Objective and feasibility

The component pages define the network's variables and constraints — the feasible set. This page defines what is optimised over it: the objective, and the feasibility relaxation used to diagnose networks that have no feasible point. Unlike the component pages this is a formulation page, not a data object. Symbols are defined in Notation.

Objective

The default snapshot objective minimises total active-power dispatch cost rate, summed over every dispatchable element (generators, the voltage source, IBRs) and every phase:

\[\min \; \sum_{e} \sum_{k} \textcolor{red}{c_{e,k}}\; P_{e,k}/1000,\]

where $\textcolor{red}{c_{e,k}}$ (currency/kWh, from each element's per-phase cost array) is the energy price of phase $k$, and $P_{e,k}$ is that phase's injected active power in watts — the same bilinear expression the element defines ($P_{e,k}=\Delta v^r\,c^r + \Delta v^i\,c^i$). The factor $1/1000$ converts W to kW, so this snapshot objective is a cost rate in currency/h. For a multi-period monetary objective, multiply every snapshot rate by its duration in hours before summing.

Sign convention

\[P_{e,k}\]

is the power injected into the network by the element, uniformly across generators, IBRs, and the voltage source (each stamps $+I$ into KCL). Therefore:

• a positive cost minimises that element's injection;
• a negative cost maximises it.

The voltage source is not special: for the slack, positive injection means importing from the grid, so a positive source cost is the grid import price (and export, a negative injection, is credited at the same price). Maximising system exports is a positive slack cost with free DERs.

The cost is linear in the dispatch and is added exactly — there is no polynomial/quadratic term. A cost must be a per-phase vector; a scalar is rejected.

Feasibility relaxation

A constant-power OPF can be infeasible: the load/generation specification may be irreconcilable with the network physics. To diagnose such cases, a feasibility-relaxed variant adds an elastic slack current and minimises it, instead of cost.

Elastic slack

At every ungrounded, non-source terminal a free slack current $\textcolor{blue}{s_{i,p}}=s^r_{i,p}+\textcolor{brown}{j}\,s^i_{i,p}$ is added directly into that terminal's KCL:

\[\kappa^{\Re}_{i,p} + s^r_{i,p} = 0, \qquad \kappa^{\Im}_{i,p} + s^i_{i,p} = 0.\]

Where a slack pair is present, it can absorb any residual in that terminal's KCL. This does not guarantee that the complete relaxed NLP is feasible or that a local solver will converge: fixed source voltages, hard bounds, device equalities, or terminals without slacks can still conflict.

Objective

The cost objective is replaced by the squared magnitude ($\ell_2^2$) of all slack injections:

\[\min \; \sum_{i,p} \big( (s^r_{i,p})^2 + (s^i_{i,p})^2 \big).\]

Interpretation

The relaxation retains the standard OPF's hard constraints — voltage bounds, bus/line angle limits, and every device current limit — while enlarging the feasible set only through the nodal-current slacks. Consequently, for a successfully converged solve:

• An independently residual-checked zero-slack point demonstrates numerical feasibility under the represented loading and constraints.
• Non-zero slack at a locally optimal relaxed point identifies where that solve paid to violate KCL and by how much current. It is diagnostic evidence, not a proof that the original nonconvex problem has no zero-slack solution.
• Voltages still respect their hard bounds. When those hard constraints remain mutually consistent, an unreachable operating bound commonly surfaces as residual current. Contradictory hard constraints can instead leave even the relaxed problem infeasible (for example, a fixed source voltage contradicting a bound on the same terminal).

Implementation in BMOPFTools

• Objective — objective.jl:_add_objective! accumulates the per-phase linear terms for generators, the voltage source, and IBRs into one QuadExpr and sets @objective(model, Min, …).
• Feasibility — solve_feasibility_opf (feasibility_opf.jl) reuses the full OPF build (_add_voltage_and_bus_bounds!, _add_device_constraints!), then adds one (cs_r, cs_i) pair per KCL node into the accumulators and sets the $\ell_2^2$ objective. Post-solve, extract_feasibility! reports per-terminal slack_injections and the scalar L2 norm total_slack_magnitude_A, consumed by diagnose_infeasibility.
• Degeneracy tie-break — for Yd/Dy transformers the delta circulation current is unobservable; a tiny ($-10^{-6}$) linear term on the delta-side current selects one numerical representative. Consequently the raw solver objective is an implementation metric, not exactly the physical squared-slack norm; use the reported SI slack fields for interpretation.
• Warm start — when a case lacks useful voltage bounds, level-aware start values seed LV buses near their nominal (≈250 V) rather than the source voltage.

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