Robust Power Flow Control of Grid-tied Inverters Based on the Uncertainty and Disturbance Estimator

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1 2016 American Cntrl Cnference (ACC) Bstn Marritt Cpley Place July 6-8, Bstn, MA, USA Rbust Pwer Flw Cntrl f Grid-tied Inverters Based n the Uncertainty and Disturbance Estimatr Yeqin Wang, Beibei Ren, and Qing-Chang Zhng Abstract In this paper, an uncertainty and disturbance estimatr (UDE)-based cntrl is prpsed t achieve accurate pwer flw cntrl fr grid-tied inverters (GTI). The pwer delivering dynamics is intrduced at first after cnsidering bth frequency dynamics and vltage dynamics. Then the UDE algrithm is adpted t regulate bth utput vltage amplitude and frequency fr accurate real pwer and reactive pwer cntrl. With the adptin f the UDE methd, the mdel uncertainty (e.g., pwer angle) and external disturbance (e.g., variatins f grid frequency and grid vltage) can be cmpensated autmatically. Mrever, this UDE-based dynamic pwer flw cntrl can achieve self-synchrnizatin withut an extra synchrnizatin unit (e.g., a phase-lcked-lp) when the inverter is cnnected t the grid. In additin, the prpsed cntrller is easy fr implementatin and parameter tuning thrugh the designs f desired tracking errr dynamics and UDE filters, while having the flexibility and perfrmance f advanced cntrl methdlgies. The asympttic stability f the clsed-lp system is analyzed. Experimental results are prvided fr validatin. I. INTRODUCTION Nwadays, renewable energies, such as wind energy, slar energy, wave tidal energy and fuel cells, play very imprtant rles in energy area. Accrding t REN21 s Renewables 2015 Glbal Status Reprt, the renewable electricity shares 22.8% f glbal electricity prductin, with the wind pwer abut 3.1%, and the slar PV (phtvltaic) abut 0.9%. Accrding t the reprts f U.S. Department f Energy, wind energy is targeted t prvide 20% f U.S. electricity needs by 2030, and slar pwer is targeted t prvide 14% f U.S. electricity needs by 2030 and 27% by Mst f the renewable energies are cmprised f variable-frequency ac surces (such as wind turbines), high frequency AC surces (such as small gas turbines), r DC surces (such as slar PVs). The DC/AC cnverters, als called inverters, are needed t interface these renewable energies with the publicutility grid [1. Thugh renewable energies are quite ppular, there are still sme challenges faced by the inverter cntrl fr grid integratin. Fr example, the renewable energies, such as wind pwer, slar PV are nt stable energies in different wind cnditins r sunlight cnditins; the current Yeqin Wang is with the Natinal Wind Institute, Texas Tech University, Lubbck, TX , USA. ( yeqin.wang@ttu.edu). Beibei Ren is with the Department f Mechanical Engineering, Texas Tech University, Lubbck, TX , USA. ( beibei.ren@ttu.edu). Qing-Chang Zhng is with the Department f Electrical and Cmputer Engineering, Illinis Institute f Technlgy, Chicag, IL 60616, USA. ( zhngqc@ieee.rg). injected int the grid shuld be clean sinusidal t keep system stable and grid friendly [1; the grid variatins (frequency variatin r vltage variatin), even thugh they are small, als affect the system stability. The vectr cntrl, riginally prpsed fr the cntrl f electrical machine [2, is a ppular cntrl algrithm fr three-phase grid-tied inverters (GTI) [3, [4, [5 t cnvert DC pwer t AC grid. In the vectr cntrl, the current regulatin in d q reference frame is adpted t generate vltage reference. An extra synchrnizatin unit, e.g. the phase-lcked-lp (PLL), is required fr transfrmatins f current and vltage between the a b c reference frame and the d q reference frame. Hwever, the vectr cntrl has its fundamental limitatins in pwer flw cntrl fr three-phase GTI. Firstly, the frequency dynamics are nt cnsidered in the vectr cntrl, which makes it difficult t analyze the angle and frequency stability between GTI and the grid [6. Secndly, the PLL dynamics can affect the stability f GTI, particularly in weak grids [6, [7. 