Extremum seeking (ES) was invented in

Transcription

1 EXTREMUM SEEKING FOR WIND AND SOLAR ENERGY by Azad Ghaffari, Mirolav Krtic and Sridhar Sehagiri APPLICATIONS Extremum eeking (ES) wa invented in 1922 and i one of the oldet feedback method Rather than regulation, it purpoe i optimization For thi reaon, application of ES have often come from energy ytem The firt noted publication on ES in the Wet i Draper and Li' application to park timing optimization in internal combution engine 1 In the enuing decade, ES ha been applied to ga turbine and even nuclear fuion reactor Renewable energy application have brought a new focu on the capabilitie of ES algorithm In thi article we preent application of ES in two type of energy converion ytem for renewable energy ource: wind and olar energy The goal for both i maximum power point tracking (MPPT), or, the extraction of the maximum feaible energy from the ytem under uncertainty and in the abence of a priori modeling knowledge about the ytem For the wind energy converion ytem (WECS), MPPT i performed by tuning the et point for the turbine peed uing calar ES Performing MPPT for the photovoltaic (PV) array ytem entail tuning the duty cycle of the DC/DC converter employed in the ytem uing multivariable ES Experimental reult are provided for the photovoltaic ytem Increaing availability of energy torage device inteni- 2 MARCH

2 continuou ocillatory perturbation, alo known a a probing function More importantly, ES doe not merely monitor the direction of the output repone but exploit the meaured repone to etimate the gradient of the power map and update the control input in proportion to the gradient of the power map 3 6 and the implicity of hardware implementation In addition to a probing ignal, the ES algorithm employ only an inte- The amplitude and frequency of the probing function in ES the frequency election i not a complicated a the election ytem, it i enough to elect the ES probing frequency reaonably maller than the highet frequency that can pa article more ophiticated cheme like the Newton-baed 7 deired cloed-loop performance, for example, for peeding ytem that need careful parameter election, particularly the ampling period and perturbation amplitude In contrat, a tabilizing inner-loop control only a few retriction in electing the probing frequencie uniform in peed for multiple module Thi paper i organized a follow The next ection introduce both gradient and Newton-baed ES cheme Subequently, a Scalar gradient-baed ES i combined with THE BASICS OF EXTREMUM SEEKING A gradient-baed ES for multi-input tatic map i hown in Figure 1 The algorithm meaure the calar ignal y(t) = Q( (t)), where Q () i = [ 1, 2,, n ] T The map ha a unique maximum point at = [ 1*, *,, n* ] where ( * ) = 0, 2 ( * ) = H < 0, of the unknown map around it maximum point Gradient etimation i helped by the ignal S(t) + θ ˆθ Q( ) K y Ĝ M(t) FIGURE 1 The gradient-baed ES for a tatic map 2 with nonzero perturbation amplitude a and with a gain matrix K that i diagonal Some retriction are impoed on the probing frequencie, i the unknown map, Q () 3 4 H Thi the Newton-baed ES algorithm Figure 2, 14 MARCH 2014

3 Focu on Dynamic Sytem & Control θ Q( ) y For Region II MPPT the turbine power i related to the wind power a S(t) + ˆθ K ΓĜ Ĝ Γ =ργ ργĥγ M(t) FIGURE 2 A Newton-baed ES for a tatic map 7 The N(t Ĥ 5 N(t) where T t i the rotor torque, t i the turbine peed, and C p i the non-dimenional power the turbine power to the wind power The turbine peed can be ued to change C p, which reult in power control and optimization The MPPT algorithm in ub-rated power region hould be able to guide the WT to it MPP regard- wind peed,v w, and the turbine peed, t The wind peed i a diturbance input and the turbine peed can be manipulated to Figure 4 8 Heian a H^ (t) = N(t) y(t (t the tranient = ^ *, = H -1 i Becaue they are determined by K and, H the (local) rate i uer-aignable inner-loop control WIND ENERGY CONVERSION SYSTEMS W Figure 3) where R i the blade length, a i air denity, and V w i wind peed 6 7 Power Inner-Loop Control Deign for WECS amplitude,v om o, to I II III IV Sub-rated power Rated power V cut in V rated V cut out Wind peed FIGURE 3 Typical power curve of WT including four operating region MARCH

