Power Flow Theory

This reference explains the physics and mathematics behind power flow analysis — the foundational calculation for all grid studies.

Why Power Flow Matters

Power flow analysis answers the fundamental question: Given generation and load at each bus, what are the voltages and line flows throughout the network?

Every other grid analysis builds on power flow:

• OPF optimizes generation subject to power flow equations
• Contingency analysis runs power flow for each outage scenario
• State estimation reconciles measurements with power flow physics
• Planning studies evaluate future scenarios via power flow

Understanding power flow deeply is essential for interpreting results and debugging convergence issues.

Physical Intuition

Kirchhoff's Laws

Power flow is governed by two physical principles:

1. Conservation of charge (Kirchhoff's Current Law): Current entering a node equals current leaving
2. Conservation of energy (Kirchhoff's Voltage Law): Voltage drops around any closed loop sum to zero

In power systems terms:

• Power injected at each bus must equal power flowing out on connected branches
• Voltage differences across branches drive power flow

Why Does Power Flow?

Consider two buses connected by a transmission line:

    Bus 1              Bus 2
    V₁∠θ₁ ────────── V₂∠θ₂
           R + jX

Power flows from higher to lower voltage angle. The real power transfer is approximately:

$$P_{12} \approx \frac{V_1 V_2}{X} \sin(\theta_1 - \theta_2)$$

Key insights:

• Angle difference drives real power: Larger θ₁ - θ₂ means more MW flowing
• Reactance limits flow: Higher X means less power for the same angle difference
• Voltage magnitudes matter: Lower voltages reduce power transfer capability

For reactive power, voltage magnitude difference is the driver:

$$Q_{12} \approx \frac{V_1}{X} \left( V_1 - V_2 \cos(\theta_1 - \theta_2) \right)$$

This is why generators raise voltage to export reactive power and lower it to absorb.

AC Power Flow

AC power flow solves the exact nonlinear equations relating voltages, angles, and power injections.

Complex Power

At each bus, complex power injection is:

$$S_i = P_i + jQ_i = V_i \cdot I_i^*$$

where $V_i$ is complex voltage and $I_i^*$ is the complex conjugate of current.

The Admittance Matrix (Y-bus)

The network is characterized by the admittance matrix $\mathbf{Y}$, where:

$$\mathbf{I} = \mathbf{Y} \cdot \mathbf{V}$$

For a network with $n$ buses, $\mathbf{Y}$ is an $n \times n$ sparse complex matrix:

Off-diagonal elements ($i \neq j$):

$$Y_{ij} = -y_{ij} = -\frac{1}{R_{ij} + jX_{ij}}$$

where $y_{ij}$ is the series admittance of the branch connecting buses $i$ and $j$.

Diagonal elements:

$$Y_{ii} = \sum_j y_{ij} + y_i^{\text{shunt}}$$

Sum of all admittances connected to bus $i$, plus any shunt elements (capacitors, reactors).

Power Balance Equations

Substituting $\mathbf{I} = \mathbf{Y} \cdot \mathbf{V}$ into $S = V \cdot I^*$ gives the power flow equations:

Real power at bus $i$:

$$P_i = \sum_j |V_i| |V_j| \left( G_{ij} \cos(\theta_i - \theta_j) + B_{ij} \sin(\theta_i - \theta_j) \right)$$

Reactive power at bus $i$:

$$Q_i = \sum_j |V_i| |V_j| \left( G_{ij} \sin(\theta_i - \theta_j) - B_{ij} \cos(\theta_i - \theta_j) \right)$$

where $G_{ij} + jB_{ij} = Y_{ij}$ (conductance and susceptance).

These $2n$ equations ($n$ real power, $n$ reactive power) must be satisfied simultaneously.

