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to the loop and the second to the magnets inside the loop: Floop=Fs Fmag=Rloop φloop. (1) where Rloop is the loop reluctance matrix. The MMF Fs is linked to ...
Validation of a Magnetic Network-Based Dynamic Model of Permanent Magnet Linear Synchronous Machine Built by Finite Reluctance Approach C. Bruzzese, D. Zito

A. Tessarolo

Dept. of Astronautical, Electrical, and Energy Engineering University of Rome - Sapienza Rome, Italy [email protected]

Dept. of Engineering and Architecture University of Trieste Trieste, Italy [email protected]

Abstract—The finite reluctance approach is very well suited for fast modeling and dynamic simulation of permanent magnet synchronous machines with inverter feeding. This paper shows validating simulations, in both steady-state and transient conditions, carried out for a permanent magnet linear motor designed for on board ship load actuation. The basics of the method are resumed. The simulations obtained by using the finite reluctance approach are compared with FEM simulations, showing good agreement. Keywords—Dynamic model, dynamic simulation, linear machine, magnetic network, reluctance.

T

I. INTRODUCTION

he finite reluctance approach (FRA) has been recently introduced and detailed in [1]. The FRA takes inspiration from a more general theory about the finite formulation of electromagnetic field, developed by Tonti [2]. The theory in [2] is based on the concept of 'cell' as basic element for the decomposition of a given magnetic domain. Paper [1] exploits the concept of cell for a practical implementation in the framework of a finite-reluctance-based approach to the construction of magnetic networks useful for electrical machine dynamic simulations. The FRA allows easy model set-up and fast dynamic simulations of machines with complex air gaps and flux distributions, for which usually the more heavy and slow timestepping finite element method (TSFEM) is used. Surface permanent magnet machines or reluctance machines are naturally targeted for analysis by magnetic network-based models [3-7], and particularly by the method used here (FRA). These machines have usually very large and variable or irregular air gaps, very high teeth and magnets, and moreover important magnetic phenomena happen in regions outside the air-gap, especially in the large slots of machines with fractional-slot winding. The literature using magnetic networks for electrical machine simulation often relies on limiting assumptions such as: a) predefined paths for the magnetic flux are customarily set by the user; b) a clear and systematic rule for magnetic domain decomposition in elementary reluctance elements is often absent; c) the procedure for actual reluctance calculation is often unclear, especially in presence of non-rectangular elements or when motion effects must be taken in account. The discussion of all these aspects and their solution by FRA is presented in [1]. This paper instead shows dynamic simulations of a particular permanent magnet linear synchronous machine (PMLSM) designed for on board ship steering [8]. Steady state simulations are shown and compared with TSFEM results. Also dynamic simulations are shown, supposing the PMLSM

supplied by multiple inverters, and with a sinusoidal position control law. The FRA features (simple and clear model set-up, model flexibility, and fast simulations) are validated and will be used in future works for machine inverter control set-up and for analysis of machine faulty scenarios [9]. II. DESCRIPTION OF THE INVERTER-FED PMLSM A. Machine Description The PMLSM is a linear double-sided moving-magnet machine with surface-mounted magnets. The innovative machine has been appositely designed for rudder steering on board ship [8, 9]. The stator winding is a fractional-slot type, with twelve coils and ten magnets per side (Fig. 1). The PMLSM is thought for high-force low-speed operation.

Fig. 1. PMLSM with inverter feeding.

B. Inverter Supply and Control The PMLSM is subdivided in four magnetically and electrically independent units (modules), Fig. 1 [9]. Four inverters are used for module supply, connected to a common grid. The four inverters are individually self-controlled and only synchronized by means of the mover position feedback signal and the reference control signal. This arrangement grants the inherent machine modularity and structural redundancy, and, finally, the fault tolerant operation feature. Fig. 2 shows the vector control scheme used for the simulations reported in the following.

Fig. 2. Vector control scheme used for each inverter.

