Application of 3-D lumped parameter model in MHD generator design

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 1, MARCH 2000

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Application of 3-D Lumped Parameter Model in MHD Generator Design Alberto Geri, Guiseppe Maria Veca, Senior Member, IEEE, and Alessandro Salvini

Abstract—Magnetohydrodynamic (MHD) energy conversion consists of electrical energy generation directly from high enthalpy of ionized gas. MHD generators are devices having a channel in which ionized gas (plasma) is blown, where a high magnetic field is present. The interaction between the conductive gas and the magnetic field makes the conversion. To describe and to design the MHD plants, circuital models of MHD generators are advantageous for an easier investigation approach to the interface between generators and power plants; they become extremely useful when a 3-D analysis is necessary for easier computation taking into account 3-D effects due to spatial distribution of electromagnetic fields. In this paper we describe some design applications of the three-dimensional (3-D) lumped parameter model for MHD generators. Index Terms—Circuit model, energy conversion, generator, inverter, magnetic field computation, MHD, SC Magnets.

I. 3-D LUMPED PARAMETER RECALL HE 3-D circuital model developed by the authors and its validation tests have been described in several papers [1]–[6]. Here the model is briefly recalled before showing the way in which it may be used for the estimation behavior of MHD generators when excited by nonuniform magnetic fields (owing to the presence of industrial size magnets) or to design their connection to inverter groups [6]. The advantage of the 3-D lumped parameter model resides in both its three-dimensional and its circuital nature:

T

1) 3-D nature is effective for investigations in which 3-D effects appearing for gas conductivity, Hall parameter and, in particular, magnetic flux density having a spatial distribution, must be taken into account. 2) Circuital nature becomes helpful to designers in applications such as the interface between generators and power plants, parallel between MHD group and alternator, stability analysis and similar application. In all these studies, in fact, it is very useful to represent MHD generators by means of an equivalent Thevenin bipole [7]. 3) Circuital approach permits the minimization of computation costs due to implementation of differential set of equations in 3-D which describe the MHD process in Manuscript received March 23, 1998; revised July 29, 1999. A. Geri and G. M. Veca are with the Dipartimento di Ingegneria Elettrica, Universitá di Roma “La Sapienza,” Via Eudossiana 18, 00184 Rome, Italy. A. Salvini is with the Dipartimento di Ingegneria Elettrica, Universitá Degli Studi Roma Tre, Via Della Vasca Navale 84, 00146, Rome, Italy. Publisher Item Identifier S 0885-8969(00)02216-6.

Fig. 1. Equivalent electrical network of each brick of the plasma. One or more ideal voltage source components can be absent. The parameters are defined by Resistances: i = (1= ) (1i=21j 1k ); Faraday generators: Ei = [vj Hk vk Hj ](0 1i =2); Hall generators: EHi = ( =H )[Jj Hk Jk Hj ](1i=2); with i = x; y; z , j = y; z; x, and k = z; x; y . Where 1i are the edge lengths, J (x; y; z ) v (x; y; z ) and H (x; y; z ) are the current density, the gas velocity and the magnetic field vector respectively.  (x; y; z ) is the electrical conductivity; (x; y; z ) is the Hall parameter and 0 = 4 1007 [H/m].

R

1

0

0

0

an analytical way (equation of momentum, equation of energy, equation of mass conservation [1]). The proposed 3-D electrical network is shown schematically in Fig. 1. In the active region, ionized gas is modeled by small volumes (bricks) containing an electrical node at each center of gravity (indicated with P (x; y; z ) in Fig. 1). Six electrical branches, parallel to the axes of a Cartesian space x; y; z , start from this node. On each branch a resistance, a controlled source and an independent source are present. Passive components take into account all electric macroscopic characteristics of the plasma, independent voltage sources take into account Faraday’s law (when ionized gas flows through a magnetic region an electrical field and a corresponding voltage, both arise), while controlled voltage sources are according to the Hall effect (when current flowing in the plasma interacts with excitation magnetic flux density generates an additional electrical field). Finally, since there are two categories of MHD generators: linear and disk, so called because of their channel shape, the model operates the partitioning and the parameter evaluation in a different way for the two cases [1].

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 1, MARCH 2000

Fig. 2. Advanced designs of some multiple dipole magnetic structures [8].

