# Data-Driven Constraint Handling in Multi-Objective Inductor Design

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Procedure of Inductors for DC-DC Converters

## 3. Statement of the Design Problem

- Lower bound, ${L}_{\mathrm{op}}\ge {L}_{\mathrm{min}}$;
- Upper bound, ${L}_{\mathrm{op}}\le {L}_{\mathrm{max}}$;
- Inductance drop limit, ${L}_{\mathrm{op}}\ge {k}_{\mathrm{sat}}\xb7{L}_{0}$;

## 4. Multi-Objective Optimisation Approach

- An outer loop (network), where randomly generated configurations are introduced in the population (exploration);
- An inner loop (clonal selection), where the population is improved by means of local mutations (exploitation).

- Further details about the optimisation algorithm can be found in Appendix A.

- 1.
- First, the geometric consistency is checked (e.g., non-negative area of the core’s windows). An inconsistent configuration is discarded and replaced by a new one if it was randomly generated. Otherwise, if the configuration is a clone (mutated from a feasible configuration), the mutation is gradually reduced until the solution is feasible again.
- 2.
- 3.
- The remaining candidate solutions are evaluated in terms of objectives (volume and total losses).

- The clouds of points in Figure 4 represent all the design configurations explored during a complete run of the optimisation procedure on the test problem, in the objectives space (volume V and total losses P, on the x and y axes, respectively). The colour depends on the feasibility of the configurations and distinguishes those configurations compliant with the design constraints (i.e., good, orange points) from those that are unfeasible (i.e., bad, blue points). Two graphs are reported to further distinguish between parent configurations (Figure 4a) and clones (Figure 4b). It can be noticed that only a small percentage of the configurations generated randomly (parents) is feasible. Clones, instead, have higher chances of complying with the design constraints since they originate from local mutations of feasible configurations.

## 5. Surrogate Modelling of Constraints for Pre-Selection

#### 5.1. AIS-Based Classifier

#### 5.2. Effects of the Classifier on the Optimisation

- True positives: feasible configurations that are correctly classified as positive.
- False positives: unfeasible configurations wrongly classified as positives. Since they are evaluated with the FP during the optimisation, the incorrect classification implies unnecessary calculation that slows down the procedure.
- True negatives: unfeasible configurations correctly classified as negative, which can be correctly disregarded in the optimisation process.
- False negatives: feasible configurations wrongly classified as negatives, which are therefore disregarded during the optimisation. This is the worst case since it implies discarding solutions potentially belonging to the Pareto Front.

#### 5.3. Evaluation of the Classifier’s Performance

- The high percentage of true negatives (around 97%) in the randomly generated configurations considerably reduces the number of FP evaluations in this phase. Despite the high rate of false positives, the low number of total positives identified by the classifier among the parent results in a significant reduction in the unnecessary evaluations of the FP.
- The good performance in the classification of negatives among parents is motivated by the presence of wide areas of unfeasible configurations in the design space (Figure 5a).
- As previously discussed (Figure 5b), the classification task is much harder in the proximity of the interface between the feasibility region and the rest of the design space. Most configurations generated from local mutations are located in this area. This explains the poor performance of the classifier in the clones. In particular, while around 200,000 FP evaluations are saved by correct classification of unfeasible configurations (true negatives), an equivalent number of feasible configurations are wrongly disregarded.
- The percentage of false positives in clones is smaller than in parents. However, around 150,000 configurations are unnecessarily evaluated.

## 6. Evaluation of the Surrogate Modelling Approach

- Execution of VIS optimisation method without a classifier ($\mathrm{VIS}$).
- Application of the classifier only on randomly generated configurations (${\mathrm{VIS}}_{\mathrm{AIS}}$).
- Application of the classifier both on randomly generated and locally mutated configurations. (${\mathrm{VIS}}_{\mathrm{AIS},\mathrm{clone}}$).