III. RESUME OF THE FINITE RELUCTANCE METHOD Only a fast resume of the theory of paper [1] is given here, for commodity of the reader. The reader will find in [1] a more complete treatise about the FRA method. A. Domain Decomposition and Reluctance Calculation The FRA is a special procedure which does not require a strict and customary pre-setting of the flux paths. The FRA is based on the definition of a complementary mesh interlaced with the reluctance mesh and on unique and unambiguous rules to carry out the reluctance mesh and for reluctance calculation. A tooth of the PMLSM (Fig. 3) is treated as a modular magnetic domain. A reluctance network can be obtained for this domain by the following steps: 1. A set of complementary nodes or co-nodes (C) is first chosen on the external boundary of the magnetic domain and in customary points inside, Fig. 3-a. The co-nodes define the 'complementary mesh'. 2. Auxiliary nodes (A) are added. These nodes are classified in horizontal nodes (AH) and vertical nodes (AV). A horizontal node is midway between an upper and a lower co-node, whereas a vertical node is midway between a conode on left and one on right. The auxiliary nodes define the actual size of the cells forming the 'cell-mesh' or 'material mesh'. In first approximation, a cell represents a rectangular piece of virtually homogeneous material. 3. A centroid can be found inside each cell, which is called fundamental node (or material node or simply 'node'), Fig. 3-b. The node is given two unique matrix indexes (i,j), and it is also given all the physical properties of the cell (permeability μ, coercivities Hcx, Hcy, turn density ns, and circuit index c). If the cell's material is not homogeneous, averaged properties can be used, and the cell can be treated anyway as an homogeneous piece. Motion effects can be thus easily embedded. However, a more refined model can be obtained in case of irregular cells containing different materials, if each of the above listed property is split in four components, as μx+, μx-, μy+, μy- for the permeability. The nodes constitute the 'nodal mesh', and each node is connected to the mesh through four reluctances and four MMFs due to cell's permanent magnet properties, Fig. 4. 4. Finally, the nodal mesh defines the flux loops as shown in Fig. 3-b. Each flux loop is always limited by four nodes, and corresponds to a unique central co-node.

Fig. 4. The cell (i,j) centered on the material node (i,j), and the four reluctances R1(i,j)-R4(i,j), the four magnet-produced MMFs Mm1(i,j)-Mm4(i,j), and the four current-produced MMFs Ms1(i,j)-Ms4(i,j) associated to the cell/node. Note that the cell is associated to an irregular element of the complementary mesh.

Fig. 4 shows a cell centered on a material node. The cell is bounded by four auxiliary nodes, and can be split in four subcells as shown in Fig. 5 for calculation of the four reluctances R1, R2, R3, R4 associated to the node (i,j). The four sub-cells are always rectangular, and represent pieces of virtually homogeneous materials with a uniform field inside. B. MMF and Voltage Balances Flux loop equations are used for the magnetic network. The loop MMF is split in two terms, one due to the currents linked to the loop and the second to the magnets inside the loop: F loop= F s F mag = Rloop φ loop

(1)

where Rloop is the loop reluctance matrix. The MMF Fs is linked to the stator currents is by: F s= N is .

(2)

The same connection matrix N links the stator flux linkages and the loop fluxes circulating in the magnetic network: t

ψ s=N φ loop .

(3)

The stator flux linkages are elected as state variables by: ψ˙ s=v s −R s i s

(i , j) (i , j)

R3(i,j)

(i , j)

(i , j+1)

(i , j+1)

R4(i,j)

(b)

(i+1, j) (i+1, j+1)

(d) (i+1, j)

Fig.3. (a) Identification of co-nodes (X) and horizontal (red) and vertical(green) auxiliary nodes, with forming of the complementary mesh (material mesh). (b) Identification of fundamental nodes (blue) as centroids of the complementary mesh and formation of the nodal mesh (reluctance mesh) with flux loops.

R1(i,j) (i, j)

(i , j)

(i+1 , j)

(i+1 , j)

(i , j+1)

(i , j)

(a)

(i , j)

(i, j+1)

(i , j)

R2(i,j) (i , j)

(i , j)

(i , j+1)

(i , j)

(i , j)

(4)

(i+1 , j)

(i+1, j+1)

(c)

Fig. 5. (a)-(d): The four sub-rectangles obtained splitting the cell (i,j), and used for calculation of reluctances R1(i,j)-R4(i,j).

where vs represents the terminal voltages and Rs is the resistance matrix. is is finally carried out by using: −1

t −1 i s=  N t R−1 loop N  ⋅ψ s − N Rloop F mag  .

(5)

C. Electromagnetic Force The electromagnetic force produced by the PMLSM is carried out by using the magnetic co-energy, and is made up of two components, i.e. the electrodynamic and the cogging force: F em=

Fig. 6. TSFEM model of the PMLSM simulated in no-load condition.

FRA

 

∂Wc d ψ mag  x d  mag  x =it − =F ed  F cog . (6) ∂x i s d x dx

The magnet energy can be obtained by solving the network in no-load conditions for any position x, and then using: mag  x=∫ 

2

H 1 d =∑ R b φ 2b 2 b 2

(7)

(a)

where the summation is extended to all the branches (index b).

