Extremum Seeking for Wind and Solar Energy Applications

Transcription

1 Proceeding of the 11th World Congre on Intelligent Control and Automation Shenyang, China, June 29 - July Extremum Seeking for Wind and Solar Energy Application Mirolav Krtic and Azad Ghaffari Department of Mechanical and Aeropace Engineering Univerity of California, San Diego La Jolla, CA {krtic and aghaffari}@ucdedu Sridhar Sehagiri Department of Electrical and Computer Engineering San Diego State Univerity San Diego, CA ehagir@engineeringduedu Abtract Invented in 1922, extremum eeking (ES) i one of the oldet feedback method However, it purpoe i not regulation but 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 In both area the goal i maximum power point tracking (MPPT), ie, 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) we perform MPPT by tuning the et point for the turbine peed uing calar ES For the photovoltaic (PV) array ytem, we perform MPPT by tuning the duty cycle of the DC/DC converter employed in the ytem uing multivariable ES For the photovoltaic ytem we provide experimental reult (Abtract) Keyword energy harveting; wind energy; olar energy; nonlinear dynamical ytem; adaptive control; optimization; power control (key word) I INTRODUCTION Increaing availability of energy torage device motivate the effort to harvet maximum feaible power from renewable ource, particularly wind turbine (WT) and photovoltaic (PV) ytem Renewable ource operate under a wide range of uncertain environmental parameter and diturbance For example, uncertain quantitie uch a wind peed in WT and olar irradiance in PV module affect the repective power map and the maximum power point (MPP) However, the power map i alo a function of a control input the turbine peed in WT and the terminal voltage in the PV module The power map of WT ha a unique MPP with repect to turbine peed at each level of wind peed Likewie, the power map of a PV module ha a unique MPP with repect to terminal voltage at each level of olar irradiance The proce of governing a WT or PV module to it MPP i know a maximum power point tracking (MPPT) The conventional perturb and oberve (P&O) technique do o by a combination of adding a tep perturbation to the control ignal and monitoring the direction of change in power [2-8] Mot technique derived from P&O are baed on dicrete analyi and require a delicate balance between the amplitude of the control input tep perturbation and the poible change in environmental parameter Moreover, the ampling frequency need to be carefully elected with repect to the repone time of the ytem to the tep perturbation Since the ytem i not linear, the ampling frequency i alo a function of the tep ize and of the magnitude of change in environmental parameter [2-4, 6] Extremum eeking (ES) i an attractive alternative to P&O technique for olving MPPT problem in wind and olar ytem A a model-free, real-time optimization approach, ES i well uited for ytem with unknown dynamic or thoe that are affected by high level of uncertainty or external dynamic, like WT and PV ytem Similar to P&O technique, ES employ perturbation [9-21] However, intead of employing a dicrete tep perturbation, ES ue a 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 [9] ES ha the dual benefit of rigorouly provable convergence [9-13] and the implicity of hardware implementation [14-21] In addition to a probing ignal, the ES algorithm employ only an integrator, a well a optional high-pa and low-pa filter The amplitude and frequency of the probing function in ES influence the preciion of the MPPT algorithm However, the frequency election i not a complicated a the election of the ampling frequency in P&O technique For dynamic ytem, it i enough to elect the ES probing frequency reaonably maller than the highet frequency that can pa the ytem without ignificant attenuation ES guide the ytem to it MPP regardle of the magnitude of change in environmental parameter, a long a the change are low While the power map hape define the convergence rate of the conventional gradient-baed ES, we alo preent in thi article more ophiticated cheme like the Newton-baed ES [13] to alleviate the iue of unymmetrical tranient In ome cae we need an inner-loop control to achieve deired cloed-loop performance, for example, for peeding up the convergence rate and alleviating magnetic aturation in WT ytem [21] Combining a dicrete MPPT method uch a P&O with a continuou inner-loop control create a hybrid ytem that need careful parameter election, particularly the ampling period and perturbation amplitude [22] In contrat, ES can be applied without modification to any ytem with a tabilizing inner-loop control When dealing with a multivariable power map, uch a a cacade PV configuration with one converter per module, uing a decentralized MPPT architecture i not the mot efficient option For multivariable MPPT, the complexity of /14/$ IEEE 6184

