Lesson 7 of 8

Introduction to Power Flow

8 min read

Power flow (also called load flow) is the single most-run calculation in power systems engineering. It answers one question for an entire network at once: given how much power every generator is producing and every load is consuming, what is the voltage — magnitude and angle — at every single bus? Everything from planning new lines to real-time grid operation depends on solving this problem, over and over.

The three bus types

At every bus in the network, four quantities matter: voltage magnitude |V|, voltage angle θ, active power P, and reactive power Q. In a power flow study you never know all four at any bus in advance — you specify two and solve for the other two. Which two are known depends on what kind of bus it is:

Slack (swing) bus: |V| and θ are specified (θ is the reference, usually 0°). P and Q are solved for. There is exactly one per network — it absorbs whatever generation/load imbalance the rest of the system can't account for, including losses.
PV bus (generator bus): P and |V| are specified — the generator's power output and its voltage setpoint. Q and θ are solved for.
PQ bus (load bus): P and Q are specified (the load's demand). |V| and θ are solved for. Most buses in a real network are PQ buses.

Why it can't be solved with simple linear algebra

The power injected at a bus depends on voltage magnitudes and angles through sine and cosine terms — the power flow equations are fundamentally nonlinear. That single fact is why power flow needs iterative numerical methods rather than a direct solve: you can't just invert a matrix and get the answer the way you can for a simple linear circuit.

How it's actually solved

Two iterative methods dominate the field, and both work the same conceptual way: start from a reasonable guess (often calling every voltage 1.0 p.u. at 0°), calculate how far that guess is from satisfying the power balance at every bus, then adjust and repeat until the mismatch is smaller than a chosen tolerance.

Gauss-Seidel: The older, conceptually simpler method. Updates one bus voltage at a time using the latest available values. Reliable but converges slowly on large networks.
Newton-Raphson: The industry-standard method today. Uses derivatives (a Jacobian matrix) to take much larger, smarter steps toward the solution — typically converging in only a handful of iterations even for large networks.

This is what runs under the hood

Every time you hear about a "power flow study" — or later, when you run your first simulation in this course using pandapower — this is the exact calculation being performed automatically. Understanding bus types and why the problem is iterative is what makes the simulation output make sense, instead of feeling like a black box.

Key takeaways

Power flow finds the voltage magnitude and angle at every bus, given known power injections.
Every bus is one of three types — slack, PV or PQ — depending on which two of |V|, θ, P, Q are known.
The equations are nonlinear, so they're solved iteratively rather than directly.
Newton-Raphson is the standard modern method; Gauss-Seidel is the older, simpler one.