Modelling short- and open-circuit faults in permanent magnet synchronous machines using Modelica

Transcription

1 Modelling short- and open-circuit faults in permanent magnet synchronous machines using Modelica Paolo Giangrande, Luca Papini, Chris Gerada Power Electronics Machines and Control Research Group, Faculty of Engineering, University of Nottingham, Nottingham NG72RD, UK Published in The Journal of Engineering; Received on 15th February 2016; Accepted on 26th February 2016 Abstract: Electrical machine models are indispensable for describing and predicting the machine behaviour in several operating conditions. Modelling fault conditions are an attractive research area, since fault simulations allow to figure the behaviour of the electric drive and prevent damages to its components. In this study, winding failures in permanent magnet synchronous machines PMSMs) are considered. In particular, short-circuits and open-circuits failures have been taken into account via two simple PMSM models. These models have been created using Modelica as modelling language. Feasibility of the proposed models is investigated in simulation using Dymola environment) under the following fault conditions: single-phase open-circuit, single- and three-phase short-circuits. Moreover, models validation has been carried out through experimental tests, injecting windings failures into a fault-tolerant PMSM. 1 Introduction Developing accurate simulation models for electrical machines allows predicting motor drives behaviour and analysing particular operating conditions e.g. short- and open-circuit faults), that would otherwise be too risky to test on actual machine, without compromising its integrity. Among electrical machines, permanent magnet synchronous machines PMSMs) are characterised by an excellent efficiency, together with a high-power density, due to the PMs presence [1]. These features make PMSMs widely used in several industrial applications, where they have largely replaced both DC and wound field synchronous machines. However, the presence of PMs is source of concern in the event of fault conditions, since fault protection is not straightforward, as happens for wound field synchronous machines, where the produced flux can be electrically controlled [2]. Fault conditions affect PMSM behaviour in terms of both developed electromagnetic torque and provided electrical quantities i.e. voltages and currents). In particular, during a fault condition, torque oscillations and/or braking torque have/has a direct impact on the drivetrain, whereas high voltage and/or current values might damage power electronic devices and/or the machine itself. To prevent damages to either electronic, electrical or mechanical components of the PMSM drive, modelling fault conditions are of paramount importance for having knowledge of the drive response under fault conditions. Moreover, such information might be used to improve the machine design toward a fault-tolerant approach [3] or implement appropriate control strategies for reducing the fault-effects impact [4]. The main winding failures, which may occur within the PMSM, are: winding short-circuits and winding open-circuits. In this paper, two models, one for short-circuits and the other for open-circuits faults, are presented. These models have been elaborated using Modelica as modelling language and they have been simulated adopting Dymola package. Modelica is an object-oriented equationbased language, for modelling multi-physical systems [5], whereas Dymola is one of the available Modelica simulation environments. The idea to implement two models, one for each specific fault condition, has been exploited in order to keep the model as simple as possible, since a good model is a wise trade-off between realism and simplicity. Models description is provided in this paper, together with simulation and experimental results. Indeed, several fault conditions have been simulated and simulation results are given, in order to demonstrate the models feasibility. Model validity is experimentally confirmed by the verification of the simulation results with those obtained through a fault-tolerant PMSM drive test bench. 