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Active power between machines, 200∠0° and 200∠15° V

Short answer

Active power per phase P = (200×200/10) sin(0°−15°) = −1035.28 W, so 1.035 kW flows from machine B to A (three-phase 3.106 kW). B is generator, A motor because V_B leads V_A by 15°.

Two 3-phase synchronous machines A and B are connected by a link having series impedance of 0 + j10 Ohms/phase. The terminal voltages at machines A and B are 200∠0° V and 200∠15° V respectively. Compute the magnitude and direction of active power flow through the link.

For a purely reactive link the real power transfer is

P=∣VA∣∣VB∣Xsin⁡(δA−δB)P = \frac{|V_A||V_B|}{X}\sin(\delta_A - \delta_B)

and it flows from the machine with the leading angle to the one with the lagging angle.

Calculation (per phase)

PA→B=200×20010sin⁡(0∘−15∘)=4000×(−0.2588)=−1035.28 W\begin{aligned} P_{A\to B} &= \frac{200\times200}{10}\sin(0^\circ - 15^\circ)\\ &= 4000\times(-0.2588) = -1035.28\ \text{W} \end{aligned}

The negative sign means the power actually flows from B to A.

Check with phasors:

IA→B=200∠0∘−200∠15∘j10=5.221∠−172.5∘ ASA=VAI∗=200×5.221∠172.5∘=−1035.28+j136.30 VASB=VBI∗=−1035.28−j136.30 VA\begin{aligned} I_{A\to B} &= \frac{200\angle0^\circ - 200\angle15^\circ}{j10} = 5.221\angle-172.5^\circ\ \text{A}\\ S_A &= V_AI^* = 200\times5.221\angle172.5^\circ = -1035.28 + j136.30\ \text{VA}\\ S_B &= V_BI^* = -1035.28 - j136.30\ \text{VA} \end{aligned}

So A receives 1035.28 W and B delivers 1035.28 W. The link has no resistance, so there is no real-power loss. Each machine supplies 136.30 var to the link (I2X=27.26×10=272.59I^2X = 27.26\times10 = 272.59 var).

QuantityPer phaseThree-phase
Real power B → A1035.28 W3105.83 W
Reactive loss in link272.59 var817.78 var

Answer: About 1.035 kW per phase (3.106 kW three-phase) flows from machine B to machine A. B acts as the generator and A as the motor, because VBV_B leads VAV_A by 15°.

✓ Answer checked by Claude Opus 5.5 (high effort)

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