Multiobjective tabu search algorithm for the optimal design of a thermo-acoustic magneto-hydro-dynamic electric generator

Abstract

The aim of the proposed work is to study the optimal design of an innovative Thermo-Acoustic-Magneto-Hydro-Dynamic electric generator, with particular reference to the Magneto-Hydro-Dynamic section. A multi-objective optimization algorithm, which makes use of a Tabu Search meta-heuristic, has been developed to this purpose. Thermo-Acoustic and Magneto-Hydro-Dynamic energy conversion processes give a great advantage by converting energy without solid moving components. This makes the cited technologies very interesting for the low weight, the low maintenance costs, and the high conversion efficiency. The design of the generator has to be optimized by considering conflicting objectives, i.e., maximizing the output power, minimizing the applied electrical voltage, and minimizing mass and size of the device. Therefore, a multi-objective vectorial optimization approach is mandatory. A fully vector scheme has been implemented that takes under control both the Pareto optimality of the solutions, and the uniformity in the Pareto front sampling.

Keywords

Magneto-hydro-dynamics thermo-acoustics multi-objective optimization tabu search

1. Introduction

During the last decades, stochastic techniques have been widely investigated in order to face optimization problems with multiple conflicting objectives. These problems are called Multiobjective Optimization Problems (MOPs), or vector optimization problems. A large number of multiobjective stochastic optimizers, based on local or global search methods, have been developed in the last decades for the optimal design of electromagnetic devices [1, 2, 3, 4]. These algorithms are commonly referred to as evolutionary algorithms, including Genetic Algorithms, Tabu Search (TS) methods, Ant Colony algorithms, and Particle Swarm Optimization. Among them, TS methods are meta-heuristic iterative procedures that guide a local search avoiding trapping in local minima. These methods, starting from a potential optimum, tend to investigate in depth its immediate neighbors aiming to find an enhanced solution. As a drawback, the local search risks to be trapped in a suboptimal region, like a local minimum. However, the TS technique is able to improve the performances of the cited optimum search method by allowing worsening moves and avoiding the return to the already examined solutions. Some scalarization techniques have been proposed in literature to adapt the inherent scalar schema of the TS to the vector optimization problems [5, 6]. Conversely, in [2] an innovative fully vector multiobjective TS algorithm has been developed and the performance of the proposed algorithm has been successfully compared with other existing TS algorithms on some analytical and electromagnetic MOPs, whose Pareto front is available from the literature [7, 8, 9, 10].

In a Thermo-Acoustic-Magneto-Hydro-Dynamic (TA-MHD) electric generator, the absence of solid moving parts and the exploitation of a quasi-static working fluid result in high power-to-mass ratio, high energy conversion efficiencies, and low maintenance requirements. Depending on the specific application, which the generator is oriented to, different device parameters should be chosen in order to optimize the multiple, even conflicting, design objectives. Hence, different regions of the Pareto front could be of interest for the designer. For example, in Space applications, the limitation of mass and the space-saving are strategic, so as for automotive field, but in this case higher level of power have to be considered; on the other hand, in nuclear plants and in nuclear fusion, the main target is to maximize the flow of power, while larger size and high values of exciting voltage are allowed; finally, in renewable energy harvesting, a low level of power can be accepted, provided that the excitation voltage and the size are limited.

In this paper, the fully vector Multi-objective TS (MO-TS) proposed in [2] has been used to customize the optimal design of the Magneto-Hydro-Dynamic (MHD) section of an innovative TA-MHD electric generator [11]. Some simplifying assumptions do not allow to exactly estimate the device efficiency, which is beyond the scope of the present work. However, the proposed optimization approach is enough flexible to allow the introduction of further requirements and constraints.

2. Principle of functioning of the TA-MHD generator

Thermo-Acoustic (TA) engines can convert high temperature heat into acoustic power with high efficiencies and without moving parts. They have a competitive efficiency, a limited mass and promise to be low maintenance [12, 13, 14]. The TA phenomenon takes place in a duct containing a gas when an enough steep gradient of temperature acts between its ends in longitudinal direction. In order to obtain such effect, we need a heat source and a stack inside the duct with a large interface-surface. The gradient of temperature affects the outlet power, whereas the frequency of the acoustic wave depends on the length of the duct [15].

