II.3 Optimization of Hydropower Systems
II.3.2 Heuristic Algorithms
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.
next generation, mutated, and combined stochastically based on their assigned fitness levels,. This attempts to balance exploration of solutions from new areas of the design space and exploitation of solutions already found in regions of high fitness. This process terminates when stopping criteria has been reached; examples of these criteria include a maximum number of generations or solutions, a satisfactory fitness level, or a population homogeneity level being reached.
One of the earliest introductions of genetics algorithms in the water resources literature comes fromEsat and Hall(1994), where GAs were used to solve the “four-reservoir problem.” This prob- lem concerns a system of four reservoirs, with both parallel and series connections, operated over twelve 2 hour periods (a total of 24 hours), searching for optimal releases with constraints related to flood control and turbine capacities. The authors concluded that as system size increases, com- putational expense for DDDP increases exponentially while the expense of GAs increase linearly.
Wardlaw and Sharif (1999) solved the same “four-reservoir problem” as well as a more complex 10-reservoir problem, testing sensitivities to various GA settings.Oliveira and Loucks(1997) com- bined a genetic search algorithm with simulation models to determine optimal operating policy rules for several multireservoir systems, focusing on satisfying joint water demands and joint en- ergy requirements. Similarly, Suiadee and Tingsanchali (2007) used a combined simulation-GA optimization model to determine optimal monthly reservoir rule curves for a single reservoir in Thailand, with the objective function equal to the maximum net system benefit subject to irriga- tion constraints and the monthly releases computed by the simulation model. Ahmed and Sarma (2005),Chang and Chang(2001), andCheng et al.(2008) each employed various forms of GA for determining optimal reservoir operations.
GAs have been used in combination with surface WQMs. Kerachian and Karamouz(2007) de- termined optimal operating rules for the Ghomrud Reservoir-River system in Iran for water quality management using a stochastic GA-based conflict resolution technique. A one-dimensional WQM simulating thermal stratification and water quality at releases from different outlets was used, as well as simulation of pollutants in the downstream river. This one-dimensional model was based on the existing Ghomrud HEC-5Q model, which could not be easily linked to the optimization model.
Ostfeld and Salomons(2005) andHuang and Liu(2010) coupled hybrid GAs and ANN models for calibration of surface water quality CE-QUAL-W2 models. Ostfeld and Salomons(2005) reduced computational time by implementing a “hurdle race” approach which halts CE-QUAL-W2 simula-
tions early if a threshold is not met during simulation, whileHuang and Liu(2010) combined a GA with a local search method to improve the results while reducing expense. Dhar and Datta(2008) linked a CE-QUAL-W2 model with an elitist GA to determine optimal reservoir operation policy with the aim of maintaining water quality downstream of the reservoir while minimizing the storage deviation from target storage. The authors employed this method on a hypothetical reservoir on the upstream end of the Middle Willamette River in Oregon, USA for daily operating decisions over a 10 day management period. They concluded with the note that “[d]evelopment of parallel code or use of metamodels (e.g. ANNs) may be very useful in reducing the CPU time” and that those modifications would “make it feasible to solve larger and more complex real-life optimal reservoir system operation problems.”
Simulated Annealing (SA)
First introduced by Kirkpatrick et al. (1983), SA is a global search method which emulates the annealing process in glasses and metals to find optimal solutions for large systems. Using a temperature parameter, simulated annealing solves an optimization problem by theoretically max- imizing strength and minimizing brittleness. Early water resources applications of this technique were for groundwater management problems, with the first reservoir operations optimization appli- cation performed by Teegavarapu and Simonovic (2002). They used the technique to optimize a four-reservoir system for hydropower and irrigation needs, including a simulation model for com- puting reservoir states during optimization. They solved a weekly problem on a half-day timestep and showed that SA provides similar results to a mixed integer NLP problem. Then they expanded the decision space by solving for hourly operations over a weekly horizon, which the SA algorithm was able to solve in a computationally feasible manner. Tospornsampan et al. (2005) compared the performance of using simulated annealing and GAs for determining monthly operations over 3 years for a multi-reservoir system with diversions, with the goal of minimizing irrigation deficits.
Their results showed SA to be more efficient than GA for their application, generating higher quality solutions and requiring less computational time.Li and Wei(2008) also found SA to perform better than GA while optimizing a 3-reservoir system in series for electricity generation maximization.
Of the methods they tested, the authors determined that their improved GA-SA algorithm produced the highest quality solutions at a lower computational time than the traditional unimproved GA-SA
algorithm. Chiu et al.(2007) also employed a hybrid GA-SA for optimizing the operation scheme of a single reservoir in Taiwan, concluding that the method results in superior performance as well as reduced computational time due to parallel analyses.
