Burak Urazel

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 intervals 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.

A Security-constrained Economic Power Dispatch Technique Using Modified Subgradient Algorithm Based on Feasible Values and Pseudo Scaling Factor for a Power System Area Including Limited Energy Supply Thermal Units

Abstract A security-constrained power dispatch problem with non-convex cost function for a lossy electric power system area including limited energy supply thermal units fueled under a take-or-pay agreement is formulated. Then, an iterative solution method based on a modified subgradient algorithm based on feasible values and common pseudo scaling factor for limited energy supply thermal units is proposed and used to solve the problem. In the iterative proposed solution method, a modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the common pseudo scaling factor for limited energy supply thermal units is employed to adjust the amount of fuel consumed by the limited energy supply units during the considered operation period. Since all equality and inequality constraints in the non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles (the off-nominal tap settings), they are taken as independent variables except for those belonging to the reference bus. Load flow equations are added to the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, and off-nominal tap setting 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 given in this article before application of the modified subgradient algorithm based on feasible values to the optimization model. The proposed technique is tested on an IEEE 30-bus test system with a non-convex total cost rate curve. A portion of the dispatch problem considered in this article, where only a loading of the system in a subinterval is considered, was also solved by some other recently reported dispatch techniques. First, minimum total fuel cost rates and solution times obtained from the modified subgradient algorithm based on feasible values versus other techniques are compared, and the outperformance of the modified subgradient algorithm based on feasible values in terms of both total cost rate and solution time is demonstrated. After that, the optimal total fuel cost value and solution time of the system for the operation period for the cases where the fuel constraint is ignored and not ignored are presented, and it is shown that adding the fuel constraint into the dispatch problem further decreases the optimal total fuel cost for the considered dispatch problem. Finally, the modified subgradient algorithm based on feasible values was applied to the fuel constrained dispatch problem directly. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an intervals 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.

Application of modified subgradient algorithm based on feasible values to security constrained economic dispatch problem with prohibited operation zones

A security constrained economic dispatch problem with prohibited operation zones for a lossy electric power system is formulated. An iterative solution method that is based on modified subgradient algorithm operating on feasible values is employed to solve it. Bus voltage magnitudes and phase angles, off-nominal tap settings and susceptance values of svar systems are taken as independent (decision) variables in the solution algorithm. Since load flow equations are added into the model as equality constraints, actual power system loss is used in solution of the optimization model. The proposed technique is tested on IEEE 30-bus test systems. The minimum total cost rates and the solution times obtained from F-MSG algorithm and from the other techniques are compared, and the outperformance of the F-MSG algorithm with respect to the other methods in each test system is demonstrated.

Solution to Security Constrained Environmental Pumped-Storage Hydraulic Unit Scheduling Problem by Genetic Algorithm

A lossy electric power system area that contains thermal units and a pumped-storage (p-s) hydraulic unit is considered in this paper. The cost function, which is weighted combination of the total fuel cost and the total emission cost of the thermal units, in an operation cycle, is minimized under some possible electric and hydraulic constraints. The dispatch technique that is based on genetic algorithm considers minimum and maximum reservoir storage limits of the p-s unit, upper and lower active and reactive generation limits of the thermal units, upper and lower active pumping/generation power limits of the p-s unit, maximum transmission capacities of the transmission lines, and upper and lower limits of the bus voltage magnitudes in a considered power system. The proposed dispatch technique was tested on an example power system that has 12 buses with five thermal units and a p-s hydraulic unit. The same dispatch problem is solved via an iterative solution method based on modified subgradient algorithm operating on feasible values (F-MSG) and pseudowater price just for comparison purpose. It is seen that the solution technique based on the F-MSG algorithm and pseudowater price gives similar results with the proposed algorithm based on genetic algorithm.