Optimal Design of a Novel Permanent Magnetic Actuator using Evolutionary Strategy Algorithm and Kriging Meta-model

Abstract

The novel permanent magnetic actuator (PMA) and its optimal design method were proposed in this paper. The proposed PMA is referred to as the separated permanent magnetic actuator (SPMA) and significantly superior in terms of its cost and performance level over a conventional PMA. The proposed optimal design method uses the evolutionary strategy algorithm (ESA), the kriging meta-model (KMM), and the multi-step optimization. The KMM can compensate the slow convergence of the ESA. The proposed multi-step optimization process, which separates the independent variables, can decrease time and increase the reliability for the optimal design result. Briefly, the optimization time and the poor reliability of the optimum are mitigated by the proposed optimization method.

1. Introduction

Permanent magnetic actuator (PMA) has many advantages, such as its short operation time, compact structure, high reliability, high repeatability, and low maintenance costs [1-8]. Hence, the PMA has received much attention lately for the application of a circuit breaker [1-4, 9]. Many researchers have been conducted the research on the PMA as reported in [1, 2, 5-7, 10-17] for several decades. The PMA requires a rare earth permanent magnet for high efficiency. However, the cost of a rare earth permanent magnet is steadily increasing at present, which accounts for a considerable part of the total cost of a PMA [4]. To solve this problem, we proposed a novel PMA and its optimal design method.

The proposed PMA, termed as a separated permanent magnetic actuator (SPMA), uses less permanent magnet than the existing PMA. An optimal design method using an evolutionary strategy algorithm (ESA), a kriging metamodel (KMM), and the multi-step optimization is proposed in this paper because the trial-and-error process for the design of the SPMA increases the time needed as well as the cost while introducing the reliability problem which is whether the designed result is an optimum or not.

The ESA experiences the problem of slow convergence to the global optimum [18]. To address this problem, the KMM, which is known as an effective method for the approximation of complex and nonlinear functions [19], was combined with the ESA. Hence, an effective search of the global optimum is possible while decreasing the optimization time and increasing the reliability. Furthermore, we proposed the multi-step optimal design method, which optimizes the independent variables step-by-step, because if all of the variables are optimized at once, time and reliability problems can be occurred.

 

2. Structure and Working Principle of the SPMA

Fig. 1 shows the comparison between the conventional PMA and the proposed SPMA. If the mover is located at the top, it is in a closed state. If the mover is located at the bottom, it is in an open state. The PMA and SPMA can maintain a closed state or an open state only using the permanent magnet without any electrical power consumptions. When the closing or opening coil is excited by current, a movable electrode connected to the mover by a shaft moves and the circuit is opened or closed.

Fig. 1.Comparison of the calculated magnetic flux density of the closed state between the PMA and the SPMA using same amount of permanent magnets

As demonstrated in Fig. 2 (a), the SPMA maintains the closed state by using the upper and the lower permanent magnets. When the opening coil is excited as described in Fig. 2 (b), the mover starts to move. As presented in Fig. 2 (c), after the mover arrives to the opening position, the opening coil is turned off and the open state can be maintained by using only the lower permanent magnet.

When the closing coil is excited as shown in Fig. 3 (b), the mover starts the motion and then arrives to the closing position as elucidated in Fig.3 (c). After the mover arrives to the closing position, the closing coil is turned off and the closed state can be maintained by using both the upper and the lower permanent magnets without any electrical power consumption.

The holding force acts as a load to the actuator. Hence, a minimal holding force is essential. A larger value of the holding force is required for the closed state as compared to the open state. However, the conventional PMA cannot generate a different value of the holding force in the open state and the closed state [20, 21]. Moreover, the efficiency for the generation of the holding force is low because permanent magnet are located symmetrically only in the middle of the actuator. To address these problems with the PMA, we propose the novel. The opening holding force of the SPMA can be controlled independently by controlling the ratio between the upper magnet and the middle magnet. The efficiency of the SPMA is higher than the PMA with the following reasons: the magnetic flux of the PMA is concentrated in the mover, but the high magnetic flux area of the SPMA is in the air gap; the magnetic flux path of the SPMA is shorter than that of the PMA. Hence, the size and the cost of the coil, the mover, and the capacitor for the SPMA can be reduced significantly.

Fig. 2.Calculated result by using FEM for the opening of the SPMA: (a) Closed state; (b) Excitation of the opening coil; (c) After finishing the opening operation, the open state is maintained by the lower permanent magnet.

Fig. 3.Calculated result by using FEM for the closing of the SPMA: (a) Open state; (b) Excitation of the closing coil; (c) After finishing closing operation, the closed state is maintained by both the lower and the upper permanent magnet. Fig. 4. The equivalent circuit of the SPMA

 

3. Analysis of the SPMA

For the analysis of the SPMA, we developed an analysis tool, which is named as SNU_SPMA. The analysis method of the SNU_SPMA is as follows.