Ant Colony Optimization (ACO)
ACO is a heuristic technique based on observations of the behavioral patterns of ant colonies.
Certain ant species are capable of finding shortest paths by using pheromone communication. ACO aims to emulate the shortest path search capabilities of these species (Dorigo and St¨utzle, 2004).
Examples of ACO use in hydropower optimization applications are limited. Kumar and Reddy (2006) compared ACO to real coded GA for optimization of a multi-purpose reservoir in India and determined that the ACO algorithm converges to more globally optimal results than GA does.The developed models were used to determine operations on a monthly timestep for both short-term and long-term horizons. Optimization objectives were minimizing flood risk, minimizing irrigation deficits, and maximizing hydropower production; no water quality objectives or constraints were considered. Jalali et al.(2007) used a special version of the ACO algorithm to overcome ACO’s difficulty handling continuous problems. A random mesh of the search space was used to minimize the chance of missing the global optimum, and the algorithm is also capable of handling discrete and continuous decision variables. The algorithm was tested on a complex 10-reservoir problem, which is “beyond the capacity of traditional DP and is difficult with variants such as DDDP [discrete differential dynamic programming], but is relatively simple to solve by LP.” The system consists of reservoirs in parallel and series and was optimized over 12 operating periods with the goal of maximizing hydropower production. ACO was able to reach solutions which were 99.8% of the known global solutions. Madadgar and Afshar (2009) extended the initial ACO discrete space search method to continuous domains, improved algorithm performance and efficiency with the addition of an adaptation operator and explorer ants, and tested their algorithm on well-known benchmark problems and a single hydropower reservoir optimization problem with the objective of minimizing the sum of relative generation deficits from the installed capacity over 240 monthly operating periods.
Particle Swarm Optimization (PSO)
PSO is a technique for searching continuous nonlinear functions inspired by bird flocking and fish schooling behavior (Eberhart and Kennedy, 1995). It can solve many of the same types of problems as GAs. PSO is similar to a GA while overcoming some of GA’s challenges, including being able to retain an active memory of good solutions. Unlike a GA, there are no evolution operators. Instead, each potential solution is assigned a random velocity, and then these “particles”
are “flown through hyperspace.” There are only two variables that must be defined by the user:
maximum velocity and an acceleration constant.
Kumar and Reddy (2007) employed elitist-mutated PSO to determine operation plans for a multipurpose reservoir. Elitist-mutated PSO improves the standard PSO algorithm by adding an elitist-mutation mechanism. In their study, Kumar and Reddy applied elitist-mutated PSO to a hy- pothetical case and then to a realistic case, the Bhadra reservoir in India, which serves irrigation and hydropower generation purposes. The system was optimized on a monthly time step, for both 1 year (short-term) and 15 year (long-term) problems. This study concluded that elitist-mutated PSO performs better than both standard PSO and GAs, by yielding better solutions with fewer function evaluations. Similarly, Zhang et al.(2013) used a modified PSO approach to determine optimal hourly discharge rates for 10 cascading hydroelectric plants in a multi-reservoir system, with the goals of minimizing power deficit and uniformly distributing deficit if it should occur. This was achieved using a multi-elite guide PSO, which incorporated an archive set which preserves elite so- lutions. Multi-elite guide PSO produced improved solutions and converged quickly in comparison with other methods.
Honey Bees Mating Optimization (HBMO or MBO)
Another swarm-based algorithm is the HBMO method, which is inspired by the mating behavior of honey-bees in nature. This algorithm typically captures the bees’ genetic potentiality, environ- ment, and colony social conditions in order to converge to optimal solutions. Haddad et al.(2006) tested this algorithm on a water resources application for the first time. First it was applied to sev- eral benchmark constrained and unconstrained mathematical functions. Then the authors applied this algorithm to optimize single reservoir monthly operations over 5 years, aiming to minimize deviations between releases and target demands. They concluded that the HBMO algorithms per-
forms similarly well to GAs. More recently,Dariane and Farahmandfar(2013) applied the similar marriage in honey bees optimization (MBO) algorithm to determine 47 years of monthly operations for a three-reservoir system under irrigation and environmental flow requirements. This represented a problem with a very large number of decision variables. Their experiments revealed that MBO proved to be superior to other algorithms tested, including GA, ACO, PSO, and elitist-mutation PSO. The authors conclude by stating that “development of a hybrid algorithm consisting of MBO and any of the GA or elitist-mutation PSO algorithms could be considered in future research to further aid in solving complex optimisation problems with a large number of decision variables.”