Start the rectangular AC model from a power flow solved on the projected cone point (IndACConeStart = 2)

[erikfilias]

Problem

IndACConeStart = 1 (#212) starts the rectangular model from the rank-one projection of the W-space cone. Where the cone is tight the projection is close to an AC operating point and the start is good. Where the cone is loose it is not: the angle differences of its branches do not close around loops, and the projected voltages with the cone's dispatch violate the nodal balances by hundreds of MW. On 11 June of the Nordic calendar sample, the day whose cone has the largest loop mismatch (1.4 pu of voltage), Ipopt started from the projection with a primal infeasibility of 144 and spent its first 300 iterations repairing it, with the objective climbing to 1,357 MEUR on the load-shedding penalties before it fell. The flat start was feasible by iteration 100. The cone start was slower than the flat one on that day and on 21 May, and faster only on 19 February (entries 153 to 155 of the study log).

Solution

IndACConeStart = 2: the cone, its projection, and then an AC power flow per load level from the projected point before the model is handed to the solver. _PowerFlowFromProjection in openTEPES_ModelFormulationElectricity.py:

1. Network. The admittance matrix of the AC branches in service in the period, with the tap on the sending side and the charging of the lines that are always in service; the charging of a switched or candidate line is the constant the balance uses. The shunts enter with the susceptance and state of the cone point.
2. Injections. The active and reactive power the units, the loads, the HVDC links and the switched charging put into the AC network at the cone point, read from the cone's branch flows through the balance, so that every term the balance carries is in them.
3. Buses. A bus with a reactive-capable unit in service, or a condenser, holds the projected voltage magnitude with its reactive power free. The reference bus of each connected part of the network holds magnitude and angle. Every other bus keeps the cone's reactive injection.
4. Slack. One distributed slack per connected part takes up the active power the AC losses change. A first power flow finds its sign; the slack is then shared among the buses whose units have room that way, in proportion to that room, and the power flow is solved again from the point found. A unit at its minimum cannot back off and one at its maximum cannot add.
5. Newton. _NewtonPV: Newton-Raphson in polar form with PV buses and the slack variables, a step halved until the mismatch falls, dense, numpy only. A load level that does not converge from the projection is tried from a flat profile at the free buses, and keeps the projection if that fails too.
6. Start. The voltages, |V|², the four branch flow variables (the rectangular formulas of the flow constraints), the shunt injections, the outputs of the units that carry the slack and their second block, and the reactive outputs at the regulated buses are set from the solution, each within its bounds. What a bus's units cannot take within their bounds goes to unserved energy or reactive power not served. A flow beyond its rating or a voltage outside its band is put at the bound, which is where the solver would put it, and counted.
7. Report. The worst balance residual at the start, in MW and Mvar, with the bus and load level of the worst active one, is printed and kept in mTEPES.pACStartResidual[(p, sc, st)].

The option is validated in InputData (0, 1 or 2; 1 and 2 need IndACPowerFlow = 3), declared as a non-negative integer in DataConfiguration, and named by ReportConfiguration.

Observed behaviour

9n_AC, first two hours, IndACPowerFlow = 3, Ipopt: the three starts reach the same total cost, 0.0367991 MEUR, in 96 (flat), 107 (projection) and 104 (power flow) iterations; the power flow converges in its one load level with a slack of 13 MW and leaves the balances at a residual of 0.000 MW.

695-bus Nordic case, 11 June, 24 coupled hours with storage, Ipopt with MUMPS (data/cases/2026-06-11_calendar-ac-voltband-rect, -cone and -pf in the study repository). The cone took Gurobi 16 seconds and gave the bound 27.940062 MEUR, 5.5 per cent below the exact cost; the power flow took 8 seconds, converged in all 24 load levels, and the balances hold at its point. That point is outside the limits the cone's dispatch ignored: 276 branch-hours carry more than their rating, one 400 kV line by 1.25 GW, 24 bus-hours sit outside the voltage band, and the regulated buses need reactive power beyond capability that goes to reactive power not served. Put at their bounds, those flows leave the balance violated by 1.25 GW at that line. Ipopt still starts far better from it: the primal infeasibility falls below 1e-3 at iteration 27, against 317 from the projection and 59 from the flat start, and the run ends at the optimal exit in fewer iterations than either.

Start	Iterations	Ipopt seconds	Exit	Operation cost of generation
Flat	623	838	acceptable level	29.577277 MEUR
W-space cone, projection	838	965	acceptable level	29.577332 MEUR
W-space cone, power flow	559	767	optimal	29.577332 MEUR

The three solutions are the same: losses, hydro, branch-hours at the rating, every zonal price and the water values agree, and both power flow checks of the study reproduce the point exactly.

21 May, the other day where the projection start lost (gap 4.3 per cent): the cone needed the presolve-off retry, the power flow converged in all 24 load levels in 5 seconds with 194 branch-hours put at their rating, and Ipopt was feasible at iteration 20 against 131 from the projection and 59 from the flat start.

Start	Iterations	Ipopt seconds	Exit	Operation cost of generation
Flat	680	1161	acceptable level	27.051087 MEUR
W-space cone, projection	860	1360	acceptable level	27.051135 MEUR
W-space cone, power flow	620	1066	optimal	27.051134 MEUR

Again the same solution, reproduced exactly by both checks.

The two days where the projection was neutral or won, 15 January (gap 0.13 per cent) and 19 February (gap 7.2 per cent), run the same way. Iterations and Ipopt seconds, with the iteration at which the primal infeasibility first falls below 1e-3:

Day	Flat	Projection	Power flow
15 January	467 it, 732 s, feasible at 78	423 it, 878 s, feasible at 119	347 it, 584 s, feasible at 19, optimal exit
19 February	612 it, 811 s, feasible at 83	447 it, 618 s, feasible at 26	480 it, 669 s, feasible at 24
21 May	680 it, 1161 s, feasible at 59	860 it, 1360 s, feasible at 131	620 it, 1066 s, feasible at 20, optimal exit
11 June	623 it, 838 s, feasible at 59	838 it, 965 s, feasible at 317	559 it, 767 s, feasible at 27, optimal exit

On all four days the power flow start takes 9 to 26 per cent fewer iterations than the flat start and reaches the same solution, which both checks reproduce exactly. It beats the projection on three days; on 19 February, where the projection was already feasible at iteration 26, it is 7 per cent slower than the projection and still faster than flat. The early feasibility is the mechanism, and the flows it leaves beyond the ratings, 4 to 276 branch-hours, do not undo it. A fifth day, the coupled development day of 17 February, gives 348 iterations and 568 s to the optimal exit against 533 and 825 to the acceptable level from the flat start.

Tests

tests/test_ac_input.py: test_cone_start_reaches_the_flat_start_optimum now runs the three starts and asserts, for the power flow start, that the balance residual at the start is below 0.01 MW; test_cone_start_power_flow_converges_with_a_distributed_slack checks _NewtonPV on a two-bus network, the load bus at its injection and the slack equal to the load plus the loss.

Files

openTEPES_ModelFormulationElectricity.py (_PowerFlowFromProjection, _NewtonPV, _dSbus_dV, the call from ACConeWarmStart), openTEPES_InputData.py (the option's values and validation), openTEPES_DataConfiguration.py (the parameter's domain), openTEPES.py (the configuration report), doc/md/InputData.md, CHANGELOG.md, tests/test_ac_input.py.