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 satisfactry perfrmance [8. Thirdly, the instability f the DC-link dynamics can als affect the stability f GTI [6. Furthermre, the transfrmatin prcesses f vectr cntrl can be cmplicated due t impedance imbalance and vltage imbalance [9. An alternative cntrl strategy f the vectr cntrl, directpwer cntrl, prpsed by [10, directly cntrls the pwer switch device based n the instantaneus real pwer and reactive pwer errrs. This instantaneus cntrl methd is easily influenced by system nises, as well as the grid variatins. The drp cntrl methd, riginally prpsed fr pwer sharing amng parallel perated inverters [11, can be adpted fr GTI t achieve pwer flw cntrl [12, [13, [14, [15. Thugh sme merits have been achieved with drp cntrl methds, such as harmnic current regulatin [12, flexible peratin in either grid-cnnected mde r islanded mde [13, [15, enhancing pwer lp dynamics [14, the extra synchrnizatin units, the PLLs, are still needed in mst cases [12, [14, [15, and frequency dynamics are nt cnsidered. As the drp cntrl is based n apprximate linearizatin with the assumptins that the utput impedance f the inverter is purely inductive and the pwer angle is small [1, mst drp cntrl strategies are /$ AACC 7450

2 static [16. Inverters can als be cntrlled t behave as virtual synchrnus machines (VSM) [17, [18, [19. The self-tuning algrithms fr ptimal parameters f VSM are studied in [20, hwever, the PLL is still needed in this methd. The synchrnverter technlgy [19 has been imprved withut a dedicated synchrnizatin unit t imprve system perfrmance and reduce cmplexity in [8. Similar methds have been expanded t intrduce the inertia t the grid [6 and t imprve micrgrids stability [7. In this paper, a pwer flw cntrl strategy based n the uncertainty and disturbance estimatr (UDE) methd is develped fr GTI t achieve real and reactive pwer utput regulatins. The UDE cntrl algrithm, which was first prpsed in [21, is based n the assumptin that the uncertainty and disturbance can be estimated by using a filter with the apprpriate bandwidth. In recent years, the UDE-based cntrl demnstrated excellent perfrmance in handling uncertainties and disturbances in different systems, and was successfully applied t rbust trajectry tracking [22, a class f nn-affine nnlinear systems [23, and the variable-speed wind turbine cntrl [24. In this paper, the pwer delivering dynamics are studied base n the pwer delivering equatins. Then the UDE-based dynamic pwer flw cntrl is develped t regulate bth utput frequency and utput vltage fr accurate pwer flw cntrl with self-synchrnizatin when the inverter is cnnected t the grid. The mdel uncertainty (e.g., pwer angle) and external disturbance (variatins f grid frequency and grid vltage) can be cmpensated autmatically with this UDE-based dynamic pwer flw cntrl. Als, this cntrl apprach is easy fr implementatin and parameter tuning with the simple designs f desired tracking errr dynamics and UDE filters. The effectiveness f the prpsed cntrl apprach is demnstrated thrugh experimental studies n an experimental test rig with a single-phase Texas Instruments L-inverter. The main cntributins f this paper are highlighted as fllws: 1) The pwer delivering dynamics are intrduced based n the pwer delivering equatins with bth frequency dynamics and vltage dynamics being cnsidered. 2) The UDE methd is applied fr GTI t achieve accurate real and reactive pwer utput regulatins. The mdel uncertainty (e.g., pwer angle) and external disturbance (variatins f grid frequency and grid vltage) can be cmpensated autmatically with UDE algrithm. The self-synchrnizatin is achieved withut an extra synchrnizatin unit (e.g., the PLL). 3) The asympttic stability criteria fr GTI with the UDEbased dynamic pwer flw cntrl is prvided. 4) The UDE-based dynamic pwer flw cntrl is easy t be implemented and tuned while bringing gd rbust perfrmance. The rest f this paper is rganized as fllws. Sectin II prvides pwer delivering dynamics. In Sectin III, the UDEbased dynamic pwer flw is prpsed fr L-inverter. Effectiveness f the prpsed apprach is demnstrated thrugh experimental studies in Sectin IV, befre the cncluding remarks are made in Sectin V. II. POWER DELIVERING DYNAMICS ( ) ( ) Figure 1. Vltage surce delivering pwer t the grid. A vltage surce E δ delivering pwer t the grid V 0 thrugh an impedance θ can be mdeled as shwn in Fig. 1. The real pwer P and the reactive pwer Q received by the grid V 0 are shwn as [1 P = Q = ( EV cs δ V 2 ) Z ( EV cs δ V 2 sin δ sin θ,(1) ) sin θ EV sin δ cs θ,(2) cs θ EV where δ is the phase difference between the