IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 11, NOVEMBER Robust Power Flow Control of Grid-Connected Inverters

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1 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 11, NOVEMBER Rbust Pwer Flw Cntrl f Grid-Cnnected Inverters Yeqin Wang, Student Member, IEEE, Beibei Ren, Member, IEEE, and Qing-Chang hng, Senir Member, IEEE Abstract In this paper, an uncertainty and disturbance estimatr (UDE)-based rbust pwer flw cntrl is develped fr grid-cnnected inverters t achieve accurate pwer delivery t the grid. The mdel f pwer delivering with bth frequency dynamics and vltage dynamics is derived at first. The UDE methd is intrduced int the cntrller design t deal with mdel uncertainties (e.g., utput impedance and pwer angle), cupling effects, and external disturbances (e.g., the fluctuatin f the dc-link vltage, the variatin f utput impedance/line impedance, and the variatins f bth frequency and amplitude in the grid vltage). Als, this cntrller des nt need a vltage regulatr r a current regulatr and is easy fr the implementatin and parameter tuning thrugh the design f the desired tracking errr dynamics and the UDE filters. Experimental results are prvided t shw the effectiveness f the prpsed methd fr different disturbance rejectin scenaris, the lw-vltage fault-ride thrugh capability, and the weak grid peratin capability. The gd rbustness f the UDE-based cntrl is als demnstrated thrugh the cmparisn with tw ther cntrllers: the prprtinal integral cntrller and the active disturbance rejectin cntrller. Index Terms Fluctuatin f dc-link vltage, grid disturbances, mdel uncertainties, uncertainty and disturbance estimatr (UDE), variatin f the impedance. I. INTRODUCTION NOWADAYS, renewable energies play very imprtant rles in energy area. Mst f the renewable energies are cmprised f variable-frequency ac surces (e.g., wind turbines), high-frequency ac surces (e.g., small gas turbines), r dc surces (e.g., slar phtvltaics). The dc/ac cnverters, als called grid-cnnected inverters (GCI), are needed t interface renewable energies with the public-utility grid [1]. Thugh renewable energies are quite ppular, there are still sme challenges faced by the GCI cntrl fr grid integratin. Fr example, the renewable energies, such as wind pwer, slar phtvltaics, are unstable in changing wind cnditins r sunlight cnditins; the Manuscript received January 20, 2016; revised May 1, 2016; accepted May 14, Date f publicatin June 29, 2016; date f current versin Octber 7, This wrk was supprted in part by the Natinal Science Fundatin under Grant DGE Y. Wang is with the Natinal Wind Institute, Texas Tech University, Lubbck, TX USA ( yeqin.wang@ttu.edu). B. Ren is with the Department f Mechanical Engineering, Texas Tech University, Lubbck, TX USA ( beibei.ren@ttu.edu). Q.-C. hng is with the Department f Electrical and Cmputer Engineering, Illinis Institute f Technlgy, Chicag, IL USA ( zhngqc@ieee.rg). Clr versins f ne r mre f the figures in this paper are available nline at Digital Object Identifier /TIE variatins f the grid vltage (frequency variatin, phase variatin, r amplitude variatin), even thugh they are small, affect the stability f the GCI and grid peratin; and the fluctuating dc-link vltage als ften affects the GCI cntrl fr renewable energies [2], [3]. The vectr cntrl, riginally prpsed fr the cntrl f electrical machine drive systems [4], is a ppular cntrl algrithm fr three-phase GCI t cnvert dc pwer t ac pwer [5] [7]. In the vectr cntrl, the current regulatin in the dq reference frame is adpted t generate vltage reference cntrl signals. An extra synchrnizatin unit, e.g., the phase-lcked lp (PLL), is required fr the transfrmatins f bth current and vltage between the abc reference frame and the dq reference frame. Hwever, the vectr cntrl has its fundamental limitatins in the grid vltage variatin cnditins. First, the frequency dynamics are nt cnsidered in the vectr cntrl, which makes it difficult t analyze the pwer angle and frequency stability between the GCI and the grid [3]. Secnd, the PLL dynamics can affect the stability f the GCI, particularly in weak grids [3], [8] [10], and the cntrl respnse is als limited by the respnse f PLL [11], [12]. Third, the variatin f the dc-link dynamics can als affect the stability f the GCI in the vectr cntrl [3]. Mrever, because the PLL is inherently nnlinear and s are the inverter cntrller and the pwer system, it is extremely difficult and time-cnsuming t tune the PLL parameters t achieve a satisfactry perfrmance [12]. The drp cntrl methd fr pwer sharing amng parallelperated inverters [13] can be adpted fr the GCI t achieve pwer flw cntrl [14] [18]. Thugh sme merits are achieved with the drp cntrl methd, such as the harmnic current regulatin [14], flexible peratin in either grid-cnnected mde r islanded mde [15], [17], and enhancing pwer lp dynamics [16], the extra synchrnizatin units are still needed in mst cases [14], [16] [18]. It still has limitatins fr the drp cntrl methd t deal with variatins f the grid vltage, and extra cntrl lps (a vltage regulatr r a current regulatr) [11] are required t cmbine with the drp cntrl methd [14], [15], [17], [18]. Inverters als can be cntrlled t behave as virtual synchrnus machines (VSM) [19], [20]. The self-tuning algrithms fr ptimal parameters f the VSM are studied in [21]; hwever, the PLL is still needed in this methd. Similar t the synchrnizatin mechanism f a synchrnus machine, a pwer-synchrnizatin cntrl is prpsed fr vltage-surce cnverters in [22] withut the needs f PLL. Fllwing the wrk n the synchrnverter technlgy [20], a dedicated synchrnizatin unit is cmpletely remved in [12]. The similar IEEE. Persnal use is permitted, but republicatin/redistributin requires IEEE permissin. See standards/publicatins/rights/index.html fr mre infrmatin.

