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Optimized Design of Salient Pole Synchronous Generators O.W. Andersen Steinhaugen 43, N-7049 Trondheim, Norway [email protected] http://home.c2i.net/owand/ ABSTRACT A program for optimized design of salient pole synchronous generators, Synop, was made by the author for mainframe computers as early as in 1970. It has been improved continuously since that time and now runs on personal computers. It covers all sizes of air cooled machines from small two pole generators of a few kilowatts up to the largest hydroelectric generators. If the penalty function has the proper form and the input data make it possible, the penalty is reduced to zero as the criterion function cost+penalty is minimized. In this way the technical requirements are met, without any need for the program to be told specifically how this is to be accomplished. Keywords: Optimized design, synchronous generators, salient poles. 1 INTRODUCTION The objective is to design to the lowest cost, including the cost of losses, at the same time meeting all performance requirements automatically. A few thousand alternative designs are usually evaluated in less than 5 seconds on a modern computer. All the designs are realistic, in the sense that turns per coil are rounded to the nearest integers and numbers of slots and parallel circuits meet the requirements of winding balance. Since in this way the cost as a function of the design variables is discontinuous, the method of optimization must be able to handle such functions. It is described in the references and in more detail at the author’s web site (TRA2 user’s manual). The description here will be very brief. 2 OPTIMIZING METHOD The optimizing method is a Monte Carlo method, based on random numbers. It was first used in a transformer program made for National Industri in Norway (now ABB) in 1965. The cost of the generator plus the cost of losses is minimized. Some of the performance requirements are taken care of during the design synthesis, for others a penalty term is added to the cost for designs that do not meet the requirements. A typical penalty function is shown below. % penalty Permissible range with penalty = 0 Transient reactance Fig. 1. Penalty function for transient reactance. Fig. 2. Hydroelectric generator. The design variables are: Stator: Rotor: 1. Inside diameter 8. Flux density in pole base 2. Slot width 9. Thickness of field coil 3. Ratio slot/tooth width 10. W/cm2 surface loading 4. A/cm stator loading 5. W/cm2 surface loading 6-7. Flux densities in teeth and yoke If the customer is willing to pay extra for each percent the synchronous reactance is below a certain limit, xd is also a design variable. For each design variable, min and max limits are specified in the input, and five starting points for the optimization are based on these limits. The optimization proceeds from each starting point with small changes in the variables between successive alternative designs. The changes are partly random, but the method is such that the optimization proceeds systematically in the direction of better designs as long as possible. The best end point from the five starting points becomes the starting point for a final stage of the optimization, where the changes are smaller between successive designs, and it usually becomes possible to get even closer to the true optimum. In addition to the transient reactance, other technical requirements that are met by means of penalty functions are for temperature rise in stator and rotor, total and open circuit losses and clearance between field coils. 3 DESIGN SYNTHESIS The complete optimizing program consists basically of three parts, design synthesis, analysis and the optimizing logic. Design synthesis is the process by which a design is produced, on the basis of: A set of values for the design variables. The machine rating and other technical specifications. Design choices made by the program user (winding types, etc.). The manufacturer’s normal design practice. Since min and max limits are specified for the design variables, they should be as meaningful as possible as design parameters. It must be possible for the synthesis to proceed in a straightforward manner from a set of values, which is chosen by the optimizing logic for each alternative design. Some of the variables listed earlier will now be explained in more detail. A/cm stator loading is total current in a stator slot divided by the slot pitch (distance between slot centerlines). A higher value gives a shorter machine, but with a tendency of higher copper losses, higher excitation requirement and higher reactances. It is a very important design variable. Stator flux densities are at open circuit, since they are of importance for the open circuit core loss and don’t change much with the load. On the other hand, the flux density in the pole base is at rated load, since it is only of importance for the excitation requirement and varies considerably with the load. W/cm2 surface loading is loss per unit length divided by the surface area through which the loss is dissipated. It is directly related to the temperature rise. For the stator winding, loss