# Formulation Details

This section provides references to understand the formulations as provided by PowerModels. The list is not meant as a literature discussion, but to give the main starting points to understand the implementation of the formulations.

- Molzahn, D., & Hiskens, I. (2019). A Survey of Relaxations and Approximations of the Power Flow Equations. Foundations and Trends in Electric Energy Systems https://doi.org/10.1561/3100000012
- Coffrin, C., & Roald, L. (2018). Convex relaxations in power system optimization: a brief introduction. [Math.OC], 1–5. Retrieved from http://arxiv.org/abs/1807.07227
- Coffrin, C., Hijazi, H., & Van Hentenryck, P. (2016). The QC relaxation: a theoretical and computational study on optimal power flow. IEEE Trans. Power Syst., 31(4), 3008–3018. https://doi.org/10.1109/TPWRS.2015.2463111

## Notes on the mathematical model for all formulations

PowerModels implements a slightly generalized version of the AC Optimal Power Flow problem from Matpower, as discussed in The PowerModels Mathematical Model and presented here.

In the next subsections the differences between PowerModels' bus and branch models and those commonly used in the literature are discussed. Consideration is given to these differences when implementing formulations from articles.

### Standardized branch model

The branch model is standardized as follows:

- An idealized (lossless) transformer at the from side of the branch (immediately on node $i$) with a fixed, complex-value voltage transformation (i.e. tap and shift)
- Followed by a pi-section with complex-valued line shunt admittance, where the from and to side shunt can have different values
- A branch is uniquely defined by a tuple $(l,i,j)$ where $l$ is the line index, $i$ is the from node, and $j$ is the to node.
- Thermal limits are defined in apparent power, and are defined at both ends of a line
- Each branch has a phase angle difference constraint

Nevertheless, in the literature, the following can be observed:

- The to and from side line shunts are equal
- The line shunt admittance is a pure susceptance (equivalent to shunt conductance set to 0)
- A branch in a grid without parallel lines is uniquely defined by a tuple $(i,j)$ where $i$ is the from node, and $j$ is the to node.
- Thermal limits are defined in current (total or series), complex power limits are approximated as a regular polygon, ...
- Thermal limits are defined only at the from (or to) side
- Phase angle difference constraints are not included

Furthermore, "lifted nonlinear cuts" are used to improve the accuracy of PAD constraints for all formulations in the lifted S-W variable space:

- Coffrin, C., Hijazi, H., & Van Hentenryck, P. (2017). Strengthening the SDP relaxation of ac power flows with convex envelopes, bound tightening, and valid inequalities. IEEE Trans. Power Syst., 32(5), 3549–3558. https://doi.org/10.1109/TPWRS.2016.2634586

### Standardized bus model

The bus model is standardized as follows:

- A bus defines a complex power balance for all the sets lines, generators, loads, bus shunts connected to it, i.e. one can define multiple load and shunt components on each bus $i$

Nevertheless, in the literature, a simplified bus model is often used:

- Only a single (aggregated) load per bus is supported
- Only a single (aggregated) bus shunt per bus is supported

## Exact Non-Convex Models

`PowerModels.ACPPowerModel`

— TypeAC power flow Model with polar bus voltage variables.

The seminal reference of AC OPF:

```
@article{carpentier1962contribution,
title={Contribution to the economic dispatch problem},
author={Carpentier, J},
journal={Bulletin de la Societe Francoise des Electriciens},
volume={3},
number={8},
pages={431--447},
year={1962}
}
```

History and discussion:

```
@techreport{Cain2012,
author = {Cain, Mary B and {O' Neill}, Richard P and Castillo, Anya},
title = {{History of optimal power flow and Models}},
pages = {1--36},
url = {https://www.ferc.gov/sites/default/files/2020-04/acopf-1-history-formulation-testing.pdf},
year = {2012}
}
```

`PowerModels.ACRPowerModel`

— TypeAC power flow Model with rectangular bus voltage variables.

```
@techreport{Cain2012,
author = {Cain, Mary B and {O' Neill}, Richard P and Castillo, Anya},
title = {{History of optimal power flow and Models}},
pages = {1--36},
url = {https://www.ferc.gov/sites/default/files/2020-04/acopf-1-history-formulation-testing.pdf},
year = {2012}
}
```

`PowerModels.ACTPowerModel`

— TypeAC power flow Model (nonconvex) with variables for voltage angle, voltage magnitude squared, and real and imaginary part of voltage crossproducts. A tangens constraint is added to represent meshed networks in an exact manner.

```
@ARTICLE{4349090,
author={R. A. Jabr},
title={A Conic Quadratic Format for the Load Flow Equations of Meshed Networks},
journal={IEEE Transactions on Power Systems},
year={2007},
month={Nov},
volume={22},
number={4},
pages={2285-2286},
doi={10.1109/TPWRS.2007.907590},
ISSN={0885-8950}
}
```

