Optimization of electromagnetic devices: circuit models, neural ...

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. 3, MAY 1995

Optimization of Electromagnetic Devices: Circuit Models, Neural Networks and Gradient Methods in Concert S. Ratnajeevan H. Hoolel and M. K. Haldar

Department of Electrical Engineering National University of Singapore 10 Kentridge Crescent, SINGAPORE 0511

ABSTRACT:Optimization in designing electromagnetic products is now increasingly better understood. As opposed to classical models of magnetic circuits, today, V d i e n t techniques for mathematical optimization ave been proposed and are used. These techniques, while being expensive, are exact. More recently, artificial neural networks have been suggested, but they work best only if the data set of parameter-set, performance pairs for training the network is close to the optimal solution that we seek. In this paper, it is shown how all three methods may be used in concert to increase efficiency. The circuit model is used to generate an approximate inverse solution. Then direct finite element solutions are used to Tnerate the required training set and this is used with t e neural network to get a better solution. This solution is finally used as a starting point for the radient optimization scheme which converges quickly Because the starting point is close to the actual solution. I. INTRODUCTION The optimization of electromagnetic devices is now an increasingly mature art [l-121. It had been an iterative strategy and involved defining a performance based object function F that is at its minimum when the desired performance is best realized, and then finding that device descriptive parameter set (p) consisting of dimensions, and material properties and current densities of the various regions, that would give a minimum F((p)). Inequality constraints are incorporated into the object function using penalty terms when they are violated [6,71. Methods of finding the optimal F could be classified into three broad classes 1131. i. Search methods: At a point in the iterations, according to the evolution strategy [S. 101, arbitrary changes of fixed length are made in (p) and the object function is evaluated. The new points are likened to children, and a subset of the children with lower object function values are allowed to survive. New searches are then made using the children as a starting point. i i. Stochastic methods: Simulated annealing [9,121 is the most popular of the stochastic

On sabbatical leave from Harvey Mudd College, Claremont, Ca 91711. All correspondence on this paper to Dr. Hoole in California. Paper submitted on 4 July, 1994.

methods, and combines searching with in probability considerations. It uses a parallels to cooling in the process of annealing based on the Boltzmann probability distribution function. That is, we change from design configuration 1 to 2 with object functions F1 and F2 with a probability exp[-(F2-Fl)/(kT)]. If this probability is greater than 1, by implication F2 is lower than F1 and the change is accepted. At a given temperature, the configurations are arbitrarily changed using a random number generator and the designs are also changed if so dictated by a probability greater than 1. Subsequently, for the next round of searches, the system is "allowed to cool" by lowering the temperature. This makes uphill excursions less likely and limits the search space. i i i Gradient methods: Gradient methods [l-7, 141 employing the derivatives of the object function, search for the minimum of the object function and are the most effective [121. Although highly efficient and well established in structural engineering [151, they have been poorly understood by electrical engineers [8,121. It has been shown that unless the finite element mesh is constructed subject to certain constraints, the gradients are subject to error [3,7]. Not recognizing this, several uncomplimentary conclusions have been drawn about the method in the electrical engineering literature [8,12]. Notwithstanding these criticisms, as established outside electrical engineering, these are by far the most powerful of the optimization techniques [131. The one negative aspect of these schemes-that the finite element code has to be extended for gradients computation whereas search and stochastic schemes may be implemented with existing finite element codes, using them only to do object function evaluations-appears to be less important with a recent paper by Park et a1 [161. However, more recently, a non-iterative optimization scheme has been proposed using artificial neural networks [17-201. Here, if we knew

0018-9464/95$04.00 0 1995 IEEE

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Table I: Comparison of Zeroth, First and Second Order Gradient Methods with Different Starting Points

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Dirichlet Figure I: Symmetric portion of pole face the approximate whereabouts of the optimal solution, several direct problems are solved around this region using arbitrary combination of the parameters (p) and measuring the associated performance, say the flux density distribution (B) at given points. A neural network is then trained with (B) as input and (p) as output. The inputs and outputs of the network will be as many as there are elements in (B) and (p), respectively. The training of the network then is really the learning of the mapping (B) + (PI. Once the neural network learns this mapping, it is able to come u p with a (p) that is associated with a desired (B). Such a response from a neural network involves only some series summation and evaluation of exponents, and as such, is noniterative and quick. However, the neural network is able to do this well only if the training set is ”close” to the actual solution and even then, there is some error. When the training set is not so good in this sense, the neural network may have to “extrapolate” as opposed to “interpolate” and therefore we might expect the error to be large. In this Paper we show how good design Practice can use all these methods in concert, namely classical Circuit models Of magnetic Circuits, sophisticated optimization methods, and neural networks.