is taken as dc loss, but could have included the small addition for eddy current loss. Surface area is slot circumference below the wedge times unit length. For the field winding, the surface area is only for the outer surface, since very little heat is dissipated inward through the pole insulation and radially through the veneers. The next step is to establish the number of slots. The program should try to come as close as possible to the desired value for the stator loading, because it is such an important variable. For this reason, the desired slot pitch may have to be adjusted. Based on the slot pitch and the inside stator diameter (variable 1), an approximate number of slots can be found. It also has to be adjusted, due to the requirements of winding balance. Now the cross section of the slot can be designed, including the depth, arrangement and dimensions of the strands, and insulation allowances. The insulation is given by the voltage. The division of the bar into parallel strands is determined largely by the permissible eddy current loss. The slot depth is found by trial and error until it gives the desired value of W/cm2 surface loading (variable 5). The stator stack length is found on the basis of the desired tooth density (variable 6). At the same time, the number of ventilating ducts is determined, assuming a standard duct pitch and duct width. The outside stator diameter is found on the basis of the desired yoke density (variable 7). This now completes the synthesis of the stator. The air gap is determined from the specified synchronous reactance, and the program then proceeds inward with the pole tip, pole body, field winding and rotor rim. The width of the pole tip (pole chord) is determined from a fixed per unit pole arc, covering perhaps 70% of the pole pitch. The outer edge must be deep enough for the mechanical strength at overspeed. The pole tip can be shaped as a circular arc with a smaller radius than the stator, so that the air gap is perhaps 50% larger at the outer edge than at the center. This keeps the harmonic content in the air gap flux wave within reasonable limits, but manufacturers have different ideas about this. The width of the pole body is primarily determined by the desired flux density in the pole base (variable 8), and the depth primarily by the thickness (variable 9) and the W/cm2 surface loading (variable 10) of the field coil. The synthesis is complicated by the fact that the pole depth affects the leakage flux, and thereby the width required to carry the flux. Vice versa the width affects the available space for the coil, and thereby the depth required to get the desired surface loading. Final dimensions can therefore only be found after a number of loops and corrections. Some rounding of dimensions is normal design practice. The synthesis starts with the rounding of inside stator diameter to the nearest 10 mm and slot width to the nearest 0.5 mm. Such rounding contributes to making the criterion function discontinuous and the optimization more difficult, but is only of marginal importance in this respect compared with the adjustments of numbers of turns, slots and parallel circuits. It is convenient to have the thickness of the field coil expressed in a per unit system, so that the value one corresponds to the surface of the coil being flush with the pole tip. The program now establishes the number of parallel circuits in the stator winding. An approximate slot pitch is given by the slot width (variable 2) and the ratio slot/tooth width (variable 3). Assuming that a normal two layer one turn bar winding is used, the approximate current per circuit is calculated as the A/cm stator loading (variable 4) times half the slot pitch. Knowing this, the program can establish an approximate number of circuits from the kVA and specified min and max limits for the stator voltage (the voltage can also be fixed). The number of circuits is adjusted to the nearest number that gives a balanced winding, depending on the number of poles. In low speed generators, the optimum diameter is often such that the rotor inertia and the rim stress are both at their limits simultaneously. It is interesting to observe how this is achieved accurately and automatically, even though there is nothing specifically about this in the program. For a hydroelectric generator, the rotor synthesis is completed by finding the depth of the rim from the required inertia and the permissible stress at overspeed. It is important to note that the values for the independent variables had to be adjusted during the design synthesis in order to achieve a practical design. Therefore a distinction must be made between the actual values that were obtained, and the desired values that were aimed for. 4 DESIGN ANALYSIS Because typically more than a thousand alternative designs are evaluated, it is not practical to employ the most sophisticated and time consuming methods of analysis, such as finite element magnetic field calculations, for every one of them. However, such analysis has been used to improve conventional formulas, for example for pole leakage flux and stator leakage reactance. Every time a new design is