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| title | description | published | date | tags | editor | dateCreated |
|---|---|---|---|---|---|---|
| Help | true | 2026-05-31T11:17:09.101Z | markdown | 2026-05-31T11:17:09.101Z |
Voltage Sag Type Classification and Effect on DQ-axis PI Current Controller
1. Voltage Sag Type Definition
Voltage Sag is classified into three types based on the symmetry of the three-phase voltage vectors [1].
Let \bar{V}_S denote the sag voltage and \bar{V}_N the pre-sag (normal) voltage, where |\bar{V}_S| < |\bar{V}_N|.
1.1 Type I
A Type I sag is characterised by a reduction in magnitude of one phase while the remaining two phases retain normal symmetry:
\bar{U}_a = \bar{V}_S
\bar{U}_b = -\frac{1}{2}\bar{V}_S - \frac{1}{2}j\bar{V}_N\sqrt{3}
\bar{U}_c = -\frac{1}{2}\bar{V}_S + \frac{1}{2}j\bar{V}_N\sqrt{3}
1.2 Type II
A Type II sag occurs when one phase retains the normal voltage while the other two phases are reduced:
\bar{U}_a = \bar{V}_N
\bar{U}_b = -\frac{1}{2}\bar{V}_N - \frac{1}{2}j\bar{V}_S\sqrt{3}
\bar{U}_c = -\frac{1}{2}\bar{V}_N + \frac{1}{2}j\bar{V}_S\sqrt{3}
1.3 Type III
A Type III sag is a balanced three-phase sag in which all three phases reduce symmetrically:
\bar{U}_a = \bar{V}_S
\bar{U}_b = -\frac{1}{2}\bar{V}_S - \frac{1}{2}j\bar{V}_S\sqrt{3}
\bar{U}_c = -\frac{1}{2}\bar{V}_S + \frac{1}{2}j\bar{V}_S\sqrt{3}
2. Symmetrical Component Decomposition
Any unbalanced three-phase voltage can be decomposed into positive-sequence (V^+), negative-sequence (V^-), and zero-sequence (V^0) components using Fortescue's theorem:
\begin{bmatrix} V^0 \\ V^+ \\ V^- \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{bmatrix} \begin{bmatrix} \bar{U}_a \\ \bar{U}_b \\ \bar{U}_c \end{bmatrix}, \quad a = e^{j\frac{2\pi}{3}}
2.1 Type I — Sequence Components
Substituting Type I voltages:
V^+ = \frac{\bar{V}_S + \bar{V}_N}{2}
V^- = \frac{\bar{V}_S - \bar{V}_N}{2}
V^0 = 0
Hence, Type I contains both positive- and negative-sequence components, with magnitudes:
|V^+| = \frac{V_S + V_N}{2}, \qquad |V^-| = \frac{|V_N - V_S|}{2}
2.2 Type II — Sequence Components
Substituting Type II voltages:
V^+ = \frac{\bar{V}_N + \bar{V}_S}{2}
V^- = \frac{\bar{V}_N - \bar{V}_S}{2}
V^0 = 0
The negative-sequence magnitude in Type II:
|V^-|_{\text{II}} = \frac{V_N - V_S}{2}
Note:
|V^-|_{\text{II}} = |V^-|_{\text{I}}; however, the phase relationship differs, resulting in different vector diagrams.
2.3 Type III — Sequence Components
Substituting Type III (balanced) voltages:
V^+ = \bar{V}_S
V^- = 0
V^0 = 0
Type III contains only a positive-sequence component. No negative-sequence component is present.
3. Effect on DQ-axis PI Current Controller
In the synchronous reference frame (SRF), the three-phase voltages are transformed via the Park transformation at angular frequency \omega:
\begin{bmatrix} v_d \\ v_q \end{bmatrix} = T(\omega t) \begin{bmatrix} v_\alpha \\ v_\beta \end{bmatrix}, \qquad T(\theta) = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}
The positive-sequence component rotates at +\omega and maps to DC values in the SRF.
The negative-sequence component rotates at -\omega and therefore appears as an AC ripple at $2\omega$ in the SRF:
v_d = V^+_d + V^-\cos(2\omega t + \phi^-), \qquad v_q = V^+_q - V^-\sin(2\omega t + \phi^-)
3.1 Type III — No Ripple
Since V^- = 0:
v_d = V_S, \qquad v_q = 0
Only DC components appear in the dq-frame. The PI controller tracks the reference without steady-state error.
3.2 Type I — 2\omega Ripple
With |V^-| = \frac{V_N - V_S}{2}:
v_d = \frac{V_S + V_N}{2} + \frac{V_N - V_S}{2}\cos(2\omega t)
v_q = -\frac{V_N - V_S}{2}\sin(2\omega t)
The 2\omega (120 Hz) AC ripple in v_d and v_q cannot be suppressed by a standard PI controller, which is designed for DC reference tracking.
This leads to:
- Steady-state current tracking error
- Power factor degradation
- Increased reactive power demand
3.3 Type II — 2\omega Ripple (Same Magnitude, Different Phase)
With |V^-| = \frac{V_N - V_S}{2}:
v_d = \frac{V_N + V_S}{2} + \frac{V_N - V_S}{2}\cos(2\omega t + \Delta\phi)
v_q = -\frac{V_N - V_S}{2}\sin(2\omega t + \Delta\phi)
The ripple magnitude is identical to Type I, but the phase offset \Delta\phi differs due to the different vector configuration. The PI controller exhibits the same inability to track the 2\omega disturbance.
4. Summary
| Sag Type | Negative Sequence | 2\omega Ripple |
PI Tracking |
|---|---|---|---|
| Type III | None | None | Possible |
| Type I | \frac{V_N - V_S}{2} |
Present | Impaired |
| Type II | \frac{V_N - V_S}{2} |
Present | Impaired |
In a weak grid environment such as a ship or small harbor power system, the occurrence of Type I or Type II voltage sags introduces negative-sequence components that induce 2\omega oscillations in the synchronous reference frame. Since a conventional PI controller cannot reject these oscillations, current control performance degrades, leading to reactive power increase, cascading voltage instability, and potential overcurrent protection trips.
References
[1] M. H. J. Bollen, Understanding Power Quality Problems: Voltage Sags and Interruptions. IEEE Press, 2000.