vltage surce and the grid, ften called the pwer angle. It is nted that the pwer delivering equatins (1) and (2) are mdeled with a general vltage surce delivering pwer t the grid in Fig. 1. This general vltage surce culd be a GTI r a traditinal synchrnus generatr. The traditinal synchrnus generatrs hld the same pwer delivering characteristics with GTI, thugh the pwer delivering equatins have little difference [25. Mrever, sine functins in pwer delivering characteristics play crucial rles fr synchrnizatin and transient stability in pwer netwrk [26, [27. If the cntrller design f GTI fllws this pwer delivering equatins, the pwer delivering characteristics f GTI can btain the same as traditinal synchrnus generatrs, and the same self-synchrnizatin and transient stability f traditinal synchrnus generatrs will als be achieved fr the peratin f the GTI. Similar t the winding inductance in synchrnus generatrs, because f the utput inductr r the inductive line impedance, the utput impedance f inverters is mstly inductive and such inverters are called L inverters [1. In this case, the utput impedance f the inverter is cnsidered as purely inductive fr simplificatin with θ = 90. The real pwer and the reactive pwer can be derived frm (1) and (2) as P = EV sin δ, (3) Q = EV cs δ V 2. (4) 7451

3 Taking the derivative frm f (3) and (4) P = EV cs δ Z δ V sin δė, (5) Q = EV sin δ δ V Ėcs δ. (6) Since the pwer angle δ depends n the pwer utput f the inverter and the variatin f grid cnditins, it is quite uncertain. The pwer delivering dynamics (5) and (6) can be rewritten as where P = EV δ p, (7) Q = V Ė q, (8) p = V sin δė EV δ(cs δ 1), (9) q = EV sin δ δ V Ė(cs δ 1) (10) represent the uncertainty terms, including the nnlinearity and uncertainty f the pwer angle δ. In practice, the lumped uncertain terms p and q can be small and bunded because f the small pwer angle δ. III. UDE-BASED DYNAMIC POWER FLOW CONTROL In this paper, a pwer flw cntrl is prpsed based n the pwer delivering dynamics (7) and (8) with the UDE methd [21 fr L-inverter. The structure is shwn in Fig. 2. calculatin UDE-based cntrl (19) UDE-based cntrl (20) Figure 2. The scheme f the UDE-based dynamic pwer flw cntrl A. Cntrller design The inverter shuld accurately cntrl real pwer and reactive pwer injected int the grid independently. The cntrl bjective is t achieve accurate real pwer and reactive pwer regulatins, such that real pwer P and reactive pwer Q asympttically track their reference P set and Q set respectively. In particular, the desired tracking errrs e p = P set P and e q = Q set Q satisfy the dynamic equatins ė p = K p e p, (11) ė q = K q e q. (12) Cmbing (7) and (11), (8) and (12) respectively, then P set EV δ p = K p e p, Q set V Ė q = K q e q. Then δ and Ė need t satisfy δ = Z ( ) P set K p e p p, (13) EV Ė = Z ( ) Q set K q e q q. (14) V Accrding t the system dynamics in (7) and (8), the uncertainty terms p and q satisfy p = P EV δ, q = Q V Ė. Fllwing the prcedures prvided in [21, 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, g pf (t) and g qf (t) are the impulse respnse f strictly prper stable filters G pf (s) and G qf (s) with unity gain and zer phase shift ver the spectrum f the lumped uncertain term p and q respectively, and zer gain elsewhere. Replacing p and q with ˆ p and ˆ q in (13) and (14) results in δ = Z [ EV Ė = Z [ V P set K p e p ( P EV δ) gpf Q set K q e q ( Q V Ė) g qf,(17). (18) Then the UDE-based cntrl laws are frmulated as δ = Z { } [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 = Z { } [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 PWM (pulse width mdulatin) signals generatin can be btained after cmbining the amplitude btained frm E and the phase δ, as shwn in Figure 2. The UDE-based dynamic cntrl laws (19) and (20) are easy t be implemented and tuned with the simple designs f desired tracking errr dynamics and UDE filters. Als, there is n need with an extra synchrnizatin unit fr the UDE-based dynamic pwer flw cntrl when the inverter is 7452