2 PMSM mathematical model Space phasor theory [6] is usually adopted for modelling electrical machines, since applying Park transformation; the behaviour of three-phase machine can be described using an equivalent twophase machine. Space phasor theory considers only the fundamental component of the machine quantities such as currents, voltages and fluxes harmonic content is neglected), and it can be applied either to a balance or unbalanced machine [7]. In the case of balanced three-phase machine, Park transformation allows to pass from a time-dependent system i.e. phase variables system) to a time invariant system i.e. dq coordinate system). This operation simplifies the expression of the electrical equations and removes their time and position dependency. Moreover, the space phasors reach constant values, in steady-state conditions. Considering an unbalanced three-phase machine e.g. asymmetric fault), the model, obtained through the Park transformation, has still time dependency, even for steady-state operations. In other words, the Park transformation remains valid in unbalanced conditions, but its benefits vanish. According to these considerations and since asymmetric faults i.e. single-phase short- and open-circuit faults) will take into account by the PMSM models, they have been implemented in phase variables. PMSM models consider three-phase stator windings wyeconnected with floating neutral point) and effects as cross-coupling and saturation are neglected. Under these assumptions, the phase voltage is expressed as sum between the voltage drop across the resistance and the induced back electromotive force EMF) [8], as shown in 1) v a = R i a + e a = R i a + d dt l a v b = R i b + e b = R i b + d dt l b v c = R i c + e c = R i c + d dt l c where the subscripts a, b and c indicate the phases, v is the voltage of each phase with respect to the neutral point, R is the phase 1) Attribution-NonCommercial-NoDerivs License

2 resistance, i is the current flowing through the phase, e is the induced back EMF in the phase winding and finally λ is the flux linkage with the phase winding. The flux linkage can be considered as sum of three contributions: the flux produced by the phase current flowing through the considered winding and linked with it self-induced flux), the flux generated by the other two currents and linked with the winding under examination mutual-induced flux) and the flux linkage due to the PMs, as reported in 2) l a = L a i a + M ab i b + M ac i c + L PMa l b = M ba i a + L b i b + M bc i c + L PMb 2) l c = M ca i a + M cb i b + L c i c + L PMc where L and M are, respectively, the self- and mutual-inductances, and Λ PM is the flux linkage due to the PMs. The fluxes linkages, due to the PMs, are function of the position θ r and assuming a sinusoidal distribution, they can be expressed as shown in 3) ) L PMa = C PM cos u r L PMb = C PM cos u r 2 ) L PMc = C PM cos u r + 2 ) where Ψ PM is the magnetic flux produced by the PMs. Self-inductances are also function of the rotor position, due to the mechanical structure of the machine. Hence, assuming a sinusoidal distribution, they can be defined as reported in 4) ) L a = L l + L mav + L md cos 2u r L b = L l + L mav + L md cos 2u r + 2 ) L c = L l + L mav + L md cos 2u r 2 ) where L l is the leakage inductance of each phase winding, which describes the amount of flux not coupled to any other magnetic field, and L mav and L mδ are the magnetising inductances. In particular, L mav is the average inductance minus the leakage inductance and L mδ is the amplitude of the sinus varying part. Similarly, the mutual-inductances are expressed as shown in 5) M ab = M ba = 1 2 L mav + L md cos 2u r 2 ) M bc = M cb = 1 2 L ) mav + L md cos 2u r M ca = M ac = 1 2 L mav + L md cos 2u r + 2 ) The complete performance of PMSM under transient conditions can be determined by using 1) together with the equation of the motion 6) 3) 4) 5) T e T l B vr np = J np d dt v r 6) where T e and T l are, respectively, the developed electromagnetic torque and the load torque, J is the rotor moment of inertia, B is the friction coefficient, np is the pole pairs number and ω r is the rotor electrical speed, expressed in radians/second. To complete the PMSM mathematical model, the torque equation in terms of phase variables is provided in 7), which has been derived from the magnetic co-energy expression [9] see 7)) The presented mathematical model has been implemented using Modelica modelling language. Modelica has been preferred to other modelling languages, because it allows the creation of partial models, which can be reused [10]. This feature makes the models quite flexible and easy to be extended, for including further physical effects. Equations 1) and 6) have been built up in the Modelica Diagram layer see Figs. 1 and 2), using an object-oriented approach employing components from the Modelica Standard Library MSL), while 2) 5) and 6) have been written in the Modelica Text layer. 