Fuzzy constrained optimization of eco-friendly reservoir operation using self-adaptive genetic algorithm: a case study of a cascade reservoir system in the Yalong River, China

INTRODUCTION

Reservoir operation for both water supply and hydroelectric generation is not always rational to satisfy the full current demand because of insufficient insight into the changing inflows and fuzzy water requirements. With currently less development and utilization of hydropower resources in comparison with developed countries, China is prioritizing hydropower in the national new energy agenda. Furthermore, hydropower is clean. It provides a cheap source of electricity with few carbon emissions (Yoo, 2009). Therefore, the potential for development of Chinese hydropower resources is enormous in a long period. On the other hand, however, well-known detrimental effects arise, such as impoundment of free-flowing river habitat, blockage of fish migration, and reduced water quality in reservoirs and downstream river reaches (Postel and Richter, 2003; Wohl, 2004; Graf, 2006; Richter and Thomas, 2007; Jager and Smith, 2008). It is no longer possible to exploit water resources without taking into consideration the ecological needs.

Dam construction on rivers remains the main form of hydropower resources exploitation. Hydroelectricity is generated by implementing reservoir operation and running turbine generators. Generally, reservoir operation influences temporal flow patterns, which have profound impact on the health of the downstream ecosystem (Arthington et al., 2006; Jager and Smith, 2008). Consequently, water release from a reservoir is increasingly required to guarantee necessary ecological flows at downstream locations, and ‘ecological flows’ refers to water considered sufficient for sustaining the structure and function of an ecosystem and its dependent species. There have been many studies on reservoir release optimization that consider the ecological flow requirements in the objective or constraints of ecological demands (Gibbins et al., 2001; Arthington et al., 2006; Suen and Eheart, 2006; McCartney, 2007; Bauer and Olsson, 2008; Jager and Smith, 2008). However, the ecological target was often set as a single value of minimum instream flow requirements that eliminated the flow variability sustaining healthy and diverse aquatic communities. Also, there has been a great deal of efforts for evaluating the ecological flows and to identify the needs of maintaining downstream ecosystem integrity to protect biodiversity (Poff et al., 2003; Lytle and Poff, 2004). They demonstrated the importance of mimicking the components of natural flow regime. However, it is difficult to simulate the natural flow process in reservoir operation due to the larger number of variables. Moreover, translating general hydrologic-ecological principles and knowledge into specific operating rules remains a daunting challenge (Harman and Stewardson, 2005; Chang et al., 2010; Jing and Chen, 2011). Hydrologic methods for estimating the ecological flows require less understanding of the ecosystem processes and goals, and they are less data-intensive than hydraulic and habitat methods. Tennant (1976) suggested that 10% of the average flow provided minimum protection and 30% of average flow was satisfactory.

Uncertainties often exist in human knowledge and ecological studies (Zhang et al., 2011). A large inherent uncertainty of ecological information results from the presence of random variables, incomplete or inaccurate data, and approximate estimations instead of measurements (Salski, 2003; Ahmadi-Nedushan et al., 2006). One way of handling the uncertainty is to use fuzzy-based approaches. According to Tennant's (1976) principle, the ecological flow requirements are uncertain, and it is difficult to directly incorporate the rule into the operation programming. Fortunately, the ecological flow as Tennant suggested can be quantified by fuzzy membership function. As a result, it is viable that the ecological flow requirements be presented as fuzzy constraints when optimizing the reservoir operations (Huang and Chang, 2003; Li, 2003). Fuzzy programming considers random parameters as fuzzy numbers and constraints as fuzzy sets. Constraint violation is allowed to a degree that it is defined as the membership function of the constraint (Huang et al., 1993; Yin et al., 1999; Sahinidis, 2004; Li et al., 2007; Lv et al., 2010; Sun and Huang, 2010). Furthermore, decision makers may establish an aspiration level and a tolerable interval for the objective that they desire to achieve, and each constraint can be modeled as a fuzzy set. A fuzzy programming problem can be transformed into deterministic problems for improving the communication between the fuzzy objective and constraints and thus generating satisfactory solutions to support decision making (Maqsood et al., 2005; Nie et al., 2007).

