Evaluation of a State Observer for Frequency Estimation in a Grid Tied Photovoltaic Inverter

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1 Evaluation of a State Observer for Frequeny Estimation in a Gri Tie Photovoltai Inverter Ana Cabrera-Tobar, Oriol Gomis-Bellmunt CITCEA-UPC Universitat Politènia e Catalunya Barelona-Spain ana.abrera@itea.up.eu Joaquín Carraso, Mike Barnes Shool of Eletrial an eletroni engineering University of Manhester Manhester-Unite Kingom Abstrat A phase loke loop (PLL) is ommonly use to measure the eletrial frequeny eviations in PV gri tie inverters. However, this ontrol is affete by suen hanges of power that usually happens with this type of power generation as the solar irraiane is intermittent. Thus, the aim of this paper is to estimate the frequeny at the point of ommon oupling by using a state observer. The evaluation of the observer is evelope in DIgSILENT PowerFatory uner variable solar irraiane. The results show that the observer gives a more aurate frequeny value than a PLL when quik hanges of solar irraiane our. Inex Terms PV inverter, phase loke loop, state observer, frequeny estimator I. INTRODUCTION The performane of large sale photovoltai power plants (LS-PVPPs) is a main rawbak for transmission system operators ue to the variation of solar irraiane uring the ay. The bulk system is ommonly base on onventional power plants with synhronous generators as the base units, where the ative power an be easily manage epening on the eman. However, in LS-PVPPs the PV generator s ative an reative power epen on the variation of solar irraiane as well as other eletrial harateristis. Thus, speifi gri oes have been evelope to smooth the integration of these power plants into the eletrial system []. The requirements an be ivie into two main areas: (i) voltage an frequeny support, an (ii) ative an reative power ontrol. An aequate measurement of the voltage an frequeny at the point of ommon oupling (PCC) is an important issue to solve in orer to esign an aequate ontrol. In a PV generator, where the inverter is the basi eletroni equipment, a phase loke loop (PLL) is use to synhronize the generator with the gri by the measurement of the phase angle together with the alulation of the frequeny. However, the PLL oul present some isturbanes when phase angles hange ue to the suen injetion or reution of ative power [2]. For instane, when the solar irraiane hanges in a small perio of time, the frequeny alulation is rastially affete by the This work has been fune by the National Department of Higher Euation, Siene, Tehnology an Innovation of Euaor (SENESCYT) an by the Spanish Ministry of Eonomy an Competitiveness uner the projet ENE C4--R. hange in voltage rop aross the system impeane an oes not represent the real value at the point of ommon oupling [3]. The auray of the frequeny measurement an affet the ontrol an the response time. Some methos have been evelope to etet frequeny or phase eviation for small PV appliations (lower than MW) interonnete with a mirogri. For passive methos, the system usually measures voltage or phase istortion at the point of ommon oupling an ompares it with previous values. In ative methos, the inverter as an error value to frequeny, voltage or phase at its output, interfering with the power quality of the system. The time response of these methos are in between 43 to 5 ms, for passive, an.3 to s for ative [4]. One metho that was reently stuie for gri tie inverters is the evelopment of a state observer to estimate the phase shift an the frequeny by an aequate moel of the interonnetion between the PV inverter an the gri. The stuy evelope by [5] etails the observer esign for a gri tie inverter with an speifi LCL filter for voltage, urrent an phase shift estimation. Another stuy was evelope in [6] where the observer estimates the phase shift to be use in a PLL an thus the general ontrol is improve. In any of this researh, the observer has not been applie for the integration of PV generators with the eletrial system. Thus, this artile presents the evaluation of a state observer for frequeny estimation onsiering the variation of solar irraiane by omparing it with a PLL. This observer will