II. MAGNET DESIGN In MHD energy conversion, the amplitude of magnetic flux density should not be lower of 4 T, up to about 8 T, to obtain a reasonable value of electrical power. Such a high field can only be achieved by means of superconductive (sc) magnets because Joule dissipation could reduce the efficiency to negative values. If we consider the MHD linear generator, active length can exceed 10 m (up to 24 m in some cases) where greater electrical power is required to be produced. Thus, multiple dipole sc magnets, reproduced in Fig. 2, have been carried out projected by designers, to allow their transportation from their building to the place where an MHD generator plant is located. These magnets have a complex magnetic structure which, in spite of the efforts of the designers, creates an imperfectly orthogonal to channel axis field. Moreover the field does not have a constant value along the main axis but decays from inlet to outlet. These last two undesired characteristics cause decay of efficiency. In fact, the nonorthogonal to the main axis components of the magnetic field interact with the velocity vector creating undesired electromotive forces, while the reduction of the amplitude of the field along the channel causes a reduction of the amplitude of the emfs which create conversion. The 3-D circuit model [1]–[6] can help designers in the computation of the influence of magnetic field irregularity on MHD linear generator performance. This is fundamental for magnet shape optimization. Active and passive parameters of the circuit model (ideal voltage sources, as independent or controlled, and internal gas electrical resistances) can easily be updated by calculating the actual value of each component of the magnetic flux density vector (due to both the external magnet and the armature reaction [12], even if this last component is negligible) at each electrical node of the network [1]–[6].

Fig. 3. (a) ORM magnet, (b) filiform model of the sc dipole magnet; evaluation of the magnetic flux density components generated by the magnet, (c) along the axis (for = 0 and = 0), and (d) along the axis (for = 8 and = 0).

Y Z

X

Z

X

Y

A. 3-D Maps of Magnetic Field Component To obtain the numerical value of magnetic field components of each magnet structure, a preprocessor has been coupled to the code which builds and solves the network. Next, electrical power decay has been estimated quantifying the ratio between actual values of electrical power with those evaluated assuming an ideal magnet which can generate one only field component along the vertical axis, z and having a perfectly flat profile over all active length. The 3-D spatial distributions of the magnetic field vector components (Bx; By and Bz ) have been determined by means of a calculation code developed by the authors and linked with 3-D circuit model implementation [1]–[6]. The prepocessor is suitable for the computation of a 3-D field map due to sc coils having complex geometry within the active region of MHD channel. The magnetic field calculation procedure is thus composed of three steps: 1) simulation of each magnet coil with a filiform shape of the magnet [see Fig. 3(b)], 2) computation of magnetic field at each assigned point having coordinates internal to the active region of the MHD channel, 3) transfer of computed values to a data base used by the plasma simulator (circuital code). Thus, the designer only has to write an input data file in which he indicates elementary turns arranged in order to approximate the actual shape of the magnet. ~ , can be thus computed, by suThe magnetic flux density, B perimposing the contribution of each k th segment evaluated by means of Biot-Savart integral formulation which has been written in (1) and (2). These last equations refer to the evaluation

GERI et al.: APPLICATION OF 3-D LUMPED PARAMETER MODEL IN MHD GENERATOR DESIGN

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TABLE I POWER DECAY VERSUS 3-D MAGNETIC FIELD MAP

Fig. 4. Explanation of symbols in (1) and (2).

Fig. 5. Per unit active component of the magnetic field along the channel axis for rectangular saddle shaped magnet [1]: (I) ideal magnet with flat profile all over the active length, (II) SMSS, (III) NSMSS, (IV) ORM, (V) 3-DMM, (VI) 3-DMMCIC), and (VII) 2DMM.

of the magnetic flux density vector due to a generic ith segment, connecting the points Ai and A(i+1) , of the k th elementary turn (see Fig. 4). This segment carries the current Iscm; k equal to the ratio between the current injected into the sc magnet and the number of the elementary turns. 7

~ i; k (P ) = Iscm; k 1 10 B DM P

where: ~ i (P ) =

D

Ai; M

DAi; P

+

DA(i+1); M DA(i+1); P

1

~ i (P )

(1)



~Ai; A(i+1) d

2

d~P; M



:

(2) From the above it can be seen that the total magnetic field in each point P is espressed by (3). ~ (P ) B

