AC OPF

The AC OPF solver finds complex nodal voltages that minimize power mismatch across the network. It operates in per-unit internally for numerical stability and uses the SI-unit fixed-point iteration AC PF solver as a warm start for large systems.

Formulation¶

Given the Y-bus equation $S(V) = V \cdot \overline{Y_{bus} \cdot V}$, the solver minimizes:

where:

• $S_i^{spec}$ is the specified net power injection (positive = generation)

• $S_i^{scale}$ normalizes each residual for conditioning

• $w_{reg}$ is a voltage regularization weight

• $w_{tgt}$ penalizes deviation from regulator voltage targets

Decision Variables¶

The solver optimizes voltage magnitude $V_m$ (per-unit) and angle $\theta$ (radians) at each non-slack node. Slack nodes are held at nominal voltage and zero angle.

Per-Unit System¶

Internally, the solver converts to per-unit:

• Voltage base: Nominal phase voltage at each bus

• Power base: $S_{base} = 1\text{ MW}$

• Y-bus conversion: $Y_{pu} = Y_{SI} \cdot (V_{base}^{(i)} \cdot V_{base}^{(j)}) / S_{base}$

This absorbs transformer turns ratios into the voltage bases and improves solver conditioning.

AC PF Warm Start¶

For systems with more than 6 non-slack nodes, the solver calls the Newton-Raphson AC PF solver (solve_ac_power_flow) as a warm start. The AC PF solver:

1. Uses sparse LU factorisation with a damped Newton-Raphson update

2. Has proven convergence on all network topologies (radial, meshed, cyclic)

3. Includes its own LinDistFlow warm-start for robust initialization

4. Runs without voltage bounds — the physical power-flow solution is returned directly

When the AC PF solution satisfies the AC OPF voltage bounds ($V_{min} \leq |V| \leq V_{max}$), it is accepted directly without further refinement. If bounds are violated, the interior-point Newton-Raphson refines from the AC PF starting point using log-barrier terms and fraction-to-boundary step limiting.

This two-phase approach gives AC OPF convergence times within 2-3× of a pure AC PF solve.

Center-Tapped Transformer Support¶

The AC solver correctly handles center-tapped (split-phase) service transformers. The S2 winding is antiphase (180°) relative to the primary due to the neutral center tap. The solver:

• Initializes S2-phase node angles at $\theta = \pi$ for correct flat start

• Uses the polarity-aware Y-bus from calculate_ybus (see Y-Bus Construction)

Without the 180° initialization, Newton-Raphson diverges because the flat start (all angles at 0°) is far from the physical operating point for antiphase nodes.

Multi-Phase Slack¶

The solver supports multi-phase slack bus operation. When using optimize_ac_power_flow_from_components, all non-neutral phases of the source bus are automatically designated as slack nodes.

Usage¶

High-Level (Recommended)¶

from gdm_flow import optimize_ac_power_flow_from_components

result = optimize_ac_power_flow_from_components(
    system,
    include_loads=True,
    include_solar=True,
    include_capacitor=True,
    include_regulator_targets=True,
    include_regulator_limits=True,
)

print(f"Success:    {result.success}")
print(f"Iterations: {result.iterations}")

Low-Level¶

from gdm_flow import optimize_ac_power_flow

result = optimize_ac_power_flow(
    system,
    p_spec_w={("bus_1", "A"): -1000.0},   # 1 kW load
    q_spec_var={("bus_1", "A"): -200.0},   # 200 var load
    slack_label=[("source", "A"), ("source", "B"), ("source", "C")],
    vm_min_pu=0.95,
    vm_max_pu=1.05,
)

Extracting Source Power¶

import numpy as np

v = result.voltage
ybus = result.ybus_result.ybus
s = v * np.conj(ybus @ v)

source_bus = "my_source_bus"
idx_map = result.ybus_result.index_to_label
src_idx = [i for i, lbl in enumerate(idx_map) if lbl[0] == source_bus]

source_p = sum(s[i].real for i in src_idx)
source_q = sum(s[i].imag for i in src_idx)
print(f"Source: P={source_p:.1f} W, Q={source_q:.1f} var")

Result Object¶

PowerFlowOptimizationResult contains:

Tuning Parameters¶

Assumptions¶

• Balanced slack voltage. Slack bus phases are held at nominal voltage magnitude and reference angle (0° for A/B/C phases). No unbalanced source voltage support.

• S2 phase offset. All bus-phase nodes named S2 are assumed to be center-tapped transformer secondaries and initialized at $\theta = \pi$. Systems with S2-named phases that are not center-tapped secondaries would get incorrect initialization.

• AC PF warm start is unconstrained. The AC PF warm start runs without voltage bounds (vm_min_pu / vm_max_pu). The converged physical power-flow solution may have voltages outside the specified bounds, particularly on long LV feeders downstream of center-tapped transformers.

• Connectivity filtering. Nodes unreachable from the slack bus via Y-bus adjacency are treated as fixed at flat start and excluded from the solve. This prevents infeasibility from isolated sub-networks but means those nodes will show zero power injection.

• Sign convention. Positive power spec = generation/injection; negative = load/demand.