Efficient optimization design of dry-type air-core reactors with the improved genetic algorithm

Abstract

In this paper, we propose several approaches to improve the efficiency of the optimization design for dry-type air-core reactors. Above all, by virtue of the empirical formula of the inductance of short solenoids, a neat formula for the inner diameter corresponding to the minimum mass of the reactor, termed optimal inner diameter, is presented, and a favorable range of the inner diameter is determined accordingly. Then, an efficient optimization model is established without frequently and repeatedly adjusting the design variables. In addition, genetic algorithm for the optimization design of the reactor is improved by introducing linear scaling and a dynamic mutation operator. The proposed approaches are tested on the single-phase dry-type air-core reactor of 50 kVar with round conductors. An economic and qualified result can be obtained in a short period of time after the application of the approaches. The merits of these approaches could be more obvious for the large-capacity reactors.

Keywords

Optimization model dry-type air-core reactor genetic algorithm

1. Introduction

Dry-type air-core reactors have been extensively deployed in the power system, $e . g .$ , high-voltage direct-current (HVDC) transmission system, whose performance is critical to the operation of the entire system. To this end, related studies on the design of the reactor have been carrying out in recent decades [1, 2]. In general, an optimization design of a dry-type air-core reactor is aimed to reduce the mass of the conductors (manufacturing cost) under certain constrains, without sacrificing the performance on its inductance, package temperature-rise, and layer current-density, etc.. Previously, an optimization model based on design variables reconstruction is proposed under the equality constraints of inductance equalization, layer resistance-voltage equalization, package temperature-rise equalization, and package height equalization. Meanwhile genetic algorithm (GA) is applied to such model [3], due to the effectiveness in obtaining the derivative of the complicated objective function [4]. Later, a compromised strategy is introduced to the optimization model [5], dealing with the additional constraints. In addition, simplex method is combined with genetic algorithm to conquer the inefficient local search ability.

Despite that the present optimization models with the corresponding algorithms may yield relative economic and qualified results, the optimization design process is typically inefficient ( $e . g .$ , time-consuming for achieving an economic and qualified result), especially for the large-capacity reactors. In this paper, we improve the design efficiency of dry-type air-core reactors with round conductors by reasonably narrowing the range of inner diameter, improving the optimization model based on the compromised strategy, and introducing linear scaling and a dynamic mutation operator to genetic algorithm. To verify the effectiveness of the proposed approaches, an optimization design example is provided.

2. Range of the inner diameter

Figure 1.

A monolayer of winding of the reactor.

Figure 2.

The mass of conductors versus the inner diameter.

Dry-type air-core reactors consist of amounts of parallel-concentric layers of windings with small-diameter round-aluminum conductors. Several layers are separated by cooling ducts and encapsulated to form a package. In order to establish the relation between the mass of conductors $m_{c}$ and the inner diameter $D_{in}$ . For simplicity, we firstly consider the case of a monolayer of winding of the reactor, shown in Fig. 1, the mass of conductors $m_{c}$ can be approximately expressed as

$\displaystyle m_{c}=\rho\pi\left({\frac{d}{2}}\right)^{2}\pi D_{in}N_{pz}N$ (1)

where $\rho$ is the mass density of conductors; $d$ denotes the diameter of conductors of the reactor; $N_{pz}$ is the number of axial parallel conductors; $N$ denotes the number of turns. Here, the variable $N$ is required to be expressed by $D_{in}$ .

The ratio between the outer diameter and the height of the reactor is typically between 0.6 and 4, which pertains to that of short solenoids [6]. Thus, the empirical formula of the inductance of short solenoids could be used to derive the expression of $N$ . For the monolayer of winding, the thickness of short solenoids turns to be the diameter of conductors $d$ , and since $d$ is small enough, the empirical formula of the inductance can be expressed as

$\displaystyle L=\frac{6.4\mu_{0}N^{2}D_{in}^{2}}{3.5D_{in}+8h}$ (2)

and the height $h$ can be expressed as $h=k_{h}N_{pz}Nd$ , here $k_{h}=k_{z}\left({1+\frac{2t_{s}}{d}}\right)$ , where $k_{z}$ is the axial winding coefficient, $t_{s}$ is the unilateral insulation thickness.

