Series PID Pitch Controller of Large Wind Turbines Generator

Transcription

1 SERBIAN JOURNAL OF ELECRICAL ENGINEERING Vol. 1, No., June 015, UDC: :681.5 DOI: 10.98/SJEE150183M Series PID Pitch Controller of Large Wind urbines Generator Aleksandar D. Micić 1, Miroslav R. Mataušek Abstract: For this stable rocess with oscillatory dynamics, characterized with small daming ratio and dominant transort delay, design of the series PID itch controller is based on the model obtained from the oen-loo rocess ste resonse, filtered with the second-order Butterworth filter F bw. Performance of the series PID itch controller, with the filter F bw, is analyzed by simulations of the set-oint and inut/outut disturbance resonses, including simulations with a colored noise added to the control variable. Excellent erformance/robustness tradeoff is obtained, comared to the recently roosed PI itch controllers and to the modified internal model itch controller, develoed here, which has a natural mechanism to comensate effect of dominant transort delay. Keywords: PID itch control, Wind turbines generator, Oscillatory dynamics, Dominant dead-time, Modified internal model control. 1 Introduction Recently, synthesis on PI-based itch controller of large wind turbines generator is considered in [1, ], where the rocess dynamics is defined by the following transfer function s as as 1 a0 G() s G0()e s, G0() s, (1) 4 3 s bs 3 bs bs 1 b0 with arameters in able 1 [1], for two rocesses denoted as G () 1 s and G () s. It is suosed in [1, ] that hydraulic itch actuator AC is alied, modeled as a dead-time. hus, rocess is defined by AC ex s in series with G () 0 s, as in (1). ransfer function (1) defines a single-inut single-outut rocess, where inut is the blade itch angle and outut defines tower fore-aft deflection. Processes G () i s, i = 1,, exhibit a strong resonant resonse even by the small amount of excitation which is naturally resent in the wind [3]. 1 Faculty of echnical Sciences, University of Priština, Kosovska Mitrovica, Kneza Miloša 7, Serbia; admicic@gmail.com School of Electrical Engineering, University of Belgrade, Kralja Aleksandra 73, Belgrade, Serbia; matausek@etf.rs 183

2 A.D. Micić, M.R. Mataušek able 1 Parameters of G () 1 s and G () s. Process a a 1 a 0 b 3 b b 1 b 0 G () 1 s G (s) Design of itch controllers based on the rocess dynamics characterization (1), as in [1, ], is a comlex control roblem: to design a controller with a good erformance/robustness tradeoff for the rocess with oscillatory dynamics, characterized with small daming ratio and dominant transort delay. Investigation resented in [4] is erformed to demonstrate how to aly otimization to tune arallel PID controller for rocess (1) with strong resonant resonse. In the resent aer design of itch controllers is based on the methods roosed for series PID controller in [5] and for Modified Internal Model Control (MIMC) in [6]. In Section, control relevant models G id (s), and G mimc (s), of rocesses G () 1 s and G () s, are determined by alying a simle rocedure. In Section 3, design of PID and MIMC itch controllers are erformed based on models determined in Section. As demonstrated in Section 3, MIMC controller has a natural mechanism to comensate effect of dominant dead-time. his is the reason why the MIMC itch controller, roviding excellent erformance/robustness tradeoff, is also used to demonstrate roerties of the roosed series PID itch controller. In Section 4, closed-loo simulation results are resented and comared with results obtained by PI controllers from [1, ]. Simulations with a colored noise added to the control variable are also resented and used to simulate effect of the stochastic wind variation on the tower fore-aft deflection. Finally, simulations with strong rate constraints in actuator are used to demonstrate advantages of the series PID itch controller and PI itch controller roosed here. Models Used in the Proosed Pitch Controller Design Control relevant models G mimc (s), used for the MIMC itch controller design, are defined by mimcs s z 1 Gmimc () s Gmimc0()e s, Gmimc0() s K () s s 1 and able. Simle rocess dynamics characterization is used to obtain control relevant models () by alying fitting-by-eye technique to aroximate oenloo ste resonses of stable rocesses G () 1 s and G () s. 184

