7.4 Sequence Impedances & Positive, Negative, Zero Sequence Network Construction

7.4 Sequence Impedances & Positive, Negative, Zero Sequence Network Construction

Executive Overview: The fundamental strength of symmetrical components lies in network decoupling: in any balanced, symmetrical three-phase power system, positive-sequence currents produce only positive-sequence voltage drops, negative-sequence currents produce only negative-sequence voltage drops, and zero-sequence currents produce only zero-sequence voltage drops. Consequently, a complex three-phase power grid can be split into three independent single-phase Sequence Networks (Positive, Negative, and Zero). Sizing apparatus sequence impedances, understanding transformer zero-sequence topologies, and reducing networks to Thevenin equivalents is essential for PE Power fault analysis.

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1. Sequence Decoupling & Independent Sequence Networks

In a transposed, balanced three-phase transmission and generation system, the phase impedance matrix $[\mathbf{Z}_{abc}]$ is symmetric with equal self-impedances ($Z_s$) and equal mutual impedances ($Z_m$):

$$[\mathbf{Z}_{abc}] = \begin{bmatrix} Z_s & Z_m & Z_m \\ Z_m & Z_s & Z_m \\ Z_m & Z_m & Z_s \end{bmatrix}$$

Transforming $[\mathbf{Z}{abc}]$ into the sequence domain via similarity transformation $[\mathbf{Z}{012}] = \mathbf{A}^{-1} [\mathbf{Z}_{abc}] \mathbf{A}$ diagonalizes the matrix completely:

$$[\mathbf{Z}_{012}] = \begin{bmatrix} \mathbf{Z}_0 & 0 & 0 \\ 0 & \mathbf{Z}_1 & 0 \\ 0 & 0 & \mathbf{Z}_2 \end{bmatrix} = \begin{bmatrix} Z_s + 2Z_m & 0 & 0 \\ 0 & Z_s - Z_m & 0 \\ 0 & 0 & Z_s - Z_m \end{bmatrix}$$

Because all off-diagonal terms are zero, there is zero mutual coupling between sequence networks. The three networks operate as completely isolated circuits, coupling only at the physical point of an unsymmetrical fault or unbalanced load.

THE THREE INDEPENDENT SEQUENCE NETWORKS:

POSITIVE SEQUENCE (ACTIVE):        NEGATIVE SEQUENCE (PASSIVE):       ZERO SEQUENCE (PASSIVE):
       Z_1,th                             Z_2,th                             Z_0,th
  +---[ZZZZZZ]----+--- (F1)          +---[ZZZZZZ]----+--- (F2)          +---[ZZZZZZ]----+--- (F0)
  |               |                  |               |                  |               |
  | +             |                  |               |                  |               |
 (~) E_a          |                  |               |                  |               |
  | -             |                  |               |                  |               |
  |               |                  |               |                  |               |
 === Ref (0V)    === Ref            === Ref (0V)    === Ref            === Ref (0V)    === Ref

Sequence Voltage Governing Equations (At Fault Point $F$)

$$\mathbf{V}_{a1} = \mathbf{E}_a - \mathbf{I}_{a1} \mathbf{Z}_{1,th}$$ $$\mathbf{V}_{a2} = 0 - \mathbf{I}_{a2} \mathbf{Z}_{2,th} = -\mathbf{I}_{a2} \mathbf{Z}_{2,th}$$ $$\mathbf{V}_{a0} = 0 - \mathbf{I}_{a0} \mathbf{Z}_{0,th} = -\mathbf{I}_{a0} \mathbf{Z}_{0,th}$$

Key Principle: The Positive-Sequence Network is the ONLY active network containing an internal generated EMF source ($\mathbf{E}_a \approx 1.0\angle 0^\circ\text{ pu}$). The Negative-Sequence and Zero-Sequence networks are purely passive (no internal voltage sources).