2 6888 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 11, NOVEMBER 2016 cncept f the VSM is expanded t intrduce the inertia t the grid [3] and t imprve the micrgrid s stability [9] withut a synchrnizatin unit. In this paper, a rbust pwer flw cntrl strategy based n the uncertainty and disturbance estimatr (UDE) methd is develped fr GCI t deliver bth real pwer and reactive pwer t the utility grid. The UDE cntrl algrithm, which was first prpsed in [23], is based n an assumptin that the uncertainty and disturbance can be estimated by using a filter with an apprpriate bandwidth. In recent years, the UDE-based cntrl demnstrates its excellent rbust perfrmances in different applicatins, such as rbust trajectry tracking [24], cntrl fr a class f nnaffine nnlinear systems [25], and the variable-speed wind turbine cntrl [26]. In this paper, the mdel f pwer delivering with bth frequency dynamics and vltage dynamics is derived at first. Then, the UDE methd is intrduced int the cntrller design t achieve accurate regulatin f bth real pwer and reactive pwer, in the presence f the mdel uncertainties (e.g., utput impedance and pwer angle), the cupling effects, and the external disturbances (e.g., the fluctuatin f dc-link vltage, variatin f utput impedance/line impedance, and variatins f the grid vltage). Als, this cntrller des nt require an extra synchrnizatin unit in grid-cnnected peratin, apart frm an initial synchrnizatin. This apprach has a simple structure withut either vltage regulatin r current regulatin and is easy fr the implementatin and parameter tuning with the design f the desired tracking errr dynamics and the UDE filters. The effectiveness f the prpsed cntrl apprach is investigated by theretical analysis and demnstrated thrugh experimental studies n an experimental test rig. Thugh this methd is develped based n single-phase GCI in this paper, the results can be further extended t three-phase GCI. With the capability f handling the variatins f the grid vltage, the UDE-based rbust pwer flw cntrl is suitable fr weak grids. Als with the individual cntrl f the real pwer and reactive pwer, this apprach can be extended t ther applicatins, such as static synchrnus cmpensatrs and active pwer filters. The rest f this paper is rganized as fllws. Sectin II prvides the dynamics f pwer delivery. In Sectin III, the UDE-based rbust pwer flw cntrl is prpsed fr the GCI. Bth stability analysis and steady-state perfrmance analysis are prvided. The effectiveness f the prpsed apprach is demnstrated thrugh the experimental validatin in Sectin IV, with the cncluding remarks made in Sectin V. II. DYNAMICS OF POWER DELIVERY A vltage surce E δ delivering pwer t the grid V 0 thrugh an impedance θ can be mdeled in Fig. 1. The real pwer P and the reactive pwer Q received by the grid V 0 can be btained as [1] P = ( EV cs δ V 2 Q = ( EV cs δ V 2 ) cs θ + EV sin δ sin θ (1) ) sin θ EV sin δ cs θ (2) Fig. 1. Vltage surce delivering pwer t the grid. where δ is the phase difference between the vltage surce and the grid, ften called the pwer angle. Here, the grid vltage v is used as the reference, with the initial angle f grid vltage defined as zer. Taking the derivatives f bth (1) and (2), the dynamics f pwer delivery are expressed in P = EV δ cs δ sin θ EV δ sin δ cs θ + V Ė cs δ cs θ + V Ė sin δ sin θ (3) Q = V Ė cs δ sin θ EV δ sin δ sin θ EV δ cs δ cs θ V Ė sin δ cs θ. (4) Since the pwer angle δ depends n the utput pwer f the GCI, and the utput impedance θ always drifts with the changes f envirnments (e.g., inductance change with magnetic saturatin caused by high current, and resistance change by high temperature), they are quite uncertain. Mrever, nnlinear and cupling terms, e.g., V E sin δ sin θ and V E sin δ sin θ, exist in the dynamics f real pwer (3), and EV δ sin δ sin θ and EV δ cs δ cs θ exist in the dynamics f reactive pwer (4). The uncertainties, the nnlinearity, and the cupling effects cause difficulties fr cntrl design f pwer delivery. Fr the GCI, similar t the winding inductance in synchrnus generatrs, because f the utput LC filters r the inductance f line impedance, the utput impedance is mstly inductive. It means sin θ 1 and cs θ 0 with θ 90. In practice, the pwer angle δ is usually very small with sin δ 0 and cs δ 1. Then, the dynamics f pwer delivery (3) and (4) can be rewritten as where p = EV δ P = EV δ + p (5) Q = V Ė + q (6) (cs δ sin θ 1) EV δ sin δ cs θ + V Ė cs δ cs θ + V Ė sin δ sin θ + d pz (7)