3 Series PID itch controller of large wind turbines generator able Parameters of models () used in the MIMC itch controller design. Process/Model K z mimc G () s G () s mimc1 G () s G () s mimc Resonses of model (), comared to resonses of rocess G () s in (1), are resented in Fig. 1. From Fig. 1, and able, it follows that the oscillatory dynamics of rocesses G () 1 s and G () s is characterized by small daming ratio and dominant dead-time mimc, larger than time constant. he nonminimum-hase characteristics of G () s resonse are included in the deadtime of model G mimc (s), as demonstrated in Fig.1b. Similar values of the time constant = and daming ratio , as in the second raw of able, are obtained from the dominant comlex-conjugate ole air of G () s, at s 1 = i and s = i. Fig. 1 Unit ste resonse of G j (s), j = 1,, (dotted) and models G mimc,j (s) (solid): a) G 1 (s) and G mimc1 (s), b) G (s) and G mimc (s). Control relevant models G id (s) are obtained by alying fitting-by-eye technique to aroximate filtered oen-loo ste resonses G( s) Fbw( s) of stable rocesses G () 1 s and G () s, with oscillatory dynamics, as roosed in [5]. he second-order Butterworth filter Fbw() s is used here in the form 1 Fbw() s fbs fbs 1. (3) Models G (s) id are defined by able 3 and 185

4 A.D. Micić, M.R. Mataušek id id () id0(), id 0() [( 1 1)( 1)], 1 G s G s e G s K s s. (4) able 3 Parameters of models (4) used in the series PID itch controller design. Process/Model fb K 1 id G () s G () s id1 G () s G () s id G () s G () s id, Process resonses and filtered resonses G() s Fbw() s, with Fbw() s from (3) and able 3, are comared in Figs. and 3 to resonses of models G () s from id (4) and able 3. Correct values of arameters for model G id1 (s), denoted in [4] as G SO1 (s), are resented in able 3. Fig. Determination of model used to design series PID 1 controller: a) unit ste resonse of G 1 (s) (dotted) and filtered resonse G 1 (s)f bw (s) with fb = 1 (solid), b) filtered resonse (solid) and model resonse G id1 (s) (dashed). Fig. 3 Determination of models used to design series PID controllers: a) unit ste resonse of G (s) (dotted) and filtered resonses G (s)f bw (s) with fb = 1.41 (solidmagenta) and fb = (solid-red), b) filtered resonses and model resonses G id,1 (s) (dashed-magenta) and G id, (s) (dashed-red).(colors can be seen in electronic version), 186

5 Series PID itch controller of large wind turbines generator 3 Control System Design and uning Structure of the MIMC itch controller is resented in Fig. 4. ransfer 1 1 function ( ) G ( ) ( ) mimc0 z K Wz z W z is a discrete inverse of the model G () mimc0 s [6], defined by arameters in able and adoted samling eriod t. By alying a discretization rocedure [6], from () one obtains z a 1z a W ( z), a ex( t / ), (5) (1 a a ) z while z 1 t a1 ex( t / )cos 1 1 /, b ex t W z b z z b, (6). ( L 1) ransfer function GL z z, where L mimc t. Filter F( z) in Fig. 4, for Fr ( z) 1 defines the desired closed-loo system time constant CL [6]. In the resent aer F( z) is designed as a discrete equivalent of the second-order 1/ s 1 low-ass filter C (1 ) z F( z), ex( t / c), c 0.4. (7) z uning arameters C, and in (7), are defined by the desired value of the time constant CL, for the second-order filter equal to CL C. he adoted value of C = 0.4 s corresonds to the closed-loo system time constant obtained with PI controllers from [1, ]. Since CL 0.8 s, the samling eriod t = 0.05 s is used to define filter F(z) in (7), with 0.885, and to design MIMC j, j = 1, controllers. / z Fig. 4 MIMC itch control system structure with set-oint refilter F r (z), Zero-Order Hold (ZOH) and actuator AC. Inut and outut disturbances are denoted by d and n. 187