- For each of the presented cases, 30 optimisation executions are performed. The first result is the average number of FP evaluation calls, reported in Figure 7, in which separated results are reported for the classification of parents and clones.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

- 1.
- An initial population $Pop$ of ${N}_{\mathrm{ab}}$ configurations (antibodies $Ab$) is randomly generated.
- 2.
- A memory set $Mem$ is also created, initially empty, where all the non-dominated solutions found during the search are stored.
- 3.
- The memory set is built through a network loop, where for ${N}_{\mathrm{net}}$ iterations or until a stop criterion is reached:
- (a)
- The fitness f of all antibodies in the current population is evaluated, also including the antibodies eventually present in the memory set.
- (b)
- The current population is improved through a clonal selection loop, where for ${N}_{\mathrm{sel}}$ iterations or until a stop criterion is reached:
- i.
- The antibodies in the current population (parents) are replicated into ${N}_{\mathrm{cl}}$ clones;
- ii.
- Each clone is then locally mutated. The random mutation $\Delta $ is evaluated as follows: $\Delta =\frac{1}{\beta}\phantom{\rule{0.166667em}{0ex}}\mathcal{N}(\mu =0,\phantom{\rule{0.166667em}{0ex}}\sigma =1)\xb7\mathrm{exp}(-f)$, where $\beta $ is a mutation amplitude parameter and f the fitness of the parent (lower fitness corresponds to larger mutations);
- iii.
- The fitness of the clones is evaluated, also including the antibodies eventually present in the memory set;
- iv.
- The clone with the highest fitness is selected and it replaces the parent in the next generation if it has higher fitness.

- (c)
- The current population and the memory set are merged and their affinity is calculated.
- (d)
- Network selection is performed, removing antibodies with high affinity. The affinity threshold $\delta $ is evaluated as follows: $\delta ={N}_{\mathrm{mem},\mathrm{des}}/\sqrt{m}$, where m is the number of objectives and ${N}_{\mathrm{mem},\mathrm{des}}$ is the desired size of the memory set.
- (e)
- The memory set is updated with the non-dominated solutions, among those that survived the network selection.
- (f)
- A fraction ${k}_{\mathrm{new}}$ (with respect to the current size of the population) of randomly generated new configurations is introduced in the population to increase the exploration of the design space.

- 4.
- The final memory set is taken as the Pareto set of the problem.

- The values of the parameters adopted in all the runs of the VIS optimisation are shown in Table A1.

Parameter | Value |
---|---|

Initial population size, ${N}_{\mathrm{Ab}}$ | 50 |

Number of clones, ${N}_{\mathrm{Cl}}$ | 5 |

Desired size of the memory set, ${N}_{\mathrm{mem},\mathrm{des}}$ | 100 |

Number of network loops, ${N}_{\mathrm{net}}$ | 50 |

Number of clonal selection loops, ${N}_{\mathrm{sel}}$ | 10 |

Mutation amplitude factor, $\beta $ | 0.02 |

Fraction of new antibodies, ${k}_{\mathrm{new}}$ | 0.4 |

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**Figure 1.**Qualitative comparison of the solutions evaluated considering the operation in linear and in partial saturation conditions, in the space of the total losses and the core volume. The maximum magnetic flux density considered for the linear operation is 0.2 T.

**Figure 2.**Three-dimensional representation of the double-E core geometry. The design variables are: the width of the E core A, the height of a single E-core B, the extension of the central column F, and the air gap’s length g.

**Figure 3.**Differential inductance profile of a design configuration as a function of the output current of the converter. The bandwidth of the admissible inductance value is highlighted in green. The proposed case represents a solution operating in partial saturation.

**Figure 4.**Representation in the objectives space (total losses and core volume) of all the design configurations explored during a run of the VIS multi-objective optimisation algorithm. The orange dots represent the feasible configurations (good), while the blue dots are the unfeasible ones (bad): (

**a**) generated randomly (parents); (

**b**) generated by local mutations of the parents (clones). The total number of parents and clones evaluated and the percentages of good/bad ones are shown.

