Positive Sequence Tracking Phase Locked Loops: A Unified Graphical Explanation

Transcription

1 Positive Sequence Tracking Pase Locked Loops: A Unified Grapical Explanation L. Matakas Junior *, **, W. Komatsu *, and F. O. Martinz * * Polytecnic Scool of te University of Sao Paulo, Electrical Energy and Automation Dept., Av. Prof. Luciano Gualberto, t3, n.58, Sao Paulo, SP, Brazil ** Pontifical Catolic University of Sao Paulo, Dept. of Engineering, R. Marques de Paranagua,, , Sao Paulo, SP, Brazil Abstract-- Tis paper presents a grapical analysis wic explains several tree-pase Pase-Locked Loop () algoritms based on te detection of te positive-sequence component of te power system. It is sown tat, by employing vector algebra (dot, inner or scalar product) and space vectors concepts, several s metods presented in literature may be represented by a unique teory. Index Terms-- Pased Locked Loop (), tree pase, positive sequence tracking s, Syncronization, Utility-Connected Systems. I. INTRODUCTION Pase-Locked Loops () are metods to acieve syncronization of signals, usually employed in communication, control, automation and instrumentation systems. Wit te development of power electronic based devices connected to AC power systems, i.e. Dynamic oltage Restorers [0], Converters connected to alternative energy systems [3], FACTS [0,3] or Active Filters [], algoritms ave been widely used in power systems. In tese applications, te must assure frequency and pase angle syncronism wit te positive sequence component, even wit armonic distortion and unbalance at te grid, since tis is te component of interest for generation, transmission or power consumption. Tis paper compares and analyzes only positive sequence based tree-pase tecniques. Figure sows te topology of a tree-pase applied to power systems, considering instantaneous voltages in a,b,c system [],[2],[3]. Fig. 7 presents anoter consolidated strategy presented in literature in terms of αβ components [4], known as p- or q- [6],[7],[8]. Fig. 8 sows te most common representation of positive sequence s, wic is based on d,q syncronous reference frame [9], [0], [], [2], [3], [4], [5], [6]. Tis paper sows tat te tecniques of []-[6] are all te same and proposes an intuitive grapical explanation to tese algoritms, based on Space ectors. Moreover, comparison sows tat main differences of tese tecniques are te coordinate system implementation and te topology of filtering functions. Te stability analysis and locking conditions grapical explanations of te algoritms are also presented. II. ALGORITHM IN abc FRAME In tis paper, te is represented as a control system (Fig. ), were te control objective in is to force te dot product y(t) between instantaneous voltage vector = t = [ v t v t v t ] T a b c and te mains instantaneous voltage vector = t = [ v t v t v t ] T a b c to be null, tat is, to assure ortogonality between tese vectors. Te dot product error ε is te input of controller GC s, usually a Proportional Integral (PI) type, wose output is in general added to te mains frequency ω MAINS to minimize control effort. Te output of te Controller Block G ( s ) is te angular frequency reference ω, i.e., te frequency to be syntesized by te tree-pase oltage Controlled C Fig. representation in abc coordinate system.