XX k

~ i; k (P ) B

(3)

i

In Fig. 5, the trends of the component along the z axis of the magnetic flux density versus active length (y axis) have been reproduced (the values are in per unit). The evaluation has been repeated for each sc magnet type indicated in previous Fig. 2. All simulations refer to an MHD linear generator device having: inlet channel cross sections = 0:84 2 0:84 m outlet channel cross sections = 1:96 2 1:96 m active length = 16 m

Fig. 6. Sketch of the sc coil partitioning for the magnetic field computation: (a) cross section partitioning and (b) segmentation of the generic elementary turn. (0; 0:460; 0:775); L = magnet width; H = magnet Numerical values: C height NL = number of division of L; nH = number of division of H ; Zc ; rc = quote and radius of the magnet center of gravity C , respectively.



coil cross section = 0:63 2 1 m maximum field on the axis = 6 T: Next, in Table I, for each linear generator type (Faraday, diagonal, Hall), the various ratios between electrical power obtained with different sc magnets and both the value obtained exciting each device by means of an ideal flat profile and by means of the 3-D magnetic field map due to ORM are summarized. This is because the ORM sc magnet type is the actual magnet which can create a field closer to the ideal than the other types. The described procedure of field calculation represented by the use of (1)–(3) can easily be adapted for disk generators. In Fig. 6 an indication of the procedure is shown. In Fig. 7(a) and (b) some results are tabulated. III. CONNECTION OF MHD GENERATORS TO HV AC LINES The 3-D circuit model has also been used to investigate the behavior of the device when it is linked to HV AC lines by means

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 1, MARCH 2000

in comparison with the inverter characteristic fluctuation time (1 ms) and in the dc side of the inverter bridge smoothing inductance are posed to reduce the ripple of the MHD current. A. Determination of Thevenin Representation by the 3-D Lumped Model

Fig. 7. Evolution of magnetic flux components radial and vertical in the active channel region: (a) inlet disk channel and (b) outlet disk channel. Numercial (0; 0:460; 0:775); L = 0.34 m; H = 0.53 m; values used in simulation: C nL = 10; # = 3 ; Iscm = 1:045 107 A.



of inverter units [7]. In these last analyses it would be very convenient to have a Thevenin representation of the MHD generator available between the electrodes from which electrical power is taken. In this case a 3-D lumped parameter model can rapidly give the requested equivalent Thevenin bipole (voltage source and internal resistance shown by MHD generator) automatically taking into account 3-D effects (e.g., magnetic field) and more easily than by the direct integration of the MHD basic set of differential [9]. In addition, limiting the present observation to electrical parameter (assuming the fluid-dynamic quantities to be invariant, as a first approximation), we must consider that the electromotive forces within the plasma due to the Hall effect change their value if the external load changes. This is because of the dependence of the Hall generator to the interaction between internal currents and external magnetic field. As a consequence, Thevenin representation of an MHD device changes if the external load conditions are changed. In a more accurate observation, we must say that the variation of the Thevenin bipole values is also due to the dependence of conductivity and Hall parameter versus pressure and temperature. These last values, in fact, depending on enthalpy extraction during energy conversion, depend on external load conditions. However, in stationary conditions, the equivalent Thevenin bipole can be determined assuming the plasma internal values to be constant and equal to those of a fixed external condition. A stationary dc regime is correctly assumed if the external loads are of simple resistance, but the model can be adopted also if the external load is, for example, an inverter bridge connected to ac lines. In fact, the characteristic time of plasma discharge is smaller (100 s)

