Spiraling Impedance in Distance Relays - Behavior During Faults

Distance relays are indispensable in power system protection, detecting and isolating faults on transmission lines.

A noteworthy phenomenon during fault conditions is the spiraling behavior of impedance, observed on the complex plane due to the transient system response.

This article explores the mathematics behind this behavior, focusing on why the measured impedance spirals towards steady-state fault impedance. This spiraling trajectory on the complex plane is due to the transient response of the system, particularly the decaying DC component in the fault current.

Access this simulation: Impedance Visualizer Tool

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Definitions and Assumptions

Let’s begin by defining the relevant variables:

• Voltage Source: $V_{source}(t) = V_{peak} \sin(\omega t + \Phi)$

  where:

  • $V_{peak} = V_{nominal} \times \sqrt{2}$
  • $\omega = 2\pi f$
  • $\Phi$is the initial phase angle.

• Fault Current: $I_{fault}(t) = I_{AC}(t) + I_{{DC}}(t)$

  • AC Component: $I_{AC}(t) = I_m \sin(\omega t + \Phi - \theta)$

  • DC Component: $I_{{DC}}(t) = -I_m \sin(\Phi - \theta) e^{-\dfrac{(t - t_{{faultl}})}{\tau}}$

• Source Impedance: $Z_s = R_s + jX_s$

• Line Impedance (Fault Impedance): $Z_l = R_l + jX_l$

• Total Fault Impedance: $Z_{{total}} = Z_s + Z_l = R_{{total}} + jX_{{total}}$

• Time Constant: $\tau = \dfrac{X_{{total}}}{\omega R_{{total}}}$

• Angle of Total Fault Impedance: $\theta = \arctan\left( \dfrac{X_{{total}}}{R_{{total}}} \right)$

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Voltage at the Relay During Fault

The voltage at the relay location during the fault is: $V_{{faultl}}(t) = V_{source}(t) - I_{{faultl}}(t) Z_s$

Note that the DC component of current $I_{{DC}}(t)$ only causes a voltage drop across the resistive part $R_s$ of $Z_s$, because inductors (represented by $X_s$) block DC currents.

Therefore, we can write: $V_{{faultl}}(t) = V_{source}(t) - I_{AC}(t) Z_s - I_{{DC}}(t) R_s$

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Calculating the Instantaneous Impedance$Z(t)$

The instantaneous impedance seen by the relay is: $Z(t) = \dfrac{V_{{faultl}}(t)}{I_{{faultl}}(t)} = \dfrac{V_{source}(t) - I_{AC}(t) Z_s - I_{{DC}}(t) R_s}{I_{AC}(t) + I_{{DC}}(t)}$

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Deriving $Z(t)$ as a Function of Time

Introducing the Ratio $k(t)$

Define: $k(t) = \dfrac{I_{{DC}}(t)}{I_{AC}(t)}$

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Expressing $Z(t)$ Using $k(t)$

We can rewrite the impedance equation as: $Z(t) = \dfrac{V_{source}(t) - I_{AC}(t) Z_s - k(t) I_{AC}(t) R_s}{I_{AC}(t) (1 + k(t))}$

Simplify numerator and denominator: $Z(t) = \dfrac{\dfrac{V_{source}(t)}{I_{AC}(t)} - Z_s - k(t) R_s}{1 + k(t)}$

But since $V_{source}(t) = V_{peak} \sin(\omega t + \Phi)$ and $I_{AC}(t) = I_m \sin(\omega t + \Phi - \theta)$, we can express the ratio $\dfrac{V_{source}(t)}{I_{AC}(t)}$ as:

$\dfrac{V_{source}(t)}{I_{AC}(t)} = \dfrac{V_{peak} \sin(\omega t + \Phi)}{I_m \sin(\omega t + \Phi - \theta)} = \dfrac{V_{peak}}{I_m} \cdot \dfrac{\sin(\omega t + \Phi)}{\sin(\omega t + \Phi - \theta)}$

Assuming that $\dfrac{V_{peak}}{I_m} = |Z_{total}|$(since $I*m = \dfrac{V_{peak}}{|Z_{total}|}$), we have:

$\dfrac{V_{source}(t)}{I_{AC}(t)} = |Z_{{total}}| \cdot \dfrac{\sin(\omega t + \Phi)}{\sin(\omega t + \Phi - \theta)}$

However, since the ratio $\dfrac{\sin(\omega t + \Phi)}{\sin(\omega t + \Phi - \theta)}$ is a function of $\theta$ and $\omega t + \Phi$, it complicates the expression. For simplification, we consider steady-state conditions where this ratio approaches a constant value.

