Table 1 PQDs description and their mathematical models.

Normal operation

\(V_{\text {normal}}(t) = V_{\text {normal}} \cdot \sin (2\pi ft)\)

Voltage sag (Dip)

\(V_{\text {sag}}(t) = V_{\text {normal}}(1 - \alpha ) \cdot \sin (2\pi ft)\), where \(0.1 \le \alpha \le 0.9\)

Voltage swell

\(V_{\text {swell}}(t) = V_{\text {normal}}(1 + \beta ) \cdot \sin (2\pi ft)\), where \(0.1 \le \beta \le 0.8\)

Interruption

\(V_{\text {int}}(t) = 0\) for \(t_{\text {start}} \le t \le t_{\text {end}}\);

\(V_{\text {int}}(t) = V_{\text {normal}} \cdot \gamma \cdot \sin (2\pi ft)\) otherwise

Harmonics

\(V_{\text {harm}}(t) = V_{\text {normal}}(t) + \sum _{n=3,5,7}^{N} V_n \cdot \sin (2\pi nft + \phi _n)\)

Oscillatory transients

\(V_{\text {harm}}(t) = V_{\text {normal}}(t) + V_{\text {peak}}(t) e^{-\sigma t} \cdot \sin (2\pi f_{\text {osc}} t)\), where \(f_{\text {osc}} = 300{-}5000\) Hz

Impulse transients

\(V_{\text {impulse}}(t) = V_{\text {normal}}(t) + V_{\text {peak}} \cdot e^{\left( \frac{t - t_0}{\tau } \right) }\)

Voltage flicker

\(V_{\text {flicker}}(t) = V_{\text {normal}}(t) \left[ 1 + m \cdot \sin (2\pi f_{\text {flicker}} t) \right] \cdot \sin (2\pi f_{\text {osc}} t)\), where \(m \le 0.1\), \(f_{\text {flicker}} \le 25\) Hz

Voltage gap

\(V_{\text {gap}}(t) = V_{\text {normal}}(t) \cdot \lambda \cdot \sin (2\pi ft)\) for \(t_{\text {start}} \le t \le t_{\text {end}}\), \(\lambda \le 0.1\), \(t \le 0.2\) s

Voltage spike

\(V_{\text {spike}}(t) = V_{\text {normal}}(t) + V_{\text {spike}}(t) \cdot e^{- \frac{(t - t_0)^2}{2\sigma _{\text {spike}}^2}}\), where \(\sigma _{\text {spike}} \approx 1\) ms