Question

In case of the compensation of the power transmission lines, for the same voltage boost, the reactive power capacity of a shunt capacitor is ____________ that of a series capacitor.

• A. equal to
• B. greater than
• C. less than
• D. half of

Hint:

Series capacitors are highly effective for voltage regulation because they respond dynamically to the load current ($I^2 X_c$). They achieve the same voltage regulation as shunt options with a much smaller VA rating.

Answer 1

The Correct Option is B

Concept: Capacitive compensation is implemented in transmission lines to improve voltage profiles, reduce losses, and increase power transfer capacity. * Shunt Capacitors: Connected in parallel across the line to supply reactive power ($Q_{\text{shunt}}$) locally, raising the voltage profile by minimizing lagging power factor currents. * Series Capacitors: Connected in series with the line conductors to physically negate a portion of the line's series inductive reactance ($X_L$). This directly reduces the overall series voltage drop ($\Delta V = I \cdot X_c$).

Step 1: Formulate the mathematical expressions for reactive power capacity.
Let $\Delta V$ be the desired voltage boost required by the power network system. For a series capacitor, the reactive power injected depends on the line current ($I$) flowing through it: \[ Q_{\text{series}} = 3 \cdot I^2 \cdot X_{\text{series}} \] Since the voltage drop across the series capacitor is $\Delta V = I \cdot X_{\text{series}}$, we can substitute this expression to get: \[ Q_{\text{series}} = 3 \cdot I \cdot \left(I \cdot X_{\text{series}}\right) = 3 \cdot I \cdot \Delta V \] For a shunt capacitor, the reactive power injected depends directly on the system operating line voltage ($V$): \[ Q_{\text{shunt}} = 3 \cdot \frac{V^2}{X_{\text{shunt}}} \] The voltage boost provided by a shunt capacitor can be approximated using short-circuit dynamics as: \[ \Delta V \approx \frac{Q_{\text{shunt}} \cdot X_L}{3 \cdot V} \implies Q_{\text{shunt}} = 3 \cdot V \cdot \Delta V \cdot \frac{1}{X_L} \]

Step 2: Compare the capacities for identical voltage boost profiles.
Let's look at the basic equations for the reactive power capacities required to achieve the same voltage change $\Delta V$: \[ \frac{Q_{\text{shunt}}}{Q_{\text{series}}} = \frac{3 \cdot V \cdot \Delta V \cdot \frac{1}{X_L}}{3 \cdot I \cdot \Delta V} = \frac{V}{I \cdot X_L} = \frac{1}{\left(\frac{I \cdot X_L}{V}\right)} \] The term $\frac{I \cdot X_L}{V}$ represents the percentage inductive voltage drop across the transmission line, which is typically a small fraction (e.g., $0.1$ to $0.2$) under normal operating conditions. \[ \frac{I \cdot X_L}{V} \ll 1 \implies \frac{Q_{\text{shunt}}}{Q_{\text{series}}} \gg 1 \implies Q_{\text{shunt}} > Q_{\text{series}} \] This mathematically demonstrates that to achieve the exact same voltage boost, a shunt capacitor must have a much larger reactive power capacity than a series capacitor. Thus, the capacity of the shunt capacitor is greater than that of the series capacitor, which aligns with Option (B).