Subject: Power System Analysis

Topic: Power System Transients

Difficulty: High

A voltage having a crest value of 3000 KV is traveling on the line of 750 KV. The protective level is 1700 kV and the surge impedance of the line is 300 ohm. Calculate 1)line current before reaching the arrester 2)current through arrester 3)value of arrester resistance for this condition 4)reflected voltage 5)Verify the reflection and refraction coefficient

Reflection and Refraction Coefficient Problem

The given data -

Voltage = $ V^{'} $ = 3000 kV

Protective Level = $ V_a $ = 1700 kV

Surge Impedance of Line = $ Z_c $ = 300 Ω

To find -

i] Line current before reaching the Arrester = $ I^{'} $ = ?

ii] Current through the Arrester = $ I_a $ = ?

iii] Value of the Arrester Resistance for this Condition = R = ?

iv] Reflect Voltage = $ V^n $ = ?

v] Verification of Reflection and Refraction Coefficients

Formulae -

$$ Line\ Current = I^{'} = \frac {V^{'}}{Z_c} $$

$$ Current\ through\ Arrester = I_a = \frac {2V^{'} - V_a}{Z_c} $$

$$ Resistance\ of\ Arrester = R = \frac {V_a}{I_a} $$

$$ Reflected\ Voltage = V^n = V_a - V^{'} $$

$$ Reflection\ Coefficient = \frac {V^n}{V{'}} = \frac {R - Z_c}{R + Z_c} $$

$$ Refraction\ Coefficient = \frac {V_a}{V{'}} = \frac {2R}{R + Z_c} $$

Solution -

i] Line current before reaching the Arrester = $ I^{'} $

$$ I^{'} = \frac {V^{'}}{Z_c} $$

$$ I^{'} = \frac {3000 \times 10^3}{300} = 10^4\ A $$

II] Current through the Arrester = $ I_a $

$$ Current\ through\ Arrester = I_a = \frac {2V^{'} - V_a}{Z_c} $$

$$ I_a = \frac {2 \times 3000 \times 10^3 - 1700 \times 10^3}{300} $$

$$ I_a = \frac {4300000}{300} $$

$$ I_a = 14333\ A $$

iii] Value of the Arrester Resistance for this Condition = R

$$ Resistance\ of\ Arrester = R = \frac {V_a}{I_a} $$

$$ R = \frac {1700 \times 10^3}{14333} = 118.61\ Ω $$

iv] Reflect Voltage = $ V^n $

$$ Reflected\ Voltage = V^n = V_a - V^{'} $$

$$ V^n = 1700 - 3000 = -1300\ kV $$

v] Verification of Reflection and Refraction Coefficients

$$ Reflection\ Coefficient = \frac {V^n}{V{'}} = \frac {R - Z_c}{R + Z_c} $$

$$ L.H.S. = \frac {V^n}{V{'}} = - \frac {1300}{3000} = - 0.4333 $$

$$ R.H.S. = \frac {R - Z_c}{R + Z_c} = \frac {118.61 - 300}{118.61 + 300} = -0.4333 $$

$$ L.H.S. = R.H.S.$$

Hence, Coefficient of Reflection is Verified.

$$ Refraction\ Coefficient = \frac {V_a}{V{'}} = \frac {2R}{R + Z_c} $$

$$ L.H.S. = \frac {V_a}{V{'}} = \frac {1700}{3000} = 0.567 $$

$$ R.H.S = \frac {2R}{R + Z_c} = \frac {2 \times 118.61}{118.61 + 300} = 0.567 $$

$$ L.H.S. = R.H.S.$$

Hence, Coefficient of Refraction is Verified.

Answer -

i] Line current before reaching the Arrester = $ I^{'} $ = $10^4\ A$

ii] Current through the Arrester = $ I_a $ = 14333 A

iii] Value of the Arrester Resistance for this Condition = R = 118.61 Ω

iv] Reflect Voltage = $ V^n $ = - 1300 kV

v] Reflection Coefficient = - 0.4333 is verified.

Refraction Coefficient = 0.567 is verified.