2 Oscillator (CO). Te plant is composed by te CO and by a dot product based pase detector. Te pase detector may employ filtering actions on eiter te mains voltages [4],[5],[6], or on output of te Dot Product Block [],[2],[6] to attenuate ig order components (caused by armonics and negative sequence). A. Space vectors: Definitions and adopted notation In Fig. 2, te instantaneous mains voltages v a t, t v c t and an ortonormal basis composed by v b and te unitary vectors a, b and c, define te mains voltage space vector [5]: t = v t a v t b v c = a b c t Te zero-sequence instantaneous voltage, from (3), is equal to: ( a b c ) v0 t = v t v t v t / 3 (2) If v 0 t = 0, va t vb t vc t = 0 defines te αβ plane of Fig. 2. In tree-pase tree-wire systems, te vector is in tis plane. It is convenient to adopt te α, β, 0 basis, were te α, β vectors, wic are perpendicular between eac oter, belong to te plane and 0 is ortogonal to tis plane. Te α component is parallel to te projection of a onto te α, β plane. Te relationsip between coordinates in abc and αβ0 base is given by (3) [4]. Equation (3) is an ortogonality transformation, i.e. T M = M, in wic te amplitude and te relative position between te space vectors abc and αβ 0 are unaltered wen tis transformation is applied. In tis paper, te coordinate systems transformations from abc to αβ0 are applied in order to present te algoritm in a more didactic way. However, it sall be noted tat te implementation of te may be done in any coordinate system ( abc, αβ0 or dq0 ). In tis way, a set of positive sequence -order armonic voltages, v, v, v, wit a peak value a of, generates te vector b : c [ v ( t ) v ( t )v ( t ] T = ( t ) = a b c ) (4) Te projection of te vector α, β plane as te same amplitude, and rotates wit ω angular frequency (speed) in te counterclockwise direction (Fig. 3a). on te ( ) Fig. 3. Mains voltage armonics space vector projection on αβ-plane. (a) Positive sequence armonics. (b) Negative sequence armonics. v0 t va t 2 vα t = 2 2 vb t 3 vβ t vc t 0αβ = M abc (3) A set of negative sequence -order armonic voltages defined as va, vb, vc generates te vector, wit a peak value of : [ v ( t ) v ( t )v ( t ] T (5) = ( t ) = a b c ) Te projection of te vector on te α, β plane, as te same amplitude and rotates wit ω angular frequency (speed) in te clockwise direction (Fig. 3b). A set of zero sequence -order armonic voltages is defined as v a, v b, v c and generates te vector : o o o o [ v ( t ) v ( t )v ( t ] T (6) o o = ( t ) = a b c ) Fig. 2. Coordinate systems abc and αβ0 and vector. Te vector perpendicular to te axis. o wit variable amplitude, is always α, β plane and tus parallel to 0

3 B. Dot Product Analysis In Fig., CO block generates a set of tree, positive sequence, sinusoidal voltages va t,vb t,vc t at te angular frequency ω, associated wit te positive sequence vector : t = [ v t v t v t ] T = a b c (7) Te superscript was dropped in all signals and vectors related to te, because tey are always positive sequence ones. will always be in te αβ plane. Te dot product between vector and te mains voltage vector (Fig. 4) is given by: Te terms II, I and of (9) correspond to te dot product between vectors wit non-zero relative angular frequency, resulting in a non-zero instantaneous voltage wit null mean value. In particular, te relative angular frequency between vectors of term II is equal to 2 ω, wic is te lowest frequency tat te filters in te plant of Fig. must attenuate. Te 2 ω component will be feed-forwarded troug GC ( s ), and appear in te ω signal, producing a frequency modulation in te CO input. Tis results in distorted signals at te CO output. N N N = (8) Equation (8) is expanded as: = - 0 I II III N N N I I In (9), te instantaneous values of terms III and I, wic include mains voltage zero sequence components, are null, since te 0 (= n) vectors are parallel to te 0 axis and consequently are perpendicular to te α, β plane and to vector. Consequently, tis is not affected by zero sequence components in te mains voltages. Te term I of (9) corresponds to te dot product of syncronous vectors wit pase displacement δ = φ θ (see Fig. 4): = cos( δ ) (9) (0) Fig. 4. Dot product between and space vectors. Te filters of Fig. may be implemented by means of a PI Controller [],[2],[6], Double Syncronous Reference Frame [5], Sinusoidal Signal Integrator [4] or by te extraction of positive and negative sequence metods [6]. Some autors consider tat te loop filter implementation is done by te PI Controller and te inerent filtering action of te plant, since te CO is modeled as an integrator [9], [0], []. Te dynamic response of te is strictly related to its application and to te design of te controller and of te filter(s) of Fig.. For instance, in a Dynamic oltage Restorer [0], in te occurrence of voltage sags and swells te sould not quickly track te mains voltage, since wit suc beavior te output may exibit pase jumps. Tus, te closed loop system sould be designed to ave slow dynamic response in tis application. Fig.5. Locking Conditions in αβ0 (a) Locked (b) Lag (c) Lead (d) Stability points.