4 Turbine power, P t (kw) V w =7m/ V w =9 V w =11 V w = Turbine peed, t (rad/) FIGURE 4 Variation of the turbine power veru turbine peed for different wind peed and an auxiliary input, u 2, to achieve input-output decoupling in WECS dynamic Uing one tep of integration in front of V om the extended equation of WECS are introduced a follow: x = f (x) + g 1 u 1 + g 2 u 2, x 9, u 2 where x = [ i, i,, 0 o, V om, r, 0, t ] T where i and i are tator current, and 0 = t 0r, r = t pn 0 r dt, r i the rotor electrical frequency, u 1 = o i the electrical frequency of the tator, u 2 i an auxiliary input (voltage amplitude rate) which generate the voltage amplitude of the tator A een in Fig 4, turbine peed control power generation Decoupling oriented control (FOC) Turbine peed, x 9 x x 2 4, are introduced a meaurable output for thi reaon Feedback linearization i applied baed on the elected output Thi reult in the regulation of turbine peed, t, to it reference value ref t, while the amplitude of rotor = x 2, converge to it deired value, ref 3 + x 2 4 Wind Turbine Power Optimization To overcome challenge aociated with the conventional power control and optimization algorithm and to remove the dependence of the MPPT MPPT of WECS i employed Acce to turbine power meaurement and peed manipulation are or turbine power, it power map ha one MPP under any wind peed The propoed nonlinear control not only achieve the deired cloed- alo prevent magnetic aturation The ES cheme with inner-loop control 9 i hown in Figure 5 Shown here, the reference input of the inner-loop control are t ref and ref The MPP i parameterized by the optimal turbine peed at each wind peed, a etimated by the ES loop The other control input, ref of the rotor which prevent induction generator from magnetic aturation Combination of the Controller and WECS reult in fat dynamic, while the dynamic contained in the ES algorithm are of low and medium peed The algorithm etimate the optimal turbine peed, t ref = t * With repect to the controller-ytem' fat dynamic, thi can be conidered a contant value Simulation Reult on a WECS Model A time frame of 30 econd demontrate the and that of the conventional MPPT which i baed on P&O with an FOC in the inner loop The MPPT proce i hown in Figure 6 The extracted energy by our propoed algorithm i 236% higher than the extracted energy by the conventional MPPT and FOC Our algorithm provide perfect input-output decoupling and guarantee a larger domain of attraction, which increae performance robutne with repect to the ytem param- the competitivene of wind energy PHOTOVOLTAIC SYSTEMS Extremum eeking ha been applied to MPPT deign for photovoltaic (PV) microconverter ytem, where each PV module i coupled with it own DC/DC converter Mot exiting MPPT deign are ditributed (decentralized), ie, they employ one MPPT loop around each converter, and all deign, whether ditributed or multivariable, are gradient-baed 2 The convergence rate of gradient-baed deign depend on the Heian, which in turn i dependent on environmental condition uch a irradiance and temperature Conequently, when applied to large PV array, the variability in condition, and/or PV module degradation, reult in non-uniform tranient in the convergence to the MPP Uing a multivariable gradientbaed ES algorithm for the entire ytem intead of a calar one for each PV module, decreae enitivity to the Heian, but doe 16 MARCH 2014