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2. Sequence Impedances of Core Power System Apparatus

+---------------------------------------------------------------------------------------------------+
| POWER APPARATUS SEQUENCE IMPEDANCE COMPARISON                                                     |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Apparatus             | Positive Seq (Z_1)    | Negative Seq (Z_2)    | Zero Seq (Z_0)            |
+-----------------------+-----------------------+-----------------------+---------------------------+
| **Synchronous**       | Dynamic / Time-Varying| Approximately equal   | Much smaller:             |
| **Generator**         | $X_d''$ (Subtransient)| to subtransient:      | $Z_0 \ll Z_1$             |
|                       | $X_d'$ (Transient)    | $Z_2 \approx jX_d''$  | $Z_0 \approx 0.1 - 0.5 X_d''$|
|                       | $X_d$ (Steady-state)  | ($0.8 - 1.0 \times X_d''$) | + **$3 Z_n$** if grounded |
+-----------------------+-----------------------+-----------------------+---------------------------+
| **Two-Winding**       | Transformer Leakage   | Identical to $Z_1$:   | Governed by winding       |
| **Transformer**       | Impedance:            | $Z_2 = Z_1 = Z_{leak}$| connections and core type |
|                       | $Z_1 = Z_{leakage}$   |                       | ($Z_0 = Z_{leak}$ or Open)|
+-----------------------+-----------------------+-----------------------+---------------------------+
| **Transmission**      | Line Series Loop      | Identical to $Z_1$:   | Higher due to earth:      |
| **Line (Overhead)**   | Impedance:            | $Z_2 = Z_1$           | $Z_0 \approx 2.5 - 3.5 Z_1$|
|                       | $Z_1 = R_L + j\omega L$|                       | (Up to $4-5 Z_1$ no shield)|
+-----------------------+-----------------------+-----------------------+---------------------------+

The $3 Z_n$ Neutral Grounding Impedance Multiplier Rule

When the neutral point of a synchronous machine or transformer is connected to earth through an impedance $\mathbf{Z}_n$ (such as a neutral grounding resistor NGR or grounding reactor):

• Current flowing through the physical neutral is $\mathbf{I}N = 3 \mathbf{I}{a0}$.
• The physical voltage drop from the neutral node $N$ to true ground is: $\mathbf{V}_N = \mathbf{I}_N \mathbf{Z}_n = (3 \mathbf{I}_{a0}) \mathbf{Z}_n = \mathbf{I}_{a0} (3 \mathbf{Z}_n)$
• Because all sequence network calculations are performed on a per-phase basis using phase-a current $\mathbf{I}_{a0}$, the neutral grounding impedance MUST BE MULTIPLIED BY 3 in the zero-sequence equivalent network: $\mathbf{Z}_{0,total} = \mathbf{Z}_{0,internal} + 3 \mathbf{Z}_n$

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3. Transformer Zero-Sequence Topologies & Circuit Rules

Zero-sequence current behavior in power transformers is governed strictly by the primary and secondary winding configurations and grounding methods. Zero-sequence current requires a path to flow into the winding and a corresponding path for ampere-turn balance in the opposing winding.

+---------------------------------------------------------------------------------------------------+
| TRANSFORMER ZERO-SEQUENCE EQUIVALENT CIRCUITS                                                     |
+-----------------------------------+---------------------------------------------------------------+
| Connection Type                   | Zero-Sequence Network Equivalent Topology                     |
+-----------------------------------+---------------------------------------------------------------+
| **1. Grounded Y — Grounded Y**    | Series switch CLOSED on both sides; pass-through connection:  |
| (Yg - Yg)                         | Primary Bus o-----[ Z_0 = Z_leak ]-----o Secondary Bus        |
|                                   | (Neutral impedances add as $3Z_{np}$ and $3Z_{ns}$)           |
+-----------------------------------+---------------------------------------------------------------+
| **2. Grounded Y — Delta**         | Grounded Y side SHUNTED to zero reference bus;                |
| (Yg - Δ) *(Most Common GSU)*      | Series connection to Delta side is OPEN:                      |
|                                   | Primary Bus o-----[ Z_0 ]-----+                               |
|                                   |                               |                               |
|                                   | Secondary Bus o---[ OPEN ]----+=== Zero Ref (Ground)          |
+-----------------------------------+---------------------------------------------------------------+
| **3. Delta — Delta**              | Both sides OPEN to external lines;                            |
| (Δ - Δ)                           | Zero-sequence circulates internally in Delta loops:           |
|                                   | Primary Bus o---[ OPEN ]------+                               |
|                                   |                               |                               |
|                                   | Secondary Bus o---[ OPEN ]----+=== Zero Ref (Ground)          |
+-----------------------------------+---------------------------------------------------------------+
| **4. Ungrounded Y — Grounded Y**  | Primary side is OPEN (no ground return path);                 |
| (Y - Yg)                          | Secondary side connected to reference bus through $Z_0$:      |
|                                   | Primary Bus o---[ OPEN ]------+                               |
|                                   |                               |                               |
|                                   | Secondary Bus o-----[ Z_0 ]---+=== Zero Ref (Ground)          |
+-----------------------------------+---------------------------------------------------------------+
| **5. Ungrounded Y — Delta**       | Completely ISOLATED; both series and shunt paths OPEN:        |
| (Y - Δ)                           | Primary Bus o---[ OPEN ]                                      |
|                                   | Secondary Bus o---[ OPEN ]                                    |
+-----------------------------------+---------------------------------------------------------------+

