Busbar splitting (main contribution of the package)

Busbar Splitting (BuS) can make electrically distant parts of a substation that are electrically close. A double-busbar substation whose coupler is closed behaves as a single electrical node. If the busbar coupler is open, the two parts become two distinct nodes that are physically adjacent but electrically distinct, with independent voltage magnitudes and angles. In this formulation, every network element originally connected to the substation must then be connected to either one or the other part of the split busbar, or be disconnected in an OTS-fashion. This ideas is shown in the following figures for an AC substation:

and for both AC and DC substations:

The reconnection (or not) of the network elements is modelled through exclusivity constraint and summarized by the following figure:

constraint_exclusivity_switch is an "exclusivity" constraint including the switches connecting each grid element to the split busbar. It is either an equality ($=$ 1) or inequality ($\leq$ 1) constraint depending on whether OTS is performed or not \added{on the network elements originally connected to the split busbar}. If the network elements originally connected to the split busbar and BuS are both allowed in the same optimization problem, constraint_exclusivity_switch is an inequality constraint and the switches are both allowed to be open. As a result, the grid element is not reconnected to the split busbar i. If constraint_exclusivity_switch is an equality constraint, each grid element decoupled from the original busbar i needs to be reconnected to one part of the split busbar, and one of the two switches must be closed. The possible switching states allowed by the "exclusivity" constraint are represented in the figure above, where 1 indicates that the switch is closed, $0$ that the switch is open. Note that constraint constraint_ZIL_switch imposes that if the busbar coupler is closed, i.e. BuS is not performed, one switch connecting the network element to the original busbar will always be closed.

The three-stage workflow

BuS requires the network to be restructured before it can be optimized. There are three stages I have created in my models.

   ┌─────────────────────┐
   │                     │  AC_busbars_split / DC_busbars_split
   │   1. PREPARE DATA   │   → expands the network with auxiliary
   │                     │     buses and switches
   └──────────┬──────────┘
              │  data_split, switch_couples, extremes_ZIL
              ▼
   ┌─────────────────────┐
   │                     │   
   │    2. OPTIMIZE      │   run_acdc_BuS_AC / _DC / _AC_DC
   │                     │   → switch states as decision variables
   └──────────┬──────────┘
              │  result
              ▼
   ┌─────────────────────┐
   │                     │ 
   │  3. FEASIBILITY     │   prepare_AC_feasibility_check_*
   │        CHECK        │   then a plain AC/DC OPF
   │                     │   → verifies and prices the topology
   └─────────────────────┘

Stage 3 should be used even with each formulation as the switches are removed with the feasibility check, so one gets the exact OPF results with the optimized topology. More details about the feasibility check can be found here: AC feasibility check.

Stage 1: preparing the network

data_split, switch_couples, extremes_ZIL = AC_busbars_split(data, bus_to_be_split)
data_split, dcswitch_couples, extremes_ZIL_dc = DC_busbars_split(data, busdc_to_be_split)

bus_to_be_split is either an Int (bus_to_be_split = x) or a vector of Ints (bus_to_be_split = [x, y, z]). Both functions deepcopy their input, so the original network is left untouched.

For each busbar you nominate, the transformation:

1. duplicates the busbar into parts i and i′;
2. inserts a busbar coupler — a Zero Impedance Line (ZIL) modelled as a switch with ZIL = true — between them;
3. detaches every element attached to the busbar (generators, loads, branches, converters) and gives each its own auxiliary bus;
4. connects each auxiliary bus to both halves through a pair of switches, which constitute a switch_couple.

The optimizer then chooses, per element, which of its two switches is closed, plus whether the coupler is open or closed. The total switch count is

n_switches = Σ_b (2 · n_b) + B

where B is the number of busbars being split and n_b the number of elements attached to busbar b. This grows fast, and it is the reason splitting every busbar in a large network is not computationally efficient, nor feasible.

See Data model for the resulting dictionary layout and the meaning of every key the transformation adds.

The returned dictionary

All three elements matter, and two of them are needed again in stage 3.

• data_split — the expanded network. Pass this to the run_acdc_BuS_* functions.
• switch_couples — a dictionary of switch pairs. Each entry has f_sw and t_sw (the two switches connecting one element to the two busbar halves), bus_split (which original busbar), and switch_split (the index of that busbar's coupler). It is also stored on data_split["switch_couples"], where the model reads it from.
• extremes_ZIL — maps each split busbar to the indices of its two parts.

Variants

Only AC_busbars_split and DC_busbars_split copy their input. The others mutate, so deepcopy first if you need the original. To be fixed soon

Stage 2: optimizing

run_acdc_BuS_AC(data_split, model_constructor, optimizer; kwargs...)
run_acdc_BuS_DC(data_split, model_constructor, optimizer; kwargs...)
run_acdc_BuS_AC_DC(data_split, model_constructor, optimizer; kwargs...)

