# AC Optimal Power Flow in Various Nonlinear Optimization Frameworks

Yeah that’s not enough for sparse diff. (That’s barely sparse ). Maybe the decompression could be optimized a bit more though.

What do you consider sparse? the number of structural non-zeros is roughly 3% for the IEEE118 case:

``````(NonZeros, Entries) = (1506, 55696)
``````

And the percentage only goes down for larger cases. For the 2383 case we have:

``````(NonZeros, Entries) = (30636, 22714756)
(MaxColor, Size)    = (20, 4766)
``````

That’s roughly 0.15% of structural non-zeros. `SymbolicAD` seems to leverage this structure properly though.

I just uploaded it to Github. Just run `test.jl` and that’s it.
EDIT: I forgot to mention that right now I’m only computing the PF Hessians and Jacobians. I will compute those terms for the constraints as well, but I haven’t implemented that yet.

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Perhaps one other clarification. We’re only computing the jacobian and hessian of nonlinear (i.e., not linear or quadratic) constraints and objectives. We don’t compute it for linear or quadratic terms since they get handled by the solver.

Does that include power flow constraints? they are general nonlinear in polar coordinates.

Yes, what @odow is indicating is that these are the only equations that `SymbolicAD` is managing for building Jacobian and Hessian. The other parts of the AC-OPF problem are quadratic and have a different interface to the solver due to their simplified Jac/Hess structure.

It has been a couple of months since I made this post and I am planning re-run these test with latest versions of all packages. Please let me know of any specific package improvements that you think could impact this analysis.

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The new Optimization.jl sparsity handling is probably worth a try.

Additionally, if @YingboMa gets around to it then it might be interesting to try tearing

I literally had a draft open with reply to your original post that we added sparsity support in Optimization.jl so you might want to give it a try But we haven’t added documentation for it yet, the way to do it is pass `AutoModelingToolkit(true, true)` (the two fields are boolean for enabling sparsity for objective and constraint respectively).

I finished implementing sparsity and symbolic differentiation support in Nonconvex. Your rosetta-opf example works locally. I will make a release and update the docs shortly.

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@mohamed82008, I tried with Nonconvex v1.0.3 but I am still getting this issue of zygote crashing, could you elaborate on how I should change these calls to use your symbolic diff approach?

``````model = Nonconvex.DictModel()
...
x0 = NonconvexCore.getinit(model)
r = Nonconvex.optimize(model, IpoptAlg(), x0)
``````

First you will need Nonconvex v1.0.4. Then you can do:

``````first_order = true
options = IpoptOptions(; first_order)
sp_model = sparsify(model, hessian = !first_order)
r = Nonconvex.optimize(sp_model, IpoptAlg(), x0; options)
``````

`sparsify` will use Symbolics.jl to get the sparsity pattern of the gradients/Jacobians/Hessians and then use `SparseDiffTools` + `ForwardDiff` for the differentiation. Alternatively, you can use:

``````first_order = true
options = IpoptOptions(; first_order)
sym_model = symbolify(model, hessian = !first_order, sparse = true, simplify = true)
r = Nonconvex.optimize(sym_model, IpoptAlg(), x0; options)
``````

which will use a `Symbolics`-only approach where the gradients/Jacobians/Hessians are obtained using Symbolics.jl. The `sparse` kwarg is used to tell Symbolics to return a sparse matrix/vector for the jacobian/Hessian/gradient.

Note that these are new features so please report any issues in the repo.

2 Likes

As promised, below is a table of updated results using the latest versions of all packages. Note that GalacticOptim has been renamed Optimization.

The substantive changes follow the discussion above. In particular, NonConvex and Optimization have been updated with new AD systems based on Symbolics and now support sparse Jacobian and Hessian computations. For NonConvex, this allowed me to finish building the NonConvex OPF model and confirm that the model is correct on small networks. For Optimization, the new sparse AD system based on ModelingTookit enabled it to solve models with 73 network buses (800+ variables, 900+ constraints), which is a 2x scalability increase from what was possible with the previous dense ForwardDiff approach.

Great thanks to @mohamed82008 @Vaibhavdixit02 @odow for their help in debugging issues as we worked to update the rosetta-opf models.