5 Focu on Dynamic Sytem & Control V w State feedback λ ref ω ref t Controller u 2 ω o = u V om θ o WECS P t (V w, ω t ) a in(ωt) in(ωt) + ˆω t k ĝ Low-pa High-pa FIGURE 5 The ES algorithm for MPPT of the WECS with the inner-loop control not eliminate thi dependence The Newton-baed ES algorithm i ued, a it imultaneouly employ etimate of the gradient and Heian in the peak power tracking The convergence rate of uch a deign to the MPP i independent of the Heian, with tunable tranient performance that i independent of environmental of the propoed algorithm in comparion to exiting calar deign, a well a multivariable, gradient-baed ES Uing a multivariable gradient-baed ES MPPT deign for the micro-converter architecture, where each PV module i coupled with it own DC/DC converter, reduce the number of required enor (hardware reduction) Tranient under udden change in olar irradiance are more uniform a i the environmental temperature in comparion to a calar gradient-baed ES for each PV module True of gradient-baed deign, the convergence to MPP i dependent on the unknown Heian: it varie with irradiance, temperature, and module degradation and mimatch In comparion with the tandard gradient-baed multivariable extremum eeking, the Newton-baed ES remove the dependence of the convergence rate on the unknown Heian and make the convergence rate of the parameter etimate ueraignable In particular, all the parameter can be deigned to converge with the ame peed, yielding traight trajectorie to the extremum even with map that have highly elongated level et When applied to the MPPT problem in PV ytem, the convergence behavior under a wide range of working condition that include temperature and irradiance variation and the non-ymmetric power generation of the neighboring PV module a a reult of module degradation or mimatch FIGURE 6 Propoed algorithm, MPPT (olid red); conventional P&O with FOC (dahed blue); maximum power available to the WECS (dahed black) P t (kw) Multivariable MPPT of PV Sytem Conventionally, each DC/DC converter ha a MPPT loop to extract maximum power from the PV ytem (known a power optimizer in indutry) The output ide of the converter are connected in erie The PV ytem i connected to the power grid through a DC/AC inverter, which ha it eparate controller Two problem arie here Firt, two enor, current and voltage, are required per module which increae the levelized energy cot not addreed by thi ditributed control 80 ES w/ inner control P&O w/ FOC MPP Time () MARCH

6 (T 1, S 1 ) I 1 Converter I o1 PV 1 V 1 V o1 Multivariable MPPT (T 2, S 2 ) (Tn, Sn) Figure 7 preent a multivariable MPPT baed on an ES cheme with the following feature: A it i applied to micro-converter ytem, characterized by non-unimodal power, the deign pecialize in the iue of a a reult of partially haded condition The ue of the non-model-baed ES technique enable the deign to repond robutly even with partial knowledge of ytem parameter and operating condition the multivariable model require jut 2 enor one for the overall PV ytem current, and another for DC bu Interaction between PV module are inherent to the multivariable deign, o the tranient performance i le enitive to variation in environmental condition than a correponding calar model Gradient-Baed ES Maximizing the power generated by all PV module i equal to P = PV 2 PVn D 1 n i=1 P i = V dc I dc 10 For a micro-converter tructure including n PV module in cacade connection, there exit D * n uch that I 2 V 2 In Vn Converter D 2 Converter P (D * ) = 0, 2 P (D * ) = H < 0, H = H T D D 2 Dn I o2 Ion 11 V o2 Von P = V dc I dc V dc I dc Inverter U inv I grid AC grid FIGURE 7 Multivariable MPPT for a PV ytem One MPPT i ued for the entire ytem Temperature, and irradiance, vary all over the module The multivariable gradient-baed ES deign to MPPT of the PV ytem can be ued in Fig 7 The ES gain, K, i a poitive diagonal matrix, and the perturba- 2 and 3 In particular, the deign derive an etimate G^ of the gradient vector by adding the probing ignal S(t) to the etimate D^ = [ D^ 1,D^ 2,, D^n] T of the pule duration vector (of all the DC/DC converter) With no additional information on the Heian (and alo for implicity), we chooe the amplitude of the probing ignal to all be the ame value a It can be hown that for a proper et of ES parameter and with K > 0, the etimate D^ of the pule duration vector and the output P ettle in a mall ball around the optimal pule duration D * = [ D 1*,D 2*,,D n* ] T and the MPP P(D * ), repectively The lowet probing frequency and it correponding amplitude Since the cot function P varie with irradiance, temperature, and degradation of the PV module, o doe H K gence rate for each converter In order to alleviate the iue of unknown Heian dependent conver- verion of the multivariable Newton-baed ES The Newton-baed algorithm make the convergence rate of the parameter etimate uer-aignable In particular, all the parameter can be deigned to converge with the ame peed, yielding traight trajectorie to the extremum even with map that have highly elongated level et When applied to 18 MARCH 2014