TSFEM

IV. NO-LOAD STEADY-STATE SIMULATIONS AND COMPARISON WITH TSFEM In order to assess the validity of the proposed method, simulations by FRA have been performed and compared with TSFEM simulations. The FRA-based model was implemented in a Matlab 'script' whereas the TSFEM simulations were carried out in the Ansoft Maxwell environment. Fig. 6 shows the PMLSM finite element model (half machine, upper side), simulated in no-load condition (generator mode). Both the upper and lower machine sides were simulated with FRA. No-load steady-state simulation results are reported in Figs. 7-8. The simulated time was 500ms, with the PMLSM mover sliding at constant speed (1m/s). Thus the overall mover displacement simulated is 0.5m. The time-step in the TSFEM simulation was 1ms (fstep=1kHz), whereas the FRA time-step was much lower (0.025ms, fstep=40kHz). However, the same PC performed the FRA simulation (full machine) in 10min, and the TSFEM simulation (half) in 4h. Thus the FRA resulted 48 times faster than the TSFEM. Note that fstep could not be lowered in the FRA simulations, due to numerical stability issues. However an higher fstep grants an higher simulation fidelity especially when an inverter supply must be considered. In this sense, the speed gain of FRA with respect to TSFEM is as high as 40*48=1920.

(b) Fig. 7. No-load stator voltages vs. a): FRA. b): TSFEM.

FRA

(a)

TSFEM

The stator voltages in no-load conditions (or back-emfs) evaluated with FRA (Fig. 7-a) are quite close to those evaluated with TSFEM (Fig. 7-b). A good comparison is also obtained between the no-load stator flux linkages carried out with FRA (Fig. 8-a) and with TSFEM (Fig. 8-b), respectively. In no-load (open circuit) conditions, these flux linkages are entirely due to the permanent-magnet field, i.e. ψs=ψmag. The cogging forces simulated with FRA and with TSFEM were also compared, Figs. 9, 10. From (6), the cogging force can be evaluated by using the overall network energy ξmag in no-load conditions (energy due to the permanent magnets). Fig. 9 shows the PMLSM magnet energy calculated by FRA, by using (7), as a function of the mover position x. Since the space derivative of ξmag is needed, to avoid errors due to the evident numerical noise, the high-frequency noise has been filtered away in Fig. 9. Fig. 10-a shows the result when the filtered energy is space-derived (cogging force by FRA). Fig. 10-b shows the cogging force by TSFEM, and the comparison is quite satisfactory, especially the peak values appear very close.

(b) Fig.8. No-load stator flux linkages (ψs=ψmag). a): FRA. b): TSFEM.

V. ON-LOAD DYNAMIC SIMULATIONS WITH POSITION CONTROL The scope of the following simulations is mainly a demonstration of the FRA potentialities when dealing with a complex simulating task, such as that involving an inverter-fed machine used for a position control. Two working conditions in the following are investigated. The first concerns the beha-

FRA

MOVER DISPLACEMENT x (m) Fig. 9. Simulation by FRA, magnet energy ξmag as function of x. The bold curve is obtained by filtering away the high-frequency numerical noise.

actuates a rudder, while the ship is considered moving forward with 50% of the maximum ship speed. In this condition, the hydrodynamic load force acting on the rudder surface can be considered roughly proportional to the rudder steering angle and thus to the mover displacement with respect to the center position (x=xmax/2 in Fig. 11). Thus the load force seen by the PMLSM acts as an elastic force with equilibrium point at x=xmax/2. The load force will be maximum for x=0 and x=xmax, and will amount to 50% of the PMLSM rated force in the considered ship half-speed condition. The vector control scheme used for inverter control in the simulations was that shown in Fig. 2. The FRA approach was used with fstep=360kHz. The simulated time was 1.6s, whereas the simulation time was 4h (240min). No TSFEM simulations are available for comparison, since the simulation time was prohibitive. The second simulation scenario concerns the transient behavior of the machine when a load force step is applied. The starting load condition is the same as previously, however a force step (50% rated) is suddenly imposed, so raising the overall load force to 100% rated, and the effect on the machine variables is evaluated.

FRA

MOVER DISPLACEMENT x (m) (a)

600

Fig. 11. The PMLSM drives a rudder, following a cosine-shaped position law. The load force is represented by the hydrodynamic force acting on the rudder during the steering, and is modeled like an elastic force [9].

TSFEM

FORCE(N)

400

A. Sinusoidal Position Control With 50% of Rated Load Figs. 12-17 show simulation results obtained by FRA. Fig. 12 shows the first quarter of the cosine-shaped reference position law, which is:

200

x ref = X ref 1−cos x t 

-200 -400 -600 MOVER DISPLACEMENT x (cm) (m) (b)

Fig. 10. Cogging force Fcog. a): FRA. b): TSFEM.

vior of the machine when the PMLSM is accomplishing a normal task of position control under a relevant load (50% rated). Fig. 11 shows the overall drive simulated. The PMLSM drives a tiller via a rotary-prismatic coupling. The tiller

(8)

where Xref=xmax/2=0.25m and ωx=smax/Xref.The maximum speed of the machine smax is set to 0.25m/s. As shown in Fig. 12, the simulated time is for 0