Implementation of Induction Machine VBR Model with Optional Zero-Sequence in SimPowerSystems, ASMG, and PLECS Toolboxes

Transcription

1 Implementation of Induction Machine VBR Model with Optional Zero-Sequence in SimPowerSystems, ASMG, and PECS oolboxes M. Chapariha, F. herrien, J. Jatskeich, H. W. ommel Abstract-- In distributed generation systems, zero-sequence may be considered for accurate modeli of induction machines where for monitori and protection purposes the neutral point is either connected to ground directly or with impedance. his paper describes the so-called oltage-behind-reactance model for induction machines, wherein a zero-sequence branch is introduced in order to obtain an explicit formulation with a constant and decoupled interfaci circuit. he proposed model is demonstrated on an induction generator system implemented in three commercial electrical systems simulation toolboxes. hese toolboxes are SimPowerSystems, ASMG, and PECS which are commonly-used with MAAB/Simulink. In addition to straightforward implementation in all three toolboxes, the proposed model gies identical results with significant computational speed-up as compared to the preious implicit VBR model. he new model is also compared with the classical qd0 model which required snubber interfaci circuit. he studies demonstrate that the proposed VBR model is more accurate than the classical approaches, and has higher numerically stability and efficiency especially when the zerosequence is considered. Keywords: ac machines, induction generators, power system simulation, oltage-behind-reactance formulation. NOMENCAURE hroughout this paper, matrix and ector quantities are boldfaced, and scalar quantities are italic non-boldfaced. All the ariables are referred to the stator side usi the appropriate turns ratio. i stator actual oltage and current ectors, oltage ector from stator terminals to the abcn point n in VBR interfaci circuit e abc subtransient oltage ector stator s zero-sequence current i 0 s his research was supported by the Natural Science and Eineeri Research Council (NSERC) of Canada, under the iscoery Grant. M. Chapariha, F. herrien, J. Jatskeich, and H. W. ommel are with the epartment of Electrical and Computer Eineeri, Uniersity of British Columbia, Vancouer, BC, V6 1Z4, Canada ( {mehrdadc, francist, jurij, hermannd}@ece.ubc.ca) Paper submitted to the International Conference on Power Systems ransients (IPS013) in Vancouer, Canada July 18-0, 013., i oltage and current of the VBR interfaci r s, r r ls, qr r P r, circuit zero-sequence branch stator and rotor windi resistances stator and rotor windi leakage inductances, dr rotor oltage transformed to q and d-axes r 0, 0 r g E R S, X S aular speed of rotor aular speed of rotational reference frame number of poles resistance and inductance of the three-phase branch of VBR interfaci circuit resistance and inductance of the zerosequence branch of VBR interfaci circuit external groundi resistance external source equialent impedance I. INROUCION ECRICA ac machines are important parts of many small and large power systems. Examples of such systems include electric ehicle drie systems, distributed generators, airplane and essel electrical systems, microgrids, etc. Synchronous machines are often used as generators in thermal and hydro power plants, while induction machines are commonly used as motors in industrial, commercial, and household applications. More recently, howeer, induction machines hae also frequently been employed as smaller-scale generators in distributed energy systems. ependi on the required degree of detail, different programs and models are used to simulate electric machines. In this paper, transient simulation programs are considered. Specifically, the focus is on induction machine modeli in state-ariable-based programs such as MAAB/Simulink [1], [], for which seeral power systems and electrical circuits toolboxes are commercially aailable: SimPowerSystems [3], ASMG [4], PECS [5], etc. hese toolboxes proide graphical user interfaces and automated state-model formulation algorithms to simplify the modeli of electrical systems in Simulink enironment. Additionally, some of these toolboxes, such as PECS, hae also standalone ersions. he formulation considered in this paper can also be discretized for implementation in the nodal-analysis-based EMP-type simulation programs [6].