6 A.D. Micić, M.R. Mataušek hus, for G 1 (s), from G mimc1 (s) and t = 0.05 s, one obtains in the MIMC controller ( ) G ( ) ( ) mimc0 z K Wz z W z, 1 1( ) z z z mimc0( z) ( 0.053) G z ( ) z, (8) 9 G ( ) L z z, (9) while for G (s), from G mimc (s) and t = 0.05 s, one obtains in the MIMC controller 1 1( ) z z z Gmimc0( z) ( 0.040) z ( ) z, (10) 11 G ( ) L z z. (11) 1 MIMC controllers are defined by Fig. 4 where F(z), ( mimc0 and G z ( L ) z are defined by (7) (11). he same filter F(z) in (7) with L is used in both MIMC j, j = 1, controllers, imlemented with samling eriod t = 0.05 s. Anti-Reset Windu (ARW) imlementation from [5], resented in Fig. 5, is used for the PID itch controller with arameters K, i, d, fb defined in able 4. o obtain satisfactory set-oint following resonse this imlementation requires a set-oint refilter F r (s) = F bw (s). Fig. 5 Anti-reset windu imlementation of the series PID itch controller. For AC = ex( τs) one obtains linear case, without amlitude or rate constraints in actuator AC. Set-oint refilter F r (s) = F bw (s). Parameters of the series PID itch controller in Fig. 5 are given in able 4. hey are obtained by alying models G id (s) defined by (4), able 3 and Simle Control (SIMC) tuning rules from [7] used to determine roortional gain, integral time i and derivative time d as roosed in [7]. According to able 3, it is used 1 = in (1). Besides the Butterworth filter F bw, series PID itch controller includes a low-ass filter 1/( s f 1) in the term defining derivative action ( s 1)/( s 1), where /10. d f 188 f d

7 Series PID itch controller of large wind turbines generator K,, min{,8 },. (1) 1 1 i 1 id d K id able 4 Parameters of series PID itch controllers. Controller/Process K i d fb PID 1 /G 1 (s) PID,1 /G (s) PID, /G (s) According to results in able 4, series PID itch controllers can be imlemented also as digital controllers by using samling eriod t = 0.05 s. However, they are imlemented as continuous controllers, defined by Fig. 5, and arameters in able 4. Continuous PI itch controllers from [1, ] are imlemented as continuous controllers, in a standard way, with roortional and integral gains K and K defined by able 5. i able 5 Parameters of PI itch controllers from [1, ]. Controller/rocess 189 K K i PI 1 /G 1 (s) PI /G (s) Results of Closed-Loo Simulation As in [1, ] in linear case, actuator is modeled as the dead-time, AC ex 0.5s. Anti-Reset Windu (ARW) imlementation from [5] resented in Fig. 5, is used for the PID itch controller, with arameters K, i, d, fb in able 4. Closed-loo system resonses obtained by the PID itch controllers are comared in Figs. 6 9 with results obtained by the MIMC itch controller with Fr ( z) 1, and obtained by PI itch controllers from [1, ] defined by able 5. Parameters of the PI 1 controller are obtained in [] by unconstrained minimization of IAE r, the integrated absolute error following the unit ste setoint resonse. Parameters of the PI controller are taken from the stabilizing region in the Ki K lane [1, Fig., for Ki 0 ], as a good comromise between erformance and robustness.

8 A.D. Micić, M.R. Mataušek he desired closed-loo system time constant equal to C = 0.8 s, is satisfied almost exactly by alying MIMC controllers. For better erformance, demonstrated in Fig. 6a, and similar erformance in Fig. 7a, better robustness is obtained by the PID and MIMC controllers, as demonstrated by the robustness indices in able 6, where erformance/robustness tradeoff obtained by PI, PID and MIMC controllers is resented. Fig. 6 Closed-loo system resonses of nominal rocess G 1 (s), with controllers: MIMC 1 (solid), PID 1 (dashed) and PI 1 (dotted). Inut and outut disturbances: D(s) = 30ex( 0s)/((s+1)s) and N(s) = 0.3ex( 40s)/s. Fig. 7 Closed-loo system resonses of nominal rocess G (s) with controllers: MIMC (dash-dot), PID,1 (solid), PID, (dashed) and PI (dotted). Inut and outut disturbances: D(s) = 15ex( 15s)/((s+1)s) and N(s) = 0.15ex( 30s)/s. Performance index IAE n, in able 6, is the integrated absolute error following the unit ste outut disturbance N(s) = 1/s, obtained in simulation with the inut disturbance D(s) = 0 and set-oint R(s) = 0. Variances in able 6, are calculated as 190

