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Lagrangian relaxation has been widely used for the hydrothermal scheduling of power systems. The idea is to use Lagrangian multipliers to relax system-wide demand and reserve requirements, and decompose the problem into unit-wise subproblems that are much easier to solve. The multipliers are then updated at the high level, most commonly by using a subgradient method (SGM). Since the high level dual function is non-differentiable with many 'ridges,' SGM may zigzag across ridges resulting in slow convergence. This paper presents an algorithm that utilizes a recently developed 'reduced complexity bundle method' (RCBM) to update the multipliers at the high level. The RCBM is a kind of 'bundle method' that enjoy faster convergence compared to SGM, but has much reduced complexity as compared to a conventional bundle method. Testing results show that RCBM can find better directions, avoid zigzagging behavior, and obtain better dual and feasible solutions as compared to the SGM.

With the deregulation of electric power systems, market participants are facing an important task of bidding energy to an Independent System Operator (ISO). This paper presents a model and a method for optimization-based bidding and self-scheduling where a utility bids part of its energy and self-schedules the rest as in New England. The model considers ISO bid selections and uncertain bidding information of other market participants. With appropriately simplified bidding and ISO models, closed-form ISO solutions are first obtained. These solutions are then plugged into the utility's bidding and self-scheduling model which is solved by using Lagrangian relaxation. Testing results show that the method effectively solves the problem with reasonable amount of CPU time. Index Terms--Bidding strategies, Lagrangian relaxation.

Lagrangian relaxation has been widely used in hydrothermal scheduling. Complicating constraints are relaxed by multipliers which are usually updated by a subgradient method (SGM). The SGM suffers from slow convergence caused by the non-differentiable characteristics of dual functions. This paper presents an algorithm that utilizes the Bundle Trust Region Method (BTRM) to update the multipliers within the Lagrangian relaxation framework. The BTRM is shown to converge faster than the SGM as well as other bundle type methods in optimizing non-differentiable dual functions. The application of BTRM for solving hydro subproblems results in greatly improved convergence over the SGM. Comparing BTRM with another bundle type method in updating the high level multipliers shows that better solution can be obtained by BTRM. Keywords: Hydrothermal Scheduling, Bundle Methods, Lagrangian Relaxation, Identical Units