Therefore, $N$ can be solved from Eq. (2). Inserting $N$ into Eq. (1), we consequently get

$\displaystyle m_{c}=\frac{1}{4}\rho\pi^{2}d^{2}N_{{pz}}\frac{8k_{h}N_{{pz}}dL+% \sqrt{(8k_{h}N_{{pz}}dL)^{2}+89.6\mu_{0}LD_{{in}}^{3}}}{12.8\mu_{0}}D_{{in}}^{% -1}$ (3)

Figure 2 shows the typical dependence of $m_{c}$ on $D_{in}$ , where $L$ , $d$ , $N_{pz}$ , $\rho$ , $k_{z}$ , and $t_{s}$ are 10 mH, 3 mm, 1, 2.7 $\times$ 10 ${}^{3}$ kg/m ${}^{3}$ , 1.05, and 0.12 mm, respectively. The inner diameter corresponding to the minimum of the mass of conductors can be obtained by the derivation of Eq. (3). We regard it as the optimal inner diameter, denoted by $D_{in}^{{opt}}$ , which can be expressed as

$\displaystyle D_{in}^{{opt}}=1.7878L^{\frac{1}{3}}\left({k_{h}N_{pz}d}\right)^% {\frac{2}{3}}\mu_{0}^{-\frac{1}{3}}$ (4)

For typical dry-type air-core reactors, the number of axial parallel conductors $N_{pz}$ for each layer is the same and the inductance $L$ for each layer is approximately equal to the rated inductance $L_{N}$ . With ignoring the effect of the space between layers, and since the optimal diameter is monotonously increasing with the diameter of conductors from Eq. (4), the range of the optimal inner diameter of the reactor can be estimated by Eq. (4) at the minimum and maximum of the conductor diameters, respectively, which can be used as the range of the inner diameter for the optimization design.

3. Optimization model

3.1 The primary model

The mass of conductors, $i . e .$ , the primary part of mass of the reactor, is taken as the objective function of the optimization model, which is given by

$\displaystyle\min f(D_{in},m,N_{pk},d_{k},w_{i})$ (5)

where $k$ denotes the $k$ -th package, which runs from 1 to $m$ ; $m$ is the number of packages; $N_{pk}$ and $d_{k}$ denote the number of layers and the diameter of conductors of the $k$ -th package, respectively; $w_{i}$ is the number of turns of the $i$ -th layer.

The constraints which should be strictly satisfied are elaborated as follows:

With ignoring the resistance for each layer, the circuit equations for the reactor can be expressed as

$\displaystyle\left\{{{\begin{array}[]{*{20}c}{\sum\limits_{j=1}^{n}{w_{i}w_{j}% f_{ij}I_{j}}=L_{N}I_{N}}\\ {I_{1}+I_{2}+\cdot\cdot\cdot+I_{n}=I_{N}}\\ \end{array}}}\right.$ (6)

where $j$ represents the $j$ -th layer, which runs from 1 to $n$ ; $w_{j}$ is the number of turns of the $j$ -th layer; $f_{ij}$ denotes the geometrical coefficient of the mutual inductance between the $i$ -th layer and the $j$ -th layer [1]; $I_{j}$ denotes the current of the $j$ -th layer; $I_{N}$ is the rated current.

The maximum of the package temperature-rise $\tau_{\max}$ shall not exceed $\tau_{\lim}=80{{}^{\circ}}$ C for insulation material of class B [7].

The maximum of layer current-density $J_{\max}$ is usually limited to $J_{\lim}=$ 1.4 A/mm ${}^{2}$ in case of local overheating.

The additional constraints contain layer current-density equalization, layer resistance-voltage equalization, package height equalization, and package temperature-rise equalization. They are illustrated as follows:

$\displaystyle J_{i}=J_{j}$ (7) $\displaystyle D_{i}w_{i}J_{i}=D_{j}w_{j}J_{j}$ (8) $\displaystyle w_{k}d_{k}=w_{l}d_{l}$ (9) $\displaystyle\frac{N_{pk}J_{k}^{2}w_{k}^{0.25}d_{k}^{1.25}}{(\alpha_{k}^{in}+% \alpha_{k}^{out})}=\frac{N_{pl}J_{l}^{2}w_{l}^{0.25}d_{l}^{1.25}}{(\alpha_{l}^% {in}+\alpha_{l}^{out})}$ (10)

where $l$ represents the $l$ -th package, which runs from 1 to m; $J_{i}$ and $J_{j}$ are the current-density of the $i$ -th and $j$ -th layer, respectively; $D_{i}$ and $D_{j}$ are the mean diameter of the $i$ -th layer and the $j$ -th layer, respectively; $w_{k}$ and $w_{l}$ are the average number of turns of the $k$ -th package and the $l$ -th package, respectively; $J_{k}$ and $J_{l}$ are the average current-density of the $k$ -th and $l$ -th package, respectively; $\alpha_{k}^{in}$ , $\alpha_{k}^{{out}}$ , $\alpha_{l}^{{in}}$ , and $\alpha_{l}^{{out}}$ denote the utilization coefficients of inner and outer heat-dissipation surface of cooling ducts of the $k$ -th and the $l$ -th package, respectively.