2 θ Q h 2 (θ θ ) 2 Q(θ) θ Q( ) y a in(ωt) ˆθ 1 ˆθ Fig 1: The implet perturbation-baed ES cheme for a quadratic ingle-input map P&O algorithm increae dramatically with the ize of the input vector In contrat, ES trivially extend to multivariable MPPT, with only a few retriction in electing the probing frequencie [20] Furthermore, with ES we have the option of employing the algorithm Newton-baed verion to achieve tranient that are ymmetric relative to the peak of the MPP and uniform in peed for multiple module [19] The ret of thi paper i organized a follow In Section II we introduce ES for calar tatic map and then for dynamic ytem with multivariable map We preent both gradientand Newton-baed cheme In Section III, we combine the calar ES with a nonlinear inner-loop control developed from field-oriented control (FOC) to achieve power control and optimization in WT We preent imulation reult to how the effectivene of the propoed algorithm In Section IV we preent multivariable MPPT baed on ES for PV ytem We verify the validity of the propoed algorithm with experimental reult II THE BASICS OF EXTREMUM SEEKING Many verion of ES exit, with variou approache to the analyi of their tability The mot common verion employ perturbation ignal for the purpoe of etimating the gradient of the unknown map that i being optimized [9] To undertand the baic idea of ES, it i bet to firt conider the cae of a tatic ingle-input map of the quadratic form, a hown in Fig 1, Qθ Q θθ, where Q,h,θ are all unknown At the optimal point we have 0, ( 1 ) The uer ha to only know the ign of, namely, whether the quadratic map ha a maximum or a minimum, and ha to chooe the adaptation gain uch that gn gn The uer ha to alo chooe the frequency a relatively large compared to,, and Three different appear in Fig 1: i the unknown optimizer of the map, i the real-time etimate of, and i the actual input into the map The actual input i baed on the etimate but i perturbed by the ignal in for the purpoe of etimating the unknown gradient of the map The etimate i generated with the integrator 1/ with the adaptation gain controlling the peed of etimation The ES algorithm i ucceful if the error between the etimate and the unknown, namely the ignal ( 2 ) converge toward zero Baed on Fig 1, the etimate i governed by the differential equation in, which mean that the etimation error i governed by d in d in ( 3 ) Expanding the right-hand ide one obtain k ĝ 2 a in(ωt) d d S(t) Fig 2: The ES algorithm for a multivariable map in in mean ˆθ 2 in mean fat, mean low in fat, mean low (4) A theoretically rigorou time-averaging procedure allow replacing the above inuoidal ignal by their mean, yielding the average ytem dave d K M (t) ave ( 5 ) which i exponentially table For a ufficiently large ω, if the initial etimate 0 i ufficiently cloe to the unknown, then the input exponentially converge to a mall interval around the unknown and, conequently, the output converge to the vicinity of the optimal output A ES for Multivariable Static Map For tatic map, ES extend in a traightforward manner from the ingle-input cae hown in Fig 1 to the multi-input cae hown in Fig 2 The algorithm meaure the calar ignal, where i an unknown map whoe input i the vector The gradient i etimated with the help of the ignal in in ( 6 ) in in ( 7 ) with nonzero perturbation amplitude and with a gain matrix that i diagonal To guarantee convergence, the uer hould chooe Thi i a key condition that differentiate the multi-input cae from the ingle-input cae In addition, for implicity in the convergence analyi, the uer hould chooe / a rational and for ditinct,, and If the unknown map i quadratic, namely,, the averaged ytem i, Heian ( 8 ) If, for example, the map ha a locally quadratic peak (which implie 0), and if the uer chooe the element of the diagonal gain matrix a poitive, the ES algorithm i guaranteed to be locally convergent However, the convergence rate depend on the unknown Heian Thi weakne of the gradient-baed ES algorithm i removed with the Newton-baed ES algorithm B ES for Dynamic Sytem ES extend in a relatively traightforward manner from tatic map to dynamic ytem, provided the dynamic are table and the algorithm parameter are choen o that the algorithm dynamic are lower than thoe of the plant The algorithm i hown in Fig 3 The technical condition for convergence in the preence of dynamic are that the equilibria of the ytem,,, where, i the control law of an internal feedback loop, are locally exponentially table Ĝ 6185