11. COMBINED METHODS What we propose and demonstrate here is both simple and natural. We suggest that the initial design should be made classically. That is, using circuit models of magnetic systems. However, it is emphasized that today, dealing with circuit models is no longer a slow hand-calculation process. Sophisticated programs exist for doing this and they are routinely used in industrial practicea. In this, we a

An example is the magnetic circuit-based program suite Maxwell, developed by Dr. Wade Cole and routinely used in-house at IBM, San Jose, CA. This has no connection to the program Maxwell by Ansoft Corp., Pittsburgh, Pa.

could also be assisted by modern AI techniques in changing the parameters of a circuit model so as to move it towards the better design [211. The advantage here is constraints are easily accounted for since the design is put together by the designer and there is no danger of the computer leading us into constraint violations. But at the end of the exercise, we know that circuit models are approximate. However, this approximate design leads US to the training data for the neural network. A finite element program is used to generate (p], (B) pairs around this design and the neural network then gives us the best solution. Using this as a starting solution, the gradient technique of mathematical optimization is used to fine-tune the solution.

111. CONJUGATE GRADIENTS AND STARTING

SOLUTIONS The conjugate gradients scheme for determining a vector, progressively determines components in orthogonal directions. It has been known that in matrix solution with conjugate gradients, taking a starting solution close to the actual solution does not help in minimizing the iterations - since the iterations depend on the conjugate components to be determined, diminishing the size of the components does not diminish the number of searches [22]. However, traditionally less competitive methods like SOR that rely on progressive magnitude reduction with iterations, are benefited by knowledge of an approximate solution. A similar issue exists in minimizing object functions and needs to be resolved. In optimization by first order gradient methods - a good compromise between the simple but slow zeroth order methods and the complex but accurate second order order methods - the steepest descent and conjugate gradients are the most commonly used, and of these,

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the latter is considered superior in terms of robustness. A question then is, does knowledge of an approximate solution give the usually less effective steepest descent method any advantage? Our studies using a 2parameter minimization problem (Table 1) show that both methods benefit equally from prior knowledge of an approximate solution. The optimal solution is at (0.0,O.O) and the table compares a far-away starting point with a nearby one.

IV.TESTPROBLEM AND RESULTS Test Problem and Object: The studies here were based on optimizing the poleface of Fig. 1 so as to obtain a constant flux density of 1.1 T in the airgap. The actual problem depicts a series of alternating N and S poles under a a flat plate, with a small airgap. Figure 1 details a part of the geometry. Flux flows are shown in Figure 2. The right vertical boundary is a line of symmetry through a pole and the left one mid-way between adjacent poles. The object is to obtain 1.1 T in the gap above the pole. As a first step we want an initial design based on circuit models. Figure 2 shows the leakage paths and Figure 3 the reluctance circuit. A verification of this model by adjusting x of Figure 2 and comparison with finite elment solutions is provided in a companion paper [231. As indicated by the finite elment comparison of the model, we did indeed get about 1.1 T in the airgap, but it was much lower at the line of

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symmetry and much higher at the left end of the pole where the flux bends left to flow to the oppsite pole. When the airgap gets larger, this effect reverses, as a result of leakage from the left end of the pole and the flux density is higher at the middle than at the sides. Circuit Model: We have an approximate design now, but with a flat pole face which gives a varying flux density along the pole face, but with the average at the desired value. Thus to obtain a flat distribution, we need to shape the pole-face, raising the height where the flux density is below the desired value (to reduce the reluctance) and increasing the airgap along places where the flux density is too high (to increase the reluctance). With these qualitative considerations, the flat pole-face was shaped. Neural Networks and Gradients-based Fine-tuning Having an approximate design, forward mappings of (p) -+ (B) were obtained using finite elment analysis, where (p) represents a parametric description of the pole face height. These are scaled parameters that represent the pole-face as a quadratic spline curve. A detailed description of this fitting is given in [201. For these sets the neural network was trained to do the inverse mapping (B) + ( p ) . Detailed optimization studies of this are reported in [7]. Table 2 shows a rough approximation of the desired pole face, neural network improvement of this using training sets around the rough approximation, and finally the actual design by accurate mathematical optimization using the methods of 7.

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V.CONCLUSIONS This paper has pointed out how optimization may be efficiently accomplished by progressively using more sophisticated methods, at each stage using the result of the previous stage as a starting point since neural networks a n d mathematical optimization are done more efficiently if an approximate solution is available. This also avoids Corresponds to F i w 1

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Figure 2: Model of Leakage Paths

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Table 2: Three Parameters by Rough Approxima-tion, Neural Network and Gradient Optimization I Circuit I 0.400 I 1.o I 0.15 I Model 0.440 0.904 0.082 Neural Network Gradient 0.490 0.900 I 0.090

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the possibility of getting into 1ocal.minima of object functions that may lie far away from the optimum and helps in taking care of constraints.

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