The lumped parameter 3-D model can numerically determine both internal resistance, Ri , and the open circuit voltage, V0 , seen between the analysed two load electrodes, of the MHD generator. For evaluating Ri it is necessary to solve the 3-D network in two different ideal electrical situations of the electrodes that we want to model: open circuit (evaluating V0 ) and short-circuit (evaluating the current flowing between electrodes, Isc ). In this case we have Ri = V0 =Isc . But, because of the variability of the electrical parameters of the network according to the load conditions, we cannot work without introducing some assumptions. As a first step, the 3-D lumped parameter model could be solved by imposing a load value closer to those that we think apply to the Thevenin model in subsequent analysis. As a second step, the network can be solved again twice with the electrode pair in open circuit, and in short-circuit. But in these last two solutions the controlled voltage sources present in the branches of the network [1]–[6] and taking into account the Hall effect should be considered independent voltage sources, their value determined by the internal currents obtained in the first step of the calculation. Alternatively, it is possible to solve the network, as usual, twice by means of two different but very close values of resistance load. The proximity of the load values of the two simulations allows us to ignore the previous cited Thevenin parameter variation. It is evident that this last method could introduce numerical problems if the resistances chosen are too closely. B. Application Example A scheme which refers to an analysis concerning the Ryazan plant [10] is shown in Fig. 8. The simulation of the generator has been finalized to determine the condition that the inverter group has to have for an assigned output power. The main characteristics of the simulated generator, which allows us to determine the parameter of the equivalent 3-D lumped parameter model, are listed below: MHD generator type: diagonal type, having 1200 electrodes, inlet section 1:2 2 1:2 m2 , inlet temperature = 2650 C, inlet pressure = 0.9 Mpa, velocity = 1300 m/s, outlet section of 2:4 2 2:4 m2 , max. flux induction 6 T, total length of 24 m and electrical power of 270 MW; gas type: burnt gas from fossil fuel, seeded with 2% of K2 CO3 ; inverter units: 4 unit of 12-pulse line-commutated type and with 4 subunits in parallel to limit the current having rated voltage of 20 kV and rated current of 720 A; transformers: 4 units each for an inverter group (see Fig. 8) of rating power 300 MVA, with plant ac busbars = 20 kV, high voltage = 380 kV, and short circuit Vsc% = 12%. The sc magnet has been considered to be of ORM type (see Figs. 2 and 3). C. Numerical Computation Indicating the rms voltage at the ac side of each inverter with

Vr and with Ir the rated current of an inverter unit, we can

GERI et al.: APPLICATION OF 3-D LUMPED PARAMETER MODEL IN MHD GENERATOR DESIGN

Fig. 8.

Scheme of MHD and its inverter groups.

Fig. 9. MHD electrical power converted versus SCR firing angle [10], [11].

deduce the load voltage EL , on the dc side, seen from the MHD electrodes by:

"p

3 2 V cos( 0  0  ) 0 3!LS K I EL = K1  r  2r

#

(4)

where the factor K2 = 4 indicates that there are 4 inverter groups (as described in Fig. 8), the factor K2 = 4 takes into account that there are 4 inverter bridges in parallel for each inverter group in series. Furthermore,  is the firing delay angle, Vr and Ir are the rated voltage and current of only one inverter unit, ! is the sinusoidal pulsation (=100 at 50 Hz) while  is the angle due to the presence of an inductance Ls on the ac side due to the leakage inductance of the transformer (in calculation we have assumed Ls = 0:4 mH). The angle  is function of  and is determined by:

 =  0 a cos

"

p

0Vr cos  + !Ls 2K2 Ir Vr

!#

0 :

(5)

The power converted dc/ac:

PMHD = ELK2 Ir

83

(6)

has been assumed equal to the value of 270 MW [11]. The simulation which has been carried out has enabled us to determine the Thevenin parameters of the MHD generators (assumed as pure dc generators, i.e., assuming the smoothing inductance on dc side (see Ld in Fig. 8) ideally infinite. The values obtained are: VMHD = 125 kV (open cycle voltage) and Ri = 10:8

(internal resistance). By means of this, Thevenin representation, it has been possible to evaluate the value of the SCR firing angle referred to the 270 MW of MHD electrical power. Fig. 9 reproduces the trend of the electrical MHD power versus SCR firing angle. Because the trend has been calculated by assumed invariant the Thevenin representation the range of its validity must be assumed reliable only for a value quite close to 148.7 . Because of the variation of Thevenin parameters with changed load conditions, a family of curves, valid in small ranges around previously determined operation point should be considered.

IV. CONCLUSIONS The results of the simulations executed confirm that the 3-D model developed by the authors is suitable to help the engineer in the design of the device, i.e., when the influence on electrical performance of the generator has to be estimated as a function of the effective 3-D map of pressure, temperature and magnetic flux density, or when a circuit approach can give an useful instrument to represent the electrical behavior of the device.

