In the optimal flow problem, the generation and transmission of electric= energy is optimised, taking into account the active and reactive power gen= eration limits, demand requirements, bus voltage limits, and network flow l= imits. Due to the non-linear power flow constraints, OPF is formulated as a= non-convex optimisation problem that is generally difficult to solve.

The problem was first formulated in 1962 and a large number of optimizat= ion algorithms and relaxations have been proposed since then. A popular app= roximation, the directed-current optimal power flow (DCOPF), is based on th= e linear programming problem that is obtained through the linearisation of = the power flow equations. While DCOPF is useful in a wide variety of applic= ations, a solution of DCOPF may not satisfy the non-linear power flow equat= ions and hence the resulting solution may be infeasible and may be of limit= ed utility. Other types of algorithms that were studied for the OPF include= Newton Raphson, quadratic programming, nonlinear programming, Lagrangian r= elaxation, interior point methods, genetic algorithms, and other heuristics= . Although some of these algorithms can handle large-scale networks most th= em can only compute a local optimal usually without assurance on the qualit= y of the solution. That is because most of the algorithms rely on the= Karush-Kuhn-Tucker necessary conditions of optimality, which cannot even g= uarantee a locally optimal solution, in the non-convex problem. Alternative= ly, the OPF can be formulated as a quadratically constrained quadratic prog= ram. There, convex relaxations within second-order cone (SOCP) programming = and semidefinite programming (SDP) can be applied. In contrast to the other= proposed approaches, convex relaxations make it possible to check if a sol= ution is globally optimal. If the solution is not optimal, the relaxations = provide a lower bound and hence a bound on how far any feasible solution is= from optimality. In particular, Bai et al. proposed a semidefinite program= ming relaxation for the ACOPF for general networks, which makes it possible= to find globally optimal solutions for several well-known instances. More = recently, the moments and sum-of-square decomposition have been used to bui= ld hierarchies of improving SDP relaxations for a polynomial programming fo= rmulation of ACOPF. To overcome the computational complexity of using SDP a= nd polynomial programming, graph sparsity has been exploited to simplify th= e SDP relaxation of the OPF.

To further improve the scalability of SDP relaxations, ADMM-based comput= ation can be used to solve sparse, large-scale SDPs. Alternatively, cheaper= hierarchies are being investigated based on LP and SOCP relaxations. In th= e future, a combination of hierarchies mixing constraints from different co= nes may be envisioned.

Another issue to address is development of techniques to certify infeasi= bility of optimal power flow instances.

From an industrial point of view, dealing with incomplete data is one of= the issues models and tools have to address. Aggregations of industrial da= ta may lead to physically non-meaningful models, since some section of the = power network are not represented in the data.

**References**:

J. Carpentier, Contribution to the economic dispatch problem, Bulletin S= ociety Francaise Electriciens, vol. 3, no. 8, pp. 431-447, 1962.

X. Bai, H. Wei, K. Fujisawa and Y. Wang, Semidefinite programming for op= timal power flow problems, International Journal of Electric Power & En= ergy Systems, vol. 30, no. 6-7, pp. 383-392, 2008.

J. Lavaei and S. H. Low. "Zero duality gap in optimal power flow problem= ." IEEE Transactions on Power Systems, vol. 27, no 1, pp. 92-107, 2012.

S.H. Low . "Convex relaxation of optimal power flow, part I: Formulation= s and equivalence." IEEE Transactions on Control of Network Systems, vol. 1= , no. 1, pp. 15-27, 2014.

X. Kuang, approximating the ACOPF problem with hierarchy of SOCP problem= s, 2015.

Madani, Kalbat, Lavaei, ADMM for Sparse Semidefinite Programming with Ap= plications to Optimal Power Flow problem.

Molzahn (2013), Sufficient conditions for power flow in-solvability with= applications to voltage stability margins.

