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Thesis/Master_260531/Latex__.md

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+---
+title: Help
+description: 
+published: true
+date: 2026-05-31T11:17:09.101Z
+tags: 
+editor: markdown
+dateCreated: 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.