`PowerModels.IVRPowerModel`

— TypeCurrent voltage formulation of AC OPF. The formulation uses rectangular coordinates for both current and voltage. Note that, even though Kirchhoff's circuit laws are linear in current and voltage, this formulation is nonconvex due to constants power loads/generators and apparent power limits.

```
@techreport{ONeill2012,
author = {{O' Neill}, Richard P and Castillo, Anya and Cain, Mary B},
pages = {1--18},
title = {{The IV formulation and linear approximations of the ac optimal power flow problem}},
url = {https://www.ferc.gov/sites/default/files/2020-04/acopf-2-iv-linearization.pdf},
year = {2012}
}
```

Applicable to problem formulations with `_iv`

in the name.

## Linear Approximations

`PowerModels.DCPPowerModel`

— TypeLinearized 'DC' power flow Model with polar voltage variables.

This model is a basic linear active-power-only approximation, which uses branch susceptance values `br_b = -br_x / (br_r^2 + br_x^2)`

for determining the network phase angles. Furthermore, transformer parameters such as tap ratios and phase shifts are not considered as part of this model.

It is important to note that it is also common for active-power-only approximations to use `1/br_x`

for determining the network phase angles, instead of the `br_b`

value that is used here. Small discrepancies in solutions should be expected when comparing active-power-only approximations across multiple tools.

```
@ARTICLE{4956966,
author={B. Stott and J. Jardim and O. Alsac},
journal={IEEE Transactions on Power Systems},
title={DC Power Flow Revisited},
year={2009},
month={Aug},
volume={24},
number={3},
pages={1290-1300},
doi={10.1109/TPWRS.2009.2021235},
ISSN={0885-8950}
}
```

`PowerModels.DCMPPowerModel`

— TypeLinearized 'DC' power flow model with polar voltage variables.

Similar to the DCPPowerModel with the following changes:

- It uses branch susceptance values
`br_b = -1 / br_x`

for determining the network phase angles. - Transformer parameters such as tap ratios and phase shifts are considered.

The results should be equal to the results of matpower calculations.

`PowerModels.BFAPowerModel`

— TypeLinear approximation of branch flow model.

The implementation builds on the second-order cone relaxation of the branch flow model, but neglects the active and reactive loss terms associated with the squared current magnitude so the power flow equations become linear. Note that flow bounds are still second order cones.

```
@article{Baran1989OptimalSystems,
title = {{Optimal capacitor placement on radial distribution systems}},
year = {1989},
journal = {IEEE Transactions on Power Delivery},
author = {Baran, Mesut E. and Wu, Felix F.},
number = {1},
pages = {725--734},
volume = {4},
doi = {10.1109/61.19265},
issn = {19374208}
}
```

Applicable to problem formulations with `_bf`

in the name.

`PowerModels.NFAPowerModel`

— TypeThe an active power only network flow approximation, also known as the transportation model.

## Quadratic Approximations

`PowerModels.DCPLLPowerModel`

— Type`PowerModels.LPACCPowerModel`

— TypeThe LPAC Cold-Start AC Power Flow Approximation.

Note that the LPAC Cold-Start model requires the least amount of information but is also the least accurate variant of the LPAC Models. If a nominal AC operating point is available, the LPAC Warm-Start model will provide improved accuracy.

The original publication suggests to use polyhedral outer approximations for the cosine and line thermal lit constraints. Given the recent improvements in MIQCQP solvers, this implementation uses quadratic functions for those constraints.

```
@article{doi:10.1287/ijoc.2014.0594,
author = {Coffrin, Carleton and Van Hentenryck, Pascal},
title = {A Linear-Programming Approximation of AC Power Flows},
journal = {INFORMS Journal on Computing},
volume = {26},
number = {4},
pages = {718-734},
year = {2014},
doi = {10.1287/ijoc.2014.0594},
eprint = {https://doi.org/10.1287/ijoc.2014.0594}
}
```

## Quadratic Relaxations

`PowerModels.SOCWRPowerModel`

— TypeSecond-order cone relaxation of bus injection model of AC OPF.

The implementation casts this as a convex quadratically constrained problem.

```
@article{1664986,
author={R. A. Jabr},
title={Radial distribution load flow using conic programming},
journal={IEEE Transactions on Power Systems},
year={2006},
month={Aug},
volume={21},
number={3},
pages={1458-1459},
doi={10.1109/TPWRS.2006.879234},
ISSN={0885-8950}
}
```

`PowerModels.SOCWRConicPowerModel`

— TypeSecond-order cone relaxation of bus injection model of AC OPF.

This implementation casts the problem as a convex conic problem.