4 cnnected t the grid, and the glbal settings ω and E are just used fr initial synchrnizatin. It is wrth nting that the uncertainty terms p (9) and q (10) nly include the mdel uncertainty f the pwer angle δ in the L inverter (with θ = 90 ). If the utput impedance f the inverter is nt purely inductive, the deviatin can als be lumped int the uncertainty terms p and q, which can be estimated and cmpensated by UDE algrithm. Therefre, the UDE-based dynamic pwer flw cntrl is nt limited t GTI with purely inductive utput impedance. B. Stability analysis Therem 1. Cnsider the clsed-lp system cnsisting f GTI delivering pwer t the grid (7) and (8), and the UDEbased dynamic pwer flw cntrl laws (19) and (20). If the filters G pf (s) and G qf (s) are chsen as strictly prper stable filters with unity gain and zer phase shift ver the spectrum f the lumped uncertain terms p (9) and q (10) and zer gain elsewhere, then the clsed-lp system is asympttically stable, which guarantees stable pwer delivering f GTI t the grid. Prf: When the estimated terms ˆ p (15) and ˆ q (16) are adpted t replace p in (13) and q in (14), then the errr dynamics (11) and (12) becme where ė p = K p e p p, (21) ė q = K q e q q, (22) p p ˆ p q q ˆ q are the estimated errr f uncertainty terms. Accrding t (15) and (16), the estimated errrs are p = p L 1 {1 G pf (s)}, (23) q = q L 1 {1 G qf (s)}. (24) By substituting (23) int (21) and substituting (24) int (22), and taking the Laplace transfrmatin, se p (s) = K p E p (s) p (s) [1 G pf (s), (25) se q (s) = K q E q (s) q (s) [1 G qf (s), (26) 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) s K p, (27) E q (s) = q(s) [1 G qf (s) s K q. (28) Accrding t (9) and (10), here p and q are assumed t be bunded with p M p and q M q, as δ is very small in practical case, E and V are engineering signals, and is a nn-zer value. Therefre, p (s) M p s, q(s) M q s. Since the filters G pf (s) and G qf (s) are designed as a strictlyprper stable filter with unity gain and zer phase shift ver the spectrum f the lumped uncertain terms p and q and zer gain elsewhere, respectively, by applying the final value therem t (27) and (28), there are lim e p = lim s E p (s) t s 0 lim M p [1 G pf (s) s 0 s K p = 0, lim e q = lim s E q (s) t s 0 lim M q [1 G qf (s) s 0 s K q = 0. Therefre, the tracking errrs f the system cnverge t zer and the clse-lp system is asympttically stable. Stable pwer delivering f GTI t the grid can be achieved. This cmpletes the prf. A. Experimental setup IV. EXPERIMENTAL RESULTS T verify the UDE-based dynamic pwer flw cntrl (19) and (20), a test rig with ne Texas Instruments L-inverter delivering pwer t a AC bus as shwn in Fig. 3(a) was built, where the AC bus is generated by a pwer amplifier with a signal generatr. The circuit diagram is shwn in Fig. 3(b), where the lads cnsist f a resistr R L = 6.67 Ω and a capacitr C L = 90 µf. The lads are cnnected t AC bus directly and the inverter is cnnected t the AC bus via a switch CB. In rder t synchrnize the inverter t the AC bus at the initial state, the lad vltage V after switch CB is measured by the inverter. The parameters f the inverter are given in Table I. The inverter is cntrlled thrugh the cntrlcard f TI F28M35H52C1 micr-cntrller chip. The measurement data is sent t a desktp thrugh serial cmmunicatin. Parameters Values L 7 mh C 1 µf V DC 40 V Table I INVERTER PARAMETERS. Parameters nminal grid frequency nminal grid vltage PWM frequency B. Filter design and parameter selectin Values 60 Hz 14 Vrms 19.2 khz The filters in the UDE algrithm shuld cver the spectrum f disturbances with the unity gain and zer phase shift. Here, G pf (s) and G qf (s) are chsen as the fllwing first-rder lw-pass filters G pf (s) = 1 1 τ p s, G 1 qf (s) = 1 τ q s, (29) 7453

5 20 _ PWM Inverter (a) Lads (b) AC bus Pwer amplifier Figure 3. Experimental test rig. (a) Setup (b) Circuit diagram. Signal generatr with the bandwidth wide enugh t cver the spectrum f p and q. Furthermre, and 1 1 G pf (s) = 1 1 τ p s, 1 1 G qf (s) = 1 1 τ q s, sg pf (s) 1 G pf (s) = 1 τ p, sg qf (s) 1 G qf (s) = 1 τ q. Therefre, with the lw-pass filters (29), the UDE-based cntrl laws (19) and (20) are simplified as δ = Z [ P set (K p 1 )e p K t p e p dt,(30) EV τ p τ p 0 Ė = Z [ Q set (K q 1 )e q K t q e q dt. (31) V τ q τ q The cntrl parameters fr desired tracking errr dynamics and UDE filters are shwn in Table II. Table II CONTROL PARAMETERS FOR DESIRED TRACKING ERROR DYNAMICS AND UDE FILTERS Parameters Values K p 5 τ p 0.1 s P set 15 W C. System perfrmance 0 Parameters Values K q 10 τ q 0.05 s Q set -5 Var