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Y-bus and power flow

Bus admittance matrix, slack / PV / PQ bus classification, nonlinear power-flow equations, and the Newton-Raphson / Fast Decoupled / DC solution methods. Foundation of SCADA, planning, and electricity markets.

Senior ~12 min

Step 1 — Y_bus & power flow: I = Y_bus · V at every bus

Animation speed 0.55×

step — iter — method —

Reference notes

Power-flow analysis answers: given generation, load, and network impedances, what are the bus voltages and line flows? Use Next → to walk through the bus admittance matrix Y_bus, the three bus types (slack / PV / PQ), the nonlinear power-flow equations, and the iterative solution methods that solve them in production (Newton-Raphson, Fast Decoupled, DC).

Bus admittance matrix Y_bus

N-bus network → N×N complex matrix Y_bus.
Y_ii (diagonal) = sum of admittances connected to bus i (lines, transformers, shunts).
Y_ij (off-diagonal) = −y_ij, the negative of the direct admittance between buses i and j. Zero if no direct connection.
Sparse — each bus connects to only a few neighbors → most off-diagonal entries are zero. Sparsity enables solving 10 000-bus systems on a laptop.

I = Y_bus · V (Kirchhoff's current law, network-wide)

Bus types

Type	Specified	Computed	Use
Slack (swing)	V, θ (usually 1.0 pu, 0°)	P, Q	Power balance / reference
PV (generator)	P, V	Q, θ	Generator buses
PQ (load)	P, Q	V, θ	Load buses (most)

Exactly one slack bus per island. Most buses are PQ. PV buses get reclassified as PQ if Q hits its limit (reactive limit handling).

Power-flow equations

Substituting I = Y_bus·V into S_i = V_i·I_i^* and separating real/imaginary parts:

P_i = V_i · Σ_j V_j · (G_ij cos θ_ij + B_ij sin θ_ij)

Q_i = V_i · Σ_j V_j · (G_ij sin θ_ij − B_ij cos θ_ij)

where G_ij + j·B_ij = Y_ij, and θ_ij = θ_i − θ_j. These are NONLINEAR in V and θ — no closed-form solution. 2(N-1) equations in 2(N-1) unknowns.

Solution methods

Gauss-Seidel — walk through buses, update each V using current best estimates of others. Linear convergence. Historical; slow for large systems.
Newton-Raphson — linearize around current estimate, solve [ΔP; ΔQ] = J · [Δθ; ΔV] each iteration. Quadratic convergence: 3–5 iterations typically. Industry standard.
Fast Decoupled — NR approximation exploiting weak coupling between (P, θ) and (Q, V). Solves two smaller systems alternately. ~2× faster per iteration than full NR.
DC Power Flow — heavy linearization: V ≈ 1.0 pu, cos θ ≈ 1, sin θ ≈ θ, R ≪ X → P ≈ B · θ. Linear, no iteration. Used for planning, market simulations, contingency screening. Accurate to a few % on real-power flows; gives no Q or V information.

Convergence

Newton-Raphson convergence criterion: max |ΔP_i|, |ΔQ_i| < 10⁻³ pu (0.1 % of S_base).
Typical iterations: 3–5 from flat-start. Pathological cases (near voltage collapse): 10–20 or divergence.
Each iteration: assemble 2(N-1)×2(N-1) sparse Jacobian; solve via sparse LU factorization, ~O(N^1.5) scaling.
1000-bus power flow → under a second on a modern laptop. Same problem in 1970 → 1 hour on an IBM mainframe.
Divergence usually signals voltage collapse — operating point beyond the loadability (nose-curve) limit, no real solution exists.

Applications

Planning — long-term capacity expansion, line siting, renewable integration studies.
Operations planning — day-ahead unit commitment, hour-ahead dispatch, security-constrained economic dispatch.
Real-time SCADA — state estimation runs every 1–5 minutes at every transmission operator. Power flow with statistical weighting on noisy telemetry.
Contingency analysis — n−1 (and increasingly n−1−1, n−2) contingency screening per NERC reliability standards. Runs hundreds of power flows for each operating snapshot.
Market clearing — locational marginal prices (LMPs) at each bus = dual variables of the security-constrained economic dispatch power-flow optimization. LMP = system marginal cost + congestion + losses.

Take-away. Power flow is the foundational power-system computation. Y_bus is a sparse N×N admittance matrix; bus types (slack / PV / PQ) define which variables are specified vs solved. The nonlinear power-flow equations P_i, Q_i are solved iteratively, most often via Newton-Raphson (quadratic convergence, 3-5 iterations typical). DC power flow is a linear approximation used for planning and market clearing. Sparsity + sparse LU is what makes thousand-bus systems tractable. Used for planning, operations, real-time SCADA, contingency analysis, and electricity-market LMP calculation.

Keyboard shortcuts

The 5-bus example network shows the three bus types color-coded. The Y_bus visualization shows sparsity: orange diagonal, green off-diagonal where buses connect, dark blue zeros.