7 Focu on Dynamic Sytem & Control D D 1 DC /DC V 1 PV1 P 1 D 2 DC /DC V2 PV2 P 2 P = V dc I dc Dn DC /DC Vn PVn Pn S(t) + Dˆ K ΓĜ Ĝ Low-pa M (t) High-pa N (t) Γ = ργ ργĥγ Hˆ Low-pa FIGURE 8 Multivariable Newton-baed ES for MPPT of a PV ytem The purple part i added to the gradient-baed ES to etimate the Heian - degradation or mimatch Newton-Baed ES cally in Figure 8 5 FIGURE 9 Experimental etup 1 DC Bu 2 DC/DC Converter, 1 and 2 3 PV Panel, 1 and 2 4 CP Ocope 6 DS 1104, Simulink and Control Dek i to replace the etimation-error dynamic D = KHD D = D = H 1, that H H 1 ) in an H^ H^ = ^ 12 to H^ H Denote = H 1 = 12 i = ^ H^ i a good H - or mimatch condition MARCH

8 Power (W) Multivariable Newton Multivariable Gradient Scalar Gradient Time() FIGURE 10 Variation of power veru time The Newton algorithm how uniform and fat tranient with low teady-tate error Experimental Reult n = 2 Figure 9 Figure 10 t CONCLUDING REMARKS S 20 MARCH 2014

9 Focu on Dynamic Sytem & Control REFERENCES 1 Draper, C S, and Li, Y T, Principle of Optimalizing Control Sytem and an Application to the Internal Combution Engine, Optimal and Self-Optimizing Control, MIT Pre, 1951 Left to right: Azad Ghaffari, Mirolav Krtic, and Sridhar Sehagiri ABOUT THE AUTHORS Azad Ghaffari received hi BS degree in Electrical Engineering and MS degree in Control Engineering from KN Tooi Univerity of Technology in Tehran, Iran He received hi PhD degree in Mechanical and Aeropace Engineering from the Joint Doctoral Program between San Diego State Univerity and Univerity of California, San Diego Hi reearch interet include demand repone in power ytem, extremum eeking and it application to maximum power point tracking in photovoltaic and wind energy converion ytem, induction machine, power electronic, and liding mode control Mirolav Krtic i the Alpach endowed chair profeor at UCSD, founding director of the Cymer Center for Control Sytem and Dynamic and Aociate Vice Chancellor for Reearch Krtic i a recipient of PECASE, NSF Career, ONR Young Invetigator, Axelby, Schuck, and UCSD Reearch award, and i a Fellow of IEEE and IFAC He ha held Springer-Berkeley and Royal Academy of Engineering ditinguihed viiting profeorhip He ha erved a Senior Editor of IEEE TAC and Automatica, VP of CSS, and chair of it IEEE Fellow Committee Krtic i coauthor of ten book on nonlinear, adaptive, PDE control, and delay ytem Sridhar Sehagiri received hi BS Tech degree from the Indian Intitute of Technology, Madra, and hi MS and PhD degree from Michigan State Univerity, in 1995, 1998, and 2003 repectively, all in electrical engineering He joined the Electrical & Computer Engineering Department at San Diego State Univerity in 2003, where he i currently an Aociate Profeor Hi reearch interet are nonlinear control with application to energy ytem 2 Eram, T, and Chapman, Comparion of photovoltaic array maximum power point tracking technique, IEEE Tranaction on Energy Converion, vol 22, pp , Krtic, M, and Wang, H-H, Stability of extremum eeking feedback for general nonlinear dynamic ytem, Automatica, vol 36, pp , Ariyur, K B, and Krtic M, Real-Time Optimization by Extremum Seeking Feedback, Wiley-Intercience, Tan, Y, Neic, D, and Mareel, I, On non-local tability propertie of extremum eeking control, Automatica, vol 42, pp , Liu, SJ, and Krtic, M, Stochatic Averaging and Stochatic Extremum Seeking, Springer, Ghaffari, A, Krtic, M, and Neic, D, Multivariable Newton baed extremum eeking, Automatica, vol 48, pp , 2012 propoed A multivariable gradient-baed ES algorithm wa conidered - gradient-baed counterpart