Implementation of a Lagrangian relaxation based unit commitment problem

TLDR: The Lagrangian relaxation methodology has been used for solving the unit commitment problem as discussed by the authors, which is a class of complex combinatorial optimization problems in the power system, where the objective is to obtain an overall least-cost solution for operating the system over the scheduling horizon.

Abstract: The unit commitment problem in a power system involves determining a start-up and shut-down schedule of units to be used to meet the forecasted demand, over a future short term (24-168 hour) period. In solving the unit commitment problem, generally two basic decisions are involved. The "unit commitment" decision involves determining which generating units are to be running during each hour of the planning horizon, considering system capacity requirements including reserve, and the constraints on the start up and shut down of units. The related "economic dispatch" decision involves the allocation of system demand and spinning reserve capacity among the operating units during each specific hour of operation. As these two decisions are interrelated, the unit commitment problem generally embraces both these decisions, and the objective is to obtain an overall least cost solution for operating the power system over the scheduling horizon. The unit commitment problem belongs to the class of complex combinatorial optimization problems. During the past decade a new approach named "Lagrangian Relaxation" has been evolving for generating efficient solutions for this class of problems. It derives its name from the well-known mathematical technique of using Lagrange multipliers for solving constrained optimization problems, but is really a decomposition technique for the solution of large scale mathematical programming problems. The Lagrangian relaxation methodology generates easy subproblems for deciding commitment and generation schedules for single units over the planning horizon, independent of the commitment of other units.