Optimal Power Flow and Formulation Papers

The AC Optimal Power Flow (ACOPF) is at the heart of Independent System Operator (ISO) power markets and vertically integrated utility dispatch. ACOPF simultaneously optimizes real and reactive power. An approximated form of the ACOPF is solved in some form annually for system planning, daily for day-ahead commitment markets, and even every 5 minutes for real-time market balancing. The ACOPF was first formulated in 1962 by Carpentier.

With advances in computing power and solution algorithms, we can model more constraints and remove unnecessary limits and approximations that were previously required to find a good solution in reasonable time. Today, 50 years after the problem was formulated, we still do not have a fast, robust solution technique for the ACOPF. Finding a good solution technique for the ACOPF could potentially save tens of billions of dollars annually.

The ACOPF formulation co-optimizes real and reactive power, internalizes losses (not estimated as in the ‘DC’ model) and has explicated voltage bounds, but requires more time to solve and better data. The current-voltage formulation (IV-ACOPF) has linear network flow equations. Its non-convexities occur at injection and withdrawal busses when current and voltage are converted to real and reactive power and its thermal constraints expressed in term of current magnitude. It better models thermal constraints by limiting the line current instead of power flow.

The iterative linear approximation (ILIV-ACOPF) solves faster and is more robust than most other approaches examined. Parameter tuning can improve performance. With binary variables, for example, as in the unit commitment and optimal transmission switching problems, linear approximations can be solved faster than nonlinear models.

In this series of papers, we seek to present the ACOPF problem through clear formulations of the problem, its constraints and its parameters. We survey historical approaches to solving the problem. We also formulate and test several approaches and algorithms to solving the ACOPF. We find that rectangular formulation solves faster than the polar formulation for the larger problems. We also present an iterative approximation and test it against a set of standard nonlinear solvers.

Papers & Abstracts

Paper 1 - History of Optimal Power Flow and Formulations

In “History of Optimal Power Flow and Formulations”, we present a literature review of the AC Optimal Power Flow (ACOPF) problem over the 50 years since it was formulated in 1962, and present the major formulations of the ACOPF. We refer to the full ACOPF as an ACOPF that simultaneously optimizes real and reactive power. This paper defines and discusses the polar power-voltage, rectangular power-voltage, and rectangular current-voltage formulations of the ACOPF, as well as different forms of constraints and objective functions. (issued December 17, 2012)

Paper 2 - The IV Formulation and Linearizations of the AC Optimal Power Flow Problem

In “The IV Formulation and Linearizations of the AC Optimal Power Flow Problem”, we formulate the ACOPF in several ways, compare each formulation’s properties, and argue that the current-voltage or "IV" formulation and its linear approximations may be easier to solve than the traditional quadratic power flow "PQV" formulation. (Issued January 30, 2013)

Paper 3 - The Computational Testing of AC Optimal Power Flow Using the Current Voltage (IV) Formulations

In “The Computational Testing of AC Optimal Power Flow Using the Current Voltage (IV) Formulations”, we compare solving the IV linear approximation of the ACOPF to solving the ACOPF with several nonlinear solvers. In general, the linear approximation approach is more robust and faster than several of the commercial nonlinear solvers. On several starting points, the nonlinear solvers failed to converge or contained positive relaxation variables above the threshold. The iterative linear program approach finds a near feasible near optimal in almost all problems and starting points. (Issued January 30, 2013)

Paper 4 - Survey of Approaches to Solving the ACOPF

In “Survey of Approaches to Solving the ACOPF”, we present a background on approaches historically applied to solve the ACOPF, many which are used in our following companion study on testing and the computational performance of solution techniques. In this paper we present the associated theory in nonlinear optimization and discuss solvers and published algorithms that have been applied to the ACOPF. We provide insight into the major contributions starting from Carpentier's initial contribution in the early 1960's to present day state-of-the-art. (Issued March 29, 2013)

Paper 5 - Computational Performance of Solution Techniques Applied to the ACOPF

In “Computational Performance of Solution Techniques Applied to the ACOPF”, we present an experimental framework and statistical methods that are an improvement on current practices and in line with practices in the optimization community. We report numerical results from testing nonlinear commercial solvers with varying ACOPF formulations and initializations. Our experimental results indicate a clear advantage to employing a multistart strategy, which leverages parallel processing in order to solve the ACOPF on large-scale networks for time-sensitive applications. (Issued March 29, 2013)

Paper 6 - Exploration of the ACOPF Feasible Region for the Standard IEEE Test Set Optimal Power Flow

In "Exploration of the ACOPF Feasible Region for the Standard IEEE Test Set Optimal Power Flow", we investigate to gain some perspective on how non-convex the feasible region of the Alternating Current Optimal Power Flow ACOPF problem is. We develop a metric for comparing how infeasible different solutions are. The set-up for this examination will be to generate convex combinations of feasible points, and determine how “far” these new points are from the feasible region.

We propose a metric based on the relative two norm. This value is compared over various regions around the optimal solution. We determine how elastic the area around the global optimum is; in other words, we will determine the largest range of values the optimization variables can take, given a small perturbation from the global solution. This gives insight into the structure of the feasible region. Finally, we will examine the two dimensional relationships between the optimization variables and the objective function value. (Issued April 8, 2013)

Paper 7 - Developing Line Current magnitude Constraints for IEEE Test Problems

In "Developing Line Current magnitude Constraints for IEEE Test Problems", we present a simple method for constructing current magnitude constraints and to report on the computational properties in solving the resulting problems. This paper finds limits on the maximum allowable current magnitude that result in a feasible solution for the 14-bus, 30-bus, 57-bus, and 118-bus IEEE test problems.