3 WANG et al.: ROBUST POWER FLOW CONTROL OF GRID-CONNECTED INVERTERS 6889 Fig. 2. UDE-based rbust pwer flw cntrl. q = V Ė (cs δ sin θ 1) EV δ sin δ sin θ = EV δ cs δ cs θ V Ė sin δ cs θ + d qz (8) represent the lumped uncertain terms, including the uncertainties, the nnlinearity, and the cupling effects. d pz and d qz are the deviatins due t the deviatin f the real utput impedance in (3) and (4) frm the nminal utput impedance.in practice, the lumped uncertain terms p and q can be small and bunded because f the inductive utput impedance and small pwer angle δ in GCI. With the treatment based n the uncertain terms (7) and (8), the dynamics f pwer delivery (3) and (4) are reduced as linear equatins in (5) and (6) plus the uncertain terms, which will simplify the fllwing cntrller design. Als, the uncertain terms p and q will be estimated and cmpensated in the fllwing cntrller design. III. ROBUST POWER FLOW CONTROL In this sectin, an UDE-based rbust pwer flw cntrl is develped fr the GCI based n the dynamics f pwer delivery (5) and (6). The structure is shwn in Fig. 2. The real pwer and reactive pwer can be regulated individually. A. Cntrller Design The inverter shuld deliver bth real pwer and reactive pwer t the grid accurately. The cntrl bjective is t achieve the regulatins f bth real pwer and reactive pwer, such that the real pwer P and the reactive pwer Q asympttically track their settings P set and Q set, respectively. In particular, the tracking errrs e p = P set P and e q = Q set Q shuld satisfy the errr dynamic equatins ė p = K p e p (9) ė q = K q e q (10) where K p > 0 and K q > 0 are the cnstant errr feedback gains. Cmbining (5) and (9) and (6) and (10), respectively, then P set EV δ p = K p e p (11) Q set V Ė q = K q e q. (12) Therefre, δ and Ė need t satisfy δ = ( ) P set p + K p e p EV (13) Ė = ( ) Q set q + K q e q. (14) V Accrding t the system dynamics in (5) and (6), the uncertain terms p and q can be btained as p = P EV δ, q = Q V Ė. Fllwing the prcedures f the UDE design prvided in [23], p and q can be estimated by ˆ p = ( P EV δ) gpf (15) ˆ q = ( Q V Ė) g qf (16) where is the cnvlutin peratr, and g pf (t) and g qf (t) are the impulse respnse f strictly prper stable filters G pf (s) and G qf (s) with the apprpriate bandwidth. Replacing p and q with ˆ p and ˆ q in (13) and (14) results in δ = [ P set + K p e p ( P EV EV ] δ) gpf (17) Ė = [ Q set + K q e q ( V Q V ] Ė) g qf. (18) Furthermre, the UDE-based rbust pwer flw cntrl laws are btained as δ = { } [L 1 1 ( P EV 1 G pf (s) set + K p e p ) { } ] L 1 sgpf (s) P (19) 1 G pf (s) Ė = { } [L 1 1 ( V 1 G qf (s) Q set + K q e q ) { } ] L 1 sgqf (s) Q. (20) 1 G qf (s) The final cntrller utput v r fr generating pulse width mdulatin (PWM) signals can be btained after cmbining the frequency frm the phase δ and the amplitude btained frm E, asshwninfig. 2, where ω is the rated frequency and E is the rated vltage fr glbal settings. Here, V can be the denminatr in bth (19) and (20), since the amplitude f the grid vltage V is nnzer. It is wrth nting that the uncertain terms p (7) and q (8) include the mdel uncertainties f the utput impedance θ and the pwer angle δ fr the GCI. The variatin f utput impedance (e.g., caused by high current, r by high temperature), the additin f virtual impedance, and the change f pwer angle can be estimated and cmpensated by the UDE-based cntrl algrithms (19) and (20). The grid vltage v and the utput current i f the GCI are measured t calculate the real pwer P and the reactive pwer Q,

4 6890 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 11, NOVEMBER 2016 as shwn in Fig. 2, s that the real pwer and reactive pwer can be cntrlled individually. The GCI with the UDE-based rbust pwer flw cntrl can regulate its wn frequency and vltage based n the tracking errrs f the real pwer e p and the reactive pwer e q. When the disturbances happen in the grid vltage (e.g., variatins f frequency r amplitude), the pwer utputs P and Q will change frm the reference values P set and Q set, which causes the prpsed cntrller t regulate the frequency and vltage f the GCI till bth pwer utput tracking errrs cnverge t zer. Therefre, the UDE-based rbust pwer flw cntrl can deal with the variatins f bth frequency and amplitude in the grid vltage withut an extra synchrnizatin unit fr the grid-cnnected peratin. The change f line impedance will affect the measured grid vltage and the pwer utputs P and Q. This effect als can be cmpensated by the UDE-based cntrller. Furthermre, a PWM mdulatin unit is applied in the cntrller utput v r,asshwninfig. 