2 For some transient simulations, machine models with magnetically coupled rotor and stator equialent circuits are considered [7]. Such phase-domain representations are wellestablished but rarely used directly for numerical simulations due to the presence of ariable and coupled inductances: timearyi inductances require re-factorization/regeneration of the system matrix at each time-step and may drastically increase the computational burden of simulation. he reformulation of phase-domain models usi the rotati reference frame theory originally presented in [8] significantly reduces their computation time. hese models, herein referred to as classical qd0 models, are most frequently found in commercial simulation packages today. espite bei commonly aailable as built-in components, the qd0 models hae interfaci problems with external inductie networks [9]. he qd0 models are typically interfaced with external networks as oltage-controlled current sources in abc phase coordinates. If these current sources are connected in series with inductors, simulation programs encounter problems when formulati a proper explicit state-ariable model of the oerall system. In such cases, fictitious resistie or capacitie snubber circuits should be added to enable the formulation of input-output coupli ariables and the oerall state-ariable model. Howeer, the snubber circuits add error and can make the system numerically stiff, decreasi the numerical efficiency and stability of the whole simulation. he so-called oltage-behind-reactance (VBR) formulation offers a solution to the interfaci problem. he VBR formulation takes adantage of the reference frame transformation but leaes enough elements in abc phase coordinates to obtain direct interfaci [9] [13]. A simple VBR formulation was presented in [10] which has a constant and decoupled interfaci circuit; howeer, therein an algebraic loop is created when zero-sequence is simulated. It was recently enhanced by includi zero-sequence in the interfaci circuit instead of in the subtransient oltage equations, maki the model explicit. he improed formulation is presented in [11] where it is also extended to synchronous machines. his formulation requires less number of parameters and is claimed to be easy to implement in different simulation programs as will be erified in this paper. he proposed model is suitable for accurate modeli of induction machines with or without groundi. Compared to the qd0 model with snubbers, the VBR formulation is more accurate and is numerically less stiff. hese numerical adantages stem mostly from the direct interfaci circuit. No simulation studies are aailable in the literature to alidate the VBR formulation for induction machines gien in [11]. In this paper, this formulation (here referred to as the explicit VBR) is compared to the one in [10] (referred to as the implicit VBR-III) and its numerical adantages are shown for a sample machine-network system. he implementation method of the explicit VBR formulation is demonstrated in the three toolboxes mentioned earlier for the first time. A distributed energy system with induction generator is considered for the studies where the machine is connected to an external inductie network and the neutral point is directly grounded for protection purposes [14]; a sile-phase-toground fault happens in the system close to the generator to erify the model in a more general case includi zerosequence. ately, the VBR formulation is bei used in the simulation software programs. For example, PECS has added some ariation of VBR machine models into its library of components [5]. his paper demonstrates the simplicity of implementation of the new constant-parameter VBR formulation and confirms its accuracy and shows that it is easily implementable in different state-ariable based simulation programs. Comparison with classical qd0 model requiri snubbers shows that this model is superior for most of the general applications. II. VBR FORMUAION FOR INUCION MACHINE MOE In this paper, a three-phase wye-connected induction machine with sinusoidally distributed rotor and stator windis is considered. o be consistent with preious publications, specifically [10] and [11], the motor conention with the same ariable names are used here. For conenience, the important parameters and ariables are defined in the Nomenclature section; the others are explained in the text. he machine is originally modeled as coupled-circuits in phase-domain as shown in [7]. he classical qd0 model, whose adantages and disadantages were listed in the Introduction, is