Known Traps

A gallery of small instances that exhibit each pathology from Bounds, Branches, and Feasibility. Each entry states what it demonstrates, a minimal construction, the expected behaviour per formulation, and what to look for. A validator can use these as targeted tests: "give me the input that should expose my bug."

Trap 1 — Unbounded without bounds

Shows: §1 — a non-coercive objective over a non-compact set has no optimum.

Construction: any feeder, remove voltage magnitude bounds and generator limits, set a linear objective (e.g. minimize a single weighted injection, or maximize $\sum |V_i|$).

Expected behaviour:

Look for: after the bound-everything recipe (Diagnostics §1), voltage magnitudes pinned at the artificial bound. Adding $\underline v,\overline v$ alone makes it finite.

In BMOPFTools: start from an augmented feeder and strip its voltage/generator bounds — the inverse of the case augmentation step.

Trap 2 — Branch multiplicity (high vs low voltage)

Shows: §2 — "square" is not "unique."

Construction: the two-bus system of §2 — source $E = 1.0\angle 0$, lossless reactance $X$, constant-power load $P + jQ$ chosen so the radicand $\tfrac{E^4}{4} - X^2P^2 - XE^2Q$ is small and positive (loaded but below the nose).

Expected behaviour: two real AC solutions $V_+ \approx 1$ p.u. and $V_- \ll 1$ p.u.; an exact relaxation and a high-voltage-started nonconvex solve both return $V_+$; a flat-but-low or adversarially started Newton solve can land on $V_-$.

Look for: different solutions from different starts ⇒ multiplicity. The low-voltage cluster is the tell (Diagnostics §4).

In BMOPFTools: OPF test T1 (Validating the OPF) solves the high-voltage root V = (V_s + √(V_s² − 4RP))/2 of this same quadratic; a low-R/X reactive variant loaded toward the nose exposes the second root.

Trap 3 — Relaxation feasible past collapse

Shows: §3 and §4 — a relaxation enlarges the feasible set beyond the true AC boundary, so it can certify feasibility for a loading the network cannot serve.

Construction: the two-bus, one-generator characterization of Kocuk, Dey & Sun (2016). Pick a load just beyond the nose (radicand $< 0$).

Expected behaviour:

Look for: relaxation OPTIMAL while the AC solve is LOCALLY_INFEASIBLE; large per-branch cone gap (Diagnostics §3). This is the headline reason a feasible relaxation is only a one-sided certificate.

In BMOPFTools: check the AC side with solve_feasibility_opf — a repeatable non-zero total_slack_magnitude_A is evidence consistent with the analytically predicted collapse, not by itself a global infeasibility certificate.

Trap 4 — Loss-maximizing objective breaks exactness

Shows: §5 and the loss-maximization list — an objective that rewards current makes the relaxed $\ell_{ij}$ slack.

Construction: a small radial feeder with DER; objective = maximize PV injection (or maximize export), full bounds, SOC-BFM.

Expected behaviour: the SOC solution reports inflated currents; the cone gap $\ell_{ij} - |S_{ij}|^2/v_i$ is strictly positive on loaded branches; primal_feasibility_report against the AC model is non-empty. Switching the objective to maximize injection net of losses restores exactness.

Look for: positive cone gaps concentrated where upper voltage bounds bind (Diagnostics §3).

In BMOPFTools: adapt a radial DER feeder from the DER placement tutorial.

Trap 5 — Generator lower bounds break exactness

Shows: the standard escape-hatch conditions in §5 — exactness proofs often assume no generator lower bounds or load over-satisfaction; reintroducing a lower bound can make the relaxation inexact, exact, or feasible-while-AC-infeasible (Kocuk, Dey & Sun, 2016).

Construction: the same two-bus, one-generator system with a binding $\underline{P}^g > 0$; sweep the lower bound to move between the three outcomes.

Expected behaviour: a single small instance reproduces all three approximation outcomes as the lower bound varies — a compact regression test for any relaxation implementation.

Look for: the outcome flipping as a single parameter sweeps; tabulate cone gap vs $\underline{P}^g$.

In BMOPFTools: as Trap 3, sweeping the binding $\underline{P}^g > 0$.

Traps 1–5 are formulation / physics traps. Traps 6–9 are numerical traps — failure modes of a correct model on a local solver, the subject of Trusting the solver.

Trap 6 — Degenerate duals from an over-determined bus

Shows: Trusting the solver §4 — a constraint-qualification (LICQ) failure leaves the multipliers non-unique without corrupting the primal point.

