Formulations

Tellegen solves DC power flow, DC OPF, AC power flow, and the Jabr SOCWR relaxation through the PowerIO module boundary. The shared response has optional fields for the quantities each formulation produces. Economic fields are present only when the declared objective gives them that interpretation. Supported implicit derivatives use the contract described in the sensitivity contract.

DC power flow and DC OPF (B–θ)

The linearized power flow couples bus angles $\theta$ to injections through the susceptance-weighted graph Laplacian

$$ B = A^\top\operatorname{diag}(b)A, \qquad B\theta = p, $$

where $A$ is the branch by bus incidence and $b$ contains positive solver weights derived from the public branch susceptances. The OPF minimizes generation cost subject to the network balance and the thermal and generation limits; it is a convex quadratic program solved with Clarabel. solve_module_json is the portable entry. Rust callers that already own a DcOpfInstance can use solve_instance.

MATPOWER model 2 quadratic generator costs are read directly. Convex model 1 piecewise linear costs use one epigraph variable and one inequality per segment, so dispatch, objective values, prices, and supported implicit derivatives refer to the declared curve. Malformed and nonconvex model 1 rows are rejected before the program is assembled. At a breakpoint or an active set change, the marginal value or its derivative need not be unique; the sensitivity API reports the local KKT linearization and numerical checks identify stencils that cross a different active set.

Branch angle-difference bounds are enforced in radians after normalization. MATPOWER’s unconstrained -360/360 spelling and an unset 0/0 pair become exactly -60/+60 degrees. When a branch has no thermal rating, Tellegen synthesizes its fallback rating from that same 60 degree window and the terminal voltage bands. Explicit tighter source bounds are preserved.

AC power flow (polar)

The nodal power balance in polar coordinates,

$$ S_i = V_i \sum_j \overline{Y_{ij}} \overline{V_j}, $$

is solved by Newton–Raphson on the reduced system $\partial(P, Q)/\partial(\theta, V_m)$. Buses are typed slack / PV / PQ (PV and slack buses hold the generator voltage setpoint; PQ buses solve for both angle and magnitude), and the solve takes damped steps with a backtracking line search from the setpoint start plus a few perturbations, keeping the lowest-residual result. Select acpf in solve_module_json.

Conic SOCWR (Jabr)

The Jabr second-order cone relaxation lifts the voltage product to W-space variables $w_i = |V_i|^2$, $w^r_{ij} = \Re(V_i \overline{V_j})$, $w^i_{ij} = \Im(V_i \overline{V_j})$, with the rotated cone coupling

$$ (w^r_{ij})^2 + (w^i_{ij})^2 \le w_i w_j. $$

The relaxation is a convex lower bound on AC OPF, solved with Clarabel’s second-order cone support. It uses the same exact quadratic and convex piecewise linear generator cost representation as DC OPF. Select socwr in solve_module_json.

These formulations compile to native Rust and WebAssembly.