then obtained by transformi the abc ariables into the arbitrary reference frame. Finally, algebraic manipulation of the qd0 equations yields the VBR formulation [10]. his formulation can be directly interfaced to any external circuit while maintaini an interfaci circuit with constant and decoupled parameters. he branch oltage equation of the interfaci circuit is [10] where r ri pi eabc 3r0 i 30 pi (1) m rs rr () (3) ls m m 1 r0 rr (4) 3 0 m. (5) 3 1 he subtransient magnetizi inductance is gien by m m. (6) he terms inside the parenthesis in (1) are equal to zero if there is no zero-sequence current in the circuit (i.e. the neutral is floati). he ector e abc is the subtransient back EMF

3 oltages in abc phase coordinates which is defined as abc 1 K [ e e 0] e s q d (7) where K s is the transformation matrix from the stationary reference frame abc to the rotational reference frame qd0, and e e q d mrr m q qr qr rd (8) mrr m d dr dr rq. (9) he q- and d-axes subtransient flux linkages are defined as qr q m (10) dr d m. (11) Usi flux linkages as state ariables, the rotor dynamics are represented by the followi state equations: rr p qr ( qr mq ) ( r ) dr qr (1) rr p dr ( dr md ) ( r ) qr dr (13) where the magnetizi fluxes are calculated by i (14) mq md m qs m ds q d i. (15) he electromechanical torque is 3P e ( md iqs mqids). (16) 4 On the one hand, induction motors are ery rarely grounded. On the other hand, for protection or monitori purposes, induction generators may be grounded directly or ia impedance [14] which in turn requires the terms inside parentheses in (1). Manipulati (1) yields [10] where r i pi e abc 30rs 30 3r0 i (17) ls ls [ i i i ] i (18) [ ]. (19) he zero-sequence current i 0 s and oltage 0 s are the algebraic aerage of their correspondi three phase alues as i ( i i i ) / 3 (0) as bs cs ( ) / 3. (1) as bs cs Eq. (17) is the algebraically exact implicit VBR-III implementation; howeer, it introduces an algebraic loop when the terminal oltage is unknown, e.g. when the machine is in series with inductie elements. o aoid an implicit formulation, the stator oltage equation (1) can be separated into two sections [11] () abcn where abcn abc r i pi e (3) [ ] (4) and the elements of are r0 ( 3i0 s ) 0 p(3i0 s ) r0i 0 pi. (5) he aboe equation, (5), represents a zero-sequence branch in series with the three-phase decoupled R branch which is used to represent (3). he resulti interfaci circuit is shown in Fig. 1. his formulation, named explicit VBR, is algebraically exact and explicit and has a constant-parameter decoupled circuit interface. o summarize, the circuit shown in Fig. 1 with the elements gien by () (5), in addition to the rotor state equations gien by (1) and (13), the definitions of flux linkages gien by (10), (11), (14), and (15), and the reference frame transformation (7), compose the explicit VBR formulation. Fig. 1. Interfaci circuit for the explicit VBR formulation gien at [11]. III. MOE IMPEMENAION IN SIMUAION OOBOXES o demonstrate its ease-of-use, the explicit VBR formulation is implemented in three commonly used toolboxes in MAAB/Simulink. Fig. shows the SimPowerSystems (SPS) [3] implementation, wherein the interfaci circuit is Fig.. Implementation of VBR model with zero-sequence in SPS.

4 shown inside the box. For compactness, the machine is connected to the héenin equialent circuit of a network; howeer, the network could also be represented in detail. In this figure, the machine is grounded through the resistance r g. his resistance is not a part of the VBR model and is added externally to limit the short circuit current. If the machine is not grounded, the neutral point is left unconnected without any modification to the model. o focus on the electrical model, the mechanical system and the additional input and output ports are omitted in Fig.. For the other two toolboxes, ASMG [4] and PECS [5], the electrical network is simply replaced with one instance of ASMG-System or PECS Circuit, respectiely, which for conciseness, are not shown here. IV. COMPUER SUIES o assess the numerical efficiency of the proposed explicit VBR model, a simple distributed generation system is considered here. It is assumed that an induction generator is connected to a prime moer (e.g. a wind turbine) that in the course of the study maintains a constant speed of 1.07 perunit. he system is illustrated in Fig. 3, wherein the generator is connected to a héenin equialent circuit. he machine parameters are gien in the Appendix. o emulate a seere unbalanced transient situation, a sile-phase-to-ground