The Three Golden Rules of Transformer Zero-Sequence Circuits

1. Grounded Wye ($Y_g$): Provides a connection from the external line terminal through $Z_{leakage}$ to the internal transformer neutral node.
2. Ungrounded Wye ($Y$): An open circuit in series with that winding (blocks all zero-sequence current).
3. Delta ($\Delta$): Acts as a short circuit to the zero-sequence reference bus on the Delta winding side (providing a circulation path for zero-sequence trapped flux), but acts as an open series switch to the external line (blocks zero-sequence current from passing through to the other system).

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4. Comprehensive Worked Sequence Network Construction Example

One-Line System Diagram & Data

A three-phase, $100\text{ MVA}$ base transmission system consists of:

• Generator G1: $100\text{ MVA}, 13.8\text{ kV}$, $X_1 = X_d'' = 0.15\text{ pu}$, $X_2 = 0.15\text{ pu}$, $X_0 = 0.05\text{ pu}$. Neutral grounded through reactor $X_n = 0.03\text{ pu}$.
• Transformer T1 (Step-Up): $100\text{ MVA}, 13.8\text{ kV (}\Delta\text{)} / 138\text{ kV (}Y_g\text{)}$, $X_{leakage} = 0.10\text{ pu}$. High side solidly grounded.
• Transmission Line TL: $138\text{ kV}$, $X_1 = X_2 = 0.20\text{ pu}$, $X_0 = 0.60\text{ pu}$.
• Transformer T2 (Step-Down): $100\text{ MVA}, 138\text{ kV (}Y_g\text{)} / 13.8\text{ kV (}\Delta\text{)}$, $X_{leakage} = 0.08\text{ pu}$. High side solidly grounded.
• Motor M1: $100\text{ MVA}, 13.8\text{ kV}$, $X_1 = X_d'' = 0.20\text{ pu}$, $X_2 = 0.20\text{ pu}$, $X_0 = 0.08\text{ pu}$. Neutral grounded through reactor $X_{nm} = 0.02\text{ pu}$.

Calculate the Thevenin Sequence Impedances ($\mathbf{Z}{1,th}, \mathbf{Z}{2,th}, \mathbf{Z}_{0,th}$) looking into Bus 2 (the 138 kV high-voltage terminal of Transformer T1).

ONE-LINE SYSTEM DIAGRAM:
             T1: 13.8 kV / 138 kV             T2: 138 kV / 13.8 kV
                 (Delta - Yg)                     (Yg - Delta)
 [ G1 ]=== Bus 1 ===[ T1 ]=== Bus 2 ===[ Line ]=== Bus 3 ===[ T2 ]=== Bus 4 ===[ M1 ]
   |                           ^                                                |
  [Xn]                   (FAULT BUS F)                                         [Xnm]

Step-by-Step Sequence Network Reduction

=========================================================================================
CALCULATION WORKFLOW & SOLUTION:
=========================================================================================

Step 1: Construct Positive-Sequence Network & Compute Z_1,th at Bus 2
  Looking Left from Bus 2:
    Z_1,left = j X_G1,1 + j X_T1
             = j 0.15 + j 0.10 = j 0.25 pu

  Looking Right from Bus 2 (through Line, T2, and Motor M1):
    Z_1,right = j X_TL,1 + j X_T2 + j X_M1,1
              = j 0.20 + j 0.08 + j 0.20 = j 0.48 pu