**Figure 5.**Representation in a design sub-space identified by the number of turns, the extension of the central column and the air gap length, of all the configurations explored during a run of the VIS multi-objective optimisation algorithm. All the feasible configurations (orange dots) are compared to (

**a**) the unfeasible configurations generated randomly (grey dots); (

**b**) the unfeasible solutions generated with local mutations (blue dots).

**Figure 6.**Confusion matrix of the responses given by the classifier system and the evaluation with the FP iterative technique. The traffic lights highlight the effect of the response on the optimisation process: green stands for a correct classification, yellow for an incorrect evaluation that should not affect the output of the optimisation, and red for an incorrect classification which could negatively affect the final result.

**Figure 7.**Box plots of the FP executions in the 30 runs of the three tested approaches. The pink diamonds represent the mean value of each case: (

**a**) describes the FP evaluations for the randomly generated cells. A zoom of the results for the two classifier approach is reported in the upper right corner; (

**b**) describes the FP evaluations for the locally mutated cells.

**Figure 8.**Box plot of the total FP executions in the 30 runs of the three tested approaches. The pink diamonds represent the mean value of each case.

**Figure 9.**Pareto fronts of the analysed design case, obtained with the three proposed approaches. The reported fronts are computed by combining the solutions of the 30 executions performed in each case. In addition, the Pareto front of the training-test data set is reported. A zoom of the Pareto fronts in the region of small volumes is also reported to highlight the marked differences in this area.

**Figure 10.**(

**a**) Box plots of the reverse generational distance index for the 30 runs of the three tested approaches. The pink diamonds represent the mean value of each case; (

**b**) Box plot of the spacing index for the 30 runs of the three tested approaches. The pink diamonds represent the mean value of each case.

Input voltage | 48 V |

Output voltage | 24 V |

Output current | 10 A |

Maximum current ripple (pp) | 30% |

Switching frequency | 50 kHz |

Inductance value | 80 μH |

Variable | Lower Bound | Upper Bound |
---|---|---|

A | 30 mm | 59.2 mm |

B | 8 mm | 22.3 mm |

F | 2 mm | 8 mm |

g | 0 mm | 2 mm |

N | 6 | 100 |

Minimum inductance value, ${L}_{\mathrm{min}}$ | $0.95\xb7{L}_{\mathrm{nom}}$ |

Maximum inductance value, ${L}_{\mathrm{max}}$ | $1.2\xb7{L}_{\mathrm{nom}}$ |

Maximum inductance drop factor, ${k}_{\mathrm{sat}}$ | $0.6$ |

Maximum over-temperature, ${T}_{\mathrm{max}}$ | $100{\phantom{\rule{0.277778em}{0ex}}}^{\circ}\mathrm{C}$ |

Parents | Clones | |
---|---|---|

True positive (%) | 44 | 67 |

False positive (%) | 56 | 33 |

True negative (%) | 99 | 51 |

False negative (%) | 1 | 49 |

Total positive | $8\times {10}^{3}$ | $500\times {10}^{3}$ |

Total negative | $200\times {10}^{3}$ | $400\times {10}^{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Lorenti, G.; Ragusa, C.S.; Repetto, M.; Solimene, L.
Data-Driven Constraint Handling in Multi-Objective Inductor Design. *Electronics* **2023**, *12*, 781.
https://doi.org/10.3390/electronics12040781

**AMA Style**

Lorenti G, Ragusa CS, Repetto M, Solimene L.
Data-Driven Constraint Handling in Multi-Objective Inductor Design. *Electronics*. 2023; 12(4):781.
https://doi.org/10.3390/electronics12040781

**Chicago/Turabian Style**

Lorenti, Gianmarco, Carlo Stefano Ragusa, Maurizio Repetto, and Luigi Solimene.
2023. "Data-Driven Constraint Handling in Multi-Objective Inductor Design" *Electronics* 12, no. 4: 781.
https://doi.org/10.3390/electronics12040781