fault is assumed in the system close to the generator feeder and the groundi resistance is zero (r g = 0). Initially the machine is in steady state and the fault happens after one cycle. he fault is modeled by decreasi the phase a oltage of the equialent source to zero ( a = 0). he VBR model is directly connected to the R network similar to Fig.. Howeer, the classical qd0 model requires a snubber for interfaci as shown in Fig. 3. o minimize the error, a 10 per-unit resistie snubber is used here. he snubber adds error and makes the system numerically stiff, therefore less numerically stable. As a reference, the system (includi the machine) is represented in the synchronous reference frame and the whole model is soled usi the OE45 soler with the time-step limited to 1 μs. his model is implemented usi basic Simulink blocks. Hai a ery simple and linear system, the conersion to rotational reference frame is straightforward in this example. Howeer, for general cases e.g. when the system Fig. 3. Induction generator connected to the héenin equialent circuit of a network.. is large or includes non-linear elements such as rectifiers, such a conersion is not triial, if possible at all. he implicit VBR-III [10] is implemented in the SimPowerSystems (SPS) toolbox, as SPS coneniently allows soli algebraic loops in Simulink. he explicit VBR [11] is implemented in the SPS, ASMG, and PECS toolboxes; the three implementations gie identical results. Since the built-in qd models in the library of SPS and PECS do not include zero-sequence, the qd0 machine model is implemented usi basic Simulink blocks. he model is interfaced to an instance of PECS circuit by means of controlled current sources. o show the consistency of the explicit VBR [11] and the implicit VBR-III [10], both models are run with the OE45 soler usi the same settis. he maximum and minimum time-steps are set to 1 ms and 0.1 s, respectiely. he models are represented in per-unit and the relatie and absolute tolerances of the soler are For the qd0 model with snubbers, the stiffly-stable Simulink soler OE15s is used. Its settis are identical to those used for the VBR models. he simulation results shown in Fig. 4 demonstrate the consistency between the VBR models and the reference solution. hree windows highlighted in Fig. 4 are enlarged and shown in Fig. 5 to Fig. 7. Fig. 5 shows phase c stator current in steady-state; it shows that both VBR models yield exactly the same results. Moreoer, they hae chosen identical time-steps. For the qd0 model, the snubbers sink part of the machine output current and therefore produce some error. Comparatiely, the qd0 model with snubbers has chosen seeral times more time-steps and has a isible steady-state error. Fig. 6 and Fig. 7, which show phase c stator current and electromagnetic torque duri fault, confirm that the VBR models hae no isible error in transient as well. For the qd0 model with snubbers, the transient response has some error, although it is less isible due to the comparatiely large fault current. he stiffly-stable soler OE15s has chosen een smaller time-steps duri the transient period (after the fault) than in steady state. A quantitatie ealuation of the simulation study is summarized in able I. he time-steps and calculation data are obtained from Simulink Profiler []. he relatie error is calculated by compari the predicted trajectory with the reference solution usi the -norm of the error and normalizi the difference [15]. For example, the error for i as is gien by ~ ias ias ( ias) ~ 100% (6) ias ~ where ias is the reference solution trajectory. he aerage three-phase current error i ), which is shown in able I, ( is ealuated usi i ) ( i ) ( i ) ( i ) / (7) 3 ( as bs cs

5 Fig. 7. Magnified iew of the electromechanical torque duri the fault from Fig. 4. Fig. 4. Simulation results for the sile-phase-to-ground fault study: source oltage; source current; machine neutral current; and electromechanical torque. able I erifies that both implicit and explicit VBR formulations are algebraically identical to the reference. It also reeals the difference between the computational costs of the two VBR formulations by compari the number of subtransient oltage calculations. Practically, the implicit VBR-III is significantly slower since it requires iterations in each time-step for the algebraic loop solution (3865 calculations compared to 764 for the explicit VBR). he qd0 model is explicit but numerically stiff, thus it used seeral times more time-steps (989 compared to 110 times for the VBR models) and een more internal