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Modeling of Grid-connected Superconducting Synchronous Machines
Loïc Quéval a, Masaki Sekino b and Hiroyuki Ohsaki a a Department of Advanced Energy, Graduate School of Frontier Sciences, The University of Tokyo, Japan
b Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Japan
Abstract — We developed an accurate, efficient and flexible modeling method for the steady-state and transient analysis of grid-connected superconducting synchronous machines. On the one hand, the machine flux linkages and electromagnetic torque as function of operational parameters are obtained from static finite element analysis and stored in lookup tables. On the other hand, the machine is represented by a lumped-parameter phase-domain model. The potential of the method is demonstrated by simulating a 10 MW superconducting wind turbine generator connected to the grid through a AC/DC/AC converter under fault condition.
Keywords — Superconducting synchronous machine, modeling and simulation, phase-domain model.
I. INTRODUCTION
For conventional machines, the classical qd model based on the well-established general machine theory is able to calculate the steady-state and transient characteristics of a grid-connected machine with good precision. However, it is not always reasonable to adapt such methods for the analysis of HTS machines. For instance, typical designs for superconducting generator are air-cored with racetrack coil field windings and distributed armature windings supported by non-magnetic material [2]. In this case, typical approximations such as a sinusoidal distribution of the field winding cannot be made, and consequently the qd model is not adapted. A realistic model should notably be able to correctly take into account the space harmonics resulting from the absence of a shaped iron core. In this paper, we examine the possibility of modeling superconducting synchronous machines using the phase-domain (PD) model in association with the machine parameters obtained from a static finite element (FE) analysis. Firstly, the phase-domain mathematical model is introduced. Secondly the extraction procedure of the machine parameters from the FE analysis is detailed. Then, we show how the coupled FE phase-domain model was set up. Finally the flexibility of the method is demonstrated through simulating steady-state and transient phenomena of a grid-connected superconducting wind turbine generator.
II. PHASE-DOMAIN MODEL
It is assumed that iron losses can be neglected. In motor convention, the phase-domain model is commonly expressed in terms of physical variables as,
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v = R[ ] i +dλdt
(1)
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λ = λ(θ, i ) (2)
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J d2θ
dt 2=Tm +Te +Tdamp (3)
€
Te =Te (θ ,i) (4)
where
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v ,
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i and
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λ are respectively the instantaneous voltage, current and flux linkage vectors of the windings. [
€
R] is the resistance matrix of the windings. The flux linkage
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λ is a function of the rotor mechanical angle
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θ and the winding currents. J is the moment of inertia, and Tm, Te and Tdamp are respectively the mechanical torque, the electromagnetic torque and the damping torque. The electromagnetic torque Te depends on the rotor mechanical angle and winding currents. Note that in the general case, the flux linkage and the electromagnetic torque are not explicitly given, as they are nonlinear functions of rotor position and winding currents. When both flux linkage and electromagnetic torque are function of the rotor position only, a simple expression can be derived for them using the concept of inductance. In this case, (2) and (4) can be rewritten,
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λ = L(θ )[ ] i (5)
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Te =12iT∂ L(θ )[ ]∂θ
i (6)
where [
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L(θ ) ] is the stator and rotor self and mutual inductance matrix and depends on the rotor mechanical angle
€
θ . The flux linkage, electromagnetic torque and/or inductance matrix can be determined in various ways following the application: analytical formula, finite element calculation or measurements. In this paper, we use the finite element analysis.
We underline that working directly in the physical phase-domain model simplifies the interfacing of the machine model with the power system network as it is not necessary to go through complex parameter transformations as in the case of the qd model. Moreover internal faults in synchronous machines can be simulated by minor modifications of the equations [4].
II. FINITE ELEMENT PARAMETERS CALCULATION
By way of example, the air-cored HTS 10 MW class wind turbine generator with double layer distributed three-phase armature windings [2] shown in Fig.1 and Fig.2 is analyzed. As this design has no back iron, (5) and (6) describe correctly the machine. The rotor angle dependant inductance matrix can be accurately evaluated through static finite element analysis [5]. The stator and rotor self and mutual inductance matrix of this machine is in the form of,
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L(θ ) =
Ls M s M s Mar (θ )Ms Ls M s Mbr (θ )Ms Ms Ls M cr (θ )
Mar (θ ) Mbr (θ ) Mcr (θ ) Lr
(7)
Thus for this generator, only 3 constant inductances and 3 rotor angle dependant inductances have to be determined to characterize completely the machine. Constant inductances values are summarized in Table I. The rotor angle dependant inductances
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Mar (θ ) ,
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Mbr (θ ) and
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Mcr (θ ) are calculated from 0 to 60 degrees in steps of 2 degrees and are plotted on Fig.3. It should be emphasized that with this method, the calculation of the machine inductance matrix is performed only once for a given machine design. Afterwards, for any operating conditions, the machine steady-state and transient characteristics can be determined. Thus in comparison with a full FE model, it is particularly efficient and adapted for the design and analysis of grid-connected machines.
TABLE I GENERATOR PARAMETERS
Parameter Value Nominal power 10 MW Nominal phase-to-phase voltage 5 kVRMS Nominal frequency 1 Hz Armature self inductance Ls 0.0444 pu Armature mutual inductance Ms -0.0174 pu Field self inductance Lr 3.819.10-5 pu Armature resistance Rs 0.008 pu Field resistance Rf 0.4 pu Inertia constant H 0.62 s Friction factor F 0.01 pu Pairs of poles P 6
Fig. 1. Overview of the generator model
Fig. 2. Cross section of the generator model
Fig. 3. Rotor angle dependant inductance Mar, Mbr and Mcr.