3 ẋ = f (x,α(x,θ)) y = h(x) S(t) ˆθ K Ĝ Low-pa filter Fig 3: The ES algorithm in preence of dynamic S(t) θ ˆθ Fig 4: A Newton-baed ES for a tatic map θ K Q( ) ΓĜ M (t) Ĝ y = h l(θ) High-pa filter uniformly in and that, given the output map, there exit at leat one uch that 0 and 0, The aforementioned criteria to elect the ES parameter for a tatic map are till valid Adding the inner control affect the probing frequency and the band-pa of the filter C Newton ES Algorithm for Static Map A Newton verion of the ES algorithm, hown in Fig 4, enure that the convergence rate be uer-aignable, rather than being dependent on the unknown Heian of the map [13] The element of the demodulating matrix for generating the etimate of the Heian are given by in in in ( 9 ) The multiplicative excitation help to generate the etimate of the Heian a The Riccati martrix differential equation generate an etimate of the Heian invere matrix, avoiding matrix inverion of Heian etimate that may be ingular during the tranient y Γ = ργ ργĥ Γ Ĥ N (t) M (t) For a quadratic map, the averaged ytem in error variable, i dave d dave d ave ave quadratic ave ave ave ( 10 ) quadratic Since the eigenvalue are determined by and, and are therefore independent of the unknown, the (local) convergence rate i uer-aignable In the next ection we apply the calar gradient-baed ES to MPPT of a WT with an inner-loop control III WIND ENERGY CONVERSION SYSTEMS A chematic of a wind energy converion ytem (WECS) including wind turbine (WT), induction generator (IG), and matrix converter (MC) i hown in Fig 5 Wind turbine work in four different region a depicted in Fig 6 In Region I, the wind peed i too low for the turbine to generate power Region II, alo called the ub-rated power region, lie between the cut-in peed and rated peed Here the generator operate at below rated power The theoretical hape of thi curve reflect the baic law of power production, where power i proportional to the cube of the wind peed In Region III, the turbine limit the power output; thi occur when the wind i ufficient for the turbine to reach it rated output power Region IV i the period of tronger wind, where the power in the wind i o great that it could be detrimental to the turbine, o the turbine hut down [22] The wind power available on the blade impact area i defined a 2,, ( 11 ) where i the blade length, i air denity, and i wind peed For Region II MPPT, auming zero blade pitch angle, the turbine power i related to the wind power a,, ( 12 ) where i the rotor torque, ω i the turbine peed, and i the non-dimenional power coefficient, which i a meaure of the ratio of the turbine power to the wind power The power coefficient i a function of wind peed and turbine peed The turbine peed can be ued to change the power coefficient,, 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 regardle of the variation of the wind peed The power captured by the WT i S aa Output frequency = ω o S ba Input frequency = ω i Wind turbine Shaft and gearbox B S ca Matrix converter AC grid Wind peed = V w Blade pitch angle = β Turbine peed = ω t Turbine inert ia = J t K Gearbox ratio = n Rotor inertia = J Generator IG Rotor frequency = ωr Pole pair = p Rotor peed = ω r p v c v b v a S ab S bb S cb S ac S bc S cc v A v B v C N Fig 5: WECS including WT, gear box, IG, and MC 6186

4 Fig 6: Typical power curve of WT including four operating region Fig 7: Variation of the turbine power veru turbine peed for different wind peed The MPP move on curve which how the characteritic of the ub-rated region of WECS defined by the wind peed,, and the turbine peed, However, the wind peed i a diturbance input and we can manipulate the turbine peed to govern the turbine power to it maximum power point (MPP) in ub-rated region The variation of turbine power veru turbine peed i hown in Fig 7 for different wind peed A hown in Fig 7 under a contant wind peed the relevant power curve ha a unique MPP, which i defined by a pecific turbine peed At the MPP the following obervation are valid 0, 0 ( 13 ) A hown in Fig 5 the WT haft i modeled a a pringdamper, and i connected to the electrical generator, which in thi cae i a quirrel-cage induction generator (SCIG), via a gearbox Squirrel-cage IG are relatively inexpenive, robut, and require little maintenance When operated uing vector control technique, fat dynamic repone, and accurate torque control i obtained The generator i connected to the AC grid through a matrix converter (MC), which i a replacement for the conventional rectifier-inverter combination (AC-DC-AC), and teer the generator to it maximum