Fig. 4 shows the equivalent circuit of the SPMA, where C is the capacitance of the capacitor, Vc is the voltage at the capacitor, Tr is the switching controller, I is the current of the coil, Rcoil is the resistance of the coil, L is the inductance of the coil, Diode is the Fly-wheel diode [22].

Using the equivalent circuit and time difference method(TDM), the voltage equation and the change in the current at the nth time step, dIn, are governed respectively by: (1) and (2) while the mover is in a standstill at early in an operating stage; (3) and (4) while the mover in a motion; (6) and (7) after the mover arrived at the target position, in which is the transformer electro-motive force (TEMF), is the motional electro-motive force (MEMF), and V is the external input voltage with a zero value because the controller cuts off the power after the mover arrived at the target position.

Fig. 4.The equivalent circuit of the SPMA

The capacitor voltage will drop, as shown by (7), until the switch controller cuts off the capacitor voltage.

The equation of motion for the SPMA, the velocity of the mover and the displacement of a mover can be induced by (8, 9), and (10), respectively, where m is the mass of the mover, is the acceleration of gravity, is the magnetic force acting on the mover, is the spring load, and represents the frictional force. The magnetic force is calculated by the magneto-static field analysis.

 

4. Verification of the Analysis Method

The proposed SPMA was prototyped for a 17.5(kV), 40(kA) VCB in this research. The analysis data and the experimental values are shown in Figs. 5 and Fig. 6.

In experimental data of Figs. 5 and Fig. 6, the inclination of the displacement is changed and the oscillation is occurred from a specific point, 12mm. This point is related to the compressive spring, which is installed for the stable contact of the movable electrode with a fixed electrode and for the aid of the opening operation by using the compressive force of the spring. In case of the opening operation, the oscillation starts at 12mm because the compressive spring is fully stretched at this point and the opening velocity is decreased from that point. As shown in calculated data, the change of the inclination of the displacement is taken into account in the analysis by considering the stiffness of the compressive spring. However, the oscillation could not be considered in the analysis because the oscillation is a nonlinear vibration occurred by an instantaneous change of applied force. In case of the closing operation, similar oscillating pattern is also demonstrated because of the spring effect as the case of the opening operation.

Fig. 5.The displacement of the movable electrode and the voltage of the opening capacitor during the opening operation of the verification model

Fig. 6.The displacement of the movable electrode and the voltage of the closing capacitor during the closing operation of the verification model

The analysis data show good agreement with the experimental values such that the correctness of the proposed analysis method is confirmed. The minute difference between the experimental data and the calculated results shown in Figs. 5 and Fig. 6 is induced from the nonlinearity of the material.

 

5. Optimal Design of the SPMA

5.1 ESA combined with the KMM

The KMM creates a surrogate model, which is similar to the actual function, using data calculated through the ESA. The predicted optimum result using the KMM can be served as a candidate for the next generation of the ESA. The ESA is a stochastic algorithm. Although, the ESA can search for a solution over a large area, the convergence to the optimum result is slow. Hence, by combining the KMM and the ESA, the global optimum can be found and the optimization time can be reduced while maintaining the high reliability.

The optimal design method is proposed for the SPMA using the ESA and the KMM via the steps shown below.

Step.0. Generation of the initial population: n individuals are generated with same grid in the problem region.

Step.1. Generation of the parent set: among the population with n individuals, μ number of elite solutions are selected as members of the parent set. In this research, μ equals to one due to the use of (1+1) ESA [23]. The remaining values are used for the surrogate model using the KMM.

Step.2. Generation of children: calculate the fitness of each solution and produce λ number of new children within the evolution range, where λ equals to one [23-25].

Step.3. Annealing: if the elite solution improves compared to the prior generation, the evolution range will be increased through division with the annealing factor to prevent convergence to the local optimum, where the annealing factor is set to 0.85 empirically. If the elite solution does not improve, the evolution range will be decreased through the multiplication with the annealing factor [23].

Step.4. Shaking: the shaking step is the generation of a random solution within the search area for diversity of the solution. In a general ESA, the frequency of shaking depends on the convergence ratio. If the degree of convergence is increased, the frequency of shaking is increased [23-25]. In the ESA combined with the KMM, the shaking number grows as the iteration is increased to improve the overall accuracy of the surrogate model and to prevent falling into the local optimum.

Step.5. Reshaping: the surrogate model, the KMM, will be updated using the calculated data during the optimization step and will converge to the actual function by the reshaping process.

