IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 2, MAY2006 835 Solving the Hydro Unit Commitment Problem via Dual Decomposition and Sequential Quadratic Programming Erlon Cristian Finardi and Edson Luiz da Silva, Senior Member, IEEE Abstract—This paper presents an algorithm that achieves the hydro unit commitment in hydrothermal systems. This problem is difficult to solve since several constraints with continuous and discrete variables exist, including hydraulic coupling, storage and released flow limits of the reservoirs, and unit forbidden operation zones. The forbidden zones cause a noncontinuous operation of the generating units, making the solution of the problem more diffi- cult, due to the associated combinatorial nature. Moreover, there exists the presence of nonlinear functions that represent the tail- race level, the hydraulic losses, and the unit efficiency. To solve a problem that contains all of these characteristics is a challenging task. Within this scenario, an algorithm is presented that makes use of Lagrangian relaxation, in which some variables are artificially duplicated in order to separate the problem into simpler subprob- lems. The idea is to relax the spatial and temporal coupling present in the constraints associated with the forbidden zones. In order to solve the subproblems of nonlinear continuous nature that result from the relaxation, this paper presents a sequential quadratic pro- gramming algorithm. To update the Lagrange multipliers, an algo- rithm based on the Bundle Method is used. We assess our approach on a real-life hydroelectric configuration, proving the conceptual and practical feasibility of the proposed algorithm. Index Terms—Hydrothermal systems, hydro unit commitment, Lagrangian relaxation, sequential quadratic programming. I. INTRODUCTION T HE HYDRO unit commitment problem is an important activity that must be carried out in power systems. The objective consists of determining which units must be uti- lized to satisfy the demand and other operational constraints, with minimum operating cost. This problem is a large-scale mixed-integer nonlinear programming problem. For this reason, Lagrangian relaxation (LR) has been the solution strategy com- monly adopted [1]–[5], although other methods were proposed in literature [6]–[10]. The divide-and-conquer approach of LR, also called price decomposition, is well known. Essentially, coupling constraints are relaxed via Lagrange multipliers whose corresponding dual problem is decomposable into simpler subproblems (called local subproblems). The coordination of subproblems is then done by a master program, which finds new multi- Manuscript received February 3, 2005; revised November 23, 2005. This work was supported in part by CNPq—National Council for Scientific and Tech- nological Development and in part by CEPEL—Centro de Pesquisas de Energia Elétrica. Paper no. TPWRS-00075-2005. The authors are with Universidade Federal de Santa Catarina, Trindade, CEP 88040-900, Florianópolis, SC, Brazil (e-mail: erlon@labplan.ufsc.br; edson@labplan.ufsc.br). Digital Object Identifier 10.1109/TPWRS.2006.873121 pliers by making one iteration of the nonsmooth algorithm that maximizes the dual function. Regarding the optimum short-term hydrothermal scheduling problem, in general, there are individual thermal unit (thermal subproblem) and river catchment subproblems [hydroelectric subproblem (HS)] [11]. In hydrothermal systems with several hydro-valleys of inter- connected reservoirs, the HS is more difficult to solve than the thermal subproblems. Given that the stored water can be utilized in the following stages, the decision to generate in the present or leave water stored for future use becomes a problem coupled in time. Still, the presence of hydro-valleys of interconnected reservoirs makes the operation coupled in space, since the released outflow of a plant affects the operation of all the plants downstream. Due to the multiple use of the water, factors such as irrigation, flood control, and navigation restrict the released outflow of the plants. Additionally, the travel time of the water and the storage constraints produced by long-term operating planning models [12] must also be taken into consideration. The generating units also present a complex operational be- havior. The power output from a hydro generating unit depends on the efficiency of the turbine and generator, the net head, and the plant discharge. On the other hand, the net head is a non- linear function of the water storage and the released flow of the reservoir. The resulting efficiency of the turbine-generator is a nonlinear function of the net head and the unit outflow. Besides this, a hydro unit presents forbidden zones that do not permit an ample and continuous range of the generation. These zones re- sult from mechanical vibrations (which cause an oscillation in the power output), cavitation, and low efficiency level. This paper presents an algorithm that seeks to solve the hydro unit commitment problem, which considers the features afore- mentioned. Through the artificial variable technique, a separable dual problem is conceived, consisting of the resolution of sev- eral independent subproblems. One subproblem is of a linear nature, coupled in time and space, in which the streamflow bal- ance equations, operational limits of the reservoirs, and the con- straints supplied by long-term models are represented. Although of large scale, this subproblem can be efficiently solved by any linear programming (LP) algorithm. The remaining subprob- lems are nonlinear mixed-integer, uncoupled in time and space. In these subproblems, the unit operating constraints of a given plant and time stage are addressed, such as the forbidden zones. In order to solve them, the work carried out a strategy in ac- cordance with the number of units and zones, from an exhaus- tive enumeration of the combinations to the new decomposi- 0885-8950/$20.00 © 2006 IEEE