Coupling this system with a MHD energy conversion stage, a totally static electric generator powered by the heat can be suitably built, e.g., for space applications [11, 16]. MHD power generation systems can make the best with the available enthalpy gradient because the interaction of plasma with a magnetic field must take place at much higher temperatures than in a classic mechanical turbine. However, this kind of machines will be efficient only if the charges concentration in the gas is at an adequate level. This is usually obtained by heating the gas at high temperatures and by additively seeding it with ionizing molecules. In conventional MHD generators, the plasma crosses perpendicularly an intense magnetic field where two electrodes are immersed; under the action of the Lorentz force, by closing the circuit on a load, an electric current will flow through the fluid [17]. Conventional MHD generators present several drawbacks [11]: the need of high temperatures for the gas ionization; the need of seeding the gas with alkali materials and then recovery that seeding alkali; the need of a high magnetic field (up to 5 T), which requires the use of superconducting coils; the rapid deterioration of the electrodes, which are strictly in contact with the fluid.

Figure 1.

Schematic view of the TA-MHD generator.

The technology proposed in [11, 18] offers a solution to all the above-listed limits. The proposed innovative TA-MHD generator, whose schema is shown in Fig. 1, does not require an external magnetic field to work, because it performs the energy conversion through the induction of an alternate electrical current. In particular, it consists of two stages: the conversion of heat into mechanical energy of vibration at a first stage, and the cascading conversion of such vibration into electrical energy. The working fluid is forced to become plasma by means of an electrical discharge coming from an external high voltage generator, through the use of two electrodes placed into the fluid. As long as the applied voltage is strong, the gas can be ionized also at low temperatures and seeding is not necessary [19]. The charge carriers of opposite sign are separated by an external DC electric field, applied by means of a properly designed capacitor. The voltage, necessary to maintain separated in equilibrium the two clouds of charges of different sign, linearly depends on the surface of the plates; for this reason the shape of such plates has to be carefully designed. Once the equilibrium is reached, if a pressure wave travels along the duct, because of the thermo-acoustic effect, the clouds of charge carriers participate to the oscillation of the surrounding neutrals, giving rise to an axial alternating current inside the gas [20, 21]. Such current in turn induces an electromotive force in the toroidal coils wrapped around the duct in correspondence of the capacitor plates. The coils are connected to an electrical load where the energy conversion process ends [11].

The electric device behaves like an amperometric transformer where the primary circuit is constituted by the charge carriers oscillating inside the gas, while the secondary one is, as usual, made of copper wire and connected to the load.

3. Multi-objective tabu search (MO-TS) algorithm

A MOP is characterized by a vector of mutually conflicting objective functions. This vector problem can be also seen as a constrained optimization problem where one of the quantities is assumed as the function to be optimized, whereas the other quantities are defined as constraints of the optimization problem. The two definitions are similar and all the quantities involved can be treated as objective functions of the MOP, neglecting their division into goals and constraints.

Therefore, a MOP can be formalized as follows:

$\displaystyle\mathop{\min}\limits_{\bar{x}\in\Omega}F(\bar{x});\quad\Omega=% \left\{\bar{x}\in\left.R^{n}\right|G(\bar{x})\leqslant 0;H(\bar{x})=0\right\}$ (1)

where $\bar{{x}}$ is the vector of the $n$ real-value input variables (design variables), $\Omega$ denotes a decision variable space and $F$ is the vector of objective functions $(F_{1},\ldots,F_{K})$ . Here $G(\bar{{x}}),\;H(\bar{{x}})$ represent the problem constraints. A MOP aims to find the Pareto-optimal solutions, by seeking non-dominated solutions. In a MOP, a solution $\bar{{x}}_{d}$ is dominated by another solution $\bar{{x}}_{nd}$ if $\bar{{x}}_{nd}$ is better than $\bar{{x}}_{d}$ on all objectives, and it will be denoted, here, by $\bar{{x}}_{d}\prec\bar{{x}}_{nd}$ . A solution $\bar{{x}}_{p}$ is a Pareto-optimal solution if no objective function can be improved without worsening at least one other objective function.