power point (MPP) by controlling the electrical frequency of it' tator of SCIG, which in turn lead to a peed variation in the turbine haft Matrix converter provide bidirectional power flow, inuoidal input/output current, and controllable input power factor [7, 24] The input phae voltage of MC,, which i connected to the AC grid, i given by co co co, ( 14 ) where i the peak value of the input voltage amplitude and d ( 15 ) i the input electrical angle, where 2 i the input electrical frequency of the MC Output voltage i It i the job of the MC to create local-averaged inuoidal output phae voltage (the tator voltage of IG) and input phae current (the AC grid current) ave co co co, ( 16 ) where i the peak value of the tator voltage amplitude and d ( 17 ) i the output electrical angle where 2 i the tator electrical frequency Stator electrical frequency,, and the peak value of the tator voltage amplitude,, are control input and can be ued for power control and optimization of the WECS A Inner-Loop Control Deign for WECS In many motor drive ytem, it i deirable to make the drive act a a torque tranducer wherein the electromagnetic torque can nearly intantaneouly be made equal to a torque command In uch a ytem, peed or poition control i dramatically implified becaue the electrical dynamic of the drive become irrelevant to the peed or poition control problem In the cae of induction machine drive, uch performance can be achieved uing a cla of algorithm collectively known a field-oriented control (FOC) When flux amplitude i regulated to a contant reference value, and conidering the fact that the dynamic of are coniderably lower than the electrical dynamic, we can aume that the dynamic are linear, but during flux tranient the ytem ha nonlinear term and it i coupled Thi method can be improved by achieving exact input-output decoupling and linearization via a nonlinear tate feedback that i not more complex than the conventional FOC [24] One can manipulate tator voltage amplitude,, and it frequency,, through the MC to obtain the deired cloedloop performance for WECS By employing FOC idea we introduce an integrator and an auxiliary input,, to achieve input-output decoupling in WECS dynamic Uing one tep of integration in front of the extended equation of WECS are introduced a follow:,,, ( 18 ) where, where and are tator current, and are rotor fluxe,, d, i the rotor electrical frequency, i the electrical frequency of the tator, i an auxiliary input (voltage amplitude rate) which generate the voltage amplitude of the tator From Fig 7 we know that the turbine peed control the power generation Alo we are intereted in decoupling the rotor flux and electromagnetic torque to obtain the benefit of FOC For thee reaon, we introduce turbine peed,, and flux amplitude,, a meaurable output Baed on the elected output, we apply feedback linearization which reult in the regulation of turbine peed,, to it reference value ref, while the amplitude of rotor flux,, converge to it deired value, ref B Wind Turbine Power Optimization To overcome challenge aociated with the conventional power control and optimization algorithm and to remove the dependence of the MPPT algorithm on the ytem modeling 6187

5 V w V w u 2 =0 u 1 = ω o 1 1 V om θ o WECS P t (V w, ω o ) λ ref ω ref t Controller State feedback u 2 ω o = u V om θ o WECS P t(v w, ω t) a in(ωt) ˆω o k ĝ Fig 8: The ES algorithm for MPPT of the WECS without the inner-loop control and identification, we propoe an ES algorithm for MPPT of WECS Firt we preent ES without the inner-loop control to clarify the advantage of the inner-loop control on the cloedloop performance of the ytem In thi paper we aume that we have acce to turbine power meaurement and we can manipulate the turbine peed through the MC Furthermore, we do not have a model of the power coefficient or turbine power But, we know that the turbine power map ha one MPP under any wind peed The torque-peed characteritic of an induction machine i normally quite teep in the neighborhood of tator electrical frequency (ynchronou peed),, and o the electrical rotor peed,, will be near the ynchronou peed Thi mean that changing the reference value of the turbine peed,, which tranlate in variation of the electrical rotor peed eventually, reult in changing the tator electrical frequency Thu, by controlling the tator electrical frequency one can approximately control the turbine peed or vice vera We can rewrite (13) a follow Low-pa t in(ωt) High-pa t 0, 0 ( 19 ) A chematic of MPPT for WECS with extremum eeking without inner-loop nonlinear control i hown in Fig 8 A mentioned in the lat paragraph, the power i parameterized by, which