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Fig. (20). Variation of the objective function with respect to SAR iterations. During the minimization of (18) a number of local minimums with values very close to the global minimum were identified. Due to this fact the number of successes of the GA was lower than that of the SA and SAR algorithm, as the GA was trapped most of the times in local minima. The optimum distribution of the design parameters as calculated by the SA and SAR algorithms correspond to an objective function value of f «) = 677.5 $, ie. a 3.2% reduction of the sum of magnetic steel cost and PV of future no load loss as it was pre-evaluated by the industrial optimization scheme [48]. Furthermore, the SAR algorithm exhibits a 20% reduction in objective function calls comparing to the classic SA algorithm. Figure 20 depicts the variation of the objective function with respect to the SAR iterations and Fig. (21) illustrates a detail of the computed flux density distribution with the 3D FEM model of the multiple grade lamination wound core with the optimum variables.

Figure 20 (20). Variation of the objective function with respect to SAR iterations. During the minimization of (18) a number of local minimums with values very close to the global minimum were identified. Due to this fact the number of successes of the GA was lower than that of the SA and SAR algorithm, as the GA was trapped most of the times in local minima. The optimum distribution of the design parameters as calculated by the SA and SAR algorithms correspond to an objective function value of f «) = 677.5 $, ie. a 3.2% reduction of the sum of magnetic steel cost and PV of future no load loss as it was pre-evaluated by the industrial optimization scheme [48]. Furthermore, the SAR algorithm exhibits a 20% reduction in objective function calls comparing to the classic SA algorithm. Figure 20 depicts the variation of the objective function with respect to the SAR iterations and Fig. (21) illustrates a detail of the computed flux density distribution with the 3D FEM model of the multiple grade lamination wound core with the optimum variables.

Related Figures (21)