be teste with a PV generator moel uner variable solar irraiane. The paper is ivie in four main setions: the PV generator moel, its ontrol, the observer esign an the simulations orresponing to variable ambient onitions. Lastly, some onlusions are presente an isusse. II. PV GENERATOR MODEL AND CONTROL In LS-PVPPs, PV generators an present ifferent topologies suh as entral, string an multistring [7]. For the present stuy, the onfiguration stuie is the entral type, whih presents the most ost-effetive solution [8]. In this onfiguration a PV array is onnete to a single stage PV inverter. The output of the inverter is interonnete to a transformer through an LCL filter (see Fig.) /8/$3. 28 IEEE

2 PV array DC apaitor PV inverter Filter Transformer BUS DC BUS AC BUS AC BUS AC Lf Lf Gri i Lf PCC Lg ig LV Cf LV MV v Cf vp vg Fig.. PV generator in entral onfiguration iph i ip Rs ipv Fig. 3. Gri onnete LCL filter onsiering stationary frame A. PV array moel Rp Fig. 2. Simplifie moel of the PV solar ell The eletrial equivalent iruit of a PV solar ell orrespons to Fig. 2, where the total urrent (i pv ) epens on the photogenerate urrent (iph) (2), the ioe saturation urrent (i o ) an the parallel urrent (i p ) [3]. i pv = i ph i o ( e ( vpv.nparrs.ipv Ns.A.k B.T/q ) ) i p.n ser () R s is the series resistane, N s is the number of solar ells onnete in series, A is the ieality fator, k B is the Boltzman onstant, T is the solar ell s temperature, q is the eletrial harge of an eletron, N ser an N par are the PV panels onnete in series or in parallel respetively. The photogenerate urrent hanges proportionally to the solar irraiane (G) an it is given by: G i ph i phst (2) G st i phst an G st are the photogenerate urrent an the solar irraiane at stanar test onitions (solar irraiane of W/m 2 an an ambient temperature (T a ) of 25 o C). The point of operation for eah solar irraiane epens on the ontrol evelope in the PV inverter. The moel of this is explaine in the following setion. B. PV inverter moel an ontrol For the PV generator onfiguration, an inverter of one stage of onversion from to a is hosen. The output voltage of the PV inverter is obtaine by Sinusoial Pulse With Moulation (SPWM), where the a voltage epens on the moulation inex (m a ) an the voltage value at its input terminals. Aitionally, a filter LCL is onnete at the output of the PV inverter helping to reue the harmonis that the PV generator injets into the eletrial system. Aitionally, the moel of the inverter interonnete with the gri is illustrate in Figure 3 an an be efine by a state vetor, where the state variables for the system is given by: x = [i v p i g ] T (3) - vpv v p is the voltage at the point of ommon oupling, i an i g are the onverter an the gri urrents, respetively. The state-spae moel of the gri tie inverter an be expresse as follows: t i v p i g jω g L = C f jω g C f x jω g L g L v L g v g, (4) y = [ ] x. (5) In stationary referene frame, the gri voltage is efine as: v g = e jθg v g (6) θ g is the phase angle an equal to θ g = ω g t. For the ynami ontrol of the PV generator, the three phase voltages an urrents are transforme to the rotating iret-quarature frame. The main tasks of the inverter are relate to the ontrol of power an synhronization. In this ase, the ontrol of the PV generator onsists of a maximum power point traker (MPPT), outer voltage ontrol, inner urrent ontrol, voltage moulation, phase loke loop an a frequeny estimator (Fig. 4). The MPPT ontrol traks the voltage to have the maximum ative power at eah solar irraiane an temperature. For the PV inverter ontrol, the algorithm use is the one known as a Perturb an Observe. The outer ontroller efines the i referene (i ref ) relate to the ative power an the voltage ontrol, an the i q referene (i ref q ) relate to the reative power. The voltage ontrol has the task to follow the voltage referene an to maintain this as onstant as possible insie the voltage limitation. In this regulation, a PI ontroller is use: its ynamis epens on the MPPT time response plus the ambient onitions. In this ase, the reative power ontrol only ats when a the a voltage is varie more than ±.p.u. The inner urrent ontrol is the one that gives the signal referenes (v an v q) to the voltage moulation blok whih has to transform this voltage referenes to three phase voltages with the orresponing moulation tehnique (Fig.5). The PLL struture is a feebak ontrol system that has the task to ajust the phase angle between the gri referene (q) frame an the onverter (q) referene frame (Figure 7). To o this, the ontroller has to align the gri voltage phasor

