Area Interchange Control

What it does

Net-interchange control: each participating PSY.Area is driven so that its net active-power exchange with the rest of the system meets a scheduled target $PDES_a$ (derived from the system's PSY.AreaInterchange records). Control is embedded in the AC solver — the schedules are met at convergence, not by an outer sweep — so it composes with reactive-power limits, distributed slack, and multi-period solves. Both AC boundary branches and two-terminal HVDC (LCC and VSC) boundary links count toward each area's net interchange.

Terminology

The names below are PSS/E's, kept so that engineers comparing against a PSS/E case can map between the two directly.

When to use it

• Planning-case parity: multi-area interconnection cases (PSS/E-style area interchange control, control mode 1) where solved tie flows must honor the scheduled interchange, not just system-wide balance.
• Seams and transfer studies: hold every other area to schedule while one schedule is varied, so the solved state isolates the transfer of interest.
• Schedule feasibility screening: an unenforceable schedule is relaxed rather than failing the solve, and the results table reports the achieved-vs-scheduled gap as the infeasibility certificate.
• Multi-period studies: each time step re-solves its own adjustments against per-step injections; relax decisions never leak across steps.

Enabling it

Set the flag on a polar power flow; NR, TR, LM, and both Fast Decoupled variants are supported:

pf = ACPowerFlow{NewtonRaphsonACPowerFlow}(; area_interchange_control = true)
solve_power_flow(pf, sys)

Area interchange control is polar-only. GradientDescentACPowerFlow and RobustHomotopyPowerFlow are rejected at construction (gradient descent has no natural home for the interchange border; robust homotopy would require exact second-order curvature for every tie term, which is not maintained), as is any non-polar formulation.

Formulation

Each enrolled area contributes one state and one residual row, appended as a tail after the LCC/VSC tails:

• State$\Delta P_a$: an active-power adjustment injected at the area's slack (ISW) bus. It enters that bus's P-balance row in the same way the distributed-slack term does.
• Residual$r_a = NI_a - PDES_a$, where $NI_a$ is the sum of metered active power over the area's boundary ties, signed positive out of the area.

Solving drives every $r_a$ to the same tolerance as the bus mismatch rows, so the schedules hold exactly at convergence while $\Delta P_a$ reports how much the area's slack generation moved to achieve them.

If the ISW bus hits a reactive limit and is switched PV → PQ mid-solve, area control keeps working and nothing needs to be re-enrolled. $\Delta P_a$ couples into the P-balance row at index 2 * slack_bus_ix - 1, which does not depend on bus type, so the Jacobian entry stays put across the flip. $\Delta P_a$ is also applied directly to the residual vector rather than folded into the bus's accumulated net injection, precisely so it is counted exactly once after the bus becomes PQ. What is checked at enrollment (not mid-solve) is that the ISW bus is voltage-regulating to begin with: an area whose slack bus normalizes to PQ is de-enrolled.

Ties

AC ties are in-service AC branches whose endpoints lie in different areas (a ThreeWindingTransformer decomposes into its star-node windings, so a winding crossing the boundary is a correctly metered tie). Which end of an AC tie is metered follows the branch's ext["metered_end"] ("from"/"to", defaulting to from-metered when the key is absent), the same rule DC ties use; an unrecognized value warns and falls back to from-metered rather than being silently coerced. The tie-flow kernel reads each corridor's admittances directly from the aggregate Y-bus. Because a nodal Y-bus diagonal sums every device at a bus, a per-tie diag_pollution correction, cached at enrollment, recovers the corridor's own self-admittance; this stays exact under a controlled tap on the corridor itself.

DC ties are two-terminal HVDC links (LCC or VSC) whose converter buses lie in different areas. The metered-terminal converter active power enters $NI_a$ with the same out-of-area sign convention as AC ties, and metering follows the same ext["metered_end"] rule. The Jacobian carries the exact cross-derivatives of $r_a$ with respect to the DC state: for an LCC, ∂P/∂(terminal voltage, tap, thyristor angle) at the metered terminal; a VSC's metered power is linear in its converter-power state. Series FACTS and generalized network elements are not supported models in PowerSystems/PowerFlows, so ties through them are out of scope by definition, not a gap in the tie enumeration.

How each solver carries the interchange border

• Newton–Raphson / Trust Region: the $\Delta P_a$ columns and $r_a$ rows are embedded directly in the augmented Jacobian.
• Levenberg–Marquardt: the same augmented rows feed the least-squares normal equations; no separate machinery.
• Fast Decoupled, fixed-Jacobian variant: the frozen Jacobian is the augmented one, so the border is factored once with everything else.
• Fast Decoupled, classic B′/B″ variant: a bordered-Schur substep corrects $\theta$ and $\Delta P_a$ each cycle against the fixed B′ factor. The step uses only the $\partial r_a/\partial\theta$ coupling (DC-tie and voltage cross-terms are dropped, consistent with the decoupling approximation), but the target is evaluated with the exact residual — including DC ties — so schedules are still met exactly at convergence.

Enrollment guards

An area is enrolled only if it can be embedded-controlled. At PowerFlowData construction each candidate is checked, and any failure emits a @warn and de-enrolls that area (its schedule then falls back to the island's REF/distributed slack). An area must:

• have exactly one in-service, voltage-regulating slack bus that survives network reduction;
• not contain a network reference (REF/swing) bus;
• lie within a single electrical island;
• hold less than AREA_SLACK_ABSORPTION_LIMIT (0.9) of system slack-participation weight;
• have at least one in-service tie to another area.

A tie-endpoint tap or switched shunt that is not a corridor member is a diag_pollution staleness hazard; it is flagged with a @warn (net-interchange tracking then assumes it holds its enrollment-time value) rather than de-enrolled.

Infeasibility: greedy relax

A schedule can be unenforceable given the network and tie capacity. On a non-converged time step with areas still enrolled, the driver de-enrolls the area with the largest residual gap $|r_a|$, emits an @error naming it (with its achieved net interchange, AC and DC ties included), and re-solves with the rest — repeating until the solve converges or no controlled area remains. Relaxation is never silent. Relax decisions are per time step: the full enrollment (including the DC-tie set and all solver caches) is restored before the next time step's own attempt, and likewise re-grown on a later re-solve of the same data.

Results

solve_and_store_power_flow! / write_results add an "area_interchange_results" DataFrame, one row per area enrolled at construction:

For a relaxed area, ni_solved - pdes is its infeasibility certificate.

beyond_limits compares the machines' summed active_power_limits against the slack bus's whole solved active output — its scheduled injection plus its distributed-slack share plus$\Delta P_a$ — so the two effects compound rather than being checked in isolation. An area whose $\Delta P_a$ fits on its own can still be flagged once its slack-participation share is added. The flag never clamps anything; it is a diagnostic.