--- title: "Sequence components decoupled for power flow" subject: "Power System Analysis II" chapter: 4 key: "semester_5/power_system_analysis_ii:sequence-decoupled-complex-power" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/sequence-decoupled-complex-power site: BCT Engineering --- # Sequence components decoupled for power flow Power System Analysis II · Chapter 4: Unbalanced System Analysis - Asked in: 2075 Chaitra, 8 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/sequence-decoupled-complex-power - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-4 ## Question Show that the sequence component are decoupled for complex power flow in a power system. ## Short answer Complex power expressed in sequence quantities is S = 3Va0Ia0* + 3Va1Ia1* + 3Va2Ia2*, with no cross-sequence terms because AᵀA* = 3I. Each sequence network contributes independently to total power, proving decoupling. ## Answer The complex power of a three-phase system splits into three separate terms, one for each sequence, with **no cross terms** between different sequences. So each sequence network carries its own power independently; this is what "decoupled for power flow" means. #### Proof Complex power in phase quantities: $$ S = V_aI_a^* + V_bI_b^* + V_cI_c^* = V_p^TI_p^* $$ With $V_p = AV_s$, $I_p = AI_s$, where $A = \begin{bmatrix} 1&1&1\\1&a^2&a\\1&a&a^2 \end{bmatrix}$: $$ S = (AV_s)^T(AI_s)^* = V_s^T\,A^TA^*\,I_s^* $$ $A$ is symmetric, so $A^T = A$; and $a^* = a^2$, so $A^* = \begin{bmatrix} 1&1&1\\1&a&a^2\\1&a^2&a \end{bmatrix}$. Using $1+a+a^2 = 0$, $a^3 = 1$: $$ A^TA^* = \begin{bmatrix} 3 & 1+a+a^2 & 1+a^2+a \\ 1+a^2+a & 1+a^3+a^3 & 1+a^4+a^2 \\ 1+a+a^2 & 1+a^2+a^4 & 1+a^3+a^3 \end{bmatrix} = \begin{bmatrix} 3&0&0\\0&3&0\\0&0&3 \end{bmatrix} $$ Hence $$ S = 3V_s^TI_s^* = 3V_{a0}I_{a0}^* + 3V_{a1}I_{a1}^* + 3V_{a2}I_{a2}^* $$ #### Why this shows decoupling - Each term contains a voltage and a current of the **same** sequence only. Terms like $V_{a1}I_{a2}^*$ or $V_{a0}I_{a1}^*$ do not appear, because the off-diagonal elements of $A^TA^*$ are zero. - Physically, a positive sequence voltage acting with a negative (or zero) sequence current produces only an oscillating power whose sum over the three phases is zero; it gives no net power. - Therefore the power delivered by the positive sequence network ($S_1 = 3V_{a1}I_{a1}^*$), negative sequence network ($S_2$) and zero sequence network ($S_0$) can be calculated separately and simply added: $$ S = S_0 + S_1 + S_2 $$ - Together with the diagonal sequence impedance matrix of symmetrical equipment (voltage of one sequence depends only on current of the same sequence), this shows the three sequence networks are fully decoupled for both voltage drop and power. #### Example A balanced system has only positive sequence: $S = 3V_{a1}I_{a1}^*$. A generator feeding an unbalanced load delivers useful power mainly through the positive sequence network; power in the negative sequence network appears as losses (for example, extra rotor heating). The factor 3 appears because $A$ is not unitary. With the power-invariant form $\frac{1}{\sqrt3}A$, $S = V_s^TI_s^*$ exactly. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The product A^T A* equals: A. Unit matrix B. Zero matrix C. 3 times the unit matrix D. A^-1 Answer: C. Off-diagonals vanish because 1+a+a^2=0. (Key point: Key matrix) 2. Terms like Va1 Ia2* are absent from S because: A. Va1 is zero B. Ia2 is zero C. They cancel with Va2 Ia1* D. Off-diagonal elements of A^T A* are zero Answer: D. Only same-sequence products remain. (Key point: No cross terms) 3. Positive voltage with negative current gives: A. Oscillating power summing to zero over three phases B. Steady net power C. Reactive power only D. Double power Answer: A. No net power from cross terms. (Key point: Physical) 4. Total complex power is: A. S = S1 only B. S = S0 + S1 + S2 C. S = S1 - S2 + S0 D. S = S1 S2 S0 Answer: B. Each sequence network's power can be added. 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