The proposed MO-TS algorithm is able to build a list of non-dominated solutions deeply exploring regions of the parameters space that look promising, and moving away from regions that do not look promising. A Pareto-optimal list P is introduced in the proposed method. The values of the archive are dynamically updated and used in the optimization process: at each iteration, some solutions in the Pareto-optimal list P could become dominated by the new solution, and then they have to be removed. The length of this list is finite.

In order to overcome the limit of the TS schema, which is inherently scalar, the vector optimization problem is formalized by assigning to each visited solution $\bar{x}_{i}$ two functions to be maximized:

$\displaystyle f_{1}\left(\bar{x_{i}}\right)=\frac{1+N\left(\bar{x}_{d}\prec% \bar{x_{i}}\right)}{1+N\left(\bar{x}_{nd}\succ\bar{x_{i}}\right)}$ (2) $\displaystyle f_{2}\left(\bar{x_{i}}\right)=\left(exp\left(-0.5\left(\frac{d-% \lambda}{\gamma}\right)^{2}\right)-exp\left(-0.5\left(\frac{d}{\gamma}\right)^% {2}\right)\right)$ (3)

The first one ( $f_{1}$ ) allows storing a set of solutions that are not dominated by those already present in the list. Referring to the list P of the Pareto optimal solutions presently found, $N(\bar{x}_{d}\prec\bar{x}_{i})$ is the number of such solutions that are dominated by the current solution $\bar{x}_{i}$ , $N(\bar{x}_{nd}\succ\bar{x}_{i})$ is the number of solutions dominating $\bar{x}_{i}$ . The aim of the algorithm is to find solutions not dominated, therefore solutions with $N(\bar{x}_{nd}\succ\bar{x}_{i})=0$ .

The second function ( $f_{2}$ ) has two terms, the first one that favors the new points at a certain distance from the nearest point of P, the second one that penalizes the points that are very close to a point of P. Some parameters in this function have to be set: $d$ is the smallest distance between the visited solution $x_{i}$ and the points of the list P, whereas $\lambda$ and $\gamma$ determine the sampling step of the Pareto front. In fact, Eq. (3) is the combination of two Gaussian functions with the same variance $\gamma$ and different mean value ( $\lambda$ in the first one and 0 in the latter). Then, $f_{2}$ penalizes the solutions close to those already found, whereas it privileges those with a distance equal to $\lambda$ . The $f_{2}$ is null for the solutions far from the already found ones. Therefore maximizing $f_{2}$ one favorites a uniform sampling of the Pareto Front.

4. MHD section design optimization

In [11], the coarse sizing of the MHD section is performed, in analytic terms, by assuming the physical quantities derived from the TA section as inputs. As a result, the relations among the variables of the system are assessed, allowing defining the set of design parameters and the objectives. In the present work, the feasibility ranges of the design parameters and the target values are assumed on the basis of the power requirements for future deep Space and Mars science missions stated by NASA [22], which envisage Radioisotope Thermoelectric Generator systems with a specific power of 9–10 [W/kg] and a power of 100 [W] or greater.

In [23], the applicability to the present context of the data concerning the assumed velocity and pressure profiles has been validated by means of a Finite Element Method analysis. In the present paper, the MOP has been used to optimize only the MHD section of the generator, i.e., the size of the duct, the characteristics of the stack, which works also as capacitor plate, and the core of the secondary winding.

Figure 2.

Schematic view of the MHD section to be optimized. In the right side the capacitor plates configuration.

In Fig. 2, the schematic view of the MHD section to be optimized is shown. In particular, the geometrical design parameters are the thickness of the torus, $R_{x}$ , the radius of the duct $R_{d}$ , and the distance $\delta$ between the sheets of the capacitor’s plates. The capacitor length $l$ is set on the basis of the required thermal gradient, by assuming ends temperatures of 827 ${}^{\circ}$ C and 127 ${}^{\circ}$ C respectively [24]. Concerning the thickness $L$ of the torus, it cannot be used as degree of freedom, because the optimization process tends to maximize it without any bound. A value has been set, such that the magnetic saturation of the core should be avoided, independently on the other design parameters. To a first approximation, the induction $B$ is assumed to be uniformly distributed in the cross section of the core. In the real case, a number of influencing factors have to be taken into account. For example, the thickness $R_{x}$ of the core and the ratio between the widths $L$ and $l$ of core and plate should cause a non-uniformity of $B$ ; on the other hand, the local saturation of the core should improve the homogeneity of the distribution. All these aspects are beyond the scope of this work, and they have been neglected.

