Since the application of Taguchi based regression rate optimisation framework for elec- trical machine design has not been attempted in literature, it is essential to ensure the framework is correctly implemented and functioning properly. According to [154–156], a successful implementation of the TRBB method exhibits the following attributes:
• The number of iterations increases with the RR value.
• The performance results using different RR values should show a clear convergence or close correlation of the OEC.
• There should be a clear convergence between the OEC optimum and trial mean. • A stable performance point should be identifiable before the termination of the
optimisation.
• The TBRR optimisation should demonstrate the ability to recover if the OEC drops of an optimum region.
In addition to the above attributes the TBRR method must realise similar OEC per- formances for each topology case when using the selected RR values as in Table 5.4. To determine if the implemented TBRR framework demonstrates the above-mentioned attributes, a case study including four rotor topologies is devised as described in Table 5.4.
Table 5.4: Machine optimisation case study for validating the TBRR framework. PM rotor topology Spoke, Radial, V- and U-type
Cage slot Parallel-tooth slot (fixed parameters)
OEC MAX(PF · η)
Outer noise OA L4 OA
Regression rate 50, 55, 65, 75, 85, 95 (static)
For the optimisation, the OEC is formulated to maximise the product of the power factor (PF) and efficiency (η). To achieve this objective the PM duct design parameters will be included in an L9 OA and exposed to outer noise factors through an L4 OA. The
cage slot design is fixed in this study. For each PM duct topology the optimisation will be repeated using the regression rates specified in Table 5.4 with a parameter specific CV value of 1% of its original metric range.
5.2.1
Parameter Range Checking Method
One important aspect of the TBRR framework is to check if the newly calculated level parameters for the subsequent iteration (e.g. L1 to L3 of parameter Pn) are within
be approached. In this dissertation two methods are proposed, which are described as follows. The flow-charts of both methods are given in Fig. 5.2.
• Boundary substitution method: if either L1 or L3 is out of bounds, the optimum (Lopt) is still placed in L2, but the out-of-bounds level parameter is replaced by the
relevant boundary. The LD used for the next iteration (if needed) is replaced with LDn+1=(L2new-L3new)/2 to account for the regular LD between the newly set level
parameters.
• Optimum substitution method: Lopt is placed in the slot of the out-of-bounds
level parameters with the remaining two parameters calculated as in Fig. 5.2(b). For both methods, if all the level parameters are within range the normal placement procedure is used.
(a) (b)
Figure 5.2: Proposed out-of-bounds parameter verification and handling methods: (a) bound- ary substitution method, (b) optimum substitution method
To further assess the proposed methods, both methods are implemented and evaluated in the case study defined in Table 5.4. It was found that the level parameters only moved out-of-bounds during the fist couple of iterations. This is mainly because the level differences are still high during the initial iterations (e.g. if the best performing parameter is either L1 or L3 in the initial iterations, then in the subsequent iterations L1 or L3 will likely be out-of-bounds). As a result of this the boundary substitution method causes a non-uniform LD between two parameters for the next iteration. Since the use of the TBRR optimisation method requires the LD between two levels for a parameter to be equal, using boundary substitution method within the TBRR framework in some cases leads to poor recoverability and premature termination. The optimum substitution method is shown to generate consistent results as the LD is reduced at an uniform and controlled tempo. Thus, the optimum substitution method is a preferred method for checking and/or correcting parameter boundaries in the TBRR framework.
5.2.2
Comparison with the Known Attributes of TBRR
Framework
In this section the results of the implemented case study are synthesised and evaluated against the known attributes of the TBRR method. It is found that for all the PM rotor topologies the RR value directly influences the number of iterations (see Fig. 5.3(a)). Secondly, for each RR value, the OEC shows clear convergence before the optimisation is
completed. In all cases, the OEC converged before the minimum LD value was reached. The number of iterations required for the OEC to converge at each RR value is indicated in Fig. 5.3(b) for different PM rotor topologies. Both Fig. 5.3(a) and (b) show the same exponential increase in iterations with the increase of RR as presented in Fig. 3.4. The increase in iterations is mainly due to the decreased tempo of regression of the parameter range. From the two figures, it can be observed that for all the optimisation cases, convergence was achieved before the minimum LD was reached. Thus, it can be inferred that a stable performance point was identified in each implementation of the TBRR framework.
(a) (b)
Figure 5.3: The number of iterations as a function of RR value for: (a) total number of iterations, (b) required number of iterations at which the OEC converges
In Fig. 5.3(a) for each topology, the number of iteration for RR ∈ {0.5, 0.65} is prac- tically the same and only start to differ significantly when RR > 0.85. The difference is due to the OEC requirement when LDmin has been reached. For the optimisation to ter-
minate, the current iteration’s OEC value has to be within a specified minimum range of the previous iteration. This requirement ensures that a robust optimum point is located. The possibility also exists that additional iterations are required when using a larger RR. For each iteration, both the OEC and the trial average are calculated. The OEC in- dicates the performance of the current optimum (as determined by the Taguchi method) while the trial average indicates the performance of the trial range. The trial average is calculated using the same approach as for the optimum trial’s OEC. However, each main OA trial is calculated individually before calculating the average of the main OA. For the L9×L4 OA configuration, the nine trials’ OEC are calculated before the trial average can
be determined.
The LD is reduced after each iteration, so is the parameters’ trial range. The range is reduced towards the optimum conditions, thus, the trial average should regress closer to the OEC performance. For a successful implementation of the TBRR method, the trial average converges with the OEC as in Fig. 5.4(a) and (b) with the trial average never exceeding the OEC. The opposite usually indicates a failed implementation of the TBRR framework.