3 Analysis and implementation of windings faults Example of electrical machine libraries, developed using Modelica, can be found in [11 13], but fault conditions in PMSMs i.e. turn-to-turn short-circuit) are considered only in [13]. The PMSM models presented in this paper consider short- and open-circuit faults, which have been implemented by means of electrical switches. This approach has been already adopted by Winkler and Gühmann [14] for modelling faults in induction machine IM). The switches are driven by Boolean parameters and allow opening and closing of the phase windings. 3.1 Short-circuit faults The major fault class for PMSMs is represented by the short-circuit faults. These faults can occur in case of physical damage of the connection cables between the machine and the power converter or when the winding insulation fails, due to thermal stress. Three-phase short-circuit may intentionally be induced in response to an asymmetric fault detection, in order to reduce both the risk of PMs demagnetisation and the braking torque [15]. Main consequences arise by a short-circuit fault are the risk of irreversible demagnetisation of the PMs, the increasing of the temperature, which could lead to multiple winding failures, and the developing of braking torque and/or torque oscillations, which may cause mechanical failures to the drivetrain. In Fig. 1, the Modelica Diagram layer of the PMSM model, which takes into account the short-circuit faults, is shown. Positive and negative plugs positiveplug and negativeplug ) are adopted as electrical interfaces to electrically supply the machine. According to 1), each phase winding is modelled using a resistance in series with a driven voltage source, which provides the induced back EMF. Electromagnetic torque calculated using 7) is applied to the rotor inertia block inertiarotor ) through the torque source block torque ). Finally, the developed torque is available to the mechanical flange flange ), where the load torque is applied. Three normally open switches one per phase) have been connected in parallel to each phase and they are used for injecting short-circuit faults. These switches are provided by MSL IdealClosingSwitch ) and they are driven by a Boolean parameter, which represents the fault injection. In particular, the normally open switch is closed when the Boolean parameter has high logical level. { [ T e = np L md i 2 a sin 2u ) r i 2 b sin 2u r + 2 ) i 2 c sin 2u r 2 ) 2 i a i b sin 2u r 2 ) ) 2 i b i c sin 2u r 2 ic i a sin 2u r + 2 )] [ ) C PM i a sin u r + ib sin u r 2 ) + i c sin u r + 2 )]} 7) Attribution-NonCommercial-NoDerivs License

3 Fig. 1 Modelica Diagram layer of PMSM model, which takes into account short-circuit faults The presented model allows simulating single- and three-phase short-circuit faults. Short-circuit between one terminal of the machine and the neutral point will occur when only one normally open switch is closed and this condition is indicated with the name single-phase short-circuit. On the other hand, short-circuit among all three terminals of the machine will occur when the three normally open switches are simultaneously closed and such condition is called three-phase short-circuit. The former fault leads to an unbalanced operation condition asymmetrical fault), whereas the latter keeps the system balanced symmetrical fault), in spite the fault injection. 3.2 Open-circuit faults Another topology of winding failures is represented by the opencircuit faults. It may be produced by one or more broken connecting cables or by an internal winding ruptures [16]. Fig. 2 reports the Modelica Diagram layer of the PMSM model, which includes opencircuit faults. Adopted electrical and mechanical interfaces are the same as discussed in the previous section, as well as the model structure. Main difference of this model is the employment of normally closed switches one per phase) for injecting open-circuit faults. These switches are connected in series with each phase Fig. 2 Modelica Diagram layer of PMSM model, which takes into account open-circuit faults Attribution-NonCommercial-NoDerivs License