3 i i Lf Lg ig vg v C v Cf vp iab A system vab G Ta v_min MPPT ontrol v_max Vab ab q PLL OBSERVER pll,fpll obs, fpll ab q G f iab vq iq f Inverter moel ma_max vq* vref Inner Current Controller iq* iq_max* V_meas vmpp ma_min Outer ontroller vab Pa, Qa iq_min* Referenes P, Q Fig. 4. PV generator s general ontrol iref(g,ta,v) i iq iqref(g,ta,v) ε u k(s) 2 f Lf 2 f Lf εq uq kq(s) m mq v X X v vq v g vq g vq*= vq εpll ωpll θpll /s kpki/s Park transformation vab Fig. 6. Phase lok loop ontroller iagram Fig. 5. Inner urrent ontrol of the PV inverter with one of the two q axis. In this ase, it is aligne with the -axis whih means that the vq g =. With this new angle, then the VSC beomes synhronise with the gri, so vq = in steay state [9]. Then, the eletri quantities - voltages an urrent- an use the PLL angle to be transforme from the three phase frame to q referene frame an vie-versa. The feebak ontrol struture of the PLL uses ommonly a PI ontroller whih ompensates the error between the referene voltage vq = an the new vq onsiering the new phase angle (Fig. 6). The output voltage of the inverter is iretly affete by the hanges of power as it is esribe in eq. 7 an eq.8 [9]. In aition, P an Q epens on the solar irraiane, temperature q β γ δ V θ Vg Fig. 7. q iagram for AC sie inverter an voltage aoring to the apability urves esribe in []. Thus, the output voltage presents small variations when ig ρ α

4 hanges of solar irraiane our that affets to the information given by the PLL (frequeny an phase angle) [3]. v = 2 v q = 3 L f v g 2 3 L f v g. P(G, T 2 a, V ) t. Q(G, T a, V ) t 3 L fω g v p 2 3 L fω g v p.q(g, T a, v )v g (7).P(G, T a, v ) (8) Taking into onsieration this behaviour, a frequeny estimator is ae to the PV inverter ontrol. This estimator has the main task of supporting the frequeny an the phase measure by the PLL when quik hanges of solar irraiane our. The objetive of this estimator is to alulate the frequeny an the phase shift at eah instant onsiering the real interation between the PV inverter an the gri through the LCL filter. The esign of this frequeny estimator is explaine in the following setion. C. Frequeny estimator In a PV gri tie inverter, the vq p is impose to be zero, however when isturbanes our this value oul hange. Then, as it is shown in Fig. 5, the i q of the onverter is iretly affete by the a voltage, the gri frequeny, the LCL filter, an the hanges of solar irraiane an voltage. Thus, the frequeny estimator will be onstrute on the premise of this relationship (i q vs f). The esign of the frequeny estimator is base on the work evelope in [5] whih has two parts: (i) a Luenberger observer an (ii) a aaptative ontroller. The Luenberger observer has the struture represente in Fig. 8, where the variable to observe is the urrent in q frame. Every time the solar irraiane hanges, the real (i ) an the imaginary omponents (i q) present some perturbations. The imaginary omponent i q is the one that presents a relationship with the angular spee perturbation eah time the solar irraiane varies in large proportion. Due to this relationship a PI aaptative ontroller is linke with the Luenberger observer. The objetive of this is to reue the error of the estimate i q when perturbation ours. Then, the phase angle an the frequeny are estimate by: ˆθ g = ˆω g t, (9) ˆf = 2π ˆωg. () Following [5], the mathematial analysis of the observer, together with the linearization to get the transfer funtion i(s)/ω(s) is explaine in Appenix. Aitionally, the equations to get k p an k i are also etaile. The frequeny estimator onstrute is then teste in DigSilent PowerFatory together with the PV inverter ontrol explaine previously, the following setion explains the stuy ase an the results. TABLE I PV PANEL AND ARRAY CHARACTERISTICS PV panel harateristis PV array