REFERENCES [1] A. Geri, A. Salvini, and G. M. Veca, “Performance evaluation of MHD generators: The lumped parameter model and its validation,” IEEE Trans. on Energy Conversion. [2] , “MHD linear generator modeling,” IEEE Trans. on Applied Superconductivity, pt. I, vol. 5, no. 2, pp. 465–468, June 1995. , “Magnet for MHD linear generator,” IEEE Trans. on Magnetics, [3] pt. II, vol. 30, no. 4, pp. 1847–1850, July 1994. [4] A. Geri, A. Salvini, G. M. Veca, and B. Zaporowski, “Comparison between numerical integration and 3-D circuital approach for analysing the performance of MHD linear generators,” in Symposium on Engineering Aspects of Magnetohydrodynamics (SEAM 33nd), Tullahoma, TN, June 13–15, 1995, pp. IV.3-1–IV.3-8.L. [5] A. Geri, A. Salvini, and G. M. Veca, “A numerical computation of the current density vector and the Lorentz Force in a MHD disk generator,” Progress in Astronautics and Aeronautics, vol. 182, pp. 497–507, 1998. , “Effects of magnetic field characteristics on SC MHD linear gen[6] erator performance,” Latvian Journal of Physics and Technical Sciences, vol. 3, pp. 10–16, 1996. [7] Mohan et al., Power Electronics: J. Wiley & Sons, 1989. [8] G. M. Lia, I. Montanari, and P. L. Ribani, “Design of superconducting MHD saddle shaped magnet using an optimization procedure,” in Conference Proceedings International Workshop on MHD Superconducting Magnets, Bologna, Italy, Nov. 13–15, 1991, pp. 9–13. [9] R. J. Rosa, Magnetohydrodynamic Energy Conversion. Washington: Hemisphere Publishing Corporation, 1987. [10] A. E. Scheindlin, “Ryazan pilot MHD power plant,” Magnetohydrodynamics, vol. 2, no. 2-3, pp. 117–124, 1989. [11] A. Geri, A. Salvini, and G. M. Veca, “Connection of MHD generator to high voltage AC lines,” in Conference Proceedings UPEC ’96, vol. 1, Iraklio, Greece, Sept. 18–20, 1996, pp. 322–325. [12] L. De Rosa, A. Geri, and G. M. Veca, “Three-dimensional magnetic field analysis in an MHD device using a discrete element model,” Journal of Magnetism and Magnetic Materials, vol. 101, pp. 283–285, 1991.

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Alberto Geri was born in Temi, Italy, on August 4, 1961. He received a University degree in Electrical Engineering from the University of Rome “La Sapienza” in 1987. He began academic activity in 1989 as a Researcher of electrical science at University of Rome “La Sapienza.” He was appointed Professor of electrical science at the Military Academy of Rome “Scuola Trasporti e Materiali” in 1989. He was appointed substitute Professor of electrical science at the University of Rome “La Sapienza” in 1993. He began research activity in 1982 and his interests include MHD energy conversion, low frequency electric and magnetic field computation, high frequency magnetic device modelization, and nonlinear electromagnetic problems related to lightning protection systems and grounding systems. These activities are described in more than seventy papers presented at international conferences or published in various international journals.

Giuseppe Maria Veca (SM’88) was born in Rome, Italy, on September 27, 1942. He received a University degree in electrical engineering from the University of Rome “La Sapienza” in 1966, and in nuclear engineering from the University of Rome “La Sapienza” in 1969. He began academic activity in 1967 as an Assistant Professor. In 1973, he became Associate Professor of electrical science. In 1986, he became a full Professor of electrical science at the University of Rome “La Sapienza.” He was elected an IEEE Senior Member in 1988. He has extensive experience in superconductivity applied to power systems, problems related to MHD application, and in EMC/EMI analysis. These activities are described in more than ninety papers presented at international conferences or published in various international journals.

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 1, MARCH 2000

Alessandro Salvini was born in Rome, Italy, on August, 16 1962. He received the University degree in Electrical Engineering from the University of Rome “La Sapienza” cum laude. Since 1993, he has been engaged in the teaching of electrical engineering to officers of the transport corps of the Italian Army at Military Academy of Rome “Scuola Trasporti e Materiali.” In 1994, he was nominated Researcher at the University of “Rome Tre” where he began his academy activity. His scientific interests include applied superconductivity, MHD energy conversion, magnetic device modelization, and protection and safety devices. His scientific activity is evidenced by more than twenty papers published in international journals and conference proceedings.