Alternatively, recognizing that $Z\_{{total}} = Z_s + Z_l$, we can write: $Z(t) = \dfrac{(Z_s + Z_l) - Z_s - k(t) R_s}{1 + k(t)} = \dfrac{Z_l - k(t) R_s}{1 + k(t)}$

Therefore: $Z(t) = Z_l \left( \dfrac{1}{1 + k(t)} \right) - R_s \left( \dfrac{k(t)}{1 + k(t)} \right)$

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Analyzing $k(t)$ and Its Time Dependence

Expression for $k(t)$

From the definitions of $I_{AC}(t)$ and $I_{DC}(t)$:

$k(t) = \dfrac{I_{{DC}}(t)}{I_{AC}(t)} = \dfrac{ -I_m \sin(\Phi - \theta) e^{ -\dfrac{ (t - t_{{faultl}}) }{ \tau } } }{ I_m \sin( \omega t + \Phi - \theta ) } = \dfrac{ -\sin(\Phi - \theta) }{ \sin( \omega t + \Phi - \theta ) } e^{ -\dfrac{ (t - t_{{faultl}}) }{ \tau } }$

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Time-Dependent Behavior of $k(t)$

• Exponential Decay: The term $e^{ -\dfrac{ (t - t_{{faultl}}) }{ \tau } }$ decays exponentially from its initial value at $t = t_{{faultl}}$.
• Oscillation: The denominator $\sin( \omega t + \Phi - \theta )$ causes $k(t)$ to oscillate.

Therefore, $k(t)$ is a time-varying function that decreases in magnitude over time due to the exponential decay and oscillates due to the sinusoidal denominator.

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Final Expression for $Z(t)$

Combining the expressions:

$Z(t) = Z_l - \left( \dfrac{ k(t) }{ 1 + k(t) } \right) ( Z_l + R_s )$

This equation shows that $Z(t)$ is composed of:

1. Steady-State Impedance Component: $Z_l$

2. Time-Dependent Decreasing Term: $\Delta Z(t) = -\left( \dfrac{ k(t) }{ 1 + k(t) } \right) ( Z_l + R_s )$

This term decreases over time as $k(t)$ decays to zero.

Thus, the impedance can be written as:

$Z(t) = Z_l + \Delta Z(t)$

Where $\Delta Z(t)$ is a time-dependent decreasing factor that causes $Z(t)$ to spiral towards $Z_l$.

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Demonstrating the Spiraling Behavior

Initial Conditions

At $t = t_{faultl}$:

-$k(t_{faultl})$ is at its maximum magnitude. -$\Delta Z(t_{{faultl}})$ is significant, causing $Z(t_{{faultl}})$ to deviate from $Z_l$.

Time Evolution

As $t \rightarrow \infty$:

-$e^{ -\dfrac{ (t - t_{faultl}) }{ \tau } } \rightarrow 0$ -$k(t) \rightarrow 0$ -$\Delta Z(t) \rightarrow 0$ -$Z(t) \rightarrow Z_l$

Spiral Trajectory

• The oscillatory nature of $k(t)$ due to the sinusoidal term causes $\Delta Z(t)$ to change in both magnitude and angle over time.
• The exponential decay ensures that the amplitude of $\Delta Z(t)$ decreases over time.
• Combined, these effects cause $Z(t)$ to spiral towards $Z_l$ on the complex plane.

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Visualization on the Complex Plane

By plotting $Z(t)$ over time on the complex plane:

• Real Part ($\text{Re}[Z(t)]$) and Imaginary Part ($\text{Im}[Z(t)]$) vary due to $\Delta Z(t)$.
• The trajectory forms a spiral that converges to the steady-state impedance $Z_l$.

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Implications for Distance Relays

• Distance relays calculate impedance to determine if a fault is within their protected zone.
• The spiraling impedance can cause the relay to overreach or underreach if not properly accounted for.
• Protective algorithms often include filters or phasor estimation techniques to mitigate the effects of the decaying DC component.

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Conclusion

The decaying DC component in the fault current causes the calculated instantaneous impedance $Z(t)$ to change over time. The spiraling behavior of $Z(t)$ on the complex plane is due to the combination of decreasing DC current and the oscillatory nature of the AC current. As the DC component decays, the system transitions from transient to steady-state conditions, and the calculated impedance converges to the steady-state fault impedance $Z_l$.

Understanding this behavior is crucial for the design of distance relays, ensuring they can accurately detect and isolate faults without being misled by transient phenomena.