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# Voltage Sag Type Classification and Effect on DQ-axis PI Current Controller

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## 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 $|\bar{V}_S| < |\bar{V}_N|$.

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### 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:

$$\bar{U}_a = \bar{V}_S$$

$$\bar{U}_b = -\frac{1}{2}\bar{V}_S - \frac{1}{2}j\bar{V}_N\sqrt{3}$$

$$\bar{U}_c = -\frac{1}{2}\bar{V}_S + \frac{1}{2}j\bar{V}_N\sqrt{3}$$

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### 1.2 Type II

A Type II sag occurs when one phase retains the normal voltage while the other two phases are reduced:

$$\bar{U}_a = \bar{V}_N$$

$$\bar{U}_b = -\frac{1}{2}\bar{V}_N - \frac{1}{2}j\bar{V}_S\sqrt{3}$$

$$\bar{U}_c = -\frac{1}{2}\bar{V}_N + \frac{1}{2}j\bar{V}_S\sqrt{3}$$

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### 1.3 Type III

A Type III sag is a balanced three-phase sag in which all three phases reduce symmetrically:

$$\bar{U}_a = \bar{V}_S$$

$$\bar{U}_b = -\frac{1}{2}\bar{V}_S - \frac{1}{2}j\bar{V}_S\sqrt{3}$$

$$\bar{U}_c = -\frac{1}{2}\bar{V}_S + \frac{1}{2}j\bar{V}_S\sqrt{3}$$

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## 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:

$$\begin{bmatrix} V^0 \\ V^+ \\ V^- \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{bmatrix} \begin{bmatrix} \bar{U}_a \\ \bar{U}_b \\ \bar{U}_c \end{bmatrix}, \quad a = e^{j\frac{2\pi}{3}}$$

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### 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:

$$|V^+| = \frac{V_S + V_N}{2}, \qquad |V^-| = \frac{|V_N - V_S|}{2}$$

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### 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.

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### 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.

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## 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:

$$v_d = V^+_d + V^-\cos(2\omega t + \phi^-), \qquad v_q = V^+_q - V^-\sin(2\omega t + \phi^-)$$

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### 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.

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### 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

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### 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.

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## 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.

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## References

[1] M. H. J. Bollen, *Understanding Power Quality Problems: Voltage Sags and Interruptions*. IEEE Press, 2000.