i etimated by the ES loop The other input for WECS which generate the voltage amplitude ha been et to zero which mean the tator voltage ha a contant peak amplitude The probing frequency, Ω, need to be elected at leat 10 time maller than the highet frequency that can pa the ytem without ignificant attenuation The band-pa of the filter hould alo be le than 10% of Ω The ES gain,, and alo need to be reaonably mall The turbine power meaurement i fed into the ES cheme The optimization parameter for ES without the inner-loop control, Fig 8, i the electrical frequency of IG tator, Stability of ytem dynamic i required for convergence of ES algorithm to it peak point It i alo required that the ES algorithm operate more lowly than the WECS ytem dynamic A previouly mentioned, ince WECS in Fig 8 without the inner- loop controller how a low tranient, the entire ytem ha a lengthy convergence proce which reult in low power efficiency We employ the propoed nonlinear control to achieve the deired cloed-loop performance, including fater repone time (high power efficiency), and preventing magnetic aturation Our propoed ES cheme with the inner-loop control i hown in Fig 9 In thi cae, the reference input of the inner-loop control are and ref We know that the MPP i parameterized by the optimal turbine peed at each a in(ωt) in(ωt) ˆω t k ĝ Low-pa t High-pa t Fig 9: The ES algorithm for MPPT of the WECS with the inner-loop control wind peed which i etimated by the ES loop The other control input, ref, define the level of the flux linkage of the rotor which prevent IG from magnetic aturation Combination of the Controller and WECS include fat dynamic and ES algorithm contain low and medium peed dynamic The ES algorithm etimate the optimal ref turbine peed, which can be conidered a a contant valuewith repect to the fat dynamic of the controller-ytem The ES cheme etimate the gradient of the turbine power,, by injecting a mall perturbation, in, which i very low with repect to the dynamic of the controller- ytem and it amplitude i enough mall in comparion to The high-pa filter remove the DC part of the ignal The multiplication of the reulting ignal by in create an etimate of the gradient of the cot function, which i moothed uing a low-pa filter When i larger than it optimal value the etimate of the gradient,, i negative and caue to decreae On the other hand, when i maller than then 0 which increae the toward It hould be noted that i mall enough in comparion to the lowet dynamic of the controller-ytem, with an order le than 10% C Simulation Reult on a Wind Turbine Model A we mentioned earlier repone time of the ES deign without the inner-loop i coniderably low which reult in a very low power efficiency However, we preent one imulation that compare the repone of the deign without the inner-loop a hown in Fig 8 to our propoed algorithm a hown in Fig 9 which illutrate the role of the inner-loop control Alo, we compare the performance of our propoed algorithm to the conventional algorithm including FOC and MPPT baed on perturb and oberve (P&O) technique We how a time frame of 30 to viualize the difference between our propoed algorithm and the two other algorithm Fig 10 how the wind regime applied to the WECS The MPPT proce i hown in Fig 11 The extracted energy by our propoed algorithm i 236% higher than the extracted energy by the conventional MPPT and FOC A we expected, Fig 10: Variation of wind peed veru time 6188

6 C, 10 kw/m 2 25 C, 10 kw/m 2 PV module Power (W) C, 10 kw/m 2 25 C, 08 kw/m 2 25 C, 06 kw/m 2 25 C, 04 kw/m 2 Fig 11: MPPT, (olid red) our propoed algorithm, (dah-dot green) ES without inner-loop, (dahed blue) conventional P&O with FOC, and (dotted black) maximum power available to the WECS the power efficiency of the ES deign without the inner-loop i low The propoed algorithm combine two well-known control algorithm namely, feedback linearization baed on the FOC concept and extremum eeking, to achieve MPPT in a WECS operating in Region II Our algorithm provide perfect inputoutput decoupling and guarantee a larger domain of attraction, which increae performance robutne with repect to the ytem parameter The improved efficiency alo increae the competitivene of wind energy IV PHOTOVOLTAIC SYSTEMS Extremum eeking ha been applied to MPPT deign for photovoltaic (PV) micro-converter 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 gradientbaed [2-4] 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 environmental condition and/or PV module degradation reult in non-uniform tranient in the convergence to the MPP Uing a multivariable gradient-baed ES algorithm for the entire ytem intead of a calar one for