AC Power Flow (Fixed-Point Iteration)

The AC PF solver implements an OpenDSS-style fixed-point iteration for distribution systems. Unlike the AC OPF which optimises voltage magnitudes within bounds, the AC PF solves the standard power-flow equations: given fixed P/Q injections at PQ buses and a fixed voltage at the slack bus, find the voltage magnitude and angle at every bus.

Formulation¶

The solver works directly in SI units (volts, amps, siemens) — no per-unit conversion — to avoid the ill-conditioning that arises when nominal voltages span 120 V–40 kV (a 331:1 ratio causing Y-pu diagonal ratios exceeding 325 million).

Fixed-Point Iteration¶

At each iteration, the current injection at each non-slack node is computed from the specified power and current voltage:

I_i^{(k)} = \frac{\overline{S_i^{\text{spec}}}}{\overline{V_i^{(k)}}} \quad \forall i \notin \text{slack}

(1)

The updated voltage is obtained by solving the linear system:

V_{\text{ns}}^{(k+1)} = Y_{\text{ns}}^{-1} \left( I_{\text{ns}}^{(k)} - Y_{\text{ns,slack}} \cdot V_{\text{slack}} \right)

(2)

where $Y_{\text{ns}}$ is the Y-bus submatrix for non-slack nodes and $Y_{\text{ns,slack}}$ couples non-slack nodes to the fixed slack voltages.

An acceleration factor $\alpha = 0.5$ damps the update:

V_{\text{ns}}^{(k+1)} = V_{\text{ns}}^{(k)} + \alpha \left( V_{\text{ns,calc}} - V_{\text{ns}}^{(k)} \right)

(3)

Convergence Criterion¶

Convergence is measured by the maximum relative voltage change across all non-slack nodes:

\max_i \frac{|V_i^{(k+1)} - V_i^{(k)}|}{V_{\text{base},i}} < \text{tolerance}

(4)

Initial Voltage Estimate¶

Rather than starting from a flat 1.0 pu profile, the solver builds an initial voltage by solving $V = Y^{-1} \cdot I_{\text{source}}$ with loads modeled as constant impedances. This direct solve accounts for all transformer ratios and connections, giving a physically correct starting point.

Key Features¶

SI-unit formulation — avoids per-unit ill-conditioning across multi-voltage-level systems (120 V–40 kV)
Sparse LU factorisation — a single scipy.sparse.linalg.splu factorisation reused across all fixed-point iterations
OpenDSS-style iteration — no Jacobian construction; each iteration is a fast back-substitution
Direct initial solve — $V = Y^{-1} \cdot I$ with constant-impedance loads provides a physically correct warm start
Neutral node support — explicit neutral (N) nodes on split-phase secondary buses held at 0 V reference
Split-phase angle initialization — correct S1/S2 angle initialization for center-tapped transformers with propagation through secondary networks
Connectivity detection — unreachable nodes are automatically treated as slack to prevent singular systems
External warm-start (QSTS) — accepts v0_complex from a previous timestep for time-series simulation

Usage¶

Low-level interface¶

from gdm.distribution import DistributionSystem
from gdm_flow import solve_ac_power_flow

system = DistributionSystem.from_json("model.json")

result = solve_ac_power_flow(
    system,
    p_spec_w={("bus_5", "A"): -20_000.0},
    q_spec_var={("bus_5", "A"): -5_000.0},
    max_iterations=100,
    tolerance=1e-6,
)

print(result.success, result.iterations)
print(result.max_mismatch_pu)

Component-based interface¶

from gdm.distribution import DistributionSystem
from gdm_flow import solve_ac_power_flow_from_components

system = DistributionSystem.from_json("model.json")

result = solve_ac_power_flow_from_components(
    system,
    include_loads=True,
    include_solar=True,
    include_capacitor=True,
    load_scale=1.0,
    solar_scale=1.0,
)

print(f"Converged: {result.success}")
print(f"Iterations: {result.iterations}")
print(f"Max voltage change: {result.max_mismatch_pu:.2e} pu")

Result Object¶

ACPowerFlowResult contains:

Field	Type	Description
`success`	`bool`	Whether solver converged within tolerance
`message`	`str`	Convergence status message
`ybus_result`	`YBusResult`	Y-bus matrix and node mapping
`voltage`	`np.ndarray`	Complex bus voltages in SI volts
`voltage_pu`	`np.ndarray`	Per-unit voltage magnitudes
`power_injection`	`np.ndarray`	Complex power injection at each bus (W + j·var)
`iterations`	`int`	Number of fixed-point iterations
`max_mismatch_pu`	`float`	Final maximum per-unit voltage change

AC PF vs AC OPF¶

Aspect	AC PF	AC OPF
Method	Fixed-point iteration (current injection)	Nonlinear least-squares (optimisation)
Units	SI (volts, amps, siemens)	Per-unit
Slack bus	Fixed at nominal voltage	Adjusted within bounds
Voltage bounds	None — reports actual voltages	Enforced via `vm_min_pu` / `vm_max_pu`
Regulator targets	Not modeled	Soft voltage targets via penalty
Use case	Classical power flow baseline	Voltage regulation studies

Both solvers share the same Y-bus construction and produce compatible result objects, making them easy to compare side-by-side.