2, t cnvert the dc vltage t the ac vltage. This might intrduce the disturbances f the fluctuating dc-link vltage int the system. The fluctuating dclink vltage als can be treated as external disturbances and handled by the UDE-based rbust pwer flw cntrl (19) and (20). There is n requirement t measure the dc-link vltage, as lng as the dc-link vltage is high enugh t deliver pwer t the grid. In practice, the dc-link vltage can be measured fr ther purpses, such as the prtectin. In additin, an inner-current-lp cntrl can be added t the cntrller utput v r shwn in Fig. 2 fr ther purpses, such as the virtual impedance design [27] and the current prtectin [1]. Hwever, the added virtual impedance will nt affect the perfrmance f pwer delivery with the prpsed methd. The related experimental result will be shwn in Sectin IV. B. Stability Analysis Therem 1. Cnsidering the GCI delivering pwer t the grid with the dynamics (5) and (6), and the UDE-based rbust pwer flw cntrl laws (19) and (20), if the lumped uncertain terms p (7) and q (8) are bunded, then the clsed-lp system is stable in the sense f bundedness, i.e., the pwer-tracking errrs e p and e q are bunded fr all t 0. Prf. Cnsidering the Lyapunv functin candidate V (t) = 1 2 (e2 p + e 2 q ). Taking the derivative f V (t), alng with (9) (12), V (t) = e p ė p + e q ė q ( = e p P set EV ) δ p +e q ( Q set V ) Ė q. With the UDE-based cntrl laws (19) and (20), and alng with (15) and (16), ( V (t) = e p K p e p + ˆ ) p p +e q ( K q e q + ˆ q q ) = K p e 2 p K q e 2 q e p p e q q (21) where p p ˆ p and q q ˆ q are the estimated errrs f the uncertain terms. Accrding t (15) and (16), the estimated errrs are represented as p = p L 1 {1 G pf (s)} (22) q = q L 1 {1 G qf (s)}. (23) Since the filters G pf (s) and G qf (s) are strictly prper stable filters with the apprpriate bandwidth, if the lumped uncertain terms p (7) and q (8) are bunded, the estimated errrs p and q are bunded. By applying Yung s inequality t (21), there is V (t) K p e 2 p K q e 2 q e2 p p e2 q q ( = K p 1 ) ( e 2 p K q 1 ) e 2 q p q c 1 V (t)+c 2 (24) where c 1 = min{2k p 1, 2K q 1} > 0, and c 2 0 is the upper bund fr p q. Then, slving (24) gives 0 V (t) V (0)e c 1 t + c 2 (1 e c 1 t ) (25) c 1 [ where V (0) = 1 2 e 2 p (0) + e 2 q (0) ] is the initial value. When t, the first term n the right-hand side f (25) will decay t 0, and the secnd term is bunded with c 2 c 1. This means V (t) is bunded fr all t 0. Therefre, the clsed-lp system is stable in the sense f bundedness, which indicates that the pwer-tracking errrs e p and e q are bunded fr all t 0. The bunds f e p and e q can be adjusted thrugh the design f feedback gains K p, K q and the filters G pf (s) and G qf (s). When the estimatins f the uncertain terms p and q are accurate enugh, e.g., the filters G pf (s) and G qf (s) have the unity gain ver the bandwidths f p and q, respectively, the estimatin errrs p and q will be clse t zer. Then, the pwer-tracking errrs e p and e q will cnverge t zer. C. Steady-State Perfrmance The steady-state perfrmance f pwer delivery fr the GCI will be analyzed in this sectin. When the estimated terms ˆ p (15) and ˆ q (16) are adpted t replace p in (13) and q in (14), the errr dynamics (9) and

5 WANG et al.: ROBUST POWER FLOW CONTROL OF GRID-CONNECTED INVERTERS 6891 (10) becme ė p = K p e p ( p ˆ p ) = K p e p p (26) ė q = K q e q ( q ˆ q ) = K q e q q. (27) By substituting (22) int (26), substituting (23) int (27), and taking the Laplace transfrmatin se p (s) = K p E p (s) p (s)[1 G pf (s)] (28) se q (s) = K q E q (s) q (s)[1 G qf (s)] (29) where E p (s), p (s), E q (s), and q (s) are the Laplace transfrm f e p, p, e q, and q, respectively. Then E p (s) = p(s)[1 G pf (s)] (30) s + K p E q (s) = q (s)[1 G qf (s)]. (31) s + K q It is wrth nting that p in (7) and q in (8) are nrmally small with the small pwer angle δ and inductive utput impedance. They can be assumed bunded, i.e., lim s p(s) <, lim s q (s) <. s 0 s 0 If the filters G pf (s) and G qf (s) are designed as the strictlyprper stable filters with G pf (0) = 1 and G qf (0) = 1, by applying the final value therem t (30) and (31), there are lim e p = lim s E p (s) t s 0 = lim s 0 s p (s)[1 G pf (s)] s + K p = 0 lim e q = lim s E q (s) t s 0 = lim s 0 s q (s)[1 G qf (s)] s + K q = 0. Hence, the mean values f tracking errrs fr bth real pwer and reactive pwer cnverge t zer and accurate pwer delivery t the grid is achieved. IV. EXPERIMENTAL VALIDATION A. Experimental Setup Althugh the majr uncertainties and disturbances are cnsidered in the theretical develpment, sme practical issues cannt be fully cnsidered. In rder t verify the real-time perfrmance f the UDE-based rbust pwer flw cntrl (19) and (20), a test rig with ne experimental inverter delivering pwer t a simulated grid prvided by a cntrlled inverter shwn in Fig. 3(a) is built up. The circuit diagram is shwn in Fig. 3(b), where a lad cnsisting f a resistr R L = 40 Ω and a capacitr C L = 45 μf is cnnected t the cntrlled inverter t Fig. 3. Experimental test rig. (a) Setup. (b) Circuit diagram. TABLE I NOMINAL VALUES OF INVERTER PARAMETERS Parameters Values Parameters Values L 7 mh Grid vltage 110 V rms R 1 Ω Grid frequency 60 Hz C 1 μf DC-link vltage 300 V DC prvide the base pwer cnsumptin. The experimental inverter is cnnected t the grid simulatr via a switch CB1 and a line resistance R line = 2 Ω. The line resistance R line is bypassed by a switch CB2 t mimic the change f line impedance. The inverter parameters are given in Table I. Here, the impedance f experimental inverter includes