CNECs, Thresholds and Margins

Everything the optimizer maximizes is defined on this page, and none of it is difficult. It is nevertheless where a RAO is most likely to be quietly wrong, because a threshold arrives in one of four units, may be one-sided, may be stated per terminal, and is tightened by a reliability margin before anyone sees it. Get any of that wrong and every subsequent number stays perfectly self-consistent while describing a limit nobody wrote.

From a threshold to a pair of bounds

A CNEC carries a list of thresholds. Each states a min, a max, or both, in one of

The ampere conversion is the one worth writing out, because dropping the \(\sqrt{3}\) is a 73% error that still looks entirely plausible:

\[ P_{pu} ;=; \frac{\sqrt{3}; U_{rated}; I}{S_{base}}, \qquad\qquad I ;=; \frac{P_{MW}\cdot 10^{6}}{\sqrt{3}; U} \]

with \(U\) line-to-line, in volts. ratings::current_to_power_pu exists as a named function purely so that \(\sqrt{3}\) has one home.

percentImax is a fraction and not a percentage, despite the name — 1.0 means 100%, as the reference's own ThresholdAdder javadoc states. Reading it as a percentage makes every such threshold a hundred times too tight.

Thresholds accumulate, they do not replace

A CNEC with several thresholds is bounded by the tightest of each side, taken independently:

\[ f^{+} ;=; \min_t\ \bigl(\text{max}_t\bigr) - \text{FRM} - \text{charge}, \qquad f^{-} ;=; \max_t\ \bigl(\text{min}_t\bigr) + \text{FRM} + \text{charge} \]

Each side remembers the voltage its own binding threshold named, so an ampere margin can be converted back at the right voltage even when a CNEC's thresholds disagree about which that is.

FRM is the CRAC's reliability margin (frm), a fixed MW cushion held back from every limit to cover the difference between the model and the network. charge is the headroom already consumed by reactive flow, and is zero under DC — see the AC case below.

The bounds are not assumed symmetric

A CRAC routinely writes a one-sided threshold. min: -1500, max: null means no more than 1500 A in the reverse direction, and nothing at all in the forward one. Collapsing that to \(\vert F \vert \le 1500\) invents a constraint nobody wrote, in the direction the flow is most likely to go.

So Bounds starts at \((-\infty, +\infty)\) and only ever narrows, and a side that no threshold mentions stays infinite and contributes no row to the LP. A CNEC whose thresholds are all inexpressible — a percentImax threshold on a CNEC whose \(i_{max}\) the CRAC never stated, say — constrains nothing at all, which is better than constraining it with a number nobody wrote.

The margin

\[ m(c) ;=; \min\bigl(, f^{+} - F,;; F - f^{-} ,\bigr) \]

against the signed flow \(F\), not its magnitude. For a symmetric pair of bounds this reduces to the familiar

\[ m(c) ;=; \text{limit} - \vert F \vert \]

and for a one-sided pair that identity does not hold. The margin is the quantity to trust; CnecResult keeps limit_mw alongside it for display only, and the two disagree by construction on a one-sided CNEC.

The sign carries the verdict: \(m < 0\) is an overload, exactly, and by exactly the amount reported.

Two ends of one flow

CnecResult reports flow_mw at side one and side_two — active power and current — at the other, and the pair follows one rule: both are measured in the same direction along the branch. Power entering at side one, power leaving at side two. So the two differ by the branch's own losses, which is the comparison anyone asking for both ends wants, and under DC they are equal to the last bit because a linear model has no losses to spend.

The alternative reading — report the power entering at each end, which is what powsybl's Terminal::getP gives and what the AC evaluator computes internally — negates side two, and the two figures then differ by roughly twice the flow rather than by a fraction of a megawatt. Both readings agree on magnitude, so nothing but a per-side expectation from outside can tell them apart.