each PV module, while decreaing the enitivity to the Heian, doe not eliminate thi dependence [20] We ue the Newton-baed ES algorithm that imultaneouly employ etimate of the gradient and Heian in the peak power tracking [19] The convergence rate of uch a deign to the MPP i independent of the Heian, with tunable tranient performance that i independent of environmental condition We preent experimental reult that how the effectivene of the propoed algorithm in comparion to exiting calar deign, and alo to 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), and it reult in more Voltage(V) Fig 12: Characteritic of a typical PV module under varying temperature and irradiance Peak power and the optimal terminal voltage varie with change of temperature and irradiance uniform tranient under udden change in olar irradiance and environmental temperature in comparion to a calar gradient-baed ES for each PV module However, a i true of gradient-baed deign, the convergence to MPP i dependent on the unknown Heian, and 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 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 the MPPT problem in PV ytem, the method offer the benefit of uniform 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 A Photovoltaic Module and Power Extraction A i clear from Fig 12, the power-voltage ( ) characteritic of a typical PV module ha a unique peak (, ) which depend on temperature and irradiance, It i the job of the MPPT algorithm to automatically track thi MPP In many grid-tied PV ytem (including our current work), thi i done by mean of a eparate DC/DC power electronic tage controlled by an MPPT algorithm like ES that erve two function: (i) regulating the output DC voltage at a (near) contant value, and (ii) extracting maximum power by forcing the PV module output to equal Fig 13 how thi etup for a DC/DC converter tage, whoe output voltage i maintained contant a dc The ratio between the input voltage,, and output voltage, dc, can be controlled by changing the duty cycle of the tranitor witch in DC/DC converter, d, which erve a the control input From Fig 12, it follow that at the MPP (, ), the power atifie 0, 0 ( 20 ) An MPPT algorithm baed on ES i hown for a ingle PV module in Fig 13 Since 0, a poitive gain, 0, guarantee the convergence of the ES toward the MPP, 6189

7 a in(ωt) d DC/ DC Converter ˆd k ĝ Fig 13: The ES algorithm i applied to MPPT of a ingle PV module V Low-pa filter (T, S ) PV module P(V, T, S ) High-pa filter 2 a in(ωt) current, and the DC bu voltage Thi i a ignificant reduction in hardware cot Moreover, interaction between PV module are inherently part of the multivariable deign, and hence the tranient performance i le enitive to environmental variable variation than a correponding calar deign C Multivariable ES for MPPT of PV Sytem 1) Gradient-Baed ES We want to maximize the power generated by all PV module which i equal dc dc ( 21 ) A typical power map of a PV ytem including two erie module under tandard tet condition, 1000 W/m 2 and 25 C, i hown in Fig 16 Maximum power point happen at D %57 %57 We can generalize the above obervation for the micro-converter tructure, ie, there exit D uch that 0, 0, ( 22 ) Accordingly, we can ue the multivariable gradient-baed ES deign hown in Fig 17 to MPPT of the PV ytem in Fig 15 The ES gain,, i a poitive diagonal matrix, and the perturbation ignal are defined a (6) and (7) Fig 14: Ditributed MPPT algorithm for a PV ytem Each module ha a eparate MPPT Temperature,, and irradiance,, varie all over the module provided that the initial condition, 0, i reaonably cloe to it optimal value, B Micro-Converter Architecture Traditionally, PV array have been connected to the AC power grid through centralized DC/AC power converter (or inverter) Thee are now giving way to ditributed architecture that connect each module to a dedicated converter/inverter The micro-converter configuration, where each PV module i connected to it own DC/DC converter, i hown chematically in Fig 14 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 It i aumed that the DC voltage at the input ide of the inverter i held contant at dc Two problem arie here Firt, the cheme of Fig 14 require two enor per module, current and voltage, which increae the levelized energy cot Second, the coupling effect between PV module i not addreed by thi ditributed control We preent a multivariable MPPT baed on ES cheme a hown in Fig 15 with the following feature: It i applied to micro-converter ytem, and hence deal with the cae of non-unimodal power characteritic, and deal pecifically with the iue of module mimatch (for example, poibly different irradiance level a a reult of partially haded condition) The ue of the non-model-baed ES technique make the deign robut to partial knowledge of the ytem parameter and operating condition A oppoed to calar deign, our multivariable deign only require 2 enor in all, for the overall PV ytem Fig 15: Multivariable MPPT for a PV ytem One MPPT i ued for the entire ytem Temperature,, and irradiance,, varie all over the module Fig 16: A typical power map of two cacade PV module veru pule duration, Level et how the power in Watt 1000 W/m 2, and 25 C 6190