4 C. Locking Conditions As stated before, te controller of Fig. must force te filtered dot product to zero, resulting in =, sown in figure 5a. vector lags by 90 (see Fig. 5a). However, since te majority of applications employs a vector syncronized and in pase wit te mains positive sequence voltage, te vector / / must be obtained leading by 90 (Fig. 5a). Fig. 6 introduces te OUT block to generate / /. If leads / / (Fig. 5b), δ > 90 and te mean value of te dot product is negative. Consequently, te error ε(t) in Fig. is positive, resulting in an increment of te CO frequency, accelerating. If lags / / Fig. 6. Generating a output vector / / parallel to. (Fig. 5c), te same analysis is valid and will decelerate. Tus, it can be noted tat δ = 90 is a point of stable equilibrium and δ = 270 or δ = 90 is an unstable equilibrium point (Fig. 5d). III. ALGORITHM IN αβ0 FRAME Te in αβ0 frame is directly obtained from te abc frame presented in Fig.. Fig. 7 sows te control topology of te in te αβ0 coordinate system. Te coordinate transformation (3) is applied to te abc mains voltages, resulting in te αβ0 coordinates vα, t vβ, t v t, of te mains vector αβ 0 : 0 αβ 0 t = vα t α vβ t β v0 t 0 v t v t v t T = α β 0 Applying te above procedure to te CO Fig..7. representation in αβ0 coordinate system.

5 Fig.8. representation in αβ coordinate system including two possible OUT blocks. outputs v t,v t,v t a b c, results in te αβ0 coordinates vα, t v β, t v 0 t of te vector αβ 0 (2): t = v t α v t β v t 0 αβ 0 α β 0 T v α t vβ t v0 t = (2) Item IIA as sown tat te abc/αβ0 coordinate transformation (3) preserves amplitudes and relative position (i.e., angles) between space vectors, resulting in αβ 0 = abc =, and αβ 0 = abc =. Te dot product αβ 0. αβ 0 as exactly te same value of te dot product calculated in abc frame in Fig., because of te ortogonality property of T matrix. Tus, te in Fig. 7 as te same beavior of te one in Fig.. Since, as proved by te Dot Product analysis of item IIB, te CO as no zero sequence components, tis is not affect by zero sequence components of te mains voltages, and te in Fig. 7 can be redrawn as sown in Fig. 8. Te CO directly generates te αβ components and te amplitude must be corrected by a factor 3 / 2 to respect (3) (te amplitude of must be 3 / 2 ). Fig. 8 also includes te OUT block, wic allows generation of an output vector parallel to in abc or in αβ coordinates systems. Tis is often called p- by some autors, because of te dot product similarity wit te instantaneous active power p(t) [6],[7][8]. Reference [8] calculates p(t) using te abc coordinates system. Te signals v t, v t α β are called as fictitious currents and te y(t) signal (Fig. 8) becomes a fictitious active power p( t ). Te control objective is to acieve pt = 0, tat is, to assure ortogonality between mains and voltages vectors (null dot product). Tus, te explanation developed in Topics II and III is also valid for tis tecnique. Reference [7] discusses te possibility of using te fictitious reactive instantaneous power signal. In tis case te vector will be parallel to te mains vector. Tese vectors will ave te same or opposite signals depending on te positive or negative feedback signal of Fig. 8, respectively. Te fictitious reactive power can be easily calculated by: q( t ) = v ( t )v ( t ) v ( t )v ( t ) (3) β α α β In (3), q > 0 for lagging. I. ALGORITHM IN DQ0 FRAME Tis metod is extensively cited in te literature [9], [], [2], [3], [4], [5], [6], and is known as Syncronous Reference Frame SRF-. Fig. 9 sows te control topology of te in te dq0 coordinate system. Two coordinate transformations are made: abc to αβ0 (3) and ten αβ0 to te syncronous reference frame dq0 (Park transform). Te zero sequence components are