For each test problem, one single limit is applied to all lines that makes the optimal solution without these limits infeasible. For each problem we develop a ‘tight’ and a ‘loose’ constraint. We solve the resulting problem using the current voltage formulation. Different test problems exhibit different characteristics. Including these constraints in the ACOPF increases the solution time between 2 to 20 times and costs (objective function) up to 25 percent. (Issued April 8, 2013)”, we present a simple method for constructing current magnitude constraints and to report on the computational properties in solving the resulting problems.

This paper finds limits on the maximum allowable current magnitude that result in a feasible solution for the 14-bus, 30-bus, 57-bus, and 118-bus IEEE test problems. For each test problem, one single limit is applied to all lines that makes the optimal solution without these limits infeasible. For each problem we develop a ‘tight’ and a ‘loose’ constraint. We solve the resulting problem using the current voltage formulation. Different test problems exhibit different characteristics. Including these constraints in the ACOPF increases the solution time between 2 to 20 times and costs (objective function) up to 25 percent. (Issued April 8, 2013)

Paper 8 - A Computational Study of Linear Approximations to the Convex Constraints in the Iterative Linear IV-ACOPF Formulation

In "A Computational Study of Linear Approximations to the Convex Constraints in the Iterative Linear IV-ACOPF Formulation", we test the effect of preprocessed and iterative linear cuts on the convex constraints (maximum voltage and maximum current) in iterative linear approximation of the current voltage (IV) formulation of the ACOPF (ILIV-ACOPF) problem. The nonconvex constraints are linearized using iterative first order Taylor series approximation. The ILIV-ACOPF is solved as a single linear program or a sequence of linear programs. The ILIV-ACOPF model is tested on the 14-, 30-, 57- and 118-bus IEEE test systems.

The execution time and the quality of the solution obtained from the ILIV-ACOPF are compared for different test systems and benchmarked against the equivalent nonlinear ACOPF formulation. The results show execution time up to 8 times faster and solutions close to the nonlinear solver. The performance of different algorithmic parameters varies depending on the test problem. As the number of preprocessed cuts for each bus and line increases, the relative error decreases. The results indicate that 16 to 32 constraints are the best number of preprocessed constraints in a trade off between accuracy and solution time. The marginal value of more than 32 and maybe 16 preprocessed constraints in this setting is negative. Nevertheless, the number of preprocessed constraints can be set based on the performance and requirements of the specific problem being solved.

Using iterative cuts often results in faster convergence to a feasible solution. When solving without using iterative cuts, the solution is within 18% of the best-known nonlinear feasible solution; with iterative cuts, the solution is within 2.5% of the best-known nonlinear solution. At 16 or 32 preprocessed constraints, solution time is up to about 8 times faster than the nonlinear solver (IPOPT). (issued June 12, 2013)

Paper 9 - Testing Step-size Limits for Solving the Linearized Current Voltage AC Optimal Power Flow

In "Testing Step-size Limits for Solving the Linearized Current Voltage AC Optimal Power Flow", we seek to improve the performance of the iterative linear program approximation to the current voltage AC optimal power flow (ILIV-ACOPF). By adding a set of constraints that limit the differences between the real and imaginary voltages of successive major iteration solutions, we limit the error in the linear approximation, and we seek to decrease the time to solve and increase the robustness of the procedure. The primary motivation is that the iterative linearization procedure sometimes exhibits periodic behavior ("bouncing" between two solutions). This behavior may add to the solution time or result in a failure to converge. Generally, the step-size constraints improve performance of the iterative linear approximation procedure, but the best parameters of the step-size constraint are problem dependent.

Although the convergence tests are different, the linear procedure is considerably faster than the nonlinear solver The tradeoff between the iterative linearization and the nonlinear solver was speed compared to greater accuracy. Increasing the preprocessed cuts from 16 to 32 increases the solution time. As the problem size gets bigger, we see diminishing returns to the number of preprocessed constraints. The tighter tolerance for convergence takes longer, but does not seem to have a major impact on the optimal solution value, except when there is no step-size constraint, where a more restrictive tolerance sometimes results in the linear program not converging. We find that step-size constraints decrease the time to solve and increase the robustness of the procedure. Solution times were up to six times faster using step-size limits. (issued June 13, 2013)

Paper 10 - Optimal Transmission Switching Using the IV-ACOPF Linearization

In "Optimal Transmission Switching Using the IV-ACOPF Linearization", seek to investigate the performance of transmission switching using the iterative linear program approximation to the Current Voltage AC Optimal Power Flow IV-ACOPF. Several different methods of using this switching are investigated to find a method that addresses the MIP challenges and is both fast and accurate. We consider opening only one line, opening up to five lines, and progressively opening one line at a time. The linear method with switching is much faster than the nonlinear ACOPF and generally finds solutions within 1% of the nonlinear ACOPF.

Paper 11 - A Note on the Computational Testing of AC Optimal Power Flow Using the Current Voltage Formulations

In "A Note on the Computational Testing of AC Optimal Power Flow Using the Current Voltage Formulations", we solve a set of IEEE test problems on a different hardware platform with the same software and compare the results to those in O’Neill, Castillo and Cain 2012. The GAMS code to test the solvers is unchanged. The problems were originally solved on a 32 bit machine. The new hardware is 64 bit.