Superconducting magnetic energy storage (SMES) system has the ability to mitigate short time voltage fluctuation and sag effectively. Variable Energy Cyclotron Center has taken up research and development program to built SMES system for its accelerator facility. This will drastically reduce the downtime of the facility due to unexpected power fluctuation, sag, etc. Optimization of superconducting solenoids has been studied for past few decades [1-5]. The published works were mostly dealing with the relationship between geometrical parameters of coil and magnetic field. Unlike previous works, this paper gives an analytical description of the optimization problem in terms of coil parameters and aims to minimize both ac loss and radiation heat in leak since in present scenario, price of Nb-Ti based superconductor has reduced considerably and it hardly preset the total cost of SMES device. In general, the SMES system is operated in DC energy storage mode. But, if energy is charged or discharged, the current and magnetic field are changed. A time varying magnetic field causes dynamic loss especially the ac loss in the stabilizer, superconducting cable, all metallic parts, etc.

Therefore, the design goal of SMES coil is to minimize ac loss to reduce the dynamic heat load to the cryogenic refrigeration and coil or cryostat surface area to reduce static heat load into the cryostat since both the parameters have a direct impact on operating cost of the device. In this study, we have considered the solenoidal type of SMES coil since it has the advantage of high energy storage density and simplest configuration. The paper presents multi-variable non-linear constrained global optimisation using Differential Evolution (DE) method of geometrical and operating parameters (B_m, I) to be met for a given stored energy level and power rating, maximum allowable circumference or hoop stress of the coil, and maximum allowable discharge voltage across the SMES coil. The corresponding optimal design of 5MJ, 0.5 MW SMES coil using Rutherford cable and comparative analysis is given.

The geometry of a solenoid is completely defined by its inside radius (a), shape factor ratio α = b/a, and β = l/a with outside radius(b) and total length of 21 as shown in Fig 1.

The inductance (L) of the air core solenoid coil given by

Where, N is the total number of turns of the solenoid; 0{α,β) is a geometry dependent factor given by Welsby [8] as permeability in free space.

The energy stored in a solenoid type winding is given by Wesche4

Peak field to center field ratio is defined as,

The winding volume is given by,

The center field B₀ and peak field B_m of the solenoid can be written as

Here K(α,β) is given [9] as

Here, higher order correction terms K₂, K₄, and K₆ are as given by Feng6 as

Basic work in designing any superconducting coil is to get characteristics of the wire such as critical current density (J_c) vs external magnetic field (B). Considering the space or filling factor(λ) of the coil, the safety factor (ζ), from the short sample critical characteristics of the superconducting cable (i.e.j_cvs B), average overall or operating current density (J) is determined in terms of coil peak field (B_m) as follows:

Usually safety factor ζ is kept roughly around 30 %, while space factor (λ) depends on winding details such as insulation thickness, etc. Once the superconducting cable along with its insulation and winding scheme is fixed, space factor can be determined. For our study with Rutherford cable as specified in Table. 1 and interlayer spacer for liquid helium passage, space factor is determined to be around of 0.85.

The variation of overall current density with magnetic field can be fitted in terms of J-B/J by analytical expression as

Here, p and q are the parameters fitted from J vs Bn/J plot. Substituting from equation (8), current density can be expressed as,

For example, Nb-Ti alloy based Rutherford cable as specified in Table.1 has engineering current density Jat 4.2 K as shown in Fig 2. The current density (J) can be fitted in the form of equation (11) with fitting parameters as, p =7.6322 and q = 1.42619x1(T⁸

For solenoid type coil, maximum stress occurs in the median plane (z=0) of inner layer of the winding. Regarding coil as composite conductor/insulation structure, hoop stress developed at the median plane of inner layer of winding (i.e. r = a & z = 0) of an isotropic solenoidal coil as given by Wilson [7] is,

Figure 2. Critical characteristics of the superconducting wire

Axial magnetic field outside a finite size solenoid in its median plane (z=0) can be approximated [8] as

While designing a SMES coil of a given rating, it is extremely important that the voltage developed across the coil leads is kept within safe value so that no electrical discharge, damage in insulation, etc. occurs.

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Optimization for Solenoid-type Superconducting Magnetic Energy Storage Coil

Voltage developed across the leads at time Instant t=0 of discharge can be written as,

If t is the carry over period and n (=0.9) is the efficiency of SMES coil, maximum voltage developed towards the end of carry over period can be written as,

Where A_c is conductor cross-section and T is decay period of current corresponding to rated power of SMES coil.

The design goal is to minimize the dynamic loss in the superconductor winding and steady state loss into the cryostat.

Based on the foregoing iterative DE optimisation algorithm, Table 2 shows the results of three groups of 5 MJ capacity.

The optimal design study for a 5 MJ system using Rutherford type cable as specified earlier has been carried out as an example. The dynamic and steady state loss for a given stored energy of SMES coil with constraint of maximum allowable voltage across the coil terminals and hoop stress in the inner layers is as shown in Fig. 3. It is observed that higher the power rating, both the loss term increases.

The corresponding optimized coil parameters (a, α, β) are as shown in Fig. 4. It is interesting to note that the optimised coil diameter and conductor volume decreases with increasing power rating for a given stored energy of the coil.

It is also quiet expected to observe that the peak magnetic field in the coil decreases with increase power rating of the coil as evident from Fig. 5 since this reduces the dynamic loss.

Effect of allowable hoop stress on the coil design is very important and higher the allowable hoop stress reduces the conductor volume and overall losses as shown in Fig. 6.

Throughout this process of optimisation, maximum allowed voltage across the coil terminals is assumed to be 1.2 kV, which is safe enough against any insulation breakdown or discharge. In fact, SMES coil of higher stored energy level and rating are made to operate at higher allowed voltages, and hoop stresses. These two criteria again reduce the total loss as well as coil volume. The mathematical formulation of the problem is completely analytical and provides optimized solution that minimizes both the static heat load in to the cryostat and dynamic heat load due to ac loss during the magnetic field transients. It is also important to note that both the losses decrease with higher allowable hoop stress in winding. Even though sufficient safety margin of 30 % has been incorporated in determining engineering current density, extensive quench calculation needs to be done in order to finalize the coil protection scheme from any thermal instability. The optimization methodology described is used to design a 5 MJ SMES coil for the first phase of SMES program in our centre. In the process of optimization, ac loss due to axial field transients is only considered since the loss contribution due to radial field distribution is significantly lower w.r.t. axial field. Though, we have considered here Rutherford type cable, however, other type superconducting cable can also be considered and above methodology can be utilized.

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