the parasitic resistance and the filter capacitr. As mentined earlier, these effects can be dealt with by the prpsed cntrller. The PWM frequency fr pwer electrnic devices is set as 19.2 khz fr bth inverters. Bth inverters are cntrlled thrugh the Texas Instruments cntrl- CARD with F28M35H52C1 micrcntrller unit. B. Selectin f Cntrl Parameters The guidance f the parameters selectin in the UDE-based rbust pwer flw cntrl (19) and (20) is prvided as fllws. The cnstant feedback gains are chsen with K p = K q =20 1 t achieve the desired tracking errr dynamics. Then, and 1 K q are time cnstants fr the step respnse. The UDE filters G pf (s) and G qf (s) in (19) and (20) are chsen as the fllwing secnd-rder lw-pass filters: G pf (s) = ω 2 p s 2 + ω p Q p s + ω 2 p, G qf (s) = ω 2 q K p s 2 + ω q Q q s + ω 2 q In rder t get a better estimatin f uncertain terms p and q and achieve the desired tracking errr dynamics, the bandwidths f UDE filters G pf (s) and G qf (s) shuld be bigger than the bandwidths f the desired step respnse, K p and K q, respectively, and wide enugh t cver the spectrum f the lumped.

6 6892 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 11, NOVEMBER 2016 uncertain terms p and q. The filters parameters are chsen as ω p = ω q =25.1, Q p = Q q =1. C. System Perfrmance 1) With DC-Link Vltage Disturbances and Impedance Disturbances: In this scenari, the grid vltage and frequency are kept in nminal value. Three types f disturbances, i.e., change f dc-link vltage, additin f virtual impedance, and change f line impedance, are cnsidered. First, the change f dc-link vltage is studied. Initially, the switch CB1 is OFF and switch CB2 is ON. The simulated grid is set with the nminal frequency 60 Hz and the nminal vltage 110 Vrms. The grid simulatr is cntinuusly wrking during the whle experimental test thrugh the cntrlled inverter. The experimental inverter starts at t =0s. Then, the vltage v after CB1 is measured by the experimental inverter fr the initial synchrnizatin using the zer-crssing methd [1] befre t =3s. Therefre, the first 3 s are mitted frm the plts. The initial synchrnizatin is disabled after the experimental inverter is cnnected t the simulated grid with switch CB1 ON. The initial values f P set and Q set are set t zer. Then, we have the fllwing. 1) The real pwer P set = 200 W and the reactive pwer Q set = 100 Var are applied at t =5s. 2) The real pwer P set changes t 100 W at t =10s, and the reactive pwer Q set changes t 50 Var at t =15s. 3) The dc-link vltage f the experimental inverter changes frm 299 t 270 V DC at t =20s and returns t 299 V DC at t =25s. 4) The test is stpped at t =30s. The same initial settings and step 1 are als applied in ther cases. The system respnses are shwn in the left clumn f Fig. 4. After the experimental inverter is cnnected t the grid simulatr, the real pwer and reactive pwer remain at zer. After t =5s, the real pwer and reactive pwer reach the set pints with P = 200 WinFig. 4(a) and Q = 100 Var in Fig. 4(c) within 0.5 s, and the vershts are almst zer. The change f real pwer causes a psitive spike in the inverter frequency, as shwn in Fig. 4(b), since the experimental inverter shuld generate a phase-lead t increase the pwer delivered t the simulated grid. Then, the inverter frequency settles dwn quickly. The real pwer and reactive pwer als have fast respnses (within 0.5 s) with the set-pint changes at t =10 s and t =15 s, respectively. The inverter frequency has a negative spike in the set-pint change f real pwer with the similar reasn as mentined earlier. The changes f vltage E, thugh are very small befre t =20s, exist in different pwer settings, as shwn in Fig. 4(d), in rder t meet the desired pwer references. When the dc-link vltage changes at bth t =20s and t =25s, as shwn in Fig. 4(e), the real pwer and reactive pwer keep almst unchanged, but the vltage E changes by abut 10%, as shwn in Fig. 4(d), t cmpensate the changes f the dc-link vltage (abut 10%). The respnse f the vltage E is nt fast (abut 2 s), because the changes f the dc-link vltage are nt fast either. Secnd, the additin f virtual impedance is studied. The real pwer P set = 200 W and the reactive pwer Q set = 100 Var are still applied at t =5 s. A virtual impedance R v =2Ω with feedback current is added at t =10 s and then remved at t =20 s t mimic the variatin f utput impedance. The system respnses are shwn in the middle clumn f Fig. 4. Thugh the reactive pwer has sme big spikes (abut 20%) after bth t =10 s and t =20 s, it cnverges t the set pint quickly (abut 0.5 s). The inverter vltage E ges up with the effect f adding virtual impedance. The real pwer and the inverter frequency nly have small spikes with adding and remving f virtual impedance, and settle dwn quickly. Third, the change f line impedance is studied with Switch CB2 ON at t =10s and then OFF at t =20s. The system