haraterists V o 58.8 [V] P array.6 [MW] I s 5. [A] N ser 5 I mpp 4.68 [A] N par 75 V mpp 47 [V] T min, T max -7 [ C] k v.45 [/ C] G max [W/m 2 ] L f.3 [p.u] C f.4[p.u] L g.5 [p.u] ω go 2π 5[ra/s] TABLE II GAIN CONTROLLERS Controller kp Unit ki Unit Observer.8 [H.ra/Vs 2 ]. [H.ra/Vs 2 ] PLL (A) [ra/vs] 3 [ra/vs 2 ] PLL (B) 2 [ra/vs] 3 [ra/vs 2 ] III. SIMULATIONS AND RESULTS A PV generator of.6 MVA was simulate in DIgSILENT Power Fatory as part of a LS-PVPP of 2 MVA. The PV generator s harateristis is summarize in Table I. The omplete moel an ontrol of the PV generator explaine in the previous setions is implemente in DIgSILENT PowerFatory together with the esigne frequeny observer. For the analysis, the LS-PVPP is onnete with a gri that has a short iruit ratio equal to five. The PV generator together with the frequeny observer is teste uner two stuy ases senarios: Stuy ase A: Fast hange of solar irraiane from to W/m 2 an from to W/m 2 (Fig.9 (a)). Stuy ase B: One ay of solar irraiane (Fig. (a)) In eah of these ases, the frequeny alulate by the PLL an the observer are obtaine. Aitionally, two PLLs with ifferent banwiths are use. The first one is use for gri synhronization (PLL (A)) an the seon one only to alulate the frequeny gri (PLL(B)). For eah PLL an the observer, the gains use are summarize in Table II. The results are illustrate in Fig. 9(b) an Fig. (b) for stuy ase A an B respetively. For quik hanges of solar irraiane (Stuy ase A) from high irraiane to low, the frequeny measure by the PLL (A), presents some perturbations ue to its quik ynamis. The time that the PLL uses to stabilize is.6 s. After this time the frequeny ame bak to be the same as the gri is imposing. With PLL (B) the perturbations reue but the time of stabilization is almost the same as PLL(A). The maximum perturbation has a value of.2 p.u. The observer, however, presents a frequeny value with lower perturbations (<.p.u) an a time response of.72 s. But, when the irraiane goes from a low to a high value, the frequeny alulate by any of the strategies is similar an the perturbations are lose to p.u. This behaviour is ue to the MPPT that slows the ynamis of the PV generator when quik hanges of solar irraiane our. The power moves smoothly thanks to the voltage ontrol. However, when solar irraiane goes

5 vp.e j pll A,B x C iq vp.e j pll A,B,L ˆ x C ˆ iq ε ˆ iq Im{.} ε ˆ i q ˆ kp ki/s wobs /s ˆ g iq ˆ iq ˆ /2 fobs Fig. 8. Frequeny estimator for gri tie PV inverter (a) (a) (b) Fig. 9. Stuy ase A. (a) Solar irraiane variation (b) Estimate an alulate frequeny (b) Fig.. Stuy ase B. (a) Solar irraiane variation (b) Estimate an alulate frequeny from high to low, the MPPT by itself annot ontrol the quik hange of power. For stuy ase (B), the frequeny estimate or alulate by any of the strategies (PLL or observer) oes not show high perturbations. The ynamis of the MPPT ontrol, permits that the variation of solar irraiane oes not affet instantaneously the a power supplie by the PV generator. However at 3:3 pm in Fig. (b), a perturbation an be seen when the solar irraiane goes from 84 to 42 W/m 2, in ten minutes. Due to this hange, the PV generator presents a ramp rate lose to 2 MW/min. This hange of power makes the a voltage present some perturbations an thus the PLL (A or B) is affete. The frequeny alulate by these two PLLs presents some eviation from the real value. Meanwhile, the frequeny estimate by the observer oes not present these perturbations an is equal to the one impose by the eletrial system. IV. CONCLUSIONS This paper introue the moelling an the ontrol of a PV generator, where the emphasis is in the eletrial frequeny estimation. For this purpose, a omplete esign of the observer is explaine. The PV generator is teste uner ifferent ambient onitions where the frequeny is estimate by the observer an ompare with the frequeny alulate by two ifferent PLLs. The observer presents less perturbations an faster response than the PLL when quik hanges of solar irraiane our. The perturbation an the settling time are both substantially reue. However, this observer epens on the LCL filter an the gri inutane. For future work an estimation of the L g is going to be stuie onsiering weak gris.