current calculations (7070 times) than the implicit VBR subtransient oltage calculations (3865 times). As shown in able I, the largest eigenalue of the qd0 model with snubbers is seeral orders of magnitude bigger than the largest eigenalue of the VBR models. ABE I COMPARISON OF NUMERICA EFFICIENCY OF VBR FORMUAIONS FOR Simulation Index SINGE-PHASE FAU SUY Implicit VBR-III [10] Formulations Explicit VBR [11] Major ime Steps qd0 with Snubber Minor oop Calculations* Current i Prediction Error % %.861 % Fig. 5. Magnified iew of the source current in steady state from Fig. 4. argest eigenalue 199 ± j ± j * his row shows the number of subtransient oltage calculations for the VBR models and the number of injected current calculations for the qd0 model. Fig. 6. Magnified iew of the source current duri the fault from Fig. 4. V. CONCUSIONS he proposed explicit VBR model for induction machine was erified, the implementation in commercially aailable simulation toolboxes was demonstrated, and the model was compared to the preious implicit ersion VBR-III and the classical qd0 models. As shown for the three commonly used simulation toolboxes, SPS, ASMG, and PECS, the interfaci circuit and internal subtransient oltage equations for the explicit VBR formulation are simple and easy to implement. A sile-phase-to-ground fault study demonstrates that the VBR models are more accurate and numerically more

6 efficient than the qd0 model. he studies presented in this paper show that when zero-sequence is considered amo the two VBR formulations the numerical efficiency of the explicit VBR model is noticeably higher. Based on the studies presented in this paper, as well as the ones in [10] and [11], the explicit VBR model is suggested as the general purpose model for squirrel-cage induction machines in state-ariable-based simulation programs. It yields identical results to the reference and does not require snubbers when connected to an inductie network/system, while offeri high accuracy, numerical stability, and simulation efficiency. [16] Cathey, J.J.; Cain, R.K.; Ayoub, A.K., "ransient oad Model of an Induction Motor," IEEE rans. on Power Apparatus and Systems, ol.pas-9, no.4, pp , July VI. APPENIX Induction Generator Parameters [16]: 4 poles, 60 Hz, 3-phase, 460 V, 50 hp, 1705 rpm, rs Ω, Xls Ω, X m Ω, rr 0. 8 Ω, X Ω. VII. REFERENCES [1] MAAB 7 (R009a), Natick, MA: he MathWorks Inc., 009. [] Simulink 7 User s Guide, Natick, MA: he MathWorks Inc., 009. [3] SimPowerSystems User s Guide, Natick, MA: Hydro-Québec and he MathWorks, Inc., 009 [4] Automated State Model Generator (ASMG), Reference Manual, Version, PC Krause and Associates Inc., West afayette, IN, 00. [5] Piecewise inear Electrical Circuit Simulation (PECS) User Manual, Version. 3., Plexim GmbH, 01. Aailable: [6] H. W. ommel, EMP heory Book, Vancouer, BC, Canada: Microtran Power System Analysis Cooperation, May 199. [7] P. C. Krause, O. Wasynczuk, and S.. Sudhoff, Analysis of electric machinery and drie systems, second edition, NJ: Wiley-IEEE Press, 00. [8] R. H. Park, "wo-reaction theory of synchronous machines generalized method of analysis-part I," rans. of the American Institute of Electrical Eineers, ol.48, no.3, pp , July 199. [9]. Wa, J. Jatskeich, V. inaahi, H. W. ommel, J. A. Martinez, K. Strunz, M. Rioual, G. W. Cha, and R. Iraani, Methods of Interfaci Rotati Machine Models in ransient Simulation Programs, IEEE rans. Power eliery, ol.5, no., pp , Apr [10]. Wa, J. Jatskeich, and S.. Pekarek, "Modeli of Induction Machines Usi a Voltage-Behind-Reactance Formulation," IEEE rans. Energy Coners., ol.3, no., pp.38-39, Jun [11] M. Chapariha,. Wa, J. Jatskeich, H. W. ommel, and S.. Pekarek, "Constant Parameter R-Branch Equialent Circuit for Interfaci AC Machine Models in State Variable Based Simulation Packages," IEEE rans. on Energy Coners., ol.7, no.3, pp , Sep. 01. [1]. C. Aliprantis, O. Wasynczuk, and C.. Rodriguez Valdez, "A Voltage-Behind-Reactance Synchronous Machine Model With Saturation and Arbitrary Rotor Network Representation," IEEE rans. Energy Coners., ol.3, no., pp , Jun [13] A. M. Cramer, B. P. oop, and. C. Aliprantis, "Synchronous Machine Model With Voltage-Behind-Reactance Formulation of Stator and Field Windis," IEEE rans. Energy Coners., ol.7, no., pp , Jun. 01. [14] (01, Sep. 8). Handbook of General Requirements for Electrical Serice to ispersed Generation Customers. Consolidated Edison Company of New York, Inc., New York, NY, Mar [Online]. Aailable: [15] W. Gautchi, Numerical Analysis: An Introduction, Birkhauser, Boston, Massachusetts, 1997.