`PowerModels.QCRMPowerModel`

— TypeThe "Quadratic-Convex" relaxation of the AC power flow equations. Recursive McCormik relaxations are used for the trilinear terms (i.e. QCRM).

```
@Article{Hijazi2017,
author="Hijazi, Hassan and Coffrin, Carleton and Hentenryck, Pascal Van",
title="Convex quadratic relaxations for mixed-integer nonlinear programs in power systems",
journal="Mathematical Programming Computation",
year="2017",
month="Sep",
volume="9",
number="3",
pages="321--367",
issn="1867-2957",
doi="10.1007/s12532-016-0112-z",
url="https://doi.org/10.1007/s12532-016-0112-z"
}
```

`PowerModels.QCLSPowerModel`

— TypeA strengthened version of the "Quadratic-Convex" relaxation of the AC power flow equations. An extreme-point encoding of trilinar terms is used along with a constraint to link the lambda variables in multiple trilinar terms (i.e. QCLS).

```
@misc{1809.04565,
author="Kaarthik Sundar and Harsha Nagarajan and Sidhant Misra and Mowen Lu and Carleton Coffrin and Russell Bent",
title="Optimization-Based Bound Tightening using a Strengthened QC-Relaxation of the Optimal Power Flow Problem",
year="2018",
Eprint = "arXiv:1809.04565",
}
```

The original model derivation is available in,

```
@Article{Hijazi2017,
author="Hijazi, Hassan and Coffrin, Carleton and Hentenryck, Pascal Van",
title="Convex quadratic relaxations for mixed-integer nonlinear programs in power systems",
journal="Mathematical Programming Computation",
year="2017",
month="Sep",
volume="9",
number="3",
pages="321--367",
issn="1867-2957",
doi="10.1007/s12532-016-0112-z",
url="https://doi.org/10.1007/s12532-016-0112-z"
}
```

`PowerModels.SOCBFPowerModel`

— TypeSecond-order cone relaxation of branch flow model

The implementation casts this as a convex quadratically constrained problem.

```
@INPROCEEDINGS{6425870,
author={M. Farivar and S. H. Low},
title={Branch flow model: Relaxations and convexification},
booktitle={2012 IEEE 51st IEEE Conference on Decision and Control (CDC)},
year={2012},
month={Dec},
pages={3672-3679},
doi={10.1109/CDC.2012.6425870},
ISSN={0191-2216}
}
```

Extended as discussed in:

```
@misc{1506.04773,
author = {Carleton Coffrin and Hassan L. Hijazi and Pascal Van Hentenryck},
title = {DistFlow Extensions for AC Transmission Systems},
year = {2018},
eprint = {arXiv:1506.04773},
url = {https://arxiv.org/abs/1506.04773}
}
```

Applicable to problem formulations with `_bf`

in the name.

`PowerModels.SOCBFConicPowerModel`

— Type## SDP Relaxation

`PowerModels.SDPWRMPowerModel`

— TypeSemi-definite relaxation of AC OPF

Originally proposed by:

```
@article{BAI2008383,
author = "Xiaoqing Bai and Hua Wei and Katsuki Fujisawa and Yong Wang",
title = "Semidefinite programming for optimal power flow problems",
journal = "International Journal of Electrical Power & Energy Systems",
volume = "30",
number = "6",
pages = "383 - 392",
year = "2008",
issn = "0142-0615",
doi = "https://doi.org/10.1016/j.ijepes.2007.12.003",
url = "http://www.sciencedirect.com/science/article/pii/S0142061507001378",
}
```

First paper to use "W" variables in the BIM of AC OPF:

```
@INPROCEEDINGS{6345272,
author={S. Sojoudi and J. Lavaei},
title={Physics of power networks makes hard optimization problems easy to solve},
booktitle={2012 IEEE Power and Energy Society General Meeting},
year={2012},
month={July},
pages={1-8},
doi={10.1109/PESGM.2012.6345272},
ISSN={1932-5517}
}
```

`PowerModels.SparseSDPWRMPowerModel`

— TypeSparsity-exploiting semidefinite relaxation of AC OPF

Proposed in:

```
@article{doi:10.1137/S1052623400366218,
author = {Fukuda, M. and Kojima, M. and Murota, K. and Nakata, K.},
title = {Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework},
journal = {SIAM Journal on Optimization},
volume = {11},
number = {3},
pages = {647-674},
year = {2001},
doi = {10.1137/S1052623400366218},
URL = {https://doi.org/10.1137/S1052623400366218},
eprint = {https://doi.org/10.1137/S1052623400366218}
}
```

Original application to OPF by:

```
@ARTICLE{6064917,
author={R. A. Jabr},
title={Exploiting Sparsity in SDP Relaxations of the OPF Problem},
journal={IEEE Transactions on Power Systems},
volume={27},
number={2},
pages={1138-1139},
year={2012},
month={May},
doi={10.1109/TPWRS.2011.2170772},
ISSN={0885-8950}
}
```