The charge density is a critical parameter because, for a given frequency and amplitude of the TA vibration, the higher is the charge density the higher the ionic current in the gas and then the outlet power. In order to confine a sufficient quantity of charge carriers in the gas, the plates of the capacitor have to be characterized by a large surface/volume ratio. This property can be obtained by adopting sieve shaped plates, obtained by means of a set of parallel sheets, electrically connected among them by a peripheral ring (see the right side of Fig. 2). At the same time, the distance between parallel sheets has to guarantee the transversal profile of velocity be totally developed, thus the distance $\delta$ between two consecutive sheets must be a multiple $k$ of the thickness of the viscous boundary layer $\delta_{v}=\sqrt{\frac{\eta}{\pi f\rho_{0}}}$ , where $\eta$ is the dynamic viscosity, $\rho_{0}$ is the equilibrium density, and $f$ is the frequency of the inlet acoustic wave.

The confinement of the charges inside the plate of the capacitor is carried out as it occurs in a porous electrode of a double layer ultracapacitor [20, 25, 26]. The applied electrical field attracts the charges inside the cavities, where an equipotential wall surrounds them. Inside the cavity, the charge tends to distribute on the wall, in order to nullify the electrical field in the cavity itself. Once the gas is set in vibration, the charge is distributed between a stagnant layer and a hydrodynamically mobile layer [27]. The stagnant layer does not take part to motion, and then the charge inside this layer does not participate to the transmission of power. The quantity of charge that stay within the stagnant layer depends on several parameters, such as the applied voltage, the number of available charges, the frequency of the vibration and the properties of the gas. The determination of such design parameters is beyond the scope of this work, so we will assume, to a first approximation, the entire charge participates to the motion of the neutrals.

Table 1 reports the design parameters of the MOP together with their feasible regions.

Table 1

Design parameters of the MOP

Design parameter	Description	Range	Units
$R_{d}$	Duct radius	[7 $\div$ 30]	[cm]
$R_{x}$	Torus thickness	[1 $\div$ 30 ]	[cm]
$k$	Ratio between the capacitor plates thickness $\delta$ and the viscous boundary layer $\delta_{v}$	[4 $\div$ 100]	[a.u.]
$f$	Frequency of the inlet acoustic vibration	[10 $\div$ 1000]	[Hz]

In order to limit the computational costs and to fulfill the construction constraints, some of the design parameters have been fixed. Table 2 reports the values of the other parameters involved in the design of the device, which have been set from literature as the best choice for a specific aerospace application [11].

Table 2

Other design parameters set from literature

Fixed parameters	Description	Value	Units
$\mu_{r}$	Torus relative permeability (mu-metal)	5 $\times$ 10 ${}^{4}$	[a.u.]
$\rho$	Torus mass density (mu-metal)	8747	[kg/m ${}^{3}$ ]
$\rho_{0}$	Helium Density	7.8	[kg/m ${}^{3}$ ]
$\eta$	Helium Dynamic Viscosity	1.9912 $\times$ 10 ${}^{-5}$	[kg/(m $\cdot$ s)]
$v_{0}$	Charges velocity [15]	30	[m/s]
$\sigma$	Average charge density confined inside the capacitor	5	[C/m ${}^{3}$ ]
$l$	Capacitor length	1	[cm]
$L$	Torus length	15	[cm]

Three conflicting objective functions have been selected: the output power $P$ delivered to the loads; the voltage $E_{0}$ to be applied to the plates of the capacitor in order to maintain the cloud of charges in equilibrium into the gas; the mass $M$ of the device.