6 Fig.9 representation in dq coordinate system. not considered in te implementation, according to previous discussions in items IIB and III. Te αβ/ dq conversion of te mains voltages is carried out using te CO signals. Tis results in always parallel to te d axis, as sown in figure 0. Te dq rotates in te couterclockwise direction around 0 axis. Figure a sows te mains voltage positive sequence vector projections v d and v q into d and q axes, respectively. Forcing yt ( ) = v d = 0 (Fig. 9) is equivalent to force te dot product v d = d to zero, wic is assured only if is perpendicular to d. Tis condition is sown in Fig. b and is acieved by rotating te dq reference frame wit angular speed ω, until becomes ortogonal to d, wic, according to te explanation of Topic II, is equivalent to set / / =d (and = -q ). As a consequence, te same previous grapical explanation is valid to Syncronous Reference Frame s. Some variations of tis strategy include forcing v q = 0 [4] (Fig. c) or te use of positive feedback of y F (t) in te control loop [5][0](Fig. d). Tis last case is very convenient because te mains positive sequence vector will be parallel to te axis d. Fig. 0 relationsip between abc, αβ 0 and dq0 coordinate systems. Te problems related to te operation wit te presence of armonics and negative sequence components, discussed in item IIB, are minimized by implementing filters. Fig.. Locking Conditions in dq0: (A) Coordinate system (B) Locked to v d (C) Locked to v q (D) Locked to v q, positive feedback.

7 . PLANT MODEL Considering te plant and te control system approac of Fig., a model of te plant (CO pase detector) will be necessary wen designing te controller. A simple metod to obtain tis unified model is ten developed. Te locally averaged value of y(t) in Fig. is: ( δ ) y( t ) = ( t ) ( t ) cos (4) - Assuming tat: te amplitudes of te CO signals in Fig. are unitary. Tis means tat te amplitude of te vector is 3/2 ; - te peak amplitude of te mains voltages positive sequence is t. Tis results in a mains voltage, positive sequence vector, wit amplitude 3/2. If (4) is linearized around te operating point δ = π 2, y(t) is rewritten as: 3 π y δ (5) 2 2 From Fig. 4, δ = φ θ, can be expanded to: [ φ t ) θ ( t )] [ ω( t ) ( t )] δ = dt (6) ( 0 0 ω p were φ ( t0 ), φ( t0 ) are te initial values of te pase angles of te positive sequence of main voltages and of te signals, respectively. From (5) and (6): 3 yt t 2 π θ ( t0) ϕ ( t0) 2 [ ω t ω t ] dt II = I (7) If te offset described by term I of (7) is neglected, te plant can be represented by an integrator and a gain 3, as sown in Fig. 2. oltage sags, swells and 2 unbalances will affect te gain of te plant, unless a normalization by te a gain and by te positive sequence amplitude is done. Fig. 2. Linearized model of te plant. For te αβ (or p-), te model is exactly te same, if te CO signals are multiplied by 3/2 as discussed in item III and sown in figure 8. It is important to note tat almost all referenced autors in tis paper use te factor 2/3 instead of 2 / 3 in (3). Tis and oter small variations found in te references only affect te plant gain, wic can be easily calculated by te reasoning presented above. For te dq presented in figure 9, using te αβ/ dq conversion presented in figure 0, te d axis is equal to te vector, and as unitary amplitude. Unless te voltages are corrected by a factor 3/2, te gain for te plant presented in figure 9 is 3 / 2 (instead of 3/2 as sown in figure 2. I. CONCLUSION Tis paper as presented a grapical metod wic explains several tree-pase algoritms applied to power systems, aiming to unify tese tecniques. It as been sown tat main differences rely on adopted coordinate system, coice of te feedback variable and on filtering tecniques. It was also presented filtering strategies and a unified plant model for te circuits from [] to [6]. Teir plants linearized model only differs by a gain, wic depends on te employed coordinate transformation formula, and on implementation details. II. REFERENCES [] M.S. Padua, S. M Deckmann, F. P. Marafão, Frequency- Adjustable Positive Sequence Detector for Power Conditioning Applications, in Proceedings of Power Electronics Specialists Conference, vol., pp , June [2] L. Matakas Jr., F.O. Martinz, A.R.Giaretta, M.Galassi, W. Komatsu, A grapical approac to a Posistive-Sequence based algoritm, in Proceedings of Brazilian Automation Conference, vol., pp , Oct. 2006, (in Portuguese). [3] W. Pipps, M.J. Harrison, R.M.Duke, Tree-Pase Pase- Locked Loop Control of a New Generation Power Converter, in Proceedings of Conference on Industrial Electronics and Applications, 2006, pp.-6, May 2006.

