The Flow of Power (Part II: Power Flow Solutions and Optimization)

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Per-phase analysis assumes balanced three‑phase lines so inductance and capacitance decouple and each line reduces to a pi-model with Z and Y per meter; transformers map phase voltages (delta/wye gives a √3 and 30° shift) and networks assemble into an admittance matrix so Y bus relates I and V linearly. Power flow is quadratic in voltages; DC power flow linearizes it. OPF is a nonconvex QCQP; common relaxations lift to W=vvH and drop rank-1 (SDP relaxation recovers global optima when W is rank‑1), and radial grids admit efficient exact SOCPs (tree networks admit SOCP exactness). Real‑time control uses gradient steps plus measurements, with tracking error set by problem variation, conditioning and approximation.

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1. 0:00Start
2. 21:46Transformer model Single-phase (non-ideal) transformer
3. 22:28Transformer model 3.phase ideal transformer
4. 27:19Generator model
5. 27:44Putting everything together
6. 33:03Network admittance matrix
7. 36:09Bus injection model
8. 47:39Solution strategy
9. 50:16Motivations
10. 58:58Tracking performance