In [11], by minimizing both the magnetic induction $B$ and the ionic current $I$ in the gas (Helium), the following relation between the theoretical output power $P_{0}$ and design parameters has been derived:

$\displaystyle P_{0}=\frac{\pi\mu_{r}{\mu_{0}\omega SR_{d}^{4}\sigma}^{2}v_{0}^% {2}}{2({R_{d}+R}_{x})}$ (4)

where $\mu_{r}$ is the relative magnetic permeability of the torus, $\omega$ is the radial frequency of the inlet acoustic wave, $S$ is the cross-section of the torus of length $L$ and thickness $R_{x}$ , $R_{d}$ is the radius of the duct, $\sigma$ is the average charge density confined inside the capacitor and $v_{0}$ is the maximum amplitude of the sinusoidal velocity of the charge carriers.

However, the actual power to the load is lower than $P_{0}$ , due to the viscous power losses $P_{L}$ inside the channels of the capacitor plates composed of parallel sheets spaced $\delta$ :

$\displaystyle P_{L}=\frac{32\pi R_{d}^{2}\eta l\left(\frac{v_{0}}{\sqrt{2}}% \right)^{2}}{\delta^{2}}$ (5)

where $\eta$ is the Helium dynamic viscosity. Therefore the output power to be optimized is:

$\displaystyle P=P_{0}-P_{L}$ (6)

Moreover, starting from the capacitance of the plates-gas system, the voltage $E_{0}$ to be applied to the plates, in order to maintain the cloud of charges in equilibrium into the gas, has been derived:

$\displaystyle E_{0}=\frac{\delta R_{D}}{2\varepsilon_{0}\beta}\sigma$ (7)

where $\varepsilon_{0}$ is the vacuum electric permittivity, and $\beta$ is the ratio between the active interface-surface and the lateral surface of the plates of the capacitor, which is related to the surface/volume ratio of the capacitor. It can be demonstrated that $\beta$ is related to the duct radius $R_{d}$ and to the distance $\delta$ among the sheets according to the following relation:

$\beta=\frac{16R_{d}}{\pi^{2}\delta}+1$

Figure 3.

2-dimensional Pareto Front.

Here, once the average charge density $\sigma$ confined inside the capacitor is fixed, the needed voltage grows with the distance between the capacitor layers, while it is limited by the value of $\beta$ . By applying a high voltage, a large quantity of charge carriers participates to generate the electrical current, with benefit to the output power. On the other hand, the dielectric strength of the electrical insulating coating puts an upper bound to the voltage that can be applied. In order to limit the applied voltage, the two plates of the capacitor should have a high surface/volume ratio, which allows to increase the capacitance, and then to reduce the voltage, without increasing the size of the device. High $\beta$ means high outlet power, hence the optimization procedure will certainly tend to maximize $\beta$ . Low values of the distance between the sheets of the capacitor (then a high number of layers constituting the plates) are positive for the applied voltage minimization because this is directly proportional to $\delta$ . In order to increase the charge stored in the plates, and then the released power, we are interested to maximize the number of sheets inside the duct. However, the velocity profiles during the vibration represent a strict constraint to the value of $\delta$ . If the gap between two sheets is less than the viscous boundary layer, the motion of charges is strongly limited, therefore concentrating an excessive quantity of charge might be not advantageous.

Finally, once selected the building materials, mass and size limiting become of basic interest for some applications such as aerospace. Actually, the total mass of the device could be roughly considered as coincident to the mass of the magnetic core, so that:

$\displaystyle M=\rho\cdot S\cdot 2\pi\left(R_{d}+\frac{R_{x}}{2}\right)$ (8)

where $\rho$ is the mass density of the material constituting the torus. Once chosen the material for the magnetic core, the mass density is set and the mass of the device will depend approximately only on the medium radius $(R_{d}+\frac{R_{x}}{2})$ and on the cross-section $S$ of the toroidal core.

5. Results

All the calculations in this paper have been performed using an Intel Core i7-6560U CPU with 16 GB of RAM running under Windows 10 Professional. A first bi-dimensional optimization problem has been solved considering as objective functions the output power $P$ to be maximized and the voltage $E_{0}$ to be minimized. As previously described, these are two conflicting objectives.