One consequence worth stating because it looks like an inconsistency: current_a and side_two.1 are magnitudes, since a current is one and a threshold is compared against one. A flow reported in amperes is signed, by the active power at that terminal — the reference divides the signed megawatts by (\sqrt{3},U) — so -1444.0 A names a direction rather than a smaller number. The evaluator keeps the magnitude and the caller applies the sign, which is why the two live apart.

One margin, in two units

CnecResult reports margin_mw and margin_a, and the second is not a unit conversion of the first at some convenient voltage — it is the same margin expressed in the units the binding threshold was written in, converted at the voltage that threshold named. Two CNECs at different voltages convert differently, which is precisely why the two units can rank the same pair of candidate networks in opposite orders:

A 400 kV CNEC and a 225 kV one with equal MW margins do not have equal ampere margins.

That is why the objective's unit is a setting at all — see the LP's objective unit.

Concretely, the ampere margin is

\[ m_A(c) ;=; f^{+}_A ;-; I \]

— the limit expressed in amperes less the current — and not \(m_{MW}\) divided by \(\sqrt{3},U\). For a threshold already written in amperes the two are the same number, because the charge above has taken reactive flow and the voltage deviation off the megawatt limit and converting back undoes exactly that. For a threshold written in megawatts they are not: \(\text{limit} - \vert P \vert) carries neither effect and \(I\) carries both.

The voltage that turns a megawatt limit into amperes is the network's nominal, not the CRAC's nominalV. A CRAC states nominalV to say what an ampere threshold was written against; a megawatt threshold was written against nothing, and the reference reads the CNEC's nominal voltage off the network. On a UCTE 380 kV line whose CRAC says 400 that is 150 A on a 2000 MW limit — enough to change which remedial action the search takes when the objective is in amperes.

A worked margin

Take the network from the chapter introduction and put a single CNEC on the BE2–DE1 tie, with a two-sided megawatt threshold of ±700 MW, no reliability margin, and no contingency:

"thresholds" : [ { "unit" : "megawatt", "max" : 700.0, "min" : -700.0, "side" : 1 } ]

The flow, by hand

The phase shifter's reactance is clamped to nearly zero, so BE1 and BE2 are one electrical node separated only by the shift \(\alpha\). Set that aside for a moment — take \(\alpha = 0\) — and the network reduces to three nodes B, D, F with

\[ b_{BD} = 144.4, \qquad b_{BF} = 2 \times 144.4 = 288.8, \qquad b_{DF} = 144.4 \]

and injections \(P_B = +5\), \(P_D = -10\), \(P_F = +5\) p.u. The radial DE1–DE2 branch carries nothing and can be dropped. Taking \(\theta_B = 0\), the two remaining DC equations are

\[ \begin{align*} 288.8,\theta_D - 144.4,\theta_F &= -10 \\ -144.4,\theta_D + 433.2,\theta_F &= +5 \end{align*} \]

Since \(144.4/433.2 = 1/3\) exactly, eliminating \(\theta_F\) gives \(240.667,\theta_D = -8.3333\), so

\[ \theta_D = -0.034626\ \text{rad}, \qquad \theta_F = \frac{5 + 144.4,\theta_D}{433.2} = 0 \]

\(\theta_F = 0\) exactly: by symmetry France sits at Belgium's angle, its 500 MW flows straight to Germany over the DE1–FR1 line, and Belgium's 500 MW flows over BE2–DE1. So with the shifter at neutral,

\[ F_{BD} = 144.4 \times (0 - (-0.034626)) = 5.000\ \text{p.u.} = 500\ \text{MW} \]