  Combine parallel branches:
    Z_1,th = Z_1,left || Z_1,right
           = (j 0.25 * j 0.48) / (j 0.25 + j 0.48)
           = (-0.120) / (j 0.730)
           = j (0.120 / 0.730)
           = +j 0.1644 pu

Step 2: Construct Negative-Sequence Network & Compute Z_2,th at Bus 2
  Because machine and line negative-sequence impedances equal their positive-sequence
  counterparts (X_2 = X_1):
    Z_2,left = j X_G1,2 + j X_T1 = j 0.15 + j 0.10 = j 0.25 pu
    Z_2,right = j X_TL,2 + j X_T2 + j X_M1,2 = j 0.20 + j 0.08 + j 0.20 = j 0.48 pu

    Z_2,th = Z_2,left || Z_2,right = +j 0.1644 pu

Step 3: Construct Zero-Sequence Network & Compute Z_0,th at Bus 2
  Analyze Transformer Topologies & Grounding Paths:

  A. Looking Left into Transformer T1 from Bus 2 (138 kV side):
     - T1 connection is Delta (13.8 kV) to Grounded Wye (138 kV).
     - The 138 kV Grounded Wye winding provides a path to the zero-sequence reference
       bus through transformer leakage reactance X_T1 = 0.10 pu.
     - The 13.8 kV Delta winding acts as an OPEN series switch, completely isolating
       Generator G1 and its neutral grounding reactor (3 * X_n) from Bus 2!
     - Therefore, looking left from Bus 2:
         Z_0,left = j X_T1 = +j 0.10 pu  (connected directly to zero reference bus)

  B. Looking Right into Transmission Line from Bus 2:
     - Transmission Line zero-sequence reactance: j X_TL,0 = j 0.60 pu.
     - At Bus 3, Transformer T2 high side is Grounded Wye (Yg), connecting to the zero
       reference bus through leakage reactance j X_T2 = j 0.08 pu.
     - The 13.8 kV Delta winding on T2 opens the series path to Motor M1, completely
       isolating Motor M1 and its neutral reactor (3 * X_nm) from the 138 kV system!
     - Therefore, looking right from Bus 2:
         Z_0,right = j X_TL,0 + j X_T2
                   = j 0.60 + j 0.08 = +j 0.68 pu  (connected to zero reference bus)

  C. Combine Parallel Zero-Sequence Branches at Bus 2:
     Z_0,th = Z_0,left || Z_0,right
            = (j 0.10 * j 0.68) / (j 0.10 + j 0.68)
            = (-0.068) / (j 0.780)
            = j (0.068 / 0.780)
            = +j 0.0872 pu

=========================================================================================
FINAL THEVENIN SEQUENCE IMPEDANCES AT BUS 2:
  • Positive-Sequence Thevenin Impedance: Z_1,th = +j 0.1644 pu
  • Negative-Sequence Thevenin Impedance: Z_2,th = +j 0.1644 pu
  • Zero-Sequence Thevenin Impedance:     Z_0,th = +j 0.0872 pu
=========================================================================================

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5. Common Exam Traps & Strategic Pitfalls

• Forgetting the $3 Z_n$ Neutral Multiplier: Inserting $Z_n$ instead of $3 Z_n$ in the zero-sequence generator or transformer model. The neutral carries $3 I_{a0}$, producing a per-phase voltage drop of $3 Z_n I_{a0}$.
• The Delta Zero-Sequence Pass-Through Fallacy: Allowing zero-sequence current to propagate through a Delta-Wye transformer. Delta winding blocks zero-sequence current from passing through to the other side, even though it provides a path to the reference bus on the Grounded Wye side.
• Assuming $Z_0 = Z_1$ for Transmission Lines: Equating transmission line zero-sequence impedance to positive-sequence impedance. Because earth has much higher resistivity than metallic conductors and mutual ground coupling occurs across all three phases, $Z_0$ is typically $2.5\text{ to }3.5$ times larger than $Z_1$.
• Including Voltage Sources in Negative or Zero Networks: Placing an internal EMF source $E_a$ inside the negative or zero sequence networks. Only the positive sequence network contains active voltage sources.