(a) (b)
Figure 5.4: OEC performance plot versus number of iterations: (a) showing clear convergences between the OEC and the trial mean, (b) showing a recovery from an optimum dip
The TBRR framework must demonstrate its optimum recoverability. One such case is shown in Fig. 5.4(b). As the TBRR method only uses the ANOM, it is possible in some cases to identify a combination that is off target. This is even more likely when the LD of a parameter is still large. For the Taguchi method to accurately predict the best param- eter combination, both the ANOM and the ANOVA are usually required. The ANOVA provides information on performance variance over the parameter’s range along with its confidence. However, the complexity of an optimisation framework can be much reduced if the use of ANOVA can be avoided. In the proposed TBRR method, this limitation can be overcome by reducing the LD over time and moving toward a region where there is little performance variance, thus, a stable mean point.
Fig. 5.5 shows the optimum performances of each topology as a function of the selected RR. It can be seen that there is a good correlation between the optimum OEC for each RR of a selected topology. The standard deviation per topology never exceeds 0.002 with a maximum percentage deviation of less than 2%. Based on the above discussion it can be inferred that the implemented TBRR framework exhibits the same behaviour and attributes as these of successfully implemented cases in literature.
5.2.3
Verifying the Use of the Dynamic Regression Rates
In this section, the use of the single and multi dynamic regression rate is investigated and discussed. The validity of using the proposed dynamic RRs will be assessed. The dynamic RR is deemed successful if the following criteria are met:
• The optimum OEC performance should have a good correlation with that of the static RR optimisations.
• The number of iterations does not exceed that of the 0.95 static RR. • The RR adjusts according to the dynamic RR formula.
• A stable RR point should be identifiable before the optimisation is terminated. • The RR must stay within the minimum and maximum RR boundaries.
For the multi RR to be deemed valid, the RR of each parameter must adhere to the above attributes.
Figure 5.5: Optimum OEC performance comparison using different RRs
Fig. 5.6 compares the optimum OECs realised by both the dynamic RR and the static RR. A close correlation is visible between the OECs from dynamic and static RR. For both the radial and V-type topologies, the multi dynamic RR realised designs that outperformed those of the 0.95 static RR while for the spoke and U-type topologies, the performances of designs are rather similar.
Figure 5.6: Optimum OEC performance comparison using different RR (both static and dy- namic)
From the results it was found that both the multi and single dynamic RRs realised optimum machines within a reasonable number of iterations. The number of iterations re- quired are provided in Table 5.5 along with the minimum and maximum iterations of the static RR. The corresponding static RR as a function of the required number of iterations (starting on the vertical axis of Fig. 5.3) is between 0.75 to 0.90 depending on the topology.
Table 5.5: Static RR vs. Dynamic RR iteration performance comparison.
Minimum Maximum Single Multi
Spoke 9 106 63 42
Radial 9 86 37 28
V-type 9 120 38 20
U-type 9 112 64 25
By plotting the RR as a function of the total number of iterations the boundaries and behaviour of the dynamic RR can be investigated. For the single dynamic RR each of the four topologies’ iterative RR plot was compiled and is presented in Fig. 5.7. It can be seen that the RR never moves out of the minimum and maximum RR boundaries even in the event of the ANOVA’s percentage contribution is below or beyond it. It can also be observed that the RR values tracks that of the ANOVA’s percentage contribution before locating a stable point required by the termination criteria. The stable RR near the termination point (of all four topologies) indicates that there exists a stable point for parameters with the highest contribution towards performance variance. This is in accordance with Taguchi’s methodology and is also an indication of locating a robust design point.
(Spoke) (Radial)
(V-type) (U-type)
For the multi dynamic RR, the same iterative RR plots are compiled. However, to gain a better understanding, a plot for each topology parameter used is compiled. Presented by Fig. 5.8 is the iterative RR plots for the U-type topology. Only the U-type topology is presented as the same behaviour was found for the other topologies. In addition to the parameter iterative RR plot, a maximum RR plot (same as for the single dynamic RR) is also compiled and presented in Fig. 5.9. The maximum plot is used to gain insight over the overall performance of the multi dynamic RR. From Fig. 5.8 and Fig. 5.9 it is clear that the multi dynamic RR has the same behaviour as that of the single dynamic RR.
O2 Rib
PMt PMw
Figure 5.8: Multi dynamic RR versus percentage contribution towards variance for each pa- rameter in the OA
In Fig. 5.9 (a), the parameters responsible for the dominant RR can be identified using Fig. 5.8 iterative parameter RR plots. The peak contributions are indicated on the plot in Fig. 5.9 (b) along with the smoothing region as a result of all the parameters. This comparison between Fig. 5.8 and Fig. 5.9 also graphically highlight the difference between the multi and single dynamic RR. For the single, the next iterations LD for each parameter is calculated using the same RR which is largely the reason for the increased number of iteration when compared to the multi dynamic RR as in Table 5.5. Since the multi dynamic RR reduced the parameters that least influence the performance variance (PMw in Fig. 5.8) faster, the overall LD is also reduced at a higher rate. This results in the parameters with higher performance variance influence to identify their stable regions faster as the lower influencing parameters are removed from the optimisation.
(a) (b)
Figure 5.9: Iterative multi dynamic RR (a) versus highest percentage contribution towards variance (b) with parameter identification
From the above discussion, it can be concluded that both the single and multi dynamic RR methods are well suited to be incorporated in the TBRR framework. From the initial investigation the only clear advantage between the two is the reduced required number of iterations of the multi dynamic RR method and the slight increase in optimum OEC performance. The optimum machines realised had similar performances to that of a static RR optimisation, however, a reduced number of iterations were required. Thus the dynamic RR methods would be well suited for the proposed weighted function OEC equation to generate the Pareto front. A conclusion regarding the dynamic RR methods can only be formulated once it has been used in more implementations.