Choose the function matching the preparation you performed. run_acdc_BuS_AC_DC requires a network prepared by both AC_busbars_split and DC_busbars_split, applied in that order —

The constraint set

For each switch, the model enforces:

For each switch couple:

The exclusivity constraint is an inequality, not an equality, which is what allows an element to be dropped entirely — busbar splitting with OTS on the affected elements. If you want to forbid that and force every element back onto one half or the other, the constraint_exclusivity_switch_no_OTS variant provides the equality form, though no shipped problem specification uses it.

The big-M reformulation

The natural way to state "these two buses have equal voltage when the switch is closed" is bilinear:

z · θ_m = z · θ_i
z · U_m = z · U_i

which would make even the linear formulations non-convex. The package instead uses the standard big-M form:

−(1 − z) · M_θ  ≤  θ_m − θ_i  ≤  (1 − z) · M_θ
−(1 − z) · M_U  ≤  U_m − U_i  ≤  (1 − z) · M_U

The constants are hardcoded per formulation file:

These are deliberately conservative. Tightening them would strengthen the LP relaxation and speed up branch-and-bound considerably; the reference paper flags optimal big-M selection as future work and points to Pineda et al. (2024) for the methodology. If you are fighting solve times on a large case, this is the highest-leverage thing to change, and it means editing the constants in src/formdcgrid/acp.jl, lpac.jl, shared.jl, and dcp.jl.

The objective

min  Σ_k c₁ₖ · Pᵍₖ  +  Σ_ZIL c_sw · (1 − z_sw)

Two things to note.

First, only the linear generation cost coefficient is used. calc_gen_cost reads g["cost"][end-1], so a quadratic term in your case data is silently ignored. This matches the paper, which sets quadratic coefficients to zero.

Second, a small penalty is charged for each open busbar coupler. Auxiliary switches are free (cost = 0.0); couplers cost 1.0 by default. This ensures the model only splits a busbar when there is a real economic benefit, rather than splitting gratuitously among equal-cost optima. If splitting never happens on a case where you expect it to, check whether this penalty is swamping a small saving — you can adjust it per switch, depending also on the generator costs:

for (id, sw) in data_split["switch"]
    !haskey(sw, "auxiliary") && (sw["cost"] > 0.01)   # couplers only
end

Stage 3: checking AC feasibility

Covered in full on the AC feasibility check page. In brief:

data_fc = deepcopy(data_split)
prepare_AC_feasibility_check_AC_busbars(
    result, data_split, data_fc, switch_couples, extremes_ZIL, data_original)
result_fc = _PMACDC.solve_acdcopf(data_fc, ACPPowerModel, ipopt; setting = s)

Reading the results [to be improved]

for (sw_id, sw) in result["solution"]["switch"]
    status     = sw["status"] < 0.1 ? "OPEN" : "closed"
    is_coupler = !haskey(data_split["switch"][sw_id], "auxiliary")
    println("switch $sw_id ($(is_coupler ? "coupler" : "element")): $status")
end

and/or

for sw_id in 1:length(data_bus["switch"])
    if !haskey(data_bus["switch"]["$sw_id"], "auxiliary") 
        println("switch $sw_id is a busbar coupler with status $(result_bus_lpac["solution"]["switch"]["$sw_id"]["status"])")
    else
        if result_bus_lpac["solution"]["switch"]["$sw_id"]["status"] > 0.9
            if data_bus["switch"]["$sw_id"]["t_bus"] == extremes_ZIL["$bus_to_split"][1]
                println("switch $sw_id, linked to element $(data_bus["switch"]["$sw_id"]["auxiliary"]) $(data_bus["switch"]["$sw_id"]["original"]) is connected to the original busbar $bus_to_split")
            else
                println("switch $sw_id, linked to element $(data_bus["switch"]["$sw_id"]["auxiliary"]) $(data_bus["switch"]["$sw_id"]["original"]) is connected to the second half of busbar $bus_to_split, which is $bus_to_split'")
            end
        end
    end
end

DC switches appear under result["solution"]["dcswitch"] with the same status key.

A busbar was actually split if and only if its coupler is open. Element switches being open only tells you which side each element picked, or — if both switches of a couple are open — that the element was dropped from the network entirely.

Limitations

• Only double-busbar configurations are modelled. Breaker-and-a-half and double-bus double-breaker arrangements are not supported.
• All switching units are treated identically. Real substations distinguish circuit breakers, disconnectors, and load-break switches, each with different operating capabilities; the model gives them all the same treatment and the same thermal rating.
• Switch ratings default to psw = qsw = thermal_rating = 100.0 p.u., which is effectively unlimited on the bundled cases. Override them on data_split["switch"] if you want them to bind.
• No N-1 security constraints. The optimized topology is not verified against contingencies. Given that splitting a busbar reduces redundancy, this is a material limitation for operational use and is listed as future work.
• No stochastic RES modelling. Single deterministic operating point only for now, but we have a paper modelling it: . The code will be added to this repository in the future.