For general awareness,

• This Zygote issue has thus far blocked using Zygote for AD on these models
• These upstream issues  appear to be the root cause blocking scaling of the ModelingTookKit approach to larger problems. It looks like it will be resolved in future Julia versions (testing was done on Julia v1.7.2).
Case Vars Cons JuMP NLPModels NonConvex Optim Optimization
case3_lmbd 24 28 1.22e-02 2.01e+01 6.19e+00 N.D. 1.77e+00
case5_pjm 44 53 1.75e-02 1.51e-01 4.28e+00 N.D. 1.99e-01
case14_ieee 118 169 6.54e-01 2.30e+01 8.77e+01 INF. 7.98e+00
case24_ieee_rts 266 315 5.38e-01 4.53e+01 5.61e+02 N.D. 2.14e+01
case30_ieee 236 348 6.16e-01 3.65e+01 5.27e+02 N.D. 2.43e+01
case30_as 236 348 6.05e-01 3.05e+01 4.88e+02 N.D. 2.25e+01
case39_epri 282 401 6.95e-02 6.00e+01 4.40e+02 N.D. 2.95e+01
case57_ieee 448 675 6.45e-01 1.05e+02 1.26e+03 N.D. 8.19e+01
case60_c 518 737 6.41e-01 2.19e+02 2.51e+03 N.D. 1.06e+02
case73_ieee_rts 824 987 6.45e-01 5.97e+02 5.74e+03 N.D. 2.68e+02
case89_pegase 1042 1649 9.08e-01 1.74e+03 N.D. N.D. N.D.
case118_ieee 1088 1539 8.36e-01 1.74e+03 N.D. N.D. N.D.
case162_ieee_dtc 1484 2313 8.72e-01 4.26e+03 N.D. N.D. N.D.
case179_goc 1468 2200 9.69e-01 7.06e+03 N.D. N.D. N.D.
case200_activ 1456 2116 7.73e-01 2.80e+03 N.D. N.D. N.D.
case240_pserc 2558 3617 3.09e+00 N.D. N.D. N.D. N.D.
case300_ieee 2382 3478 1.13e+00 N.D. N.D. N.D. N.D.
case500_goc 4254 6097 1.69e+00 N.D. N.D. N.D. N.D.
case588_sdet 4110 5979 1.55e+00 N.D. N.D. N.D. N.D.
case793_goc 5432 7978 2.01e+00 N.D. N.D. N.D. N.D.
case1354_pegase 11192 16646 6.82e+00 N.D. N.D. N.D. N.D.
case1888_rte 14480 21494 2.82e+01 N.D. N.D. N.D. N.D.
case1951_rte 15018 22075 1.16e+01 N.D. N.D. N.D. N.D.
case2000_goc 19008 29432 1.03e+01 N.D. N.D. N.D. N.D.
case2312_goc 17128 25716 8.91e+00 N.D. N.D. N.D. N.D.
case2383wp_k 17004 25039 9.37e+00 N.D. N.D. N.D. N.D.
case2736sp_k 19088 28356 7.67e+00 N.D. N.D. N.D. N.D.
case2737sop_k 18988 28358 8.01e+00 N.D. N.D. N.D. N.D.
case2742_goc 24540 38196 3.50e+01 N.D. N.D. N.D. N.D.
case2746wp_k 19520 28446 8.71e+00 N.D. N.D. N.D. N.D.
case2746wop_k 19582 28642 7.97e+00 N.D. N.D. N.D. N.D.
case2848_rte 21822 32129 1.76e+01 N.D. N.D. N.D. N.D.
case2853_sdet 23028 33154 1.26e+01 N.D. N.D. N.D. N.D.
case2868_rte 22090 32393 2.06e+01 N.D. N.D. N.D. N.D.
case2869_pegase 25086 37813 1.46e+01 N.D. N.D. N.D. N.D.
case3012wp_k 21082 31029 1.32e+01 N.D. N.D. N.D. N.D.
case3022_goc 23238 34990 1.38e+01 N.D. N.D. N.D. N.D.
case3120sp_k 21608 32092 1.26e+01 N.D. N.D. N.D. N.D.
case3375wp_k 24350 35876 1.47e+01 N.D. N.D. N.D. N.D.
case3970_goc 35270 54428 2.32e+01 INF. N.D. N.D. N.D.
case4020_goc 36696 56957 2.89e+01 N.D. N.D. N.D. N.D.
case4601_goc 38814 59596 2.84e+01 N.D. N.D. N.D. N.D.
case4619_goc 42532 66289 2.94e+01 N.D. N.D. N.D. N.D.
case4661_sdet 34758 51302 2.48e+01 INF. N.D. N.D. N.D.
case4837_goc 41398 64030 2.62e+01 N.D. N.D. N.D. N.D.
case4917_goc 37872 56917 2.78e+01 N.D. N.D. N.D. N.D.
case6468_rte 49734 75937 8.18e+01 N.D. N.D. N.D. N.D.
case6470_rte 50482 75976 4.56e+01 N.D. N.D. N.D. N.D.
case6495_rte 50426 76124 8.23e+01 N.D. N.D. N.D. N.D.
case6515_rte 50546 76290 6.09e+01 N.D. N.D. N.D. N.D.
case8387_pegase 78748 118702 6.51e+01 N.D. N.D. N.D. N.D.
case9241_pegase 85568 130826 6.62e+01 N.D. N.D. N.D. N.D.
case9591_goc 83572 130588 8.49e+01 N.D. N.D. N.D. N.D.
case10000_goc 76804 112352 5.75e+01 N.D. N.D. N.D. N.D.
case10480_goc 96750 150874 8.37e+01 N.D. N.D. N.D. N.D.
case13659_pegase 117370 170588 9.50e+01 N.D. N.D. N.D. N.D.
case19402_goc 179562 281733 1.93e+02 N.D. N.D. N.D. N.D.
case24464_goc 203374 313641 1.67e+02 N.D. N.D. N.D. N.D.
case30000_goc 208624 307752 3.54e+02 N.D. N.D. N.D. N.D.

