Cost Optimal Design of a Power Inductor by Sequential Gradient Search

Abstract

Power inductors are used for compensating VAR generated by long EHV transmission lines and in electronic circuits. For the EHV-lines, the rating of the inductor is decided upon by techno-economic considerations on the basis of the line-susceptance. It is a high voltage high current device, absorbing little active power and large reactive power. The cost is quite high- hence the design should be made cost-optimally. The 3-phase power inductor is similar in construction to a 3-phase core-type transformer with the exception that it has only one winding per phase and each limb is provided with an air-gap, the length of which is decided upon by the inductance required. In this paper, a design methodology based on sequential gradient search technique and the corresponding algorithm leading to cost-optimal design of a 3-phase EHV power inductor has been presented. The case-study has been made on a 220 kV long line of NHPC running from Chukha HPS to Birpara of Coochbihar.

Appendices

Appendix 1

[Subroutine for the Design of Power Inductor]

Input $S \leftarrow MVAR$ to be compensated $S^{'} \leftarrow 1000 \times S/2$	// rating of the inductor in KVA
Input $f,V,K,R_{w} ,K_{s}$	// frequency, line voltage, emf constant,
	// window height/width ratio, stacking factor
Input $K_{w} ,K_{c} ,B_{m} ,\delta$	// window space factor, constant depending on no of steps,
	// maximum flux density, current density (A/mm2)
$E_{t} \leftarrow K\sqrt {S^{'} }$	// emf per turn, V
$V_{p} \leftarrow V/\sqrt 3;\,\,N \leftarrow V_{p} /E_{t}$	// phase voltage, no of turns
$\phi_{m} \leftarrow E_{t} /(4.44f)$; $I = s \times 10^{6} /(3V_{p} )$	// maximum value of flux, current
$A_{i} \leftarrow \phi_{m} /B_{m};\,\,\,A_{gi} \leftarrow A_{i} /K_{s} \,\,;\,\,a_{c} \leftarrow I/\delta$	// net/gross area of iron, conductor area
Input $N_{st} ,K_{1} \ldots$	// number of core steps, corresponding length coefficients
$d \leftarrow \sqrt {A_{gi} /K_{c} };\,\,a \leftarrow K_{1} d$	// diameter of the fictitious circle round the core;
	// length of the largest core step
$A_{w} \leftarrow S^{{\prime }} /3.33fB_{m} A_{i} K_{w} \delta \times 10^{3}$	// window area
$W_{w} \leftarrow \sqrt {A_{w} /R_{w} };\,\,H_{w} \leftarrow R_{w} W_{w}$	// window width, height
$D_{c} \leftarrow W_{w} + d;\,\,W = 2D_{c} + a$	// distance between core centres, total length of the core
$A_{gy} \leftarrow A_{gi} ;\,\,\,W_{y} \leftarrow a\,\,\,\,\,\,\,$	// gross area, width of yoke
$\,H_{y} \leftarrow A_{gy} /W_{y}$ $H \leftarrow H_{w} + 2H_{y}$	// height of yoke, total height of core
Input $c_{w} ,c_{l} ,c_{h}$	// tank clearances
$t_{w} \leftarrow D_{c} + 2c_{w} ;\,\,t_{l} \leftarrow W + 2c_{l} \,\,;\,\,t_{h} = H + c_{h}$	// tank dimensions
$t_{a} \leftarrow 2(t_{l} + t_{w} )t_{h} + t_{l} t_{w}$	// tank area for radiation
Input $p_{i} ,\rho ,\rho_{i} ,\rho_{c}$	// specific iron loss, resistivity of conductor, density of iron, conductor
$V_{i} = (2W + 3H_{w} )A_{i} ;\,\,W_{i} = V_{i} \rho_{i} \,\,;\,\,\,P_{i} = W_{i} p_{i}$	// volume, weight of iron, total iron loss
$L_{mt} = \pi (D_{c} - c_{wd} )\,\,;\,\,R \leftarrow \rho L_{mt} N/a_{c} \,\,;\,\,P_{c} = 3I^{2} R$	// length of mean turn, resistance, total ohmic loss
$P_{ls} = P_{c} + P_{i} ;\,\,V_{c} = NL_{mt} a_{c} \times 10^{ - 6}$	// total loss, volume of conductor
Input $V_{a} ,\lambda_{r} ,\lambda_{c} ,d_{t}$	// velocity of air blast, dissipation by radiation, convection,
	// depth of tank wall
$\lambda_{c}^{'} \leftarrow \lambda_{c} (1 + 1.3\sqrt {V_{a} } );\,\,V_{oil} = t_{h} t_{l} t_{w} K$	// dissipation coefficient with air-blast, volume of oil
	// increased convection due to air-blast at 40 °C
$x = t_{a} \lambda_{r} \times 40;\,\,t_{t} = (p_{ls} - x)/(\lambda_{c}^{'} \times 40)$	// dissipation through radiation, total area of cooling tubes in the radiator
$Input\,s_{i} ,s_{c} ,s_{t} ,s_{o} \,\, - specific\,\,costs$
$C_{c} = V_{c} \rho_{c} s_{c}$; $C_{i} = V_{i} \rho_{i} s_{i}$	// cost of copper and iron
$C_{\tan k} \leftarrow (t_{a} + t_{t} )d_{t} \rho_{i} s_{t}$	// cost of tank
$C_{oil} = V_{oil} s_{o} /1000$	// cost of transformer oil
$C_{t} = (C_{c} + C_{i} + C_{\tan k} + C_{oil} )(1 + OH)$	// total cost along with overheads