9 Series PID itch controller of large wind turbines generator sim y 0 y ()d t t /, (13) where sim is the simulation time interval sim = 60 s and y(t) is the controlled variable resonse. Simulation is erformed with the inut disturbance D(s) = 0, set-oint R(s) = 0 and outut disturbance nt () defined by a Band-Limited White Noise (BLWN) nt ()= nw () t. It is obtained from a BLWN generator with ower b w = and samle time s = t = 0.05s. he variance of this BLWN nw ( t ) is theoretically equal to bw/ s = 0.01 [8, Aendix A]. hus, as demonstrated in able 6, all controllers guarantee low value of variance. able 6 Performance/robustness tradeoff obtained with PI, MIMC and PID controllers. sim y Controller/Process IAE n M y S M PI 1 /G 1 (s) MIMC 1 / G 1 (s) PID 1 / G 1 (s) PI /G (s) MIMC / G (s) PID,1 / G (s) PID, / G (s) Fig. 8 Closed-loo system set-oint resonses of erturbed rocess G (s), with controllers: PID, (dashed), MIMC (solid) and PI (dotted). In a) erturbed gain, G ert (s) =.04G (s), b) erturbed dead-time, greater 44%, G ert (s) = G (s)ex( 0.11s). 191

10 A.D. Micić, M.R. Mataušek It is imortant that erformance/robustness tradeoff obtained by the series PID itch controller deends weakly on the choice of the time constant fb. his is demonstrated in Figs. 7a and 7b and confirmed in able 6, for two values fb =1.41 and fb =, resulting into ractically equal erformance/robustness tradeoff for evidently different model resonses in Fig. 3b. Excellent resonses are obtained for the erturbed rocess G (s), with the roosed PID, itch controller comared in Fig. 8 to MIMC dead-time comensating itch controller and PI itch controller. High values of robustness indices M S and M in able 6, obtained by the PI 1 controller, are consequence of unconstrained otimization alied in []. Robustness indices, maximum sensitivity M S and maximum comlementary sensitivity M, are defined by the sensitivity function S(s) = 1/(1+L(s)) and comlementary sensitivity (s) = 1 S(s) as M S max Si ( ), M max i ( ). (14) Loo transfer function is defined by Ls () = G0() scs (). From Fig. 4 and AC= ex(- s), one obtains that C(s), in Ls () = G0() scs (), used to obtain robustness indices M S and M for MIMC controllers, is given by s 1 ( ) e mimc0 ( ) ( )/(1 ( ) L ( )) z ex( s t) 19 C s G z F z F z G z. (15) For the series PID controllers, from Fig. 5 and AC= ex(- s), one obtains C(s) given by s i 1 s ds 1 Cs () K e Fbw() s, (16) s i 0.1ds 1 in Ls () = G0() scs (). For PI controllers, PI 1 from [] and PI from [1], one obtains. K () i Cs K e s s. (17) Finally, to illustrate influence of wind seed variation on the tower fore-aft deflection, simulation is erformed with the set-oint R( s )=0, outut disturbance N()=0 s and with noise n wind () t acting as an unknown inut disturbance dt () nwind () t. Stochastic variation of the wind seed n wind () t is obtained from a BLWN generator with ower b w = and samle time s = 0.01 s, assed through the low-ass filter Fs ( ) = 1/(5 s + 1). Results of this simulation are summarized in Fig. 9 and able 7. Variances resented in able 7 are calculated as in (13) for the simulation time interval sim = 60 s.

11 Series PID itch controller of large wind turbines generator Random variation of the simulated wind seed acts on the blade itch angle as resented in Fig. 9a. As a result the tower fore-aft deflection is obtained as in Fig. 9b. Smaller variation of the tower fore-aft deflection is obtained by the MIMC and PID controllers, with significantly smaller control signal activity, moves u and down of the maniulated variable w(t), demonstrated in Fig. 9a. Fig. 9 A art of closed-loo system resonses of nominal rocess G 1 (s) with controllers: PI 1 (dotted), PID 1 (solid) and MIMC 1 (dashed). Only inut disturbance is active, simulating stochastic variation of the wind seed. able 7 Variances obtained with PI, PID and MIMC controllers. Controller/Process PI 1 /G 1 (s) PID 1 /G 1 (s) MIMC 1 /G 1 (s) y.3e e e-09 For almost the same resonse of the tower fore-aft deflection in Fig. 9b, much better reduction of the control signal activity is obtained by the MIMC controller in Fig. 9a. However, analyses resented until now were erformed for the linear case, for the actuator AC= ex(- s). In industrial alications constraints in actuators are inevitable. In this case advantage of the roosed PID controller in Fig. 5 is its antiwindu structure. o illustrate erformance of PI, PID and MIMC controllers in the resence of rate constraints in actuator, results of simulation with and without rate constraints are comared in Fig. 10, for rocess G 1 (s) with controllers: PI 1 PID 1 and MIMC