Fig. (1). Treanor’s wound core, extracted from patent US2960756, 1960.
Fig. (1). Treanor’s wound core, extracted from patent US2960756, 1960.
In 1980 Lin, Ellis, and Burkhardt disclosed a composite wound core having two parallel magnetic circuits of unequal mean lengths as shown in Fig. (2) [42]. The short magnetic circuit, i.e. inner part of the core, is constructed of a high permeability grain-oriented steel, having low loss but high cost unit, and the long magnetic circuit, i.e. outer part of the core, is constructed of a conventional grain-oriented steel, having higher loss but low cost unit [42]. Fig. (2). Composite wound core, extracted from patent US4205288 1980.
In 1980 Lin, Ellis, and Burkhardt disclosed a composite wound core having two parallel magnetic circuits of unequal mean lengths as shown in Fig. (2) [42]. The short magnetic circuit, i.e. inner part of the core, is constructed of a high permeability grain-oriented steel, having low loss but high cost unit, and the long magnetic circuit, i.e. outer part of the core, is constructed of a conventional grain-oriented steel, having higher loss but low cost unit [42]. Fig. (2). Composite wound core, extracted from patent US4205288 1980.
Fig. (5). Specific core loss curves of conventional, M4 0.27 mm, and high permeability, M-OH 0.27 mm, grain-oriented steels. Fig. (4). Normal magnetization curves of conventional, M4 0.27 mm, and high permeability, M-OH 0.27 mm, grain-oriented steels.
Fig. (5). Specific core loss curves of conventional, M4 0.27 mm, and high permeability, M-OH 0.27 mm, grain-oriented steels. Fig. (4). Normal magnetization curves of conventional, M4 0.27 mm, and high permeability, M-OH 0.27 mm, grain-oriented steels.
Fig. (3). Peak flux density distribution across the limb of a strip wound core.
Fig. (3). Peak flux density distribution across the limb of a strip wound core.
High permeability grain-oriented steels present improved magnetization and specific core loss characteristics in comparison with the conventional grain-oriented steel as illustrated in Fig. (4 & 5). On the other hand the cost unit of high permeability steel is higher than that of conventional grain-oriented steel. Thus, by using conventional grain- oriented steel for the inner and outer part of the wound core and high permeability grain- oriented steel for the rest part of the core, the transformer manufacturer may achieve an optimum tradeoff between first cost and PV of future no load loss.
High permeability grain-oriented steels present improved magnetization and specific core loss characteristics in comparison with the conventional grain-oriented steel as illustrated in Fig. (4 & 5). On the other hand the cost unit of high permeability steel is higher than that of conventional grain-oriented steel. Thus, by using conventional grain- oriented steel for the inner and outer part of the wound core and high permeability grain- oriented steel for the rest part of the core, the transformer manufacturer may achieve an optimum tradeoff between first cost and PV of future no load loss.
Fig. (6). Representation of the multiple grade lamination wound core geometry and variables.
Fig. (6). Representation of the multiple grade lamination wound core geometry and variables.
Fig. (9). Current probe and differential voltage probe.
Fig. (9). Current probe and differential voltage probe.
Figure 10 illustrates a detail of the wound core, the excitation coil, and the search coils. The wire used for the search coils is a solderable enameled copper wire 0.1 mm in diameter. A total of 30 search coils were inserted in the wound core for the experimental evaluation of its peak flux density distribution along the limb, yoke and comer. The overall loss of the core did not change noticeably after the coils were wound in position. Therefore, it was assumed that the flux distribution did not change much due to the search coils being introduced. Fig. (10). Detail of the wound core, the excitation coil and the search coils.
Figure 10 illustrates a detail of the wound core, the excitation coil, and the search coils. The wire used for the search coils is a solderable enameled copper wire 0.1 mm in diameter. A total of 30 search coils were inserted in the wound core for the experimental evaluation of its peak flux density distribution along the limb, yoke and comer. The overall loss of the core did not change noticeably after the coils were wound in position. Therefore, it was assumed that the flux distribution did not change much due to the search coils being introduced. Fig. (10). Detail of the wound core, the excitation coil and the search coils.
Fig. (8). Experimental setup. The experimental setup used for the local flux density and no load loss measurements of wound cores is illustrated in Fig. (8). It consists of a PC and a National Instruments (NI) 6143 data acquisition (DAQ) card (8 voltage differential inputs, 16 bit, 250 ksamples / s). The excitation coil terminals are connected to the input of the DAQ card via an active differential voltage probe (3 dB frequency of 18 MHz) and a current probe based on the Hall Effect (1 dB frequency of 150 kHz) is used for capturing the no load current. Both probes are depicted in Fig. (9). For the magnetization of the wound cores, a 23 turn excitation coil is supplied by a one- phase 230 V, 50 Hz source through a variable transformer. The no load loss was evaluated by the acquired voltage and current data with appropriate virtual instruments (VI) created with the use of LabVIEW.
Fig. (8). Experimental setup. The experimental setup used for the local flux density and no load loss measurements of wound cores is illustrated in Fig. (8). It consists of a PC and a National Instruments (NI) 6143 data acquisition (DAQ) card (8 voltage differential inputs, 16 bit, 250 ksamples / s). The excitation coil terminals are connected to the input of the DAQ card via an active differential voltage probe (3 dB frequency of 18 MHz) and a current probe based on the Hall Effect (1 dB frequency of 150 kHz) is used for capturing the no load current. Both probes are depicted in Fig. (9). For the magnetization of the wound cores, a 23 turn excitation coil is supplied by a one- phase 230 V, 50 Hz source through a variable transformer. The no load loss was evaluated by the acquired voltage and current data with appropriate virtual instruments (VI) created with the use of LabVIEW.
Fig. (12). Contours of the magnetic vector potential of a conven- tional wound core.
Fig. (12). Contours of the magnetic vector potential of a conven- tional wound core.
Fig. (14). Peak flux density nodal plot of a multiple grade lamination wound core(B =1.5T, x, =0 mm, x, =10 mm). Fig. (13). Peak flux density nodal plot of a conventional wound core(B =1.5T).
Fig. (14). Peak flux density nodal plot of a multiple grade lamination wound core(B =1.5T, x, =0 mm, x, =10 mm). Fig. (13). Peak flux density nodal plot of a conventional wound core(B =1.5T).
Fig. (11). Areas comprising the 2D model of a conventional wound core.
Fig. (11). Areas comprising the 2D model of a conventional wound core.
Fig. (19). Peak flux density distribution along line AB for B =1.5 T and x, =5 mm, x, =10mm. Fig. (18). Peak flux density distribution along line AB for B =1.4 T and x, =5mm, x, =10mm.
Fig. (19). Peak flux density distribution along line AB for B =1.5 T and x, =5 mm, x, =10mm. Fig. (18). Peak flux density distribution along line AB for B =1.4 T and x, =5mm, x, =10mm.
Fig. (15). Peak flux density nodal plot of a multiple grade lamination wound core(B =1.5T, x, =5 mm, x, =10 mm).
Fig. (15). Peak flux density nodal plot of a multiple grade lamination wound core(B =1.5T, x, =5 mm, x, =10 mm).
Fig. (17). Peak flux density distribution along line AB for B =1.5 T and x, =Omm, x, =10mm.
Fig. (17). Peak flux density distribution along line AB for B =1.5 T and x, =Omm, x, =10mm.
Fig. (16). Peak flux density distribution along line AB for B =1.4 T and x, =Omm, x, =10mm. Table 1 presents a comparison between the calculated no load loss with the FEM models of Section 4 and the respective experimental values for a number of x,, x, configurations. The considered wound core parameters are X, =24.3 mm, x, =57 mm, x, =183 mm, x, =190 mm, and the mean flux density is B =1.7 T. As can be seen from Table 1, the computed results are in good agreement with the measured ones and the error is less than 5% in all cases. Also the difference between the computed no load loss with the 2D and 3D FEM models is insignificant and is less than 1.1% in all cases.
Fig. (16). Peak flux density distribution along line AB for B =1.4 T and x, =Omm, x, =10mm. Table 1 presents a comparison between the calculated no load loss with the FEM models of Section 4 and the respective experimental values for a number of x,, x, configurations. The considered wound core parameters are X, =24.3 mm, x, =57 mm, x, =183 mm, x, =190 mm, and the mean flux density is B =1.7 T. As can be seen from Table 1, the computed results are in good agreement with the measured ones and the error is less than 5% in all cases. Also the difference between the computed no load loss with the 2D and 3D FEM models is insignificant and is less than 1.1% in all cases.
Table 1. Comparison Between Measured and Computed no Load Loss (x, = 24.3 mm) optimization with a heuristic methodology which is already used by the transformer industry [48]. The parameters of the wound cores under consideration are x, =51.3 mm, x, =57
Table 1. Comparison Between Measured and Computed no Load Loss (x, = 24.3 mm) optimization with a heuristic methodology which is already used by the transformer industry [48]. The parameters of the wound cores under consideration are x, =51.3 mm, x, =57
Fig. (21). Detail of the peak flux density vector plot of the multiple grade lamination wound core with optimum design variables x,,
Fig. (21). Detail of the peak flux density vector plot of the multiple grade lamination wound core with optimum design variables x,,
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EngineeringElectrical EngineeringOptimization (Mathematical Programming)ElectromagnetismNumerical SimulationsDesignFinite Element MethodsRenewable EnergyPower QualityOptimization (Mathematics)
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