III. COUPLED FE PHASE-DOMAIN MODEL
The phase-domain voltage, flux linkage (5), electrical torque (6) and mechanical equation of a three-phase synchronous machine have been implemented using Matlab/Simulink blocs and the SimPowerSystems toolbox. It allows performing simulation of synchronous machines connected to external mechanical and electrical systems with associated control strategies. The rotor angle dependant inductance matrix is obtained from FE analysis. Its derivative is numerically calculated using the central difference approximation. Both matrix are stored in lookup tables and retrieved in terms of rotor angle.
IV. APPLICATION EXAMPLE
A realistic estimation of the performances of a superconducting machine should include internal electromagnetic phenomena and correct identification of the machine parameters, as well as take into account the integration of the machine into the rest of the power network including control strategies. The two first points have been addressed by the proposed coupled FE phase-domain model. We show here that the interaction of the machine with the power network can be correctly characterized using the phase-domain model. As an example, we build a complete simulation model of a superconducting wind turbine generator connected to the grid through an AC/DC/AC converter. The wind turbine system’s steady-state operation and dynamic response to a voltage sag resulting from a remote fault are observed.
A. Description
An overview of the system is shown in Fig.4. The mechanical power Pmech extracted from the wind by the wind turbine is given by,
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Pmech =12ρ Cp (λ ,β) A vw
3 (8)
where
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ρ is the air density,
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Cp is the power coefficient which is a function of both tip speed ratio
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λ and blade pitch angle
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β , A is the turbine swept area and
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vw is the wind speed. The turbine efficiency is controlled by a pitch angle control system to ensure that the reference optimum power will not exceed the rated power of the generator. The drive train is a direct-drive two-mass model.
The generator used in this simulation is the 10 MW superconducting synchronous generator described in section II. The field current is regulated to 1 pu by a simple PI controller. The parameters of the generator are summarized in Table I. The wind turbine generator is connected to the grid through an AC/DC/AC power converter. The converter topology and the control design implemented are similar to the ones proposed by other authors for large scale wind turbine generators [6], [7]. The parameters of the AC/DC/AC converter are summarized in Table II. The AC/DC converter is a three-phase uncontrolled full-wave rectifier bridge in series with a DC/DC boost converter. The objective of the
DC/DC converter control is to maximize the power extracted by regulating the generator speed to its optimal value (MPPT). The DC/AC converter is an IGBT-based PWM inverter. The objectives of the DC/AC converter controls are to guarantee that all the power coming from the rectifier is instantaneously transferred to the grid by keeping the DC-link voltage constant, and to actively support the grid by regulating the reactive power fed to the grid. Finally, the generator is connected through a transformer to a 30 km, 25 kV transmission line and exports power to a 120 kV grid.
Fig. 5. Stator currents in steady-state analysis and harmonics
B. Steady-state Analysis
In steady-state conditions, the wind speed is considered to be constant at 15 m/s, the pitch angle is 8.8 deg to maintain the generator speed at 1 pu, the output active power at 10 MW, the DC voltage is regulated at 9000 V and the reactive power is kept at 0 Mvar. The generator stator current and its harmonic content are shown in Fig.5. Harmonic content is the superposition of space and time harmonics. It is clear that harmonics injected on the generator side of the AC/DC/AC converter cannot be neglected at design level.
Fig. 4. Simulation overview of a superconducting wind turbine generator connected to the grid through an AC/DC/AC converter.
Fig. 6. Simulation of the wind turbine system transient analysis.
C. Transient Analysis
To show the versatility of the proposed method, the system recovery after a remote fault is simulated. Initially the system is in steady-state conditions as described above. At t=0.4 s the positive-sequence voltage of the 120 kV grid suddenly drops to 0.5 pu causing an increase of the DC-link voltage Vdc, a decrease of the generator speed and a drop of the wind turbine output power P. Interestingly the stator current rises to almost 2 pu. After the fault elimination at t=0.5 s, the DC-link voltage, active power and reactive power are regulated to their steady-state values. The generator armature current and voltage, generator speed, DC-link voltage, active power and reactive powers fed to the grid, and grid voltage waveforms are shown in Fig.6.
TABLE II
CONVERTER PARAMETERS Parameter Value DC/DC converter resistance Rboost 5 mOHM DC/DC converter inductance Lboost 0.0012 H DC link capacitor C 0.9 F Rated DC voltage 9000 V Nominal output phase-to-phase voltage 575 VRMS Nominal output frequency 60 Hz
D. Discussion
We underline that both space and time harmonics are taken into account in these simulations. As opposed to the classical qd model, the phase-domain is able to efficiently take into account the inductance space harmonics. Other authors have shown that the coupled finite element phase-domain model is as accurate as a full finite element model [8]. But it is worth noticing that it would be challenging and computationally expensive to run such a simulation in a full finite element software environment.
IV. CONCLUSION
The proposed method permits to calculate steady-state and transient performances of grid-connected superconducting synchronous machines. The accuracy and efficiency of the model are obtained by the combination of the finite element model and the phase-domain model. The flexibility of the method is demonstrated by simulating a grid-connected superconducting wind turbine generator under fault condition.
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