made, tests versus calculations should be checked for similar machines, where this is available. For such complex machines, calculations are approximate, based on simplifications, assumptions and experience. Some correction factors must be specified in the program input. In the calculation of excitation, the hard part is the addition of saturation ampereturns. This is partly because they are very sensitive to small differences in flux density, the shape of the flux in the air gap changes with saturation and load, and ampereturns at the junction pole-rim is affected by the roughness of the surfaces that are joined together at this point. Joints between segments and local flux density variations near wedges, dovetails and ventilating ducts also complicate matters. loss is of a similar nature as described earlier for the open circuit condition. Another significant part of the stray load loss is the loss occurring in the clamping structure at the stator ends, due to flux mostly set up by the stator current in the end windings. What happens here is also of a very complex nature, but it is possible to come to some conclusions about the dependency upon diameter, pole pitch and stator loading. Windage and friction loss is a function of air gap diameter, stack length and revolutions per minute (RPM). The nature of the function can be determined from statistical analysis of tests on similar machines. Insight can be gained from the study of airflow and friction in the bearings, especially about possibilities for loss reduction, but a purely theoretical calculation of the total loss is probably hopeless. Temperature rises in stator and rotor are calculated based on equivalent thermal circuits, calculated losses and estimated airflow and speeds. Again, due to the uncertainties involved, comparisons with tests on similar machines are essential. 5 FINITE ELEMENT ANALYSIS Even more difficult is the calculation of open circuit core loss. Simply adding up loss in the stator teeth and yoke based on calculated flux densities and Epstein tests produces a loss which must be multiplied by a factor of at least 2 to get the loss in the actual machine. The discrepancy is explained partly by loss in the pole face, due to variations in flux density here caused by the movement of the pole across stator teeth and slots. Most of the loss usually occurs in the pole punchings, which are quite thick and only partly insulated from each other with naturally occurring surface layers of oxide. They are partly short-circuited by burrs and through the uninsulated amortisseur (damper) bars. Part of the loss is also due to slot frequency currents induced in the amortisseur bars themselves and is heavily affected by the ratio of slot pitches in stator and rotor. Local flux densities in the stator core contain harmonics, which add considerably to the eddy current losses in the punchings. Harmonics in air gap flux density due to the shape of the salient poles are reflected in the teeth. In the yoke, harmonics also occur because saturation changes the flux density distribution between inner and outer radius from being non-uniform near the zero points in the cycle to being nearly uniform near the peaks. Through the cycle, maximum flux density in the yoke is flattened near the teeth and peaked near the outer edge. Stray load loss is calculated and measured at short circuit. Only the part of it which is due to slot leakage flux inducing eddy currents in the stator winding can be calculated with some accuracy. The MMF from the stator winding at short circuit is of the same order of magnitude as the MMF from the field winding at open circuit. Due to the concentration of MMF in stator slots, slot harmonics are very pronounced. The resulting pole face After a run is completed, an input file for a companion special purpose finite element program is produced automatically, and the magnetic field in the air gap and between the poles can be calculated for the final design with a simple command. In this way, so-called pole shape factors can be accurately determined. Two of them are the ratios total/fundamental air gap flux and max actual/fundamental air gap flux density. Another factor relates the sinusoidally distributed stator MMF to the concentrated MMF from the field poles. Less field MMF than stator MMF is required to produce the same fundamental flux. The harmonic content of the air gap flux is analyzed with excitation from the field winding and from the stator winding in the direct and quadrature axes. The analysis can also be used to set up equivalent circuits in the two axes with branches for individual amortisseur bars. Among other things, this enables the designer to calculate more accurately subtransient reactances and analyze such phenomena as unbalanced loads and short circuits. REFERENCES 1 O.W. ANDERSEN, "Optimum Design of Electrical Machines", IEEE Transactions on Power Apparatus and Systems, Vol. 86, June 1967, pp. 707-711. 2 O.W. ANDERSEN, "Optimized Design of Electric Power Equipment", IEEE Computer Applications in Power, pp. 11-15, Jan. 1991. 3 O.W. ANDERSEN, "Teaching and Demonstration of Optimized Design", ICEM 96, Vigo, Vol. III, pp. 488-490.