respnses are almst same with the case f virtual impedance disturbances, as shwn in the right clumn f Fig. 4. Fr the UDEbased cntrl (19) and (20), the effects f virtual impedance disturbances and line impedance disturbances are similar with almst same vltage E, when R v = R line ; hwever, the real utput vltages f the experimental inverter are different fr these tw types f disturbances. Frm the studies f this scenari, it can be seen that the UDEbased rbust pwer flw cntrl can achieve fast respnse with different desired pwer settings, where the real pwer and reactive pwer can be cntrlled individually in the presence f the cupling effects. The cntrller des nt need a cntinuus synchrnizatin unit after the experimental inverter is cnnected t the grid with an initial synchrnizatin. At the same time, this UDE-based rbust pwer flw cntrl can reject the disturbances frm the fluctuating dc-link vltage, the variatin f utput impedance, the additin/remving f virtual impedance, and the change f line impedance. 2) With Grid Disturbances: In this scenari, three types f grid disturbances are cnsidered, i.e., step change f grid frequency, lw-vltage fault-ride thrugh, and cnnectin with a weak grid. Fr the change f grid frequency and the lwvltage fault-ride thrugh, they are achieved thrugh the direct vltage and frequency cntrl f the cntrlled inverter. Fr the cnnectin with a weak grid, the drp cntrl is applied in the cntrlled inverter t prvide the frequency and vltage supprt fr the simulated grid, functining as a weak grid fr the experimental inverter. First, the change f grid frequency is cnsidered. The simulated grid is initially set at the nminal frequency 60 Hz and nminal vltage 110 Vrms. With the same initial synchrnizatin and pwer settings as previus scenari, we have the fllwing. 1) The real pwer P set = 200 W and the reactive pwer Q set = 100 Var are applied at t =5s. 2) The grid frequency increases by 0.25 Hz at t =10s, and returns t nrmal 60 Hz at t =15s. 3) At t =20s, the grid frequency drps by 0.25 Hz, and returns t nrmal 60 Hz again at t =25s. 4) The test is stpped at t =30s. The system respnses are shwn in the left clumn f Fig. 5. After t =10s, the real pwer is regulated well with sme spikes

7 WANG et al.: ROBUST POWER FLOW CONTROL OF GRID-CONNECTED INVERTERS 6893 Fig. 4. Experimental results fr dc-link vltage disturbances and impedance disturbances: change f dc-link vltage (left clumn), additin f virtual impedance (middle clumn), and change f line impedance (right clumn). (a) Real pwer. (b) Inverter frequency. (c) Reactive pwer. (d) Vltage E. (e) DC-link vltage. (abut 15%) when the grid frequency changes. The inverter frequency tracks the variatin f the grid frequency well in each step change f the grid frequency. The reactive pwer and the vltage E keep almst unchanged when the grid frequency changes. Secnd, the lw-vltage fault-ride thrugh is cnsidered. The grid vltage increases by 10% at t =10s and returns t nrmal at t =15s; then, the grid vltage drps by 20% at t =20s and returns t nrmal again at t =25s. Here, the 20% drp f grid vltage is cnsidered as a lw-vltage fault cnditin. The system respnses are shwn in the middle clumn f Fig. 5. After t =10s, thugh the reactive pwer has sme big spikes (abut ±20% f spikes in 10% changes f grid vltage, abut ±25% f spikes in 20% changes f grid vltage), it cnverges t the set pint quickly (less than 0.8 s). The inverter vltage E tracks the amplitude variatin f the grid vltage well in each step change. Thugh the real pwer and the inverter frequency have sme spikes in the step changes f grid vltage, they settle dwn quickly. These spikes in bth real pwer and inverter frequency are caused by the cupling effects. The related grid vltage is shwn in Fig. 5(e). Third, fllwing the drp methd in [28], the drp cntrl with vltage drp cefficient n = and frequency drp cefficient m = π is applied in the cntrlled inverter t

8 6894 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 11, NOVEMBER 2016 Fig. 5. Experimental results fr the grid disturbances: change f the grid frequency (left clumn), lw-vltage fault-ride thrugh (middle clumn), and cnnectin with a weak grid (right clumn). (a) Real pwer. (b) Inverter frequency. (c) Reactive pwer. (d) Vltage E. (e) Amplitude f grid vltage. behave as a weak grid. Fr the experimental inverter, the settings are the fllwing. 