Coffrin, Hijazi (2015), The QC relaxation: Theoreticl and Computational = Results on Optimal Power Flow, IEEE Trans. Power Systems, Vol. PP, Issue 99=

**Software**

**OPF specific software**

1) PowerTools

PowerTools (http://hhijazi.github.io/PowerTools/) is = a mathematical programming library for Power Systems optimization. PowerToo= ls is an open source c++ mathematical modeling platform linking to state-of= -the-art optimization solvers. It implements problems such as Optimal Power= Flow, and Optimal Transmission Switching, while benefiting from cutting-ed= ge research results. These include smart on/off constraints modeling, effic= ient bound propagation, domain-specific cut generation techniques and tight= convexification procedures providing global optimality guarantees. PowerTo= ols is still under active development, future versions will include Gas exp= ansion and Unit Commitment models, decomposition methods, multi-threading, = and support to additional solvers such as Cplex, Bonmin and others. Note th= at PowerTools is already used by another open-source project for Smart Grid= simulation: http://nicta.github.io/SmartGridToolbox/

2) Matpower

- Zimmerman, R., Murillo-Sanchez, C., Thomas, R.: Matpower: Steady-state= operations, planning, and analysis tools for power systems research and ed= ucation. Power Systems, IEEE Transactions on 26(1), 2011.

- D. K. Molzahn, J. T. Holzer, B. C. Lesieutre and C. L. DeMarco, "Imple= mentation of a Large-Scale Optimal Power Flow Solver Based on Semidefinite = Programming," in IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 39= 87-3998, Nov. 2013.

- Kristiansen (2003), Utilising Matpower in Optimal Power Flow, Model Id= entification and Control, Vol 24, No 1, pp. 49-59.

- Lavei, Rantzer, Low (2011), Power Flow Optimisation using positive qua= dratic programming

3) OPFsolver

- Madani, R., Ashraphijuo, M., Lavaei, J. (2014) SDP solver for Op= timal Power Flow User'92s Manual.

- Madani, Ashraphijuo, Lavaei (2014), Promises of Conic Relaxations for = Contingency constrained OPF, IEEE Trans. Power Systems.

4) PSSE

G. P. Harrison and A. R. Wallace, "Optimal power flow evaluation of dist= ribution network capacity for the connection of distributed generation," in= IEEE Proceedings - Generation, Transmission and Distribution, vol. 152, no= . 1, pp. 115-122, 10 Jan. 2005.

5) Plexos

- PLEXOS for power systems (2013)

- Foley, A. (2010), A strategic review of electricity system models, Ene= rgy, Vol 35 (12), No. 12, pp. 4522-4530.

**General purpose solver applied to OPF**

1) CPLEX

Alguacil, N., & Conejo, A. J. (2000). Multiperiod optimal powe= r flow using Benders decomposition. Power Systems, IEEE Transactions on, 15= (1), 196-201.

Jabr, R. (2008). Optimal power flow using an extended conic quadra= tic formulation. Power Systems, IEEE Transactions on, 23(3), 1000-1008.

Alsac, O., Bright, J., Prais, M., & Stott, B. (1990). Further = developments in LP-based optimal power flow. Power Systems, IEEE Transactio= ns on, 5(3), 697-711.

2) MOSEK

R. A. Jabr, "Radial distribution load flow usin=
g conic programming," in *IEEE Transactions on Power Systems*=
, vol. 21, no. 3, pp. 1458-1459, Aug. 2006.

B. Kocuk, S. S. Dey and X. A. Sun, "Inexactness of SDP Relaxation =
and Valid Inequalities for Optimal Power Flow," in *IEEE Transact=
ions on Power Systems*, vol. 31, no. 1, pp. 642-651, Jan. 2016.

Molzahn (2015), Solution of optimal power flow problems using mome= nt relaxations augmented with objective function penalisation

3) Xpress

**Contributors**:

Dr Martin Mevissen, IBM

Dr Cedric Josz, LAAS CNRS, Toulouse

Dr Bissan Ghaddar, University of Waterloo

Dr Fabrizio Lacalandra, QuanTek