4 Fig. 3 Current control schema developed in Dymola environment) used for simulating the fault injections winding and they are available in MSL IdealOpeningSwitch ). Normally closed switch is open when the Boolean parameter, which drives the switch, has high logical level. The implemented model can simulate single- and three-phase open-circuit faults. In case of single-phase open-circuit only one normally closed switch is open), the PMSM behaviour will result in an unbalanced two-phase operation asymmetrical fault), which will affect the developed torque with an alternating torque component at twice the electrical operating frequency of the machine. Such torque oscillation may be source of mechanical stress for the drivetrain. Three-phase open-circuit three normally closed switches are simultaneously open) is less likely than the singlephase and it is even less risky, since the current in all the three phases will be null, as well as, the developed torque. 4 Simulation results The simulation results presented in this section have been carried out under the assumptions that rotor speed and current references remain the same after the fault injection. Moreover, PMSM drive is assumed operating in steady-state, when the considered fault condition is injected. PMSM drive has been simulated in Dymola environment, in order to analyse the drive response after a fault injection. Failures such as three-phase short-circuit, single-phase shortcircuit and single-phase open-circuit have been separately considered keeping the same drive parameters. The simulation model of PMSM drive is shown in Fig. 3 and it is a replica of the test bench adopted to validate the PMSM models. During the experimental test, PMSM is current controlled and it is mechanically coupled with a speed controlled IM. In simulation environment, the PMSM is still current controlled with Iq = I n and I d = 0), but the speed controlled IM has been substituted with a speed constant block connected to PMSM flange, which provide a mechanical speed of 4 krpm. PMSM is supplied through a voltage-source inverter having a DC-link voltage of 350 V dc. The inverter has been modelled using ideal switches. The parameters of the faulttolerant PMSM [17] are given in Table 1 and the same parameters have been used during the simulation tests. 4.1 Three-phase short-circuit Three-phase short-circuit fault symmetrical fault) is injected at 1 s, when the rotor speed is equal to 4 krpm. For this fault condition, simulated steady-state and dynamic responses of the PMSM are reported in Figs Before the fault injection, the phase currents have an amplitude of 40 Apk 28.3 Arms), whereas after the fault transitory their amplitude increases to 57.8 Apk 40.9 Arms), as shown in Fig. 4. Fig. 5 depicts the torque trend, and it is possible to underline that, in healthy condition, PMSM develops an electromagnetic torque equal to 9.48 Nm, whereas the braking torque value in steady-state) is equal to 2.15 Nm. Since the fault condition under analysis is symmetrical, it is worth visualising the current components in the dq rotating reference frame. These components Table 1 Fault-tolerant PMSM parameters Symbol Meaning Value P n rated power 25 kw Ω n rated speed 20 krpm I n rated current 40 A np pole pairs number 2 Ψ PM PM flux Wb R phase resistance 0.18 Ω L d d-axis inductance at I n 1.35 mh L q q-axis inductance at I n 1.5 mh J rotor inertia kg m 2 Attribution-NonCommercial-NoDerivs License

5 Fig. 4 Simulation results during three-phase short-circuit fault at 1 s and 4 krpm): phase currents Fig. 7 Simulation results during single-phase short-circuit fault on phase a at 1 s and 4 krpm): phase a current are shown in Fig. 6, where d-axis current overshoot during the transient) reaches a peak value higher than 100 Apk. Such high current along the d-axis may represent a risk for PM demagnetisation [15]. The steady-state values of d and q axes currents, after the threephase short-circuit, have been calculated according to 8) and 9) [3] I dsh = v2 r C PM L q R 2 + v 2 8) r L d L q I qsh = v r C PM R R 2 + v 2 9) r L d L q The obtained values I dsh = 57.2 A and I qsh = 8.2 A) are in agreement with those reported in Fig. 6. Fig. 5 Simulation results during three-phase short-circuit fault at 1 s and 4 krpm): electromagnetic torque 4.2 Single-phase short-circuit Keeping the same PMSM drive parameters, a single-phase shortcircuit asymmetrical fault) on the phase a has been injected at 1 s, with the rotor spinning at 4 krpm. Simulation results, regarding the steady-state and dynamic responses of the PMSM under the mentioned fault condition, are shown in Figs. 7 and 8. Fig. 6 Simulation results during three-phase short-circuit fault at 1 s and 4 krpm): dq current components Fig. 8 Simulation results during single-phase short-circuit fault on phase a at 1 s and 4 krpm): torque Attribution-NonCommercial-NoDerivs License