1) The real pwer P set = 200 W and the reactive pwer Q set = 100 Var are applied at t =5s. 2) The real pwer P set changes t 100 W at t =10s, and the reactive pwer Q set changes t 50 Var at t =15s. 3) At t =20s, the real pwer P set returns t 200 W, and the reactive pwer Q set returns t 100 Var at t =25s. 4) The test is stpped at t =30s. The system respnses are shwn in the right clumn f Fig. 5. Fr the cntrlled inverter with a simulated weak grid, bth frequency and amplitude f grid vltage change with different pwer settings f the experimental inverter. The grid frequency is less than 60 Hz, because f the psitive real pwer cnsumptin f the resistive lad, and the grid vltage is bigger that 110 V rms, because f the negative reactive pwer cnsumptin f the capacitive lad. Fr the experimental inverter, bth real pwer and reactive pwer are regulated well in different pwer settings with fast respnses. In each setting change f real pwer, the grid frequency changes with different real pwer delivery f the experimental inverter. The experimental inverter frequency fllws grid frequency very well, thugh it has sme spikes during the setting changes f real pwer, it settles dwn quickly. The setting changes f reactive pwer als affect the amplitude changes f grid vltage. The experimental inverter

9 WANG et al.: ROBUST POWER FLOW CONTROL OF GRID-CONNECTED INVERTERS 6895 vltage E fllws the changes f grid vltage; als, it is affected by the setting change f real pwer, because f the cupling effects. The UDE-based cntrl can keep reliable pwer utput regulatin in the presence f the disturbances f bth frequency and amplitude frm the grid vltage. When the grid vltage is changed, the GCI with the UDE-based rbust pwer flw cntrl can regulate the frequency and the vltage f the GCI t fllw the changes f the frequency and amplitude frm the grid vltage and deliver the required pwer accurately. This methd demnstrates its gd lw-vltage fault-ride thrugh capability and wrks well in a weak grid. D. Cmparisn With the Active Disturbance Rejectin Cntrller and the Prprtinal Integral Cntrller T evaluate the gd rbustness f the UDE-based cntrl, tw ther cntrllers, the prprtinal integral (PI) cntrller and anther ppular rbust cntrl, named active disturbance rejectin cntrller (ADRC) [29], are intrduced fr cmparisn. The sinusidal disturbances n bth frequency and amplitude f the simulated grid vltage are cnsidered t test their rbustness. The system starts with P set =0and Q set =0. Then, we have the fllwing. 1) The real pwer P set = 200 W and the reactive pwer Q set = 100 Var are applied at t =1s. 2) A sinusidal disturbance 0.2 sin (2πt) Hz is added t the nminal frequency f the grid vltage at t =4s. 3) At t =7s, a sinusidal disturbance 5.5 sin (2πt) V rms is added t the nminal amplitude f the grid vltage. 4) The test is stpped at t =12s. Fr fair cmparisn, the parameters f three cntrllers are chsen with similar transient and steady-state perfrmances fr step respnse withut disturbances. Fr ADRC, the fllwing linear extended state bserver [29] is adpted fr real pwer cntrl based n the dynamics f real pwer (5) ż p1 = z p2 + β p1 (y p z p1 )+ b p u p ż p2 = β p2 (y p z p1 ) where y p = P, z p1 is the state estimatin f real pwer P, z p2 is the state estimatin f lumped uncertain term p, b p = EV, and u p is cntrller utput fr δ. The bserver gains are chsen as β p1 =2ω p0, β p2 = ω 2 p0, with ω p0 =37.7. The final ADRC law fr real pwer is set as u p =[K p (P set y p ) z p2 ] / b p. The ADRC fr reactive pwer can have the same design with ω q0 =37.7. Fr the PI cntrller f real pwer, k pp =0.008 and k ip =0.06; the PI cntrller parameters fr reactive pwer are set as k pq =0.9 and k iq =6.4. The system respnses f three methds are shwn in bth Fig. 6 and Table II. Three methds have similar transient and steady-state perfrmances fr step respnse withut disturbance befre t =4 s, except that the PI cntrller has sme vershts n bth real pwer and reactive pwer. After t =4s, the UDE-based cntrl has the smaller tracking errr f real pwer than thse f ADRC and PI cntrller as shwn in Fig. 6(a). The rt mean square (RMS) f real pwer-tracking errr with Fig. 6. Cmparisn f three cntrl methds: UDE-based cntrl, ADRC, and PI cntrller. (a) Real pwer. (b) Inverter frequency. (c) Frequency errr. (d) Reactive pwer. (e) Vltage E. the UDE-based cntrl in steady state (10 s 12 s) is nly W, which is smaller than that f the ADRC, W, and that f the PI cntrller, W, as shwn in Table II. Als, the frequency-tracking errr between the grid frequency f g and the inverter frequency f inverter with the UDEbased cntrl is slightly smaller than that f the ADRC and

10 6896 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 11, NOVEMBER 2016 TABLE II COMPARISON OF THREE CONTROL METHODS Cnditins Cmparisn terms UDE ADRC PI Transient perfrmance f step respnse Real pwer Oversht 0% 0% 13% Settling time (s) Reactive pwer Oversht 5% 6.5% 14% Settling time (s) Steady-state perfrmance Withut disturbances The RMS f real pwer (2 4 s) tracking errr (e p ) rms (W) The RMS f reactive pwer tracking errr (e q ) rms (Var) With disturbances The RMS f real pwer (10 12 s) tracking errr (e p ) rms (W) The RMS f reactive pwer tracking errr (e q ) rms (Var) The RMS f frequency errr (f g f inverter ) rms (Hz) much smaller than that f the PI cntrller. After t =7s, the UDE-based cntrl has the smaller tracking errr f reactive pwer cmpared t the