Construction: any feeder; add a generator pinned to a single operating point ($\underline P^g = \overline P^g$) at a bus whose voltage is fixed by a hard equality (a source-like / regulated bus). Equivalently, give two same-bus generators identical cost coefficients, or a single generator a zero cost vector.

Expected behaviour: LOCALLY_SOLVED with a sane primal dispatch but non-unique or blown-up duals; convergence is slow; a zero-cost generator (non-coercive objective) can even surface as diverging iterates (NORM_LIMIT). The dispatch is unaffected — re-solving from a second start returns the same primal point with different multipliers.

Look for: large / start-dependent shadow prices; many iterations for a small problem; a primal point that is stable across starts while duals are not.

In BMOPFTools: this is exactly what the benchmark-readiness flags W.BENCH.GEN_NO_DOF, W.BENCH.GEN_ZERO_COST, and W.BENCH.GEN_DEGENERATE_COST (Methodology) catch before you solve; reproduce by augmenting a feeder with a zero-cost or pinned generator and reading the flag, then the runtime echo.

Trap 7 — Zero-voltage bifurcation / spurious all-zero solution

Shows: Trusting the solver §5–§6 — at zero voltage the bilinear power map's Jacobian rank drops and an all-zero voltage profile becomes feasible ($S = 0$ for any current).

Construction: an ungrounded or islanded sub-network; remove the strictly positive voltage floor (allow $|V| \ge 0$). The constant-power equation $p = v_r c_r + v_i c_i$ then admits $v_r = v_i = 0$ with the current free.

Expected behaviour: a sub-network collapses toward $\sim 0$ V with indeterminate currents; Ipopt churns in restoration or returns a physically meaningless point.

Look for: a cluster of terminals at near-zero voltage with large or arbitrary currents; restoration thrash that clears the moment a positive floor is restored.

In BMOPFTools: strip the positive voltage floor to reproduce; contrast with the feasibility OPF, which keeps the case's own (strictly positive) voltage bounds and relaxes only nodal current balance — the low-voltage branch is excluded by design, and violations show up as slack currents instead (see the infeasibility diagnosis tutorial).

Trap 8 — Non-smooth ZIP exponent near zero voltage

Shows: Trusting the solver §5 — a fractional-exponent load model is genuinely non-smooth as voltage → 0, with an unbounded Hessian.

Construction: a load with an exponential model $P \propto (W/V_{\text{nom}}^2)^{\gamma/2}$, $\gamma < 2$ (or a ZIP constant-current term $s = \sqrt{W}$), and $W = v_r^2 + v_i^2$ allowed to approach zero.

Expected behaviour: exploding Lagrangian-Hessian entries, vanishing step sizes, and Ipopt churning in its Restoration phase as the argument nears zero.

Look for: tiny steps and growing curvature reported in the Ipopt log on a feeder that should be easy; the symptom disappears when $W$ is floored away from zero.

In BMOPFTools: the _W_FLOOR_FRAC floor on $W$ (and on $s$) in load.jl is the built-in mitigation; removing it reproduces the trap.

Trap 9 — Redundant constraint breaks LICQ everywhere

Shows: Trusting the solver §4 — an equation already implied by the others adds no information but makes the active-constraint gradients linearly dependent at every feasible point, not just on a measure-zero set.

Construction: any feeder solving cleanly; add an explicit system-wide power-balance constraint $\sum_g S^g = \sum_d S^d + \text{losses}$ on top of full per-terminal KCL. Because summing the nodal balances already yields this identity, the added row is a linear combination of the KCL rows.

Expected behaviour: the problem is mathematically unchanged but the KKT system is singular along the redundant direction; Ipopt stalls with non-unique multipliers or terminates with diverging iterates (NORM_LIMIT), even though the primal dispatch (if it converges) is fine.

Look for: a model that was well posed becomes ill-conditioned the moment a "sanity-check" equality is added; removing that one constraint restores a clean solve. Smaller instances: a voltage-magnitude pin on an already-fixed reference bus, or a second ground on a neutral the source already fixes to zero.

In BMOPFTools: the slack source already closes the system, so a global power balance is never required — adding one reproduces the trap; deleting it is the fix. Classical transmission OPF drops the slack-bus balance for the same reason (cycle-flow formulations).

Suggested test matrix

For a formulation under validation, the minimal coverage is one instance per trap:

See also: Bounds, Branches, and Feasibility · Decision matrix · Objectives that imply loss maximization · Diagnostics & validation · Trusting the solver · References