Objectives That Imply Loss Maximization
Several objectives that read as maximizing something desirable are, mechanically, maximizing line losses or branch current. On a convex relaxation that breaks exactness; on the nonconvex model it drags the solution toward the low-voltage branch. This page is the catalogue, with safe reformulations.
The one test
An objective non-decreasing in every nodal generation — equivalently, in line losses is an important objective-side hypothesis in relaxation-exactness and high-voltage-selection results (Low, 2014; Gan et al., 2015; Yuan & Paolone, 2020). It is not sufficient on its own — the theorems additionally assume radiality, non-binding upper voltage bounds, and no binding generator lower bounds (or permitted load over-satisfaction) — but it is the condition the objective itself controls, so it is the one to screen for. For any objective not on the safe list:
At fixed loads and topology, does the optimizer prefer larger branch current anywhere — or does its optimum sit where an upper voltage bound binds? If either is yes, audit relaxation exactness and voltage-sheet selection explicitly.
The catalogue
| Objective as written | Why it rewards loss / current | Safer reformulation |
|---|---|---|
| Max PV/DER injection, max renewable dispatch, naive hosting capacity | Surplus injection becomes reverse flow → $I^2 r$, and pushes voltages up until the upper bound binds — the regime where SOC/SDP exactness is known to fail | Maximize injection net of losses; or fix injection and solve feasibility; or penalize curtailment instead of rewarding injection — then verify exactness |
| Max total load served / loadability / demand delivered | Drives straight to the nose; the loadability limit is the collapse point | Parametrize load, minimize cost/loss at each level; for true loadability use the nonconvex model + continuation |
| Max power/energy transfer, ATC, interface flow | Directly maximizes branch current → $I^2 r$ | Bound transfer as a constraint; study maximum transfer only in the nonconvex model |
| Max export to grid / feed-in / "self-sufficiency" via export | Reverse-flow current maximization | Recast as minimize net cost / minimize net import |
| Max storage throughput / arbitrage volume / EV energy delivered | Rewards cycling current; the linear charge/discharge constraints add their own documented inexactness | Minimize cost including losses; bound throughput as a constraint |
| Max reactive support / VAR injection | Reactive current also dissipates $I^2 r$ | Minimize losses subject to voltage constraints; reward constraint satisfaction, not raw $Q$ |
| Min cost with a negative cost coefficient (subsidy, feed-in tariff as negative cost) | The form is the safe one, but a negative coefficient inverts monotonicity, so min-cost silently becomes max-generation → loss-max. The trap is in the data, not the objective | Audit the sign of every cost term; if negative prices are real, switch to the nonconvex model and do not trust the relaxation |
| Max revenue / profit (price × output) | Rewards more generation and flow | As above — minimize net cost instead |
Minimize import is loss-aligned and safe. Maximize export is not — even though both sound like "less reliance on the grid." Direction of the optimization, not the English description, is what determines the monotonicity.
The negative-coefficient row is exercised directly in the OPF suite: test T4 (Validating the OPF) gives a generator a negative cost coefficient and confirms the optimum drives each phase to p_max with objective = −3·P_max/1000 ($/h, with P_max in W) — i.e. min-cost has silently become max-generation. If your own formulation reproduces that number, it is reproducing the trap, not a bug.
Two that look risky but are mostly fine
Maximize $\sum_i |V_i|$. Counterintuitively this is usually loss-reducing and often favours a high-voltage branch under the cited radial assumptions. It can drive into binding upper voltage bounds, which is a separate exactness-failure mode — the non-binding-upper-voltage-bound hypothesis of the radial exactness theorem (Gan et al., 2015). Outside those assumptions, verify the selected sheet and relaxation tightness rather than assuming either.
Minimize generation / min slack power / min cost (non-negative coeffs). The canonical safe family. Because slack power $=$ losses $+$ net load, minimizing it is loss minimization when the other injections are fixed — which is exactly why a feasibility problem is so often turned into min-slack to make it well posed and branch-favourable (§5). With other dispatchable generators free, min-slack minimizes losses plus their output, which can drive local injection against upper voltage bounds; see the min-import caveat in the decision matrix.
When loss-maximizing is the actual research question
Maximum loadability, voltage-stability margin, and worst-case transfer are legitimate questions — they are deliberately boundary-seeking. The point is not to avoid them but to model them correctly:
Solve these in the nonconvex AC model (optionally with a continuation / margin formulation) — for BMOPFTools that is solve_opf — not in a relaxation. No relaxation is a valid feasibility certificate in this regime, including any you reach via to_pmd: a relaxation will happily report an "optimal" loadability past the true collapse point. See Known traps for runnable instances that exhibit this.
See also: Bounds, Branches, and Feasibility · Decision matrix · Diagnostics & validation · Trusting the solver · References