Initially, the grid is set at the nminal frequency 60 Hz and the nminal grid vltage 14 Vrms thrugh the signal generatr. The real pwer is set at P set = 15 W, and the reactive pwer as Q set = 5 Var. Then, the grid frequency is stepped up t 60.1 Hz at t = 5 s. The grid vltage is stepped dwn t 13 Vrms at t = 10 s. The tuning f bth grid frequency and grid vltage is set by the signal generatr. Real pwer (W) Frequency (Hz) Reactive pwer (Var) Vltage E (V) (a) Real pwer (b) Inverter frequency (c) Reactive pwer 13.2 (d) Vltage E Figure 4. Experimental results The system respnse f the prpsed UDE-based dynamic pwer flw cntrl (19) and (20) with the parameters setting in Table II are shwn in Fig. 4, including real pwer respnse in Fig. 4(a), reactive pwer respnse in Fig. 4(c). The inverter frequency and vltage E are shwn in Fig. 4(b) and in Fig. 4(d) respectively. Initially, the lad vltage V is measured by the inverter fr synchrnizatin using the zer-crssing methd [1, then the inverter starts t deliver the real pwer and reactive pwer. This initial synchrnizatin is disabled when the inverter is cnnected t the grid. The real pwer and reactive pwer reach the stable status with P = 15 W in Fig. 4(a) and Q = 5 Var in Fig. 4(c) quickly. The inverter frequency is within 60 Hz in Fig. 4(b) and vltage E is abut E = 14.4 Vrms. The UDE-based dynamic pwer flw cntrl can achieve accurate real pwer and reactive pwer utput 7454

6 regulatin effectively. At t = 5 s, the grid frequency is changed t 60.1 Hz. There are sme scillatins in real pwer in Fig. 4(a) and reactive pwer in Fig. 4(c). Bth f them cnverge t the stable status within 2 s. The inverter frequency is changed t 60.1 Hz quickly after sme scillatins in Fig. 4(b) withut the grid infrmatin frm an extra synchrnizatin unit. Crrespnding respnse f the vltage E is shwn in Fig. 4(d). At t = 10 s, the grid vltage is changed t 13 Vrms. The reactive pwer in Fig. 4(c) has a small spike but cnverges t the stable status within 2 s, and the vltage E in Fig. 4(d) autmatically changes t 13.6 Vrms within 2 s. The real pwer in Fig. 4(d) and inverter frequency almst remain the same. The experimental results shw that the external grid disturbance (variatins f grid frequency and grid vltage) can be cmpensated autmatically by the UDE-based dynamic pwer flw cntrl. Als the self-synchrnizatin can be achieved withut an extra synchrnizatin unit when the inverter is cnnected t the grid in the presence f the disturbances frm the grid. V. CONCLUSION In this paper, an uncertainty and disturbance estimatr (UDE)-based dynamic pwer flw cntrl has been prpsed fr grid-tied inverters (GTI) t achieve utput pwer regulatin. The pwer delivering dynamics has been intrduced based n the pwer delivering equatins with bth frequency dynamics and vltage dynamics being cnsidered. Then the UDE algrithm has been adpted t regulate bth utput vltage and utput frequency fr accurate real pwer and reactive pwer cntrl. The asympttic stability has been analyzed fr the UDE-based dynamic pwer flw cntrl. The effectiveness f the prpsed apprach has been validated experimentally. The self-synchrnizatin can be achieved withut an extra synchrnizatin unit when the inverter is cnnected t the grid in the presence f the disturbances frm the grid. Althugh the UDE-based dynamic pwer flw cntrl was develped based n the single phase GTI in this paper, the results can be further extended t the three-phase GTI. ACKNOWLEDGMENT The authrs wuld like t thank Texas Instruments fr the dnatin f ne inverter kit TMDSHV1PHINVKIT. REFERENCES [1 Q.-C. Zhng and T. Hrnik. Cntrl f Pwer Inverters in Renewable Energy and Smart Grid Integratin. Wiley-IEEE Press, [2 S. K. Sul. Cntrl f Electric Machine Drive Systems. Wiley-IEEE Press, [3 M. Prdanvic and T. C. Green. Cntrl and filter design f threephase inverters fr high pwer quality grid cnnectin. IEEE Trans. Pwer Electrn., 18(1): , Jan [4 Z. Liu, J. Liu, and Y. Zha. A unified cntrl strategy fr threephase inverter in distributed generatin. IEEE Trans. Pwer Electrn., 29(3): , Mar [5 Q.-N. Trinh and H.-H. Lee. An enhanced grid current cmpensatr fr grid-cnnected distributed generatin under nnlinear lads and grid vltage distrtins. IEEE Trans. Ind. Electrn., 61(12): , Dec [6 M. Ashabani and Y. A.-R. I. 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