6 V. APPENDIX The full orer observer estimates the state variables x = [ i v p i g ] T, where an aitional term is inlue: L = [ l l 2 l 3 ] T. () For this appliation, the Luenberger observer is written as: ˆx t = Aˆx B v B 2 ˆv g L(y ŷ), (2) ŷ = Cˆx. (3) As the measure voltage is at the PCC, the real gri voltage an phase angle is not being measure, thus in the previous equation these variables are also estimate. The observer error e = x ˆx satisfies the equation: e t = (A LC)e B 2(e j(ˆρg) v g ˆv g ), (4) ˆv g an ˆρ orrespons to the estimate voltage amplitue an phase angle respetively. The linearization of this error an be one by using the small-signal approah. To linearize, the relationship between the phase angle an the gri frequeny is onsiere. The linearize state-spae moel of the observer is given by t [ ] ẽ = ε ρ [ Ao L o C o jb 2 v go ] [ ] [ ] [ ] ẽ B2 ε ε θ vg ε ω (5) ε ρ = ρ ˆρ, (6) ε ω = ω g ˆω g, (7) ε vg = ṽ g ˆv g. (8) From this, the relationship between the urrent onverter an the angular spee error an be etermine as: ε i(s) w(s) = C(sI A 2 ) B ω (9) A 2 = B ω = [ ], (2) [ ] Ao L o C jb 2 v go, (2) ε i(s) = î î. (22) The seletion of the full orer observer gains is evelope using pole plaement. This system has a thir orer ynamis that orrespons to the error ynamis given as follows: G iω(s) = ĩ(s) ω(s) = jv go s CL L g (s α o )(s 2 2ɛ o ω o s ωo) 2. (24) From eq.24, it an be seen that the imaginary omponent of the urrent estimation signal epens on the angular spee. The imaginary omponent is relate with the i q in q frame. So, the angular spee an be estimate from the î q value when isturbane ours (variation of solar irraiane) by the following aaptative ontroller: ˆω g = k p ε i q k i ε i q t. (25) The gains are alulate as follows: kp = 2ɛω.C f.l.l g αω 2 v go, (26) k i = ω 2.C f.l g.l αω 2 v g. (27) Then, the phase angle an the frequeny are estimate. REFERENCES [] A. Cabrera-Tobar, E. Bullih-Massagué, M. Aragüés-Peñalba, an O. Gomis-Bellmunt, Review of avane gri requirements for the integration of large sale photovoltai power plants in the transmission system, Renewable an Sustainable Energy Reviews, vol. 62, pp , sep 26. [2] N. Jaalam, N. Rahim, A. Bakar, C. Tan, an A. M. Haiar, A omprehensive review of synhronization methos for gri-onnete onverters of renewable energy soure, Renewable an Sustainable Energy Reviews, vol. 59, pp , 26. [3] A. Cabrera-Tobar an O. Gomis-Bellmunt, Performane of a small photovoltai power plant uner ifferent meteorologial onitions, in 26 IEEE 6th International Conferene on Environment an Eletrial Engineering (EEEIC). IEEE, jun 26, pp. 6. [4] C. Li, C. Cao, Y. Cao, Y. Kuang, L. Zeng, an B. Fang, A review of islaning etetion methos for mirogri, Renewable an Sustainable Energy Reviews, vol. 35, pp. 2 22, 24. [5] J. Kukkola an M. Hinkkanen, State observer for gri-voltage sensorless ontrol of a gri-onnete onverter equippe with an LCL filter, in 24 6th European Conferene on Power Eletronis an Appliations. IEEE, aug 24, pp.. [6] Y. Park, S.-K. Sul, W.-C. Kim, an H.-Y. Lee, Phase loke loop base on an observer for gri synhronization, in 23 Twenty-Eighth Annual IEEE Applie Power Eletronis Conferene an Exposition (APEC). IEEE, mar 23, pp [7] A. Cabrera-Tobar, E. Bullih-Massagué, M. Aragüés-Peñalba, an O. Gomis-Bellmunt, Topologies for large sale photovoltai power plants, Renewable an Sustainable Energy Reviews, vol. 59, pp , jun 26. [8] M. De Praa-Gil, O. Gomis-Bellmunt, J. L. Domínguez-Garía, an L. Trilla, Tehnial an eonomi omparison of various eletrial olletion gri onfigurations for large photovoltai power plants, IET Renewable Power Generation, jul 26. [9] A. Yazani an R. Iravani, Voltage-soure onverters in power systems : moeling, ontrol, an appliations. IEEE Press/John Wiley, 2. [] A. Cabrera-Tobar, E. Bullih-Massague, M. Aragues-Penalba, an O. Gomis-Bellmunt, Reative power apability analysis of a photovoltai generator for large sale power plants, in 5th IET International Conferene on Renewable Power Generation (RPG) 26. Institution of Engineering an Tehnology, 26, pp. 53 (6.) 53 (6.). et(si A LC) = (s α o )(s 2 2ɛ o ω o s ω 2 o) (23) Consiering the gains, the urrent observer is onstrute an the relationship between the urrent an the angular spee is given by.

Optimal torque control of permanent magnet synchronous machines using magnetic equivalent circuits

Optimal torque control of permanent magnet synchronous machines using magnetic equivalent circuits This oument ontains a post-print version of the paper Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits authore by W. Kemmetmüller, D. Faustner, an A. Kugi an