6 Fig. 9 Simulation results during single-phase open-circuit fault on phase a at 1 s and 4 krpm): phase currents Fig. 10 Simulation results during single-phase open-circuit fault on phase a at 1 s and 4 krpm): torque In particular, Fig. 7 depicts the trend of phase a current, and during the fault transitory the current reaches a peak of almost 80 Apk, whereas, in post-fault steady-state, the current amplitude is settled to 58.8 Apk 39 Arms). As shown in Fig. 8, the torque developed after the fault injection results highly oscillating, due to a pulsating torque component superimposed to the torque value developed during healthy condition. The pulsation torque component has amplitude equal to 18 Nm and its frequency 266 Hz) is twice the machine operating frequency 133 Hz). Such pulsating torque is a potential hazard for the drivetrain integrity and a protective control action might be required, in order to avoid or mitigate the mechanical damages. 4.3 Single-phase open-circuit In this section, the steady-state and dynamic responses of PMSM under single-phase open-circuit fault asymmetrical fault) are presented. As in the previous analysed cases, the speed used for the simulation is 4 krpm and the fault is injected into the phase a at 1 s. Since the stator winding is assumed wye-connected with floating neutral point and the current in phase a is null, due to the phase opening, the phase currents in phases b and c must be shifted by 180. The described situation can be observed in Fig. 9, where the phase currents response is depicted. After the fault injection, the currents in the healthy phases raise to 68 Apk 49 Arms), while the developed torque has a pulsation component, due to the unbalance introduced by the fault condition. The pulsating torque component is superimposed to the demanded torque in healthy condition 9.48 Nm) and it has 9 Nm amplitude and 266 Hz frequency twice the PMSM operating frequency), as shown in Fig Model validation The PMSM model taking into account short-circuit faults has been validated by comparing the simulation results with measurements Fig. 11 Fault-tolerant PMSM experimental setup for model validation Attribution-NonCommercial-NoDerivs License

7 6 Conclusions In this paper, two simple models develop with Modelica language and including winding faults in PMSM have been discussed. Simulations have been run, in order to verify the models feasibility in several fault conditions such as three-phase short-circuit, singlephase short-circuit and single-phase open-circuit. Considering the simulation results, it is possible to conclude that the presented models work properly under symmetrical and asymmetrical fault conditions. Finally, model has been validated through experimental tests, carried out on a fault-tolerant PMSM. Simulation and experimental results have been compared and a fair match between them has been pointed out. Fig. 12 Braking torque: comparison between experimental blue line) and simulation green dots) results 7 Acknowledgments This research was sponsored and financed by the European Project Actuation 2015 Modular Electro Mechanical Actuators for ACARE 2020 Aircraft and Helicopters), which is supported by the European Commission under the 7th Framework Programme. The authors are gratefully acknowledged. 8 References Fig. 13 Three-phase short-circuit current amplitude in steady-state: comparison between experimental blue line) and simulation green dots) results carried out on a fault-tolerant PMSM machine parameters are given in Table 1). The test bench adopted for performing the experimental tests is shown in Fig. 11. Designing details of the fault-tolerant PMSM and test bench features are given in [17]. During the experimental tests, three-phase short-circuits at the PMSM terminals) have been performed at several rotor speeds. For each test, braking torque and maximum short-circuit current have been measured, in steady-state conditions. The measured values have been compared with the simulation results obtained in the same operating conditions. In particular, the braking torque against the rotor mechanical speed is shown in Fig. 12, whereas Fig. 13 reports the shortcircuit current amplitude as a function of the rotor speed. Considering that some physical effects present in reality have been neglected in the PMSM model, a good agreement between experimental and simulation results is highlighted, from both the figures. In Fig. 13, the mismatch between experimental and simulation results increases, as well as the short-circuit current value increases. This behaviour is due to the magnetic saturation effect, which is not taken into account in the PMSM model. Indeed, the experimental values of short-circuit currents are lower than the simulation ones, since magnetic saturation depresses the current. 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