Appendix 2

[Cost Optimal design of a 3-Phase Power Inductor by Exhaustive Search]

Minimum cost 2,714,322/- is obtained for K = 0.43; R_w = 3.72

The design details of the cost-optimal power inductor:

• KVA-rating of the inductor = 15,000; rated line voltage = 220,000 V

• Nominal frequency = 50 Hz; connection: Y

• Conductor material: COPPER; current density = 3.2 A/sq. mm

• Core material CRGOS: flux-density = 1.7 Tesla

• Number of turns of the coil = 2933

• Current in coil = phase current = 39.368 A

• Current density = 3.2 A/sq. mm; cross section of the conductor = 12.302 sq. mm

• Net area of core iron = 0.1147357 sq. m; stacking factor = 0.972

• Gross area of core iron = 0.1180409 sq. m; 3-stepped core has been used.

• Diameter of the core circle = 0.4229 m

• Length of the core sides in m: = 0.383/0.299/0.179

• Area of the window = 0.3608 sq. m

• Window height/width ratio = 3.72; window height/width in m: 1.1709/0.3081

• Distance between core centres = 0.6909 m

• Width/height of yoke in m: 0.3827/0.3084

• Total length of core = 1.8448 m; total height of core = 1.7878 m

• Length of air-gap = 0.313 mm

• Volume of iron = 0.8264 cu. m; weight of iron = 6321.74 kg

• Iron loss/kg at this flux-density = 2.0966 W; iron loss = 13,254 W; % iron loss = 8.836E−02

• Mean length of turn = 1.7158 m; resistance of the coil = 9.0001 Ω

• Copper loss = 41,839.4 W; % copper loss = 0.27893; total % loss = 0.36729

• The tank length × width × height: 2.273 × 0.851 × 2.188

• Artificial cooling by air-blast on radiators is being used.

• The velocity of blast = 5 m/s; the blast constant (with some non-uniformity in blast) = 1.1

• The dissipation constant in = 35.234 W/sq. m/^°C

• The number of elliptical tubes (75 × 25 mm) in the radiator = 110

• The cost of sheet metal/kg = Rs. 50/-

• The weight/cost of tank: 913.4537 kg; Rs. 45,673/-

• The cost of oil/litre = Rs. 40/-

• The volume/cost of oil: 4.231074 cu. m; Rs. 169,243/-

• Volume of iron = 0.82637 cu. m; weight of iron = 6321.7 kg

• Cost of iron/kg = Rs. 150/-; cost of iron = Rs. 948,261/-

• Volume of copper = 0.1857 cu. m; weight of copper = 1652.9 kg

• Cost of copper/kg = Rs. 450/-; cost of copper = Rs. 743,816/-

• Direct cost allowing 15% labour charge = Rs. 2,193,042/-

• Selling cost allowing 25% overhead = Rs. 2,741,302/-