II.3 Optimization of Hydropower Systems
II.3.1 Classic Methods
Linear Programming (LP)
LP is one of the most popular methods for reservoir system optimization due its many advan- tages, which include efficiency, ability to solve large-scale problems, global convergence guarantee, no initial solutions required to start the algorithm, duality theory to assist in sensitivity analyses, and ease of problem setup and solution using readily available software packages (Labadie, 2004). The most notable limitation of this technique is the requirement of objective and constraint functions to be linear or linearizable and convex. These limitations can be overcome in some cases by extension methods including separable LP, successive LP, and binary, integer, and mixed integer LP; however, many reservoir systems are represented by highly nonlinear or discontinuous functions associated with reservoir hydrodynamics, power generation, and water quality. These are either not appropriate
for or cannot be efficiently solved by LP, even with extension methods.
Simple reservoir optimization problems have been solved using LP techniques. Ponnambalam et al.(1989) solved for monthly turbine releases for two reservoirs connected in series over a 40 year period, resulting in 880 decision variables and 3680 constraints. They compared the perfor- mance of simplex and interior point algorithms, concluding that the interior point method converges in far fewer steps for large problems. Crawley and Dandy(1993) used linear goal programming to identify monthly optimal operating policies for a much larger reservoir system in South Australia, with the objective of minimizing pumping costs from a nearby river for reservoir fill. The authors used separable programming to piece-wise linearize the nonlinear pumping cost curves. Needham et al.(2000) analyzed the flood-control procedures for three U.S. Army Corps of Engineers reser- voirs using a mixed integer LP model, concluding that coordinated releases may be unnecessary to minimize flood damage by showing this to be true for 8 of the 10 largest flood events on record. Ad- ditional application of optimization by LP for reservoir operations include (Martin, 1983), (Martin, 1995),Lee et al.(2006),Seifi and Hipel(2001),Ziaei et al.(2012), andMousavi et al.(2004).
Nonlinear Programming (NLP)
Because many reservoir systems cannot be realized by linear or linearizable functions, NLP techniques have been employed in previous optimization applications. NLP has the disadvantages of slow convergence, leading to large computation time requirements. There is also no guarantee of find global optima, demonstrated by NLP algorithms often converging to local optima instead.
The Karush-Kuhn-Tucker conditions for constrained nonlinear programming optimality may not be computationally feasible for many large-scale nonlinear problems (Hiew, 1987). Because of this, constrained NLP problems are often solved using penalty and barrier constraint-handling methods, which require careful choice of penalty weights and may not converge to the true feasible opti- mum. As noted byRani and Moreira(2010), software packages are available which can solve large scale nonlinear optimization problems; regardless, global optimality proves difficult for practical applications employing NLP.
This is a broad family of techniques which includes sequential linear programming (Barros et al., 2003; Grygier and Stedinger, 1985), sequential quadratic programming (Tejada-Guibert et al., 1990;Finardi et al., 2005), the augmented Lagrangian method (also known as the method
of multipliers) (Arnold et al., 1994;Naresh and Sharma, 2002;Finardi and Scuzziato, 2013), and the generalized reduced gradient method (Sale et al., 1982; Unver and Mays, 1990). All of these methods require differentiable objective and constraint functions, which may not be the case for hydropower systems due to the presence of discontinuities often associated with turbine operations.
Hiew (1987) compared various nonlinear algorithms for optimization of a system of hydropower reservoirs and concluded the sequential linear programming method to be the most efficient. Us- ing mixed integer nonlinear programming (MINLP),Teegavarapu and Simonovic(2000) optimized power generation revenues for a system of 4 hydropower plants with daily scheduling andFerreira and Teegavarapu (2012) formulated a single run-of-the-river hydropower reservoir optimization problem on a daily timestep over a 15 day operating period. They included a simplistic downstream water quality constraint to explore dam operations’ ability to counteract a downstream pollutant point source. Although formulated as a MINLP, the authors opted to solve the problem using GAs, noting that the “reduced gradient based method used initially in this study as optimization solver provided unsatisfactory (i.e., non-optimal) solutions.”
Dynamic Programming (DP)
DP methods are able to address nonconvex and discontinuous functions and their structure em- ulates the multistage decision-making process involved in reservoir system operations (Labadie, 2004). DP breaks the original problem into subproblems that are then solved in stages sequentially.
For each subproblem, an optimal cost-to-go function is developed which represents the optimal value accumulated from the current period going forward, as a function of an initial state condition.
For the majority of reservoir applications, the state consists of reservoir storage. If additional states are relevant to the constraint and objective formulations, such as the inclusion of water quality or additional reservoirs, the size of the problem grows quickly; this has been coined the “curse of di- mensionality” associated with DP. Discrete DP overcomes difficulties due to nonlinear, nonconvex, and discontinuous objective and constraint functions (Labadie, 2004).
The earliest application of determining optimal operating rules for a single multi-purpose reser- voir using deterministic DP was performed byHall et al.(1968). Their technique provided for what were considered to be “complex constraints” at the time, including time-variable flood control reser- vations; mandatory fish, wildlife, and recreational releases; and navigation minimum flows. This
resulted in an optimal schedule of releases for each month given a price schedule. Stedinger et al.
(1984) developed a stochastic DP model to define releases from a dam in the Nile River Basin based on the best inflow forecast as a hydrologic state variable, resulting in improved operations com- pared to using the proceeding period’s inflow as the state variable. Georgakakos et al.(1997) used a combination of dynamic programming and optimal control method modules to maximize firm energy generation of the Lanier-Allatoona-Carters hydropower system across multiple timescales (instantaneously, hourly, and daily).
Optimization of many linked reservoirs becomes computationally infeasible using the original DP formulation, which is the reason much of the hydropower optimization by DP literature involves modified DP approaches. Castelletti et al. (2007) employed neuro-dynamic programming, which approximates Bellman functions with ANNs, for reservoir network management. Yi et al.(2003) solved a multireservoir unit allocation problem using dynamic programming with successive ap- proximation, a technique which “replaces the original multidimensional problem with a sequence of 1D problems” and whose computational expense increases linearly with respect to the problem size. Wang et al.(2005) was able to solve a problem combining multiobjective optimization (hy- dropower, water supply, and flood control), a multireservoir system (three reservoirs in parallel), and stochastic inflows using a combination of modifications. These included a constraint technique (to transform the optimization to a single objective form) and combined decomposition iteration and simulation analysis to overcome the dimensionality problem. El-Awar et al.(1998), Yurtal et al.(2005), andZhao et al.(2014) also employed modified DP approaches to solve for optimal hydropower reservoir operations.