Technical Brief: Global Optimal Solution to AC Optimal Power Flow Problem Published

Since its first formulation in 1962, the AC optimal power flow (ACOPF) problem has been one of the most important optimization problems in electric power systems and is amongst the first that power systems engineers experience in their education.  In its most common formulation, electric power generation costs are minimized subject to network flow constraints, generator capacity constraints, line capacity constraints, and bus voltage constraints. As such, the ACOPF finds its greatest practical application in electric power systems markets where electric power is sold at the minimum cost while still securing the reliability of the electric power grid.

Unfortunately, this ACOPF formulation is a non-convex optimization problem that consequently falls into the as-yet-unsolved space of computationally complex (NP-hard) problems. Although the globally optimal solution has been elusive, the power systems field has developed a rich volume of OPF literature spanning six decades. Collectively, these works have offered numerous relaxations and approximations to the ACOPF. A dozen review articles published over that period show many hundreds of articles on the topic. The most famous approximation, the DCOPF (Direct Current Optimal Power Flow) has been at the heart of many wholesale, deregulated, “real-time” energy markets found at many independent system operators worldwide.

Such relaxations and approximations of the ACOPF result in economically suboptimal solutions that do not necessarily secure the grid. Consequently, downstream reliability procedures from control operators and automatic control systems are needed to ensure that the grid is dispatched reliably. While the potential impact on reliability can be mitigated, energy regulators have estimated that the sub-optimality costs the United States $6-19B per year. Furthermore, the sustainable energy transition necessitates renewed attention towards the ACOPF.

  1. In order to support the integration of distributed generation and energy storage, electric power systems markets are expanding beyond their traditional implementation as wholesale markets in the transmission system to retail markets in the distribution system and microgrids.  In North America, this would constitute a dramatic proliferation of the optimal power flow problem from the nine independent system operators to potentially thousands of electric distribution system utilities and operators.
  2. The radial and large-scale nature of distribution systems necessitates scalable ACOPF algorithms.
  3. Distribution systems feature a prominent role for line losses, nodal voltages, and reactive power flows which disqualifies many of the typical OPF approximations.
  4. Fourth, the integration of variable renewable energy resources further necessitates the participation of demand-side resources in two-sided markets.
  5. Finally, as the electric power grid activates these demand-side resources, it also integrates itself with the operation of other infrastructures including water, transportation, industrial production, natural gas, and heat. The non-convexity of the electric power network flow equations – as they are commonly stated – impedes the effective sector coupling of multiple infrastructure sectors.

Collectively, these reasons indicate that the ACOPF problem needs an alternative formulation and not just a new solution algorithm. Furthermore, it is of immediate importance to many grid stakeholders including transmission system operators, distribution system operators, and electric utilities. Recently in [1], a definitive solution to this once intractable problem is proposed. The heart of the solution rests upon a change of decision variables from the familiar “PQVθ” variables of active power, reactive power, voltage magnitude and voltage phase to the “IV” variables of current and voltage phasors in rectangular coordinates.

The new reformulation transforms the grid’s network flow constraints from non-convex to linear and convex. This transformation, however, is insufficient and leaves non-convexities in the objective function and the bus-voltage magnitude lower bound. To overcome these, this reformulation sheds the traditional “power flow analysis” model and instead adopts the “current-injection” model that power systems engineers use to conduct transient stability studies. This choice of model means that generator terminal voltages and lead lines are explicitly modeled. The necessary introduction of generator terminal voltage upper bounds then provides the key to overcoming the non-convexity introduced by the bus-voltage magnitude lower bounds. Similarly, the introduction of generator lead lines combined with demand-side revenue terms in the objective function serves to transform the once non-convex objective function into a convex fourth-order polynomial. The resulting formulation is a scalable convex optimization program that can be solved to global optimality in polynomial time using a Newton-Raphson algorithm.

A globally optimal solution to the ACOPF can have a profound impact on the trajectory of electricity markets worldwide. In addition to the already significant economic efficiencies in wholesale markets, its application has even greater potential in emerging electricity markets in the distribution system.  Furthermore, because it respects grid reliability constraints, it has the potential to simplify and improve control room operations.  Finally, this work opens the door to significant future work that enables the sustainable energy transition including the integration of renewable energy resources, the development of two-sided markets with elastic demand and coupling to other infrastructure sectors.

[1] Amro M. Farid, A Profit-Maximizing Security-Constrained IV-AC Optimal Power Flow Model & Global Solution”, IEEE Access, vol. 10, pp. 2842-2859, 2021.


Dr. Amro M. Farid
Thayer School of Engineering, Dartmouth College, Hanover, NH, USA
MIT Department of Mechanical Engineering, Cambridge, MA, USA

Amro M. Farid (Senior Member, IEEE) received the B.Sc. degree, in 2000, the M.Sc. degree in mechanical engineering from MIT, in 2002, and the Ph.D. degree in engineering from the University of Cambridge, U.K. He is currently a Visiting Associate Professor of Mechanical Engineering at MIT, an Associate Professor in Engineering at the Thayer School of Engineering, Dartmouth College, and the CEO of Engineering Systems Analytics LLC.  He leads the Laboratory for Intelligent Integrated Networks of Engineering Systems (LIINES) and has authored over 150 peer-reviewed publications in smart power grids, energy-water nexus, electrified transportation systems, industrial energy management, and interdependent smart city infrastructures. He holds leadership positions in the IEEE Smart Cities Technical Community, Control Systems Society (CSS), the Power & Energy Society (PES), and the Systems, Man & Cybernetics Society (SMCS).