12 A.D. Micić, M.R. Mataušek (c) (d) Fig. 10 Closed-loo system resonses of nominal rocess G 1 (s) with controllers: PI 1 (dotted), MIMC 1 (solid) and PID 1 (dashed) for R(s) = 0, D(s) = 0 and N(s) = 0.3ex( 5s)/s. In a) b) without rate constraints and in c) d) with strong rate constraints: rising rate 1s 1 and falling rate 1s 1. Finally, results of simulation with and without rate constraints are comared in Fig. 11, for rocess G 1 (s) with controllers: PID 1 and PI 1new. Controller PI 1new is roosed here. It is obtained by alying SIMC rules to simlify the Second-Order Plus Dead-ime (SOPD) model G id1 (s) to the First-Order Plus Dead-ime (FOPD) model G i1 (s), given by s K e Gi1() s, 1, id, (18) ( s 1) where arameters K, 1= and id are taken from the first row in able 3. PI 1new controller is imlemented as in Fig. 5, with d =0, time constant fb from the first row of able 3, K and i i defined by (18) and tuning rule [7] K, i min{,8 }. (19) K 194

13 Series PID itch controller of large wind turbines generator Results in Fig. 11 confirm that an effective anti-reset windu PI itch controller for rocess G 1 (s) can be easily obtained by the rocedure roosed here. Fig. 11 Closed-loo system resonses of nominal rocess G 1 (s) with controllers: PID 1 (solid) and PI 1new (dashed), for R(s), D(s) and N(s) as in Fig. 10. In a) without rate constraints. In b) with strong rate constraints as in Fig. 10. Due to sace limitations, further analyses with the PI itch controller roosed here is not resented. However, resented details makes ossible to reeat some revious simulations with the PI 1new controller, as well as to design a PI new controller for rocess G () s. It is believed that design of PI/PID itch controller roosed here is imortant for further develoment and comarative analyses of itch controllers. 5 Conclusion Better erformance/robustness tradeoff is obtained with the MIMC and PID itch controller, comared to PI itch controllers from [1, ]. It is imortant that inclusion of derivative action, in the resence of the simulated stochastic variation of the wind, results into erformance imrovement obtained with the reduced control signal activity. Better reduction of the control signal activity is obtained with the roosed MIMC controller. However, advantage of the roosed PID controller in Fig. 5 is its antiwindu structure. uning of the roosed PID and PI itch controller is simle and can be erformed exerimentally, by alying SIMC tuning rules to SOPD model G () s and FOPD model G () id i s, both determined from oen-loo rocess ste resonses memorized for different oerating regimes. 195

14 A.D. Micić, M.R. Mataušek 6 Acknowledgement A.D. Micić acknowledges financial suort from the Serbian Ministry of Science and echnology (Project III 47016). 7 References [1] J. Wang, N. se, Z. Gao: Synthesis on PI-based Pitch Controller of Large Wind urbines Generator, Energy Conversion and Management, Vol, 5, No., Feb. 011, [] J.W. Perng, G.Y. Chen, S.C. Hsieh: Otimal PID Controller Design based on PSO-RBFNN for Wind urbine Systems, Energies, Vol. 7, No. 1, Jan. 014, [3] E.A. Bossanyi: he Design of Closed-loo Controllers for Wind urbines, Wind Energy, Vol. 3, No. 3, July/Set.000, [4].B. Šekara, M.R. Mataušek: Design of PI/PID controller for control of blades angle of high ower wind turbine, ERAN Conference, Vrnjačka Banja, Serbia, 0-05 June 014,. AU (In Serbian). [5] M.R. Mataušek, B.. Jevtović, I.M. Jovanov: Series PID Controller uning based on the SIMC Rule and Signal Filtering, Journal of Process Control, Vol. 4, N. 5, May 014, [6] M.R. Mataušek, A.D. Micić, D.B. Dačić: Modified Internal Model Control Aroach to the Design and uning of Linear Digital Controllers, International Journal of Systems Science, Vol. 33, No. 1, 00, [7] S. Skogestad: Simle Analytic Rules for Model Reduction and PID Controller uning, Journal of Process Control, Vol 13, No 4, June 003, [8] A.D. Micić, M.R. Mataušek: Otimization of PID controller with Higher-order Noise Filter, Journal of Process Control, Vol. 4, No. 5, May 014,