What the shifter adds

The file leaves the PST at tap −8, which the UCTE ##R record turns into a shift of \(\alpha = 0.054387\) rad \(= 3.116°\) in gridoxide's internal convention. (A CRAC states the same angle as −3.116°, since IIDM's sign is the opposite; the conversion happens once, on the way in, and the next page works in the CRAC's frame.) A shift drives a circulating flow around the loop BE1–BE2–DE1–FR1–BE1, whose series reactance is

\[ x_{loop} = \frac{1}{144.4} + \frac{1}{144.4} + \frac{1}{288.8} = 0.017313\ \text{p.u.} \]

so the loop susceptance is \(b_{loop} = 57.76\) p.u. and the circulating flow is \(b_{loop},\alpha\):

\[ \frac{\partial F_{BD}}{\partial \alpha} = 57.76 \times \frac{\pi}{180} = 1.0081\ \text{p.u./deg} = 100.81\ \text{MW/deg} \]

against the 100.7896 MW/deg the code computes — the 0.02% gap is the phase shifter's clamped, not quite zero, reactance. So

\[ F_{BD} = 500 + 100.79 \times 3.116 = 814.0\ \text{MW} \]

The margin

\[ m = \min(700 - 814.0,\ \ 814.0 + 700) = \min(-114.0,\ 1514.0) = -114.0\ \text{MW} \]

which is what the evaluator reports:

$ gridoxide security tests/data/benchmark-grids/powsybl-open-rao/files/cases/flowbased_computation/3nodes_pst.uct \
      --crac docs/examples/pst-worked-example.crac.json

preventive / base case: 1 monitored, worst margin -114.0 MW
  OVERLOAD BE2-DE1 - preventive         flow     814.0  limit     700.0  by    114.0 MW

INSECURE: 1 overload(s) across 1 perimeter(s)

Note which bound binds: the upper one. Had the threshold been one-sided in the other direction, the margin would have been +1514 MW and this network perfectly secure — the same flow, the same element, a different question.

The AC case

Under FlowModel::Ac two things change that the linear model cannot see, and both act on an ampere threshold only — an MW threshold binds active power, which is exactly what the evaluator reports either way.

Current is measured at the voltage the bus is actually running at. A bus at 1.05 p.u. carries a given MW at 5% less current than nominal, so a DC study reading its ampere thresholds at nominal voltage is conservative there and optimistic wherever voltage has sagged.

Reactive flow consumes thermal headroom. An ampere threshold binds \(\vert S \vert\), not \(\vert P \vert\). Rather than change what a margin means, the reactive part is charged against the limit:

\[ \text{charge} ;=; \max\left(0,\ \ \sqrt{P^2 + Q^2}\ \frac{U_{written}}{U_{actual}} ;-; \vert P \vert \right) \]

taken off both bounds, since the headroom is unavailable in either direction. The two effects arrive in one expression because they are the same conversion: the flow is re-expressed as the active power that would draw the same current at the voltage the threshold was written against.

The point of doing it this way is that \(m = \text{limit} - \vert P \vert\) still holds and an MW margin remains an MW margin. The alternative — reporting a margin on \(\vert S \vert\) — would mean two runs of the same study report margins that are not comparable depending on the flow model.

Resolution: the failure that produces no error

Before any of this, the CRAC's network-element ids have to be matched to gridoxide's branch and bus indices, and that step is fallible in an interesting way. An element that fails to resolve is not a small problem:

• a CNEC that silently disappears is a constraint the optimizer will never see;
• a contingency that fails to resolve is an outage it will never simulate;
• a range action whose generators fail to resolve is a control dropped from the optimization while every margin still looks right.

All three produce a confident answer to a different question, so Resolution reports every unresolved id rather than treating a miss as an error or as nothing. Unresolved ids are legitimate — a CRAC written for a merged model names elements outside any single file — which is exactly why they have to be reported instead of either failing or being ignored.

Two matching subtleties, both of which look like sloppiness and are not:

• Ids are matched exactly, then with surrounding whitespace trimmed. UCTE element ids are fixed-width and carry padding ("BBE1AA1 BBE2AA1 1"), and a CRAC written against the same network may or may not preserve it.
• Bus ids additionally match with a trailing _generator or _load removed, because powsybl names a UCTE node's generator <node>_generator and a CRAC uses the name powsybl gave it. CNECs name branches and redispatches name buses; those are different id spaces, and resolving one against the other silently yields nothing.