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1.10 A Design Example
ОглавлениеIn this section, we will conclude this chapter with a design example. In particular, we will endeavor to design a UI‐core inductor using an optimization‐based design process. At this point, we have not studied any magnetics or magnetic devices. This will be the subject of the remainder of this book. Since we are not yet in a position to derive or understand the relationships needed, an elementary analysis will be provided and the reader is asked to simply accept these relationships at face value for the time being. It should be observed that the analysis used to derive the needed relationships is very simplistic, but this does not matter because our purpose here is only to look at the design process. We will conduct a much more detailed analysis and design in subsequent chapters.
A UI‐core inductor is depicted in Figure 1.22. Therein, the dark region is the magnetic core. The magnetic core conducts magnetic flux and is made of a U‐shaped piece (the U‐core) and an I‐shaped piece (the I‐core). Conductors (wires) pass through the middle of the U, a region called the slot, and also around the outside of the U‐core to form a winding. The slot has a width denoted ws and a height denoted ds. The width of the core (both U‐ and I‐cores) is denoted the core width wc, and the length of the core pieces is lc. The two cores are separated by an air gap g. The cross‐sectional drawing (a) is the view that one would obtain by looking to the right from the left side of the side view (b) if the side view were cut in half (in the direction into the page). The light region indicates a winding comprised of N turns of wire.
Our goal in this design will be to design an inductor that has an inductance of at least Lmn, a flux density below Bmx, and a current density below Jmx at rated dc current irt. It is desirable to minimize the mass M of the inductor and to minimize the power loss at rated current, denoted Prt. We will also constrain our designs to have a mass below Mmx and a power loss less than Pmx.
Figure 1.22 UI‐core inductor.
The free parameters in our design are the number of turns N, the slot depth ds, the slot width ws, the core thickness wc, the core depth lc, and the air gap g. Thus, our parameter vector may be expressed as
In (1.10-1), N* is the desired number of turns rather than the actual number of turns N because, as a design parameter, we will let the number of turns be represented as a real number rather than an integer. This is because for large N this variable acts in a more continuous rather than discrete fashion. The actual number of turns is calculated from the desired number as
In order to perform the optimization, we will need to analyze the device. It is assumed that windings occupy the entire slot, that the core is infinitely permeable, and that the fringing and leakage flux components are negligible. Again, for the reader unfamiliar with these terms, the analysis can be taken as a set of arbitrary mathematical equations; we will spend the rest of the book defining and developing more accurate expressions for these quantities.
With these assumptions, the mass of the design may be expressed as
In (1.10-3), ρmc and ρwc denote the mass density of the magnetic core and wire conductor, respectively, and kpf is the fraction of the U‐core window occupied by conductor. Ideally, it would be 1, but 0.7 is a very high number in practice.
The next step is the computation of loss. The power dissipation of the winding at rated current may be expressed as
In (1.10-4), σwc denotes the conductivity of the wire conductor.
There are constraints both on the inductance, flux density at rated current, and current density at rated current. These quantities may be expressed as
In (10.1‐5) and (1.10-6), μ0 is the magnetic permeability of free space, a constant equal to 4π10−7 H/m.
In order to formulate a fitness function, expressions (1.10-1)–(1.10-7) can be sequentially evaluated. Then constraint functions can be evaluated as
(1.10-8)
(1.10-9)
(1.10-10)
(1.10-11)
(1.10-12)
Keeping with (1.9-4), we find the aggregate constraint
(1.10-13)
We will consider both single‐ and multi‐objective optimization. For the single‐objective case, we will minimize mass and our fitness is given by
For the multi‐objective case, the fitness function will be taken as
In (1.10-14) and (1.10-15), we will take ε = 10−10.
For our design, let us consider a ferrite material for the core with Bmx = 0.617 T and ρmc = 4680 kg/m3, and consider copper for the wire with ρwc = 8890 kg/m3 and Jmx = 7.5 A/mm2. We will take rated current to be 10 A and take the minimum inductance Lmn to be 1 mH. Finally, let us take the maximum allowed mass as Mmx = 1kg, and the maximum allowed loss to be Pmx = 1W.
