Solution to Security-constrained Non-convex Pumped-Storage Hydraulic Unit Scheduling Problem by Modified Subgradient Algorithm Based on Feasible Values and Pseudo Water Price

Abstract

A security-constrained power dispatch problem with a non-convex cost function for a lossy electric power system area including a pumped-storage hydraulic unit is formulated. Then, an iterative solution method based on a modified subgradient algorithm operating on feasible values and pseudo water price for the pumped-storage hydraulic unit is used to solve it. In the iterative proposed solution method, the modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the pseudo water price for the pumped-storage hydraulic unit is employed to adjust the net amount of water used by the pumped-storage hydraulic unit during the considered operation period.

Since all equality and inequality constraints in the proposed non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles, the off-nominal tap settings, and susceptances values of SVAR systems, they are taken as independent variables. Load flow equations are added into the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints, and SVAR system susceptance value constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method, which does not add any extra independent variable into the model, before its application to the optimization model. Since the method does not add any extra independent variable into the model, the solution time is reduced further.

The proposed dispatch technique is tested on a sample power system that has 12 buses with 5 thermal units and a pumped-storage hydraulic unit. Optimal total cost value for the power system without any pumped-storage unit is calculated first. Later, the same optimal total cost value for the power system with the pumped-storage unit is recalculated, and the obtained saving in the optimal total cost value, due to the employment of the pumped-storage unit, is presented. The solution times for the dispatch problems with and without the pumped-storage unit is also presented. At the end, the modified subgradient algorithm based on feasible values is applied directly to the dispatch problem with a pumped-storage unit. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem with the pumped-storage unit.

Keywords:

• pumped-storage hydraulic unit scheduling problem
• security constraints
• modified subgradient algorithm based on feasible values
• non-convex cost function
• pseudo water price