Step.6. Convergence check: if a solution superior to the current elite solution is found in the reshaped surrogate model, the elite solution will be changed to the superior solution and the process will repeat steps 1 through 6 until the evolution range is under a specific value, which means the solution is converged [23].

5.2 Verification of the proposed optimization algorithm

The proposed algorithm, which is the ESA combined with the KMM, is verified through the mathematical test function (11). As demonstrated in Fig. 7, the proposed algorithm can reconstruct the function within 1% error from the real test function through 208 iterations, where the error is the average error of the function. These values are derived from the average of ten time tests.

Fig. 7.Optimization result of the test function by using the proposed algorithm, which is the ESA combined with the KMM.

5.3 Multi-Step optimization

The first optimization step is the optimization of the permanent magnet, the mover, and the stationary iron core through a magneto-static field analysis using the FEM to satisfy the required closing holding force. The variables are X1 and X2 as shown in Fig. 5.

For the opening holding force, the middle magnet can be designed prior to the optimization process because the opening holding force is generated only by the middle magnet and because the saturation does not occur by the middle magnet. The width of the permanent magnet, which has no effect on the magnetic flux density, is determined to be the minimum size, 8(mm), considering the demagnetization.

The closing coil also can be designed prior to the optimization process because saturation does not arise by the closing coil and because a high voltage drop of the closing capacitor is permitted due to its one time operation during the operating duty test, which is the estimation of the malfunction by carrying out the continuous operation of the closing and the opening. The operating duty test of the 17.5(kV)/40(kA) VCB involved an open-close-open sequence.

The variable ranges are as follows: 15

The second optimization step involves the optimization of the opening coil to meet the requirements of the velocity of the movable electrode and the voltage drop of the opening capacitor. The horizontal length of the opening coil, X3, should be shorter than the vertical length because if the horizontal length is longer than the vertical length, the end-winding effect is increased, inducing an increase in the inductance. Hence, the vertical length is determined to 1.5 times the size of the horizontal length. The variables for the second optimization step are the horizontal length, X3, and the diameter of the opening coil, X4. The variable ranges are as follows: 18

If the optimization for the SPMA is carried out by considering all variables together in an optimization process, both the magneto-static field and the dynamic characteristic have to be calculated for a model, of which dynamic characteristic analysis needs not to be calculated when the requirement of holding force is not satisfied through the magneto-static field analysis. In other words, much time is wasted by calculating the dynamic characteristic analysis using the time difference method for a useless model. In the proposed multi-step optimization method, the magneto-static field and the dynamic characteristics are analyzed in turn during the optimization of the independent variables.

Most of all, if the multi-step optimization is not used for the SPMA, the optima could not be found out without the convergence or much time is required for the convergence due to many variables. Hence, the optimization time can be reduced remarkably using the proposed multi-step optimization method.

5.4 Optimization result

As shown in Fig. 9, the length of the permanent magnet X1 and the width of the mover X2 are optimized to 19(mm) and 34(mm), respectively, satisfying the required closing holding force of 7116(N) after 346 function calls, which is the number of samples, during the first optimization step. As tabulated in Table 1, the length of the permanent magnet of the optimized model was decreased by 10% from that of a basic model which is designed via analysis of dozens models by an expert designer for SPMA. This design process of the basic model is termed as the trialand- error method in this research.

Fig. 8Variables of the SPMA for the multi-step optimization.

Fig. 9.The final KMM of the first optimization step

Fig. 10.The final KMM of the second optimization step

The width of the slot X3 and the diameter of the opening coil X4 are converged to 23.2(mm) and 2.1(mm), respectively, after 458 function calls during the second optimization step, as displayed in Fig. 10. The voltage drop of the opening capacitor in the optimized model was largely diminished, increasing the stability during the operating duty test compared to that of the basic model, as shown in Table 1.

Fig. 11.The displacement of the movable electrode and the voltage of the opening capacitor during the opening operation of the optimized model

Table 1.Comparison between the basic model which was designed using the trial-and-error method and the optimized model

Fig. 11 shows that the optimized SPMA satisfies all of the requirements. The opening velocity exceeded 1.2(m/s), which can prevent a dielectric breakdown. The voltage drop was under 10(V) for the operating duty test.

 

5. Conclusion

The remarkable finding in this paper is that for the novel PMA, which is termed the SPMA, the problems associated with the conventional trial-and-error design were solved. A fast and reliable optimal design was made possible through the proposed optimal design method which uses the characteristic analysis method, the multi-step optimization method, the ESA, and the KMM. The amount of the permanent magnet and the voltage drop of the opening capacitor were reduced compared to the basic model, leading to a decrease in the cost and an increase in the stability through rapid optimization using the proposed optimal design method. Hence, this research has significant meaning in that the proposed SPMA and the optimal design method can lead to the proliferation of a permanent magnetic actuator for CBs.