Hydro Unit Commitment via Mixed Integer Linear Programming: A Case Study of the Three Gorges Project, China

Published: 1/11/2018

The Three Gorges Project, located on the Yangtze River, is upstream from the city of Yichang in the Hubei Province of central China. It is the world’s largest hydropower system and has the three main purposes of flood protection, power generation and navigation. Regulation of forebay water level and release is another key priority of the Three Gorges Project’s operation. It is operated by the Three Gorges Corporation, though the power generation of the Three Gorges Project is under the jurisdiction of the power industry based on the day-ahead schedule. There are 32 main generating units installed and in operation in three different hydropower plants of the dam-behind type. There are seven types of units installed, 14 units are installed in the left-bank hydropower plant, 12 in the right-bank hydropower plant, and six in the underground hydropower plant. The installed capacity of 22,500 MW generated from the Three Gorges Project supplies the electricity demand for nine provinces and two cities in China. The power is transmitted to the State Grid Corporation of China and the China Southern Power Grid Co., Ltd with a constant energy price.

In this subject paper, a mixed integer linear programming model was developed for solving the hydro unit commitment problem of the Three Gorges Project, though other methods have been explored historically. The unit commitment problem is a mathematical optimization problem involving the determination of when to startup or shutdown hydro units. The problem also tackles how to schedule the online units over a specific short-term time period to minimize the operational cost of a generating company. The problem is formulated as a large-scale, mixed-integer, combinatorial, and nonlinear programming problem. Common methods developed for solving the unit commitment problem include priority listing, dynamic programming and Lagrangian relaxation. Heuristic algorithms used in solving the unit commitment problem include simulated annealing, artificial neural networks, genetic algorithms, and evolutionary programming. Methods are combined to take advantage of each one and to develop the best solution technique for the particular problem being considered. The hydro unit commitment problem has a large potential for cost savings and benefits. The major challenges in solving it include the nonlinearity of the performance curves as well as the head effect on power production.

Mixed-integer linear programming has the availability of more efficient and user-friendly software and has been used to solve the short-term hydro scheduling problem. To do so, the nonlinear unit performance curve is approximated by concave piecewise linear approximation under the assumption of constant head. A nonlinear programming model in which the complex constraints are simplified by assumptions was proposed to solve the short-term hydro scheduling problem. The assumptions include a linear power efficiency function, a linear head function, and a linear maximum total release function. As a result, the power generation function is expressed as a nonlinear function. The nonlinear programming approach can produce more benefits with negligible increases in computation time and is possible because the model does not consider the startup and shutdown statuses of units. This assumption is reasonable for short-term hydro scheduling, but not for the hydro unit commitment problem. A mixed integer nonlinear programming model has also been used for scheduling a reservoir system. The unit efficiency is estimated as a quadratic function of head and power release. The power generation of a unit can be formulated as a continuous, nonlinear and non-concave function. The mixed integer nonlinear programming model outperforms the mixed integer linear programming model in terms of accuracy.

The subject case study, The Three Gorges Project, is large-scale, real-world, and highly complicated. A practical, real-time optimization model for the hydro unit commitment problem was developed to remove the assumptions and simplifications employed in the past to increase accuracy. The model developed possesses the following key features, the maximum allowable head change could reach as much as 5.0 m/day, the relationships between head and storage as well as maximum power release and head are nonlinear, and the generating units and the operation status of each unit varies. The model addressing the hydro unit commitment problem of the Three Gorges Project focuses on accuracy and computational efficiency while still taking the individual heterogeneous generating units into account. This was done using a three-dimensional interpolation method to simulate the nonlinear power generation function of each unit.

Two nonlinearities exist in the modeling of the hydro unit commitment problem which poses a major challenge. One is the net head on a unit, the other is the power generation curves of a unit, expressed as a nonlinear function of power release and net head. The difficulties in accurate modeling of unit power generation stem from the nonlinearity and non-concavity of the unit performance curves. The model is further complicated by the tailrace water level, which is affected by the forebay water level of the Gezhouba, a reservoir immediately downstream from the Three Gorges Project. An iterative method was used to obtain the Three Gorges Project tailrace water level.

The final formulated mixed integer linear programming model was solved with a commercial optimization software package called LINGO. The branch-and-bound algorithm was called to solve the model, which has 14,816 variables, including 4,608 binary variables and 13,329 constraints. Hourly operation was considered, and a total of 24 hourly time-periods as the time horizon. Among the three scenarios considered, the narrowest final gap is 2.30%, while the widest final gap is 4.11%. The coefficient of determination for the three scenarios is above 0.98 and the MRE is below 0.0034. This validates the symmetry assumption for the unit performance curve. With a project of this magnitude, a small improvement in operation can translate into large benefits.