8 [(T 1,S 1 ),,(T n,s n )] T [(T 1, S 1 ),, (T n, S n )] T S (t) ˆD D K D 1 D C / D C V 1 PV 1 P 1 D 2 D C / D C V 2 PV 2 P 2 D n D C / D C V n PV n P n Ĝ Low-pa filter P = V dc I dc High-pa filter Fig 17: Multivariable gradient-baed ES for MPPT of a PV ytem In particular, the deign derive an etimate of the gradient vector by adding the probing ignal to the etimate 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 for 1,2,,,,,, and with 0, the etimate of the pule duration vector and the output ettle in a mall ball around the optimal pule duration and the MPP, repectively The radiu of the ball i defined by the lowet probing frequency and it correponding amplitude The linearized update equation for the etimation error i,, ( 23 ) where i the Heian of with repect to the pule duration vector, Since the cot function varie with irradiance, temperature, and degradation of the PV module, o doe H, and therefore a fixed adaptation gain reult in different (condition dependent) convergence rate for each converter In order to alleviate the iue of unknown Heian dependent convergence, we preent in the next ection a modified verion of the multivariable Newton-baed ES In comparion with the gradient-baed deign of thi ection, 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 the MPPT problem in PV ytem, the method offer the benefit of uniform convergence behavior, under a wide range of working condition that include temperature and irradiance variation, and under the non-ymmetric power generation of the neighboring PV module a a reult of module degradation or mimatch 2) Newton-Baed ES The multivariable Newton-baed ES that we propoe i hown chematically in Fig 18 A i clear from the figure, the propoed cheme extend the gradient-baed ES with the etimate of the Heian The perturbation matrix i defined a (9) The goal of the Newton-baed deign i to replace the etimation-error dynamic with one of the form, where, that remove the dependence on the Heian Calculating (etimate of ) in an algebraic fahion create difficultie when i cloe to ingularity or i indefinite To deal with thi problem, a M (t) D S(t) ˆD D 1 D C / D C V 1 PV 1 P 1 D 2 D C / D C V 2 PV 2 P 2 D n D C / D C V n PVn P n K ΓĜ Ĝ Γ = ργ ργĥ Γ Low-pa t M (t) High-pat Fig 18: Multivariable Newton-baed ES for MPPT of a PV ytem The purple part i added to the gradient-baed ES to etimate the Heian dynamic etimator i employed to calculate the invere of uing a Riccati equation Conider the following filter ( 24 ) Note that the tate of thi filter converge to, an etimate of Denote Since, then equation (24) i tranformed into the differential Riccati equation ( 25 ) The equilibria of the Riccati equation (25) are 0 and, provided ettle to a contant Since 0, the equilibrium 0 i untable, wherea the linearization of (25) around ha the Jacobian, o the equilibrium at i locally exponentially table Thi how that, after a tranient, the Riccati equation converge to the actual value of the invere of Heian matrix if i a good etimate of Linearization of the update law for the error variable reult in, 0, ( 26 ) where element of are ufficiently mall poitive number According to (26) the convergence rate of the parameter i independent of the hape of the cot function, and conequently, after tranient, when the Heian i cloe enough to it actual value, the output power converge to the MPP with the ame performance regardle of environmental or mimatch condition Fig 19: Schematic of the hardware etup Ĥ Low-pa t P = V dc I dc N (t) 6191