ADRC and PI cntrller, as shwn in Fig. 6(d). The RMS f reactive pwer-tracking errr with the UDE-based cntrl in the steady state (10 12 s) is nly Var, which is much smaller than that f the ADRC, Var, and that f the PI cntrller, Var. Therefre, the UDE-based cntrl has better rbustness than the ADRC and PI cntrller. V. CONCLUSION In this paper, an UDE-based rbust pwer flw cntrl has been prpsed fr GCI t deliver pwer t the grid accurately. Bth frequency dynamics and vltage dynamics were included in the mdel f pwer delivery. The UDE algrithm has been adpted fr bth real pwer and reactive pwer cntrl in the presence f mdel uncertainties, cupling effects, and external disturbances. This methd has a simple structure withut an extra synchrnizatin unit fr the grid-cnnected peratin, and with the easy implementatin and parameter tuning. The effectiveness f the prpsed apprach under different scenaris has been validated experimentally. ACKNOWLEDGMENT The authrs wuld like t thank Texas Instruments fr the dnatin f ne inverter kit TMDSHV1PHINVKIT. 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11 WANG et al.: ROBUST POWER FLOW CONTROL OF GRID-CONNECTED INVERTERS 6897 systems t supprt dynamic frequency cntrl, IEEE Trans. Energy Cnvers., vl. 29, n. 4, pp , Dec [22] L. hang, L. Harnefrs, and H. P. Nee, Pwer-synchrnizatin cntrl f grid-cnnected vltage-surce cnverters, IEEE Trans. Pwer Syst., vl. 25, n. 2, pp , May [23] Q.-C. hng and D. Rees, Cntrl f uncertain LTI systems based n an uncertainty and disturbance estimatr, ASME J. Dyn. Syst., Meas. Cntrl, vl. 126, n. 4, pp , Dec [24] J. P. Klhe, M. Shaheed, T. S. Chandar, and S. E. Tale, Rbust cntrl f rbt manipulatrs based n uncertainty and disturbance estimatin, Int. J. Rbust Nnlinear Cntrl, vl. 23, n. 1, pp , Jan [25] B. Ren, Q.-C. hng, and J. Chen, Rbust cntrl fr a class f nn-affine nnlinear systems based n the uncertainty and disturbance estimatr, IEEE Trans. Ind. Electrn., vl. 62, n. 9, pp , Sep [26] B. Ren, Y. Wang, and Q.-C. hng, UDE-based cntrl f variable-speed wind turbine systems, Int. J. Cntrl, 2015, [Online]. Available: di.rg/ / [27] X. Wang, Y.-W. Li, F. Blaabjerg, and P. C. Lh, Virtual-impedance-based cntrl fr vltage-surce and current-surce cnverters, IEEE Trans. Pwer Electrn., vl. 30, n. 12, pp , Dec [28] Q.-C. hng, Rbust drp cntrller fr accurate prprtinal lad sharing amng inverters perated in parallel, IEEE Trans. Ind. Electrn., vl. 60, n. 4, pp , Apr [29] Q. heng and. Ga, Predictive active disturbance rejectin cntrl fr prcesses with time delay, ISA Trans., vl. 53, n. 4, pp , Jul Yeqin Wang (S 15) received the B.Eng. degree frm the Institute f Technlgy, China Agricultural University, Beijing, China, in 2009, and the M.Eng. degree frm the Clean Energy Autmtive Engineering Center, Tngji University, Shanghai, China, in He is currently wrking tward the Ph.D. degree at the Natinal Wind Institute, Texas Tech University, Lubbck, TX, USA. His current research interests include renewable energies and pwer electrnics cntrl. Beibei Ren (S 05 M 10) received the B.Eng. degree in mechanical and electrnic engineering and the M.Eng. degree in autmatin frm Xidian University, Xi an, China, in 2001 and 2004, respectively, and the Ph.D. degree in electrical and cmputer engineering frm the Natinal University f Singapre, Singapre, in Frm 2010 t 2013, she was a Pstdctral Schlar with the Department f Mechanical and Aerspace Engineering, University f Califrnia, San Dieg, CA, USA. Since 2013, she has been an Assistant Prfessr with the Department f Mechanical Engineering, Texas Tech University, Lubbck, TX, USA. Her current research interests include adaptive cntrl, rbust cntrl, distributed parameter systems, extremum seeking, and their applicatins. Qing-Chang hng (M 04 SM 04) received the Ph.D. degree in cntrl and pwer engineering frm Imperial Cllege Lndn, Lndn, U.K., in 2004 (awarded the Best Dctral Thesis Prize), and the Ph.D. degree in cntrl thery and engineering frm Shanghai Jia Tng University, Shanghai, China, in He is the Max McGraw Endwed Chair Prfessr in Energy and Pwer Engineering in the Department f Electrical and Cmputer Engineering, Illinis Institute f Technlgy, Chicag, IL, USA. Prf. hng is a Distinguished Lecturer f the IEEE Pwer Electrnics Sciety and the IEEE Cntrl Systems Sciety. He is a Fellw f the Institutin f Engineering and Technlgy, U.K., the Vice-Chair f the IFAC Technical Cmmittee n Pwer and Energy Systems, and was a Senir Research Fellw f the Ryal Academy f Engineering, U.K. ( ) and the U.K. Representative t the Eurpean Cntrl Assciatin ( ). He serves as an Assciate Editr fr the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, IEEE TRANSACTIONS ON INDUS- TRIAL ELECTRONICS, the IEEE TRANSACTIONS ON POWER ELECTRONICS, IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, IEEE ACCESS, and the IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS.