of these models are closely adjacent and estimates (43) are verified.
(40) (41)
with the following parameters values (Tilli, A. et al. (2011)): 𝐽𝐽 = 40000kg𝑚𝑚2 , 𝜌𝜌 = 1.225 𝑃𝑃𝑅𝑅 = 200𝑘𝑘𝑘𝑘, 𝜔𝜔𝑅𝑅 = 6.7 с1 = 0.5176, с2 = 116,
𝑘𝑘𝑘𝑘
𝑚𝑚3
𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠
, 𝑅𝑅 = 13𝑚𝑚,
(42)
𝑚𝑚
, 𝑉𝑉𝑠𝑠𝑠𝑠 = 17 ,
Fig 2. Step response of nonlinear and linearized models
𝑐𝑐
𝑐𝑐3 = 0.4, 𝑐𝑐4 = 5,
𝑐𝑐5 = 21, 𝑐𝑐6 = 0.0068
6.2 Open loop identification for fluctuating wind speed simulation Simulation of object (1),(40),(41) for test signal (13) with parameters 𝛽𝛽𝑠𝑠𝑠𝑠 = 20.6245 deg , 𝑞𝑞 = 0.1, 𝛾𝛾 = 0.1 when 2311
Proceedings of the 20th IFAC World Congress 2276 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277 Toulouse, France, July 9-14, 2017
wind speed is 𝑉𝑉𝑠𝑠𝑠𝑠 added by random value is shown on Figure 3.
Algorithm 4.1 return estimates of the object linearized model parameters: 𝑎𝑎̂ = 0.04948, 𝑏𝑏̂ = 0.06405
(45)
It is practically the same values that were obtained by open loop identification.
6.4 Closed loop identification for fluctuating wind speed simulation Closed loop identification when wind speed is 𝑉𝑉𝑠𝑠𝑠𝑠 added by random value is shown on Figure 5.
Fig.3. Object input/output during open loop identification for fluctuating wind speed The shown wind speed is used for a simulation but identification procedure operates with the pitch angle (input) and rotor speed (output) data only. Algorithm 3.1 returns estimates of the object linearized model parameters: 𝑎𝑎̂ = 0.05499, 𝑏𝑏̂ = 0.06398
(44)
They do not differ significantly from estimates (43). Fig.5. Object input/output during open loop identification for fluctuating wind speed
6.3 Closed loop identification simulation The object (1) ,(40),(41) with parameters (42) with PI controller (2),(3) is simulated. Let PI coefficietns 𝑘𝑘𝑝𝑝 = 100 and 𝑘𝑘𝑖𝑖 = 10.
The pitch angle 𝛽𝛽 is formed by (20),(21). Object input and output are shown on Figure 4.
Estimates of the object linearized model parameters from the data of this test are obtained by Algorithm 4.1: 𝑎𝑎̂ = 0.04917, 𝑏𝑏̂ = 0.05793
(46)
They do not differ significantly from previous estimates. Obtained estimates obey inequations (34) and (35).
7. CONCLUSIONS Approach to first order linearized model identification based on sine wave test signal is considered in the paper. The identification procedure can be carried out in closed loop in spite of fluctuating wind speed. Results of first order linearized model identification can be the basis for PI controller tuning. Coefficients of PI controller are chosen from the condition that the roots of the characteristic equation of closed loop system of linearized model with PI controller are real-valued for wide limits of linearized model parameters. The simulation results show the high efficiency of the proposed approach. Fig.4. Object input/output during closed loop identification
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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Albert G. Alexandrov et al. / IFAC PapersOnLine 50-1 (2017) 2272–2277
ACKNOWLEDGMENT The authors wish to thank Dr. Kitch Wilson (Applied Theoretic Systems, Inc.) for valuable comments on the blade pitch angle control.
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