Short-circuit

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Short-circuit
Medium voltage network impedance
\[ I_k^{"} = \frac{S_k^{"}}{\sqrt{3} \cdot U_n} \] \[ Z = \frac{c \cdot U_n}{\sqrt{3} \cdot I_k} \] \[ X = \frac{Z}{\sqrt{1 + (R/X)^2}} \] \[ R = \sqrt{Z^2 - X^2} \]

· Sk": Initial symmetrical short-circuit power, MVA

· Ik": Initial symmetrical short-circuit current (rms), kA

· Un: Nominal voltage, kV

· Z: Network impedance

· c: Voltage factor

· X: Network reactance

· R: Network resistance

· R/X: Resistance/reactance ratio of the network


Transformer
\[ Z_T = \frac{Z_{CC}}{100} \cdot \frac{U^2}{S_T} \] \[ R_T = \frac{P_K}{3 \cdot I_T^2} \] \[ X_T = \sqrt{Z_T^2 - R_T^2} \] \[ K_T = 0.95 \cdot \frac{c_{max}}{1 + 0.6 \cdot x_T} \] \[ x_T = \frac{X_T}{U_{rT}^2 / S_{rT}} \]

· ST: Transformer apparent power, kVA

· ZT: Transformer impedance, Ω

· XT: Transformer reactance, Ω

· RT: Transformer resistance, Ω

· ZCC: Short-circuit impedance, %

· UBT: Low voltage side voltage, kV

· PK: Total transformer losses, kW

· IT: Rated transformer current, A

· KT: Transformer impedance reduction factor (2 windings)

· xT: Relative transformer reactance


Synchronous Generator

\[ Z_{GK} = K_G · Z_{G} = K_G (R_G + jX^{"}_d) \] \[ K_{G} = \frac{U_n}{U_{rG}} · \frac{c_\text{max}}{1+x''_d \cdot \sqrt{1 - \cos^2{\varphi_{rG}}}} \] \[ x''_d = \frac{X''_d}{Z_{rG}} \] \[ Z_{rG} = \frac{U_{rG}^2}{S_{rG}} \]

· ZGK: Corrected subtransient impedance of the generator, Ω

· KG: Generator impedance correction factor

· RG: Generator resistance, Ω

· ZG: Generator subtransient impedance in positive sequence, Ω

· Un: Nominal system voltage, kV

· UrG: Rated voltage generator, kV

· X"d: Generator subtransient reactance, Ω

· x"d: Relative saturated subtransient reactance

· cmax: Voltage factor

· φrG: Phase angle between IrG and UrG/√3

· ZrG: Generator impedance, Ω

· SrG: Reference apparent power of the generator, kVA



Current limiting reactor
\[ Z_R = \frac{u_{kr}}{100} \cdot \frac{Un}{\sqrt{3} · I_{rR}} \]

· ukr: Short-circuit voltage, %

· Un: Rated voltage, kV

· IrR: Rated current, A

· RR << XR


Cable impedance



\[ R = Rcc · (1 + Ys + Yp) \] \[ Rcc = \frac{L}{σ·S} \] \[ Z = \sqrt{R^2 + X^2} \]

· R: AC conductor resistance, Ω

· Rcc: DC conductor resistance, Ω

· L: Circuit length, m

· X: Conductor reactance, Ω

· YS: Increase in resistance due to skin effect, p.u.

· YP: Increase in resistance due to proximity effect, p.u.

· σ: Copper conductivity at the expected temperature, m/Ω·mm²

· S: Conductor cross-section, mm²

· Z: Conductor impedance, Ω


Conductor conductivity


\[ σ = \frac{σ_{20}}{1 + α·(θ-20)} \]

· σ: Conductor conductivity, m/Ω·mm²

· σ20: Conductor conductivity at 20ºC, m/Ω·mm²

· α: Temperature coefficient of specific resistance, 1/ºC

· θ: Conductor temperature, ºC

· Minimum short-circuit: 20 ºC

· Maximum short-circuit: 250 ºC


Short-circuit current

\[ I_k^{\prime\prime} = \frac{ \frac{c \cdot U_n}{\sqrt{3}} }{ \sqrt{R_k^2 + X_k^2}} \]

· Ik": Initial symmetrical short-circuit current, kA

· Zk, Rk, Xk: System impedance, resistance, and reactance, Ω

· c·Un/√3: Equivalent three-phase voltage source (rms value), kV

· c·Un: Equivalent two-phase voltage source (rms value), kV

· c·Un/√3: Equivalent single-phase voltage source (rms value), kV