Application of optimization techniques to reservoir operation problems has been a major focus of water resources planning and management (Chen, 2003; Ahmed and Sarma, 2005; Mousavi et al., 2005; Ngo et al., 2007; Cheng et al., 2008; Hakimi-Asiabar et al., 2010). The previous studies indicated various complexities in water resources management. Labadie (2004) reviewed various optimization methods of reservoir system operation. For maximizing hydropower generation, the objective function may be highly nonlinear. Evolutionary algorithms are capable of generating globally optimal solution for problems where traditional algorithm methods would fail to converge or be trapped in local optima and have more advantages of robust ability to solve highly nonlinear, non-convex reservoir operation problems (Labadie, 2004). In recent decades, evolutionary computation techniques, such as genetic algorithm (GA), have become popular in global optimization of water resource management with their potential in optimization techniques for complex systems (Cai et al., 2001; Huang et al., 2002; Kerachian and Karamouz, 2007; Chaves and Chang, 2008). Oliveira and Loucks (1997) used GA to evaluate operation rules for multi-reservoir systems and demonstrated that GA could be used to identify effective operating policies. Subsequently, more and more Genetic Algorithms (GAs) are applied to the optimization of reservoir operations (Cheng et al., 2002; Chang et al., 2003; Cheng et al., 2006).

It has been shown that real-coded GAs are more suitable for large dimensional search spaces than binary-coded GAs, since they are more consistent and precise and they lead to faster convergence (Baskar et al., 2003). Moreover, a binary GA requires the knowledge of bounds on parameter values bracketing the true optimal solution. A real-parameter GA does not demand this knowledge and can be used in problems where such information is not available. Nonetheless, relative computational complexity, genetic drift, inaccuracy in local search intensification, and slow rate convergence remain to be solved in GA. In order to enhance the efficiency of GA, self-adaptation feature is incorporated in the real-coded GA using special crossover operators, such as simulated binary crossover (SBX), which create offspring statistically located in proportion to the difference of the parents in the search space (Deb and Beyer, 1999; Su and Chiang, 2004; Subbaraj et al., 2011). The population diversity is introduced by making use of distribution index in the SBX operator to create a better offspring. This leads to a high diversity in population, which can increase the probability toward the global optimum and prevent premature convergence (Subbaraj et al., 2011). However, in the SBX operator, the variance of children distribution depends on the distance between the two parents. It is important to use a distribution that assigns more probability for creating near-parent solutions than solutions away from the parents (Deb and Beyer, 2001).

Therefore, one approach to potentially address the uncertainties in ecological flow requirements and the complexities in nonlinear problems is to integrate self-adaptive GA with SBX to fuzzy programming framework. Such an approach would directly incorporate the uncertainties of ecological flows expressed as fuzzy membership functions. Moreover, it possesses the excellent attributes of self-adaptive evolutionary algorithms for solving complex problems. This leads to a fuzzy constrained self-adaptive GA for implementing eco-friendly reservoir operation. It could bridge the gap between simple minimum ecological flows and the great complexity of the natural flow regime, and help decision-makers evaluate alternative operating rules. Furthermore, it will be expected to effectively achieve the optimization of an eco-friendly reservoir operation. In this paper, the methodology is successfully applied to the cascade system of Jingping reservoirs in southwest of China. The results demonstrate how the methodology performs to obtain hydropower generation and to provide ecological releases for the downstream river. It is also shown that the proposed methodology is useful for developing monthly operating rules for reservoirs and implementing eco-friendly reservoir operation. A monthly reservoir operation model for cascade reservoirs is established, aiming to obtain optimal operating schemes for maximizing hydropower generation while providing satisfactory ecological releases at downstream river reach of the cascade system of reservoirs.

METHODOLOGY

Framework

Figure 1 presents the framework of the proposed methodology. As it can be seen in the figure, a fuzzy programming model is first transformed into two deterministic sub-models to obtain the most desirable and least desirable objective values (Huang et al., 1993; Cai et al., 2001). In order to solve constrained nonlinear programming problem, the self-adaptive GA with SBX is used to search for the optimal solutions. To tackle uncertain information in the model, membership function is introduced to combine the objective function and fuzzy constraints into an integrated deterministic constrained nonlinear programming problem, and the proposed GA with SBX method is then iterated to search for the maximal membership degree. Thus, it is supposed that the optimal objective with adequate satisfaction of fuzzy constraints is achieved.

Fuzzy programming

In many real-world problems, some uncertain parameters are considered as fuzzy numbers and constraints and can rarely be acquired as deterministic values. Instead, they may be often given as subjective information and can be treated as fuzzy sets (Sahinidis, 2004). According to the seminal paper by Bellman and Zadeh (1970), flexible programming as one type of fuzzy programming deals with right-hand sided uncertainties in the constraints. Some constraint violation is allowed and the degree of satisfaction of a constraint is defined as the membership function of the constraint