3.2 Processing of equality constraints

The key problem to improve the efficiency of the optimization model is how to handle the constraints properly. In order to determine $N_{pk}$ based on package temperature-rise equalization, we should add an equation

$\displaystyle n=\sum\limits_{k=1}^{m}{N_{pk}}$ (11)

Similarly, anther equation is required to solve $d_{k}$ based on package height equalization, we introduce the average diameter of conductors $d_{av}$ , expressed as

$\displaystyle d_{av}=\sqrt{\frac{1}{n}\sum\limits_{k=1}^{m}{N_{pk}d_{k}^{2}}}$ (12)

The number of the $i$ -th layer-turns $w_{i}$ can be determined by the design method based on layer current-density equalization. In details, from Eqs (6) and (7), the calculation formula of $w_{i}$ can be written as

$\displaystyle w_{i}=\frac{L_{N}\sum\limits_{j=1}^{n}{s_{j}}}{\sum\limits_{j=1}% ^{n}{w_{j}s_{j}f_{ij}}}$ (13)

where $s_{j}$ is the cross-sectional area of conductors of the $j-$ th layer. Typically, the inner diameter of the reactor is large enough and the distribution of the number of layer-turns is relatively uniform, thus the distributions of layer current density based on Eqs (7) and (8) are almost the same. Therefore, we hold that layer resistance-voltage equalization is satisfied at the same time when $w_{i}$ is obtained by Eq. (13).

3.3 Processing of inequality constraints

Inequality constraints can be handled by penalty function method, and the objective function ultimately becomes

$\displaystyle\min\left[{{f}^{\prime}(D_{in},d_{av},m,n)+\mu_{\tau}\varphi(\tau% _{p\max}-\tau_{\lim})+}\right.\left.{\mu_{J}\varphi(J_{\max}-J_{\lim})}\right]$ (14)

where $\varphi(x)=\max\{0,x\}$ ; $\mu_{\tau}$ and $\mu_{J}$ are the penalty coefficients of temperature and current density, respectively. Since the mass of conductors would increase with larger capacity of the reactor, the penalty coefficients are required to be dependent of the capacity of the reactor to maintain the availability of the penalty. Figure 3 shows the calculation of the objective function after the processing of constraints.

Figure 3.

Flow diagram of the calculation of objective function after the processing of constraints.

4. Improved genetic algorithm

4.1 Improvement of fitness function

Fitness scaling is an important technique to improve the performance of genetic algorithm [8] and here we adopt linear scaling. Assuming that the original fitness function is $F$ and the scaled fitness function is ${F}^{\prime}$ , linear scaling can be expressed as

$\displaystyle{F}^{\prime}=aF+b$ (15)

The factors of $a$ and $b$ shall satisfy the following conditions:

$\displaystyle\left\{{{\begin{array}[]{*{20}l}{{F}^{\prime}_{{av}}=F_{{av}}=aF_% {{av}}+b}\\ {{F}^{\prime}_{\max}=kF_{{av}}=aF_{\max}+b}\\ \end{array}}}\right.$ (16)

where $F_{{av}}$ and ${F}^{\prime}_{{av}}$ are the average value of original fitness function and scaled fitness function, respectively, $F_{\max}$ and ${F}^{\prime}_{\max}$ are the maximal value of original fitness function and scaled fitness function, $k$ is a constant. Generally, at the early period, the original fitness of most individuals are with small values, namely, $F_{\max}>kF_{av}$ , yet $a<$ 1. Therefore, linear scaling can expand the search space to avoid the premature convergence phenomenon by narrowing the difference between the original-fitness values; conversely, at the late period, with $a>$ 1, linear scaling would improve the local search ability to avoid the random search phenomenon by enlarging the difference between the original-fitness values.