Some algorithm’s parameters have been empirically chosen. In particular, the Pareto-optimal list P has been sized to store 100 not dominated solutions. The algorithm stops when the list is full or when the maximum number of 200 iterations is reached. For the calculation of the function $f_{2}$ the values $\lambda=$ 100 and $\gamma=$ 33 have been selected in order to uniformly sample the Pareto front.

The MO-TS optimization procedure is able to find a sampling of the Pareto front (Fig. 3) composed by 56 Pareto optimal solutions after 200 iterations and 200 000 000 computations of objective function with a computation time of 15 min. As can be noted, the algorithm is able to find uniformly sampled Pareto optimal solutions that spam in a range of [78–5982] W, and [16.8–56.54] kV. Further fitness functions should be used to choose among the optimal solutions. All the points of the obtained Pareto front correspond in the design space to the maximum values of the first three parameters, i.e., $f$ equal to 1 kHz and both $R_{x}$ and $R_{d}$ equal to 30 cm. The only design parameter that changes is the ratio $k$ that varies between 10.9 and 20, which corresponds to a number of parallel sheets in the capacitor comprised between 1052 and 1931.

Figure 4.

3-dimensional Pareto Front.

Successively, all the three objective functions have been considered and a new optimization problem has been solved by means the MO-TS algorithm. Figure 4 shows the Pareto front in the 3-dimensional space of the objective functions.

In this case, for the calculation of the function $f_{2}$ the values $\lambda=$ 500 and $\gamma=$ 167 have been chosen and 243 Pareto optimal solutions have been obtained with a computation time of 3 h. The obtained front is still quite uniform and the optimal solutions spam in a range of [9–6763] W, [20–100] kV, and [110–1106] kg, confirming the suitability of the proposed innovative generator for aerospace applications [22].

Table 3

Values of the three objective functions and of the corresponding design parameters

Power P [W]	Voltage E ${}_{0}$ [kV]	Mass M [kg]	R ${}_{d}$ [cm]	R ${}_{x}$ [cm]	$k$	$f$ [Hz]
2985	59.89	514.3	28.5	16.9	20.49	990.65
1091.72	44.41	359.5	27.2	13	17.72	999.98
6318.4	68.64	1021	30	28	22.036	1000

In Table 3, some Pareto solutions and the corresponding design parameters values are reported. For example, as shown in Table 3, if we need an output power delivered to the board instrumentation of about 3 kW, the minimum mass of the device will be about 514 kg and the voltage to be applied to the plates of the capacitor will be about 60 kV. On the other hand, by limiting the power at 1 kW, a great advantage is obtained in terms of mass of the generator and of voltage to be applied. Finally, it is possible to obtain the maximum power with larger values of mass and volume.

If, for a fixed power, a further reduced mass of the device is requested, magnetic materials with higher permeability, such as supermalloy [28], can be used for the torus. It must be considered that, due to ferromagnetic core saturation, the achievable mass reduction will not be directly proportional to the increase of the permeability.

6. Conclusion

In the present paper, a procedure for the optimal design of a Thermo-Acoustic-Magneto-Hydro-Dynamic electric generator is described. First of all, a 2-dimensional problem have been solved in order to simultaneously maximize the generated power and minimize the voltage necessary to maintain the cloud of charges separated. Secondly, with the aim to optimize the generator for aerospace applications, the minimization of the mass of the device has been introduced as a third objective function, so that a 3-dimensional Multi-Objective Optimization Problem has been solved. Some simplifying assumptions have been adopted, which do not affect the validity of the proposed optimization procedure. The algorithm’s free parameters have been empirically chosen and the target values have been assumed bearing in mind the power requirements for future space missions stated by NASA.

The obtained results show that the optimization algorithm is able to sample the approximate optimal Pareto front with a chosen number of sampling points. The uniformity of the sampled Pareto front is good and Multi-Objective Tabu Search performs very well in terms of computational cost. The results are useful for the designers because they are the expression of a multi-purpose optimization, where one can choose different functioning points depending on the specific application. In the future, if different targets were required, in terms of lower mass and/or voltage, or higher power, different design configurations could be envisaged, or different magnetic materials could be used, with very limited work to customize the MO-TS.

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