Package Details

[b6b21f68] Ipopt v1.0.2
[4076af6c] JuMP v1.1.1
[961ee093] ModelingToolkit v8.14.0
[f4238b75] NLPModelsIpopt v0.9.1
[01bcebdf] Nonconvex v2.0.3
[bf347577] NonconvexIpopt v0.4.1
[429524aa] Optim v1.7.0
[7f7a1694] Optimization v3.6.1
[fd9f6733] OptimizationMOI v0.1.1
[c36e90e8] PowerModels v0.19.5

Ipopt was run with the default linear solver, MUMPS 5.4.1.

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Are you including the compilation time here? Yeah the RGF issues are one thing, and the other is we need to release SymbolicIR.

MTK also doesn’t do `structural_simplify` on OptimizationProblem right now, which can be a major win because then it solves a much smaller problem. @YingboMa was working on that.

Good question. In these runtime numbers I try to avoid capturing artifacts from one-time overheads, such as JIT time. In my experience, with sufficient effort, these can be avoided in production deployment settings.

In all of these implementations before I start collecting data `solve_opf("\$(@__DIR__)/data/pglib_opf_case5_pjm.m")` is executed first, to compile the relevant code execution paths. Then `solve_opf(...)` is run again on the file where runtime data is being collected.

You can have a look at exactly how the data collection is done in this script. It is not immediately obvious that pre-compilation is done there, but looking into the specific implementation files you will find it, for example, rosetta-opf/optimization.jl at cli · lanl-ansi/rosetta-opf · GitHub

Cool, looking forward to seeing the next iteration.

Thanks for this excellent source of test cases for Nonconvex. For Nonconvex, I would be interested in the results using ReverseDiff as well as SparseDiffTools instead of Symbolics.

See Using other AD packages · Nonconvex.jl and Sparse Jacobian or Hessian · Nonconvex.jl for the relevant docs sections.

@mohamed82008, If you could show me how these lines should be changed in nonconvex.jl to test these approaches then I can do testing on some example network datasets.

For SparseDiffTools, just drop the `symbolic = true`:

``````result = Nonconvex.optimize(
model,
IpoptAlg(),
NonconvexCore.getinit(model);
options = IpoptOptions(; first_order=false, sparse=true),
)
``````

For ReverseDiff, you can do:

``````using AbstractDifferentiation, ReverseDiff