Table 1.7 Domain of Design Parameters
| Parameter | N | ds (m) | ws (m) | wc (m) | lc (m) | g (m) |
|---|---|---|---|---|---|---|
| Min. value | 1 | 10−3 | 10−3 | 10−3 | 10−3 | 10−5 |
| Max. value | 103 | 10−1 | 10−1 | 10−1 | 10−1 | 10−2 |
| Encoding | log | log | log | log | log | log |
| Chromosome | 1 | 1 | 1 | 1 | 1 | 1 |
Figure 1.23 Single‐objective optimization study.
The next step in the design process is to determine the parameter space Ω. This is tabulated in Table 1.7. Some level of engineering estimation is required to select a reasonable range. However, situations where a range is incorrectly set are usually easy to detect by looking at the population distribution. We will return to this point.
We have now set forth a fitness function and a domain for the parameter vector, and so we can proceed to conduct an optimization. We will begin with a single‐objective case. To conduct this study, a MATLAB‐based genetic optimization toolbox known as GOSET was used. This open‐source code and the code for this particular example are available at no cost in Sudhoff [6].
Figure 1.23 illustrates the progression of the study, which was conducted with a population size of 1000 over 1000 generations. Therein, Figure 1.23(a) shows the gene distribution at the end of the optimization. Recall that θi is the normalized value of the ith gene. Each design is shown in encoded parameter space as a series of dots, each with its own shade (for example, a certain dark shade may correspond to design 37 of the population). Because of the large numbers of designs, it is not possible to pick one design among all the designs. However, a sense of the distribution of the gene (parameter) values in the population can readily be obtained. The horizontal coordinate of each design within its parameter window is proportional to its ranked fitness—with lower ranks toward the left side of the window for a given parameter and higher ranks toward the right side of a given window. Considerable information can be discerned from the distribution plot. For example, there seems to be more sensitivity to ds (which is tightly clustered) than to ws (which is less tightly clustered). A distribution of a gene (parameter) value at the bottom or top of the range indicates that it may be appropriate to adjust the domain of that parameter.
Figure 1.23(b) depicts the fitness versus generation. The best fitness in the population, the median fitness of the population, and the mean fitness of the population are shown. Note that for a few generations, the best fitness is zero (actually slightly < 0), but then the best fitness increases rapidly until generation 150 or so, after which the fitness climes more slowly. The median and mean fitness rise more slowly than the fitness of the best individual. Observe that there are large rapid changes in the median fitness because this is a fitness of the median individual which changes from generation to generation. The mean fitness of the population is more stable. As can be seen, the mean fitness of the population occasionally goes down; this does not happen in the case of the fitness of the most fit individual in the population because of the elitism operator.
Figure 1.24 UI‐core design.
The most fit individual in the final population is illustrated in Figure 1.24, which lists the design parameters as well as a cross‐sectional diagram. Note that N* = 25.3 maps to N = 25 from (1.10-2). The design’s mass is 0.578 kg, and the power loss at rated current is W. The inductance is right at 1 mH and the flux density is at 0.617 T. The current density of the design is 1.67 A/mm2. It would appear that our design is against all constraint limits except those on mass and current density.
At this point, the question arises regarding how we know that our design is optimal. Unfortunately, we do not. There is not an optimization algorithm known that can guarantee convergence to the global optimum for a generic problem without known mathematical properties. However, in the GOSET code used for this example, a traditional optimization method (Nelder–Mead simplex) is used to optimize the design starting from the endpoint of the GA run, and this helps to ensure a local optimum. Still, there is no guarantee that a global optimum is obtained. Therefore, the prudent designer will re‐run the optimization several times in order to gain confidence in the results. The runs can then be inspected to see if all runs converged to the same fitness. If significant variation in fitness has occurred, the use of more generations and/or a larger population size is indicated.