4.2 Improvement of mutation operator

Dynamic mutation operator can further improve the local search ability of genetic algorithm [8], and we adopt the operator

$\displaystyle x_{1}^{t+1}=x_{1}^{t}+2(r-0.5)(x_{\max}-x_{\min})\left({1-\frac{% t}{T}}\right)^{b}$ (17)

where $x_{1}^{t+1}$ and $x_{1}^{t}$ are the new and the old individuals, respectively; $r$ is a random number, $r\in\left({0,1}\right)$ , $x_{\max}$ and $x_{\min}$ are the maximum and the minimum of the individuals, respectively; $t$ and $T$ denote the number and the maximal number of iterations, respectively; $b$ is a constant, which is related to the uniformity of the individuals. It can be observed from Eq. (17) that at the early period, due to $(1-t\mathord{\left/{\vphantom{tT}}\right.\kern-1.2pt}T)^{b}\to 1$ , the dynamic mutation operator is basically the same with the static one; at the late period, due to $(1-t\mathord{\left/{\vphantom{tT}}\right.\kern-1.2pt}T)^{b}\to 0$ , it tends to local search. If new individual generated by Eq. (17) is beyond the optimization range, namely, $x_{1}^{t+1}\notin\left[{x_{\min},x_{\max}}\right]$ , then it can be produced by $x_{1}^{t+1}=x_{\min}+r(x_{\max}-x_{\min})$ .

5. Optimization result

The proposed methods are applied to the optimization design for a single-phase dry-type air-core reactor with round conductors. Whose rated capacity, rated voltage, and rated inductance are 50 kVar, 317.5 V, and 6.42 mH, respectively. Table 1 shows the result of the optimization design. It is evident that the current density for each layer and the resistance voltage for each layer are both basically equal; the distribution of package temperature-rise is basically uniform; the maximum difference between package heights is 36.57 mm and the inductance after calculation is 6.42 mH, which confirm the validity of the processing of the equality constraints. The maximum of layer current-density and the maximum of package temperature-rise are 1.39 A/mm ${}^{2}$ and 80.00 ${{}^{\circ}}$ C, respectively, which show the availability of the processing of the inequality constraints.

Table 1
Optimization result with the improved genetic algorithm

	The 4th package the package wire diameter is 3.35 mm the package temperature rise is 69.92 ${{}^{\circ}}$ C
Package#	Layer#	Diameter	The number of	Height	Resistance	Current density	Resistance voltage
		(mm)	layer-turns	(mm)	( $\Omega$ )	(A/mm ${}^{2}$ )	(V)
1	1	687.33	112.46	317.30	1.17	1.34	9.20
	2	693.77	109.57	309.14	1.16	1.36	9.19
	3	700.01	107.12	302.23	1.14	1.37	9.17
	4	706.24	105.06	296.43	1.13	1.38	9.14
	5	712.48	103.36	291.62	1.12	1.39	9.11
	The 1st package the package wire diameter is 2.73 mm the package temperature rise is 72.03 ${{}^{\circ}}$ C
2	6	775.16	91.90	295.98	0.81	1.38	8.76
	7	782.28	90.55	291.60	0.81	1.39	8.74
	8	789.40	89.54	288.35	0.81	1.39	8.73
	The 2nd package the package wire diameter is 3.15 mm the package temperature rise is 76.75 ${{}^{\circ}}$ C
3	9	852.73	83.42	284.50	0.72	1.39	8.77
	10	860.27	82.76	282.25	0.72	1.39	8.79
	11	867.80	82.43	281.14	0.72	1.39	8.83
	The 3rd package the package wire diameter is 3.35 mm the package temperature rise is 80.00 ${{}^{\circ}}$ C
4	12	931.34	81.15	276.78	0.76	1.38	9.30
	13	938.88	81.19	276.90	0.77	1.38	9.35
	14	946.42	81.57	278.20	0.78	1.37	9.38
	15	953.96	82.31	280.73	0.79	1.35	9.40

It can be calculated that the mass of (aluminous) conductors is 71.72 kg, the power loss is 1424 W. Compared with the previous optimization result (the mass: 78.93 kg, the power loss: 1436 W) [9], the mass of the conductors is reduced by 9.13 percent with the similar power loss. Notably, the long period of optimization-design time (2 hours or more) [3, 9] is reduced to only about 5 min.

6. Conclusion

Several approaches are applied to improve the efficiency of the optimization design for dry-type air-core reactors with round conductors:

The range of the inner diameter, which is derived from the empirical formula of the inductance for short solenoids, can reasonably narrow the search space of the optimization design.

The proposed optimization model can satisfy the design requirements without frequently and repeatedly adjusting the design variables.

Linear scaling and dynamic mutation operator could be introduced to improve genetic algorithm instead of combing with other algorithms.

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Efficient optimization design of dry-type air-core reactors with the improved genetic algorithm

Abstract

Keywords

1. Introduction

2. Range of the inner diameter

3.1 The primary model

4.1 Improvement of fitness function

Table 1 Optimization result with the improved genetic algorithm

References

Table 1
Optimization result with the improved genetic algorithm