PW-D2 · Power flow, faults, protection

Symmetrical components: split unbalance into three balanced sets

A balanced three-phase set needs only one phase solved, but a fault usually strikes one or two phases and breaks the balance. Yet any unbalance decomposes uniquely into three balanced sets — positive, negative and zero sequence. Drag one phase to break the balance and watch the negative and zero components grow from zero, learning the decomposition by hand.

Drag I_a and watch the three components

Among the three currents on the left, drag the tip of I_a to break the balance. The three bars on the right are the magnitudes of the positive I_1, negative I_2 and zero I_0 sequence. Balanced, only the positive exists and negative and zero are zero, but the more you unbalance the more the two grow. The zero sequence is a third of the sum of the three currents from A5.

Drag the tip of I_a to create unbalance.

Magnitudes of the three sequence components

I_1 ≈ 1.15   I_2 ≈ 0.17   I_0 ≈ 0.17

I_2 / I_1 ≈ 15%

Balanced means negative and zero = 0 (only positive). Off balance, the two grow.

A sum of three balanced sets

By the theorem of symmetrical components, any three phasors can be written uniquely as a sum of three balanced sets. The positive sequence is a balanced set rotating the original way (a-b-c), the negative sequence a balanced set rotating the other way (a-c-b), and the zero sequence a set with all three equal in magnitude and phase. Each phase is the sum of these three — I_a = I_1 + I_2 + I_0, and I_b and I_c are sums of the same three rotated by the operator a.

Why splitting makes it easy

In a balanced system the positive-, negative- and zero-sequence circuits do not interfere; they are decoupled. So each generator and line carries separate positive-, negative- and zero-sequence impedances, and an unbalanced fault is solved by connecting these three circuits according to the fault type. One hard unbalanced problem splits into three familiar balanced ones. In normal operation only positive sequence flows; negative sequence overheats generators, and zero sequence is the signal a protective relay uses to detect a ground fault.

ObserveI_a = I₁ + I₂ + I₀

Each phase is the sum of positive, negative and zero.

ChooseI₀ = (I_a + I_b + I_c) / ?

The zero sequence is a third of the sum of the three (A5).

Fill inI₁ = (I_a + a I_b + a² I_c) / ?

The positive sequence is a third of the sum weighted by operator a.

On your ownbalanced: I₂ = I₀ = ?

When balanced, negative and zero vanish.

Zero sequence · A5 returns

The zero sequence I_0 = (I_a + I_b + I_c)/3 is exactly a third of the neutral residual from A5. The three zero-sequence currents are equal in magnitude and phase, so they do not cancel but add up and flow into the neutral or earth. For zero-sequence current to flow there must be a return path such as a grounded neutral, and in a delta connection with no return path it cannot leave to the outside. This zero sequence, which was zero when balanced, rises in a ground fault where one phase touches earth and becomes the clue for protection.

Back to the first screen

Placing I_a at the balanced spot made the negative and zero bars vanish to zero, leaving only the positive, and dragging it out grew the negative and zero together. Unbalance is, in the end, a question of how much negative and zero ride on top of the positive. That zero sequence is the same quantity as the residual leaking into the neutral in A5. Splitting the unbalanced three phases into these three balanced sets lets the next unit solve every kind of fault as a connection of familiar balanced circuits.

The symmetrical components — any unbalanced three-phase set decomposes uniquely into three balanced sets: positive (rotating a-b-c), negative (rotating the other way), zero (all in phase). I_a = I₁ + I₂ + I₀, and zero I₀ = (I_a+I_b+I_c)/3 is the neutral residual of A5. Balanced gives only positive (I₂=I₀=0); an unbalanced fault brings out negative and zero — solved as three balanced positive-, negative- and zero-sequence circuits.

The next step

With the three sequence circuits in hand, the next unit (PW-D3) looks at fault types and fault current. A three-phase short uses the positive-sequence circuit alone, a single line-to-ground fault connects positive, negative and zero in series, and a line-to-line fault meshes positive with negative — each fault type connects the three sequence circuits differently to give the fault current. Whether the balanced fault gives the largest current or the most common line-to-ground fault shows up through zero sequence, the same tool solves it.