PW-D1 · Power flow, faults, protection

Power flow: the phase angle pushes the active power

Power seems to flow from the higher-voltage bus to the lower, but what pushes the active power is not voltage magnitude — it is the phase-angle difference δ between the two buses. As a pressure difference pushes water, an angle difference pushes active power. Drag the phase angle and see the transfer peak at δ = 90 degrees and fall beyond it.

Drag the phase angle and read the power curve

The two buses have equal voltage magnitude. Drag the handle to change the angle difference δ. The active power P = (V_s V_r / X) sin δ grows as δ rises, peaks at 90 degrees, and falls beyond it. The point on the curve is the power flowing now.

Drag the handle left and right to set the angle δ.

Active power transferred (P_max = V_s V_r / X)

P ≈ 0.50 P_max   (δ = 30°)

Stable region (δ < 90°)

The angle difference pushes active power

Take the two buses at the ends of a nearly pure-reactance line. Even with equal voltage magnitude, if one side’s phase leads, active power flows from that leading bus to the lagging one. Its size is P = (V_s V_r / X) sin δ, growing with the angle difference δ. Push more mechanical power into a generator and its rotor edges ahead, widening δ, and that much more power leaves into the grid. Not voltage but angle is the handle of active power.

Reactive power is pushed by magnitude

On the same two buses, reactive power is pushed the other way, by the difference in voltage magnitude. Q ≈ (V_s/X)(V_s − V_r cos δ), so reactive flows from the higher-magnitude bus to the lower. Thus raising a generator’s excitation (flux) to lift voltage magnitude steers reactive power, while raising mechanical power to advance the phase steers active power. Active by angle, reactive by magnitude — the two act like nearly independent handles.

Past 90 degrees it collapses

P = P_max sin δ reaches its maximum P_max = V_s V_r / X at δ = 90 degrees. Up to there, as load grows and δ widens more power flows, which is stable. But past 90 degrees sin δ falls, so widening the angle further sends less power, the generator accelerates, its phase slips and synchronism is lost — the steady-state stability limit. So lines reduce X with series compensation or raise voltage to lift P_max, and are operated well below 90 degrees.

ObserveP = (V_s V_rX) sin δ

Active power is V_s V_r over X times sin δ.

ChooseP × ?

What pushes the power is the angle, sin δ.

Fill inP_max → δ = ?

Maximum power is when δ is 90 degrees.

On your ownQ × V_s − V_r ?

Reactive is pushed by magnitude, V_s − V_r cos δ.

Back to the first screen

The two buses kept equal voltage magnitude throughout, yet dragging the angle δ wider started active power flowing and brought it to a peak at 90 degrees. What pushed the power was not the height of voltage but the lead and lag of phase. The curve turning downhill past 90 degrees marks the unstable region beyond, where synchronism is lost. Power-flow analysis is, in the end, setting each bus’s voltage magnitude and angle to read off the active and reactive power flowing in every line.

The power flow — between two buses, active power is pushed by the angle difference δ. On a reactance-X line, P = (V_s V_r / X) sin δ, flowing from the leading bus to the lagging one. Reactive power is pushed by the magnitude difference, Q ≈ (V_s/X)(V_s − V_r cos δ). The transfer is maximum at δ = 90°, P_max = V_s V_r/X, and beyond it the steady-state stability limit is crossed and synchronism is lost.

The next step

Power flow is the picture when the system is balanced. But a fault usually strikes only one or two phases, breaking the balance. The next unit (PW-D2) looks at symmetrical components. Decompose the unbalanced three phases into the sum of three balanced sets — positive, negative and zero sequence — and the tools of a balanced system can solve an unbalanced fault. The zero sequence met in A5 finds its place here.