9 Power (W) Multivariable Newton Multivariable Gradient Scalar Gradient 6 Fig 20: Experimental etup D Experimental Reult To how the effectivene of the propoed Newton-baed deign in Fig 18, and compare it performance with that of the gradient-baed deign in Fig 17, we preent experimental reult for a PV ytem with n = 2 cacade module Our hardware etup conit of 2 cacade PV module connected to an active load which play the role r of the DCC bu with 5 V, and i chematically hown in Fig 19 The phyical hardware etup i hown in Fig 20,, and comprie of (a) cutom-made PV module contructed uing 12 PV cell, (b) Power-Pole Board developed by the t Univerity of Minneota configured a DC/ /DC buck converter, andd (c) DS1104 R&D Controller Board to implement our MPPT algorithm inide Simulink and interact with the DC/DC converter, and generate external PWM ignal ued byy the DC/DC converter Each Power-pole board ha a current enor LA 25-NP to meaure the inductor current whichh we ue along with the capacitor ripple current meaurement to calculate the DC bu current We employ the DC bu current and DC bu voltage to meauree the power upplied to thee DC bu The election of dspace hardware i intentional and it ha been ued a our baic experimental etup to removee the difficultie attached with hardware prototyping A we mentioned in Introduction, the implementation of thee ES algorithm i not complicated Common electronic part,, like operational amplifier, reitor and capacitor can be ued to build an ES algorithm The temperature of PV module i 25 C and the module are fully expoed to the un between 0-60 and To imulate the effect of partial hading, PV 1 i covered with a platic mat from time When one module i partially haded, the overall power level decreae We not only compare the multivariable gradient-baed and a Newton-baed deign, but alo the traditional calar gradient-baed deign, that ha one MPPT loop for each converter Fig 21 how the performance of the 3 deign, andd it i clear that the Newton algorithm recover from thi power level change fater than the other 2 algorithm While the Newton method ha the leat teady-tate error and uniform repone under tep down and tep up power cenario, the calar deign ha the highet teady-tatpower decreae The multivariable gradient- baed ES perform better than the calar MPPT under partial hading condition The irradiance level of the partially haded module i error and large repone time in face of returned to normal level at t = 120 At thi point the Newton cheme how fater tranientt in comparion to the imilar tranient of the multivariable gradient-baed ES andd the ditributed ES The reult demontrate that the convergence rate of the Newton cheme doe not vary largely from tep up Time() Fig 21: Variation of o power veru time The Newton algorithm how uniform and fat tranient with low teady-tate error Fig 22: Phae portrait of the MPPT for the multivariable Newton, multivariable gradient, and calar gradient algorithm Region (a) how MPP area for full expoure to the un and region (b) how the MPP area when PV 1 i partially haded to tep down in power generation, which i not true for the gradient-baed and ditributed MPPT cheme Not urpriingly, the experimental reult are in keeping with the analytical reult Latly, the Newton-baed deign move the ytem in almot a traight linee between extrema, in contrat to curved teepet decent d trajectorie of the gradient algorithm Thi obervation i demontrated clearly in Fig 22 V V CONCLUDING REMARKS Since environmental parameter like olar irradiance and wind peed affect the power map and maximum power point of photovoltaic (PV)( and wind energy converion ytem (WECS), we propoe extremum-eeking (ES), which i a model-free real-time optimization algorithm, for maximum energy harvet orr maximum-power-point-tracking (MPPT) in uch ytem Extremum eeking i effective at guiding the WECS to it MPPP in the ub-rated power region However, the open-loop dynamic of the WECS have low left half-plane pole that make the repone time of the ES even lower In order to achieve fat cloed-loop repone and extra e feature like contant voltage-to-frequency or vector control in the ytem, we deign an inner-loop control baed on the field-oriented control (FOC) concept The combination of the inner-loop 6192

10 controller and the ES algorithm improve the performance of the WECS, a hown by the imulation For PV, we conider the micro-converter architecture, where each module i connected to it own DC-DC converter Conventional deign are calar, they: i) ignore the interaction between module, and ii) require two (enor) meaurement per module We propoe a multivariable deign that improve on each of thee apect We conider firt a multivariable gradient-baed ES algorithm, where the Heian of the power map ha a dominant role in the cloed-loop performance Next, we employ a Newton-baed ES algorithm, which remove the performance dependence of the gradient-baed deign on the Heian The Newton-baed deign ha two key/ditinguihing component: i) a perturbation matrix that generate the etimate of the Heian, and (ii) a dynamic filter to etimate the invere of the Heian Experimental reult verify the effectivene of the Newton-baed MPPT veru it (calar and multivariable) gradient-baed counterpart REFERENCES [1] CS Draper 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