8 [4] H Akagi, Ogasawara, S., K. Hyosung, Te teory of instantaneous power in tree-pase four-wire systems: a compreensive approac, in Proceedings of Industry Applications Conference, ol., pp , 999. [5] A. Nabae, H. Nakano, S. Togasawa, An instantaneous distortion current compensator witout any coordinate transformation, in Proceedings of International Power Electronics Conference, pp , 995. [6] S. M Deckmann, F. P. Marafão, M.S. Padua, Single and tree-pase digital structures based on instantaneous power teory, in Proceedings of Brazilian Power Electronics Conference, vol., pp , Sept [7] Rolim, L.G.B.; da Costa, D.R.; Aredes, M., Analysis and Software Implementation of a Robust Syncronizing Circuit Based on te pq Teory, in IEEE Transactions on Industrial Electronics, ol. 53, Issue 6, pp , Dec [8] S.A.O da Silva,. E. Tomizaki, R. Novocadlo, E.A.A. Coelo, Structures for Utility Connected Systems under Distorted Utility Conditions, in Proceedings of IEEE Conference on Industrial Electronics, pp , Nov [9]. Kaura,. Blasko, Operation of a Pase Locked Loop System Under Distorted Utility Conditions, in IEEE Transactions on Industry Applications, ol. 33, No., pp , Jan/Feb 997. [0] C. Zan, C. Fitzer,.K. Ramacandaramurty, A. Arulampalam, M. Barnes, N. Jenkins, Software Pase Locked Loop Applied to Dynamic oltage Restorer (DR), in Proceedings of IEEE Power Engineering Society Winter Meeting, ol. 3, pp , 200. [] L.N. Arruda, S.M. Silva, B.J.C. Filo, structures for utility connected systems, in Proceedings of Industry Applications Conference, ol. 4, pp , 200. [2] S. K. Cung, A pase tracking system for tree pase utility interface inverters, in IEEE Transactions on Power Electronics, ol. 5, Issue 3, pp , May [3] H. Awad, J. Svensson, M.J. Bollen, Tuning software pase-locked loop for series-connected converters, in IEEE Transactions on Power Delivery, ol. 20, Issue, pp , Jan [4] L.R. Limongi, R. Bojoi, C. Pica, F. Profumo, A. Tenconi, Analysis and Comparison of Pase Locked Loop Tecniques for Grid Utility Applications, Proceedings of Power Conversion Conference, April 2007, pp [5] P. Rodriguez, J.Pou, J. Bergas, J.I Candela, R.P. Burgos, D. Boroyevic, Decoupled Double Syncronous Reference Frame for Power Converters Control, in IEEE Transactions on Power Electronics, ol. 22, Issue 2, pp , Mar [6] H.E.P de Souza, F. Bradascia, F.A.S. Neves, M.C. Cavalcanti, G.M.S. Azevedo, J.P. de Arruda, A Metod for Extracting te Fundamental-Frequency Positive- Sequence oltage ector Based on Simple Matematical Transformations, IEEE Transactions on Industrial Electronics, ol. 56, Issue 5, May 2009, pp