Bus Classifications

Not all 4n variables (P, Q, |V|, θ at each bus) are unknowns. Bus types determine what's specified:

With one slack bus and assuming n_pv PV buses:

• Unknowns: (n-1) angles + (n - n_pv - 1) voltage magnitudes
• Equations: (n-1) real power + (n - n_pv - 1) reactive power

The system is square and (usually) solvable.

The Jacobian Matrix

The power flow equations are nonlinear, so we solve iteratively. Newton-Raphson linearizes around the current estimate:

$$\begin{bmatrix} \Delta P \ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta\theta \ \Delta|V| \end{bmatrix}$$

The Jacobian submatrices contain partial derivatives:

• $J_1 = \partial P / \partial \theta$: How real power changes with angle
• $J_2 = \partial P / \partial |V|$: How real power changes with voltage magnitude
• $J_3 = \partial Q / \partial \theta$: How reactive power changes with angle
• $J_4 = \partial Q / \partial |V|$: How reactive power changes with voltage magnitude

For example, the off-diagonal element of $J_1$:

$$\frac{\partial P_i}{\partial \theta_j} = |V_i| |V_j| \left( G_{ij} \sin(\theta_i - \theta_j) - B_{ij} \cos(\theta_i - \theta_j) \right)$$

GAT computes the full Jacobian in power_equations.rs.

Newton-Raphson Algorithm

1. Initialize: Set all |V| = 1.0 p.u., all θ = 0 (flat start)

2. Compute mismatches: Calculate P_calc and Q_calc from current V, θ

   ΔP = P_specified - P_calc
   ΔQ = Q_specified - Q_calc
   

3. Check convergence: If max(|ΔP|, |ΔQ|) < tolerance, stop

4. Compute Jacobian: Build J from current V, θ

5. Solve linear system:

   [Δθ, Δ|V|]ᵀ = J⁻¹ · [ΔP, ΔQ]ᵀ
   

6. Update: θ ← θ + Δθ, |V| ← |V| + Δ|V|

7. Repeat from step 2

Typical convergence: 3-10 iterations for well-conditioned cases.

Why Newton-Raphson?

Newton-Raphson has quadratic convergence near the solution — errors decrease as ε² each iteration. This means:

• 6 correct digits → 12 correct digits in one iteration
• Very fast once "close" to the answer

The cost is computing and factoring the Jacobian each iteration (O(n²) to O(n³) depending on sparsity).

DC Power Flow

DC power flow is a linear approximation enabling much faster solutions.

Assumptions

1. Voltage magnitudes ≈ 1.0 p.u.: |V_i| ≈ 1 for all buses
2. Small angle differences: sin(θ_i - θ_j) ≈ θ_i - θ_j, cos(θ_i - θ_j) ≈ 1
3. Lossless lines: R << X, so G ≈ 0

Simplified Equations

Under these assumptions, the real power equation becomes:

$$P_i = \sum_j B_{ij} (\theta_i - \theta_j)$$

In matrix form:

$$\mathbf{P} = \mathbf{B} \cdot \boldsymbol{\theta}$$

where $\mathbf{B}$ is the susceptance matrix (imaginary part of Y-bus, negated).

This is a linear system — no iteration needed! One matrix solve gives the answer.

Solving DC Power Flow

1. Remove the slack bus row/column from $\mathbf{B}$ (it has $\theta = 0$)
2. Solve the reduced system: $\boldsymbol{\theta} = \mathbf{B}_{\text{reduced}}^{-1} \cdot \mathbf{P}$
3. Compute branch flows: $P_{ij} = (\theta_i - \theta_j) / X_{ij}$

What DC Power Flow Ignores

• Reactive power: No Q equations, no voltage magnitude results
• Losses: I²R losses are zero (R = 0 assumption)
• Voltage constraints: Cannot check if |V| stays within limits
• VAR limits: Generator reactive limits don't apply

When to Use DC vs. AC

DC power flow typically underestimates congestion and overestimates transfer capability.

Fast Decoupled Power Flow

A middle ground between full Newton-Raphson and DC approximation.