For our single‐objective optimization problem, the optimization was re‐run a multitude of times in order to investigate the variability of the design obtained from one run to the next. We will view variation of parameters and metrics in terms of normalized standard deviations. For example, the normalized standard deviation of the number of turns is the standard deviation of the number of turns divided by the median value of the number of turns for each design, interpreted as a percentage. Conducting the optimization process 100 times yielded the following normalized standard deviations: N with a 11%, ds with a 4.4%, ws with a 17%, wc with a 6.4%, lc with a 14%, and g with a 11% standard deviation. These may seem relatively large. However, it is interesting that normalized standard deviation in mass is only 1.0%. This indicates that there is a family of designs with equally good performance. It is interesting to observe that while appreciable design variation was found, every solution determined was viable (and not that different in terms of performance metrics).
It may seem objectionable to the reader that the results are not repeatable, which arises from the use of a random set of initial designs, and stochastic operators in the GA. However, even Newton’s method will generate random variation in the solution of an optimization problem if the initial condition is selected at random. In Newton’s method, providing a consistent initial condition will of course produce a consistent final answer; however, being consistent can merely mean being consistently incorrect—which can happen if the algorithm becomes consistently trapped at a same local minimizer while missing the global minimizer.
Figure 1.25 Multi‐objective optimization results.
Figure 1.26 Sample design from Pareto‐optimal front.
Let us now turn our attention to a multi‐objective optimization of the UI‐core inductor. The only change in our approach is the replacement of (1.10-14) by (1.10-15). Using 2000 generations with a population size of 1000 yields the results shown in Figure 1.25. Figure 1.25(a) illustrates the objective space at the end of the optimization run. Each point in Figure 1.25(a) represents the objectives of a complete design. Nonviable designs have fitness values close to the origin (and are slightly negative in each axis). Viable, but dominated, designs are also apparent. The remaining designs are nondominated. In Figure 1.25(b) only the nondominated designs are shown, and the fitness elements are reciprocated so that mass and loss can be plotted directly. Again, each point represents the mass and loss of an individual design. Note the tradeoff between mass and loss. This will be a recurrent theme in this text. A sample design on the front is also indicated; this design is illustrated in more detail in Figure 1.26. The sample design has a mass of 0.75 kg, and a loss at rated current of 0.67 W. The inductance is just over 1 mH, and flux density at rated current is 0.617 T. The current density at rated current is 1.3 A/mm2.
Figure 1.27 illustrates the gene distribution of the final population of designs. In Figure 1.27(a), the genes are sorted by objective 1. This means that the genes of designs with higher mass are toward the left of the parameter window, and genes of designs with lower mass are toward the right. In Figure 1.27(b), the genes are sorted by objective 2, so that designs with the most loss are toward the left, and designs with the least loss are toward the right.
Figure 1.27 Sample design from Pareto‐optimal front.
Unlike the case of single‐objective optimization, the clustering of all values of a gene to approximately the same value is not expected in multi‐objective optimization because the parameters will vary along the front. In order to illustrate this, consider the slot depth ds. Observe that in Figure 1.27(a) it has a slightly downward slope while in Figure 1.27(b) it has a slightly upward slope. This is because as we move from a low‐mass high‐loss design to a high‐mass low‐loss design the slot depth decreases. The core depth wc can be seen to be approximately constant.
The remaining parameters undergo more interesting behaviors. Consider N*, for example. Observe that in Figure 1.27(a), the nondominated solutions fall into two groups, which are indicated with a darker shaded and lighter shaded ellipses for lower and higher mass, respectively. The direction of decreasing mass is indicated with an arrow. Observe that the designs undergo a bifurcation indicated by a black vertical arrow. This can also be seen in Figure 1.27(b), wherein the sets are again circled. Note that the direction of decreasing mass is now to the left. The designs that are not in the two groups are dominated solutions. The bifurcation of the design space is also readily apparent in the slot width ws, core length lc, and air gap g. Such bifurcations in the design space can be the result of the change of a discrete variable, or the result of the design space moving into or out of a constraint. In this example, if we replace (10.1‐1) with N = N*, the bifurcation disappears. Of course, in doing this, our problem becomes strictly mathematical in nature since N must be an integer in practice.