P-θ / Q-V Decoupling

In transmission networks:

• Real power P depends mainly on angles θ
• Reactive power Q depends mainly on voltage magnitudes |V|

This suggests solving two smaller systems instead of one large one:

ΔP = B' · Δθ      (P-θ subproblem)
ΔQ = B'' · Δ|V|   (Q-V subproblem)

Algorithm

1. Solve P-θ: Update angles using B' (constant matrix)
2. Solve Q-V: Update voltages using B'' (constant matrix)
3. Repeat until converged

Advantage: B' and B'' are factored once and reused (no Jacobian rebuild)

Disadvantage: Slower convergence than Newton-Raphson (linear, not quadratic)

Typically useful for real-time applications where speed matters more than iteration count.

Convergence Issues

Power flow doesn't always converge. Common causes and remedies:

Heavy Loading

The system may have no solution if load exceeds transfer capability.

Symptoms: Mismatch oscillates or grows Remedy: Reduce load, add generation, or check for data errors

Reactive Power Limits

Generators hitting Q limits switch from PV to PQ buses, potentially causing voltage collapse.

Symptoms: Voltages drop progressively Remedy: Add reactive support, relax limits for debugging

Bad Initial Guess

Flat start may be far from the solution for heavily loaded or unusual systems.

Symptoms: Divergence from the first iteration Remedy: Use warm start from similar case, or solve a relaxed problem first

Data Errors

Incorrect impedances, missing buses, or topology errors.

Symptoms: Immediate divergence or nonsensical results Remedy: Check input data, run gat graph islands to verify connectivity

Ill-Conditioning

Very long or very short lines create numerical issues.

Symptoms: Slow convergence, sensitivity to tolerance Remedy: Check per-unit values, consider network reduction

Power Flow in GAT

GAT implements power flow in gat-algo:

DC Power Flow

gat pf dc network.arrow --out flows.parquet

Uses sparse LU factorization of the B matrix. O(n) for radial networks, O(n^1.5) for meshed.

AC Power Flow

gat pf ac network.arrow --out flows.parquet --tol 1e-6 --max-iter 20

Full Newton-Raphson with:

• Sparse Jacobian computation
• LU factorization with AMD ordering
• Automatic PV→PQ switching at reactive limits

Implementation Details

Mathematical Derivations

Derivation of Power Flow Equations

Starting from S = V · I* and I = Y · V:

S_i = V_i · (Σⱼ Y_ij · V_j)*
    = V_i · Σⱼ Y_ij* · V_j*
    = Σⱼ V_i · V_j* · Y_ij*

Writing V_i = |V_i| · e^(jθ_i) and Y_ij = G_ij + jB_ij:

S_i = Σⱼ |V_i| · |V_j| · e^(j(θ_i - θ_j)) · (G_ij - jB_ij)

Taking real and imaginary parts gives the P and Q equations.

Derivation of DC Approximation

Starting from:

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

Apply assumptions:

1. |V_i| = |V_j| = 1
2. G_ij = 0 (lossless)
3. sin(θ_ij) ≈ θ_ij for small angles

Result:

P_i = Σⱼ B_ij · θ_ij = Σⱼ B_ij · (θ_i - θ_j)

Further Reading

Textbooks

• Grainger & Stevenson, Power System Analysis — Standard undergraduate text
• Glover, Sarma & Overbye, Power System Analysis and Design — Comprehensive coverage
• Kundur, Power System Stability and Control — Advanced dynamics

Papers

• Tinney & Hart (1967), "Power Flow Solution by Newton's Method" — Foundational paper on sparse techniques
• Stott & Alsaç (1974), "Fast Decoupled Load Flow" — The FDLF algorithm
• Van Amerongen (1989), "A General-Purpose Version of the Fast Decoupled Load Flow" — Modern FDLF

GAT Documentation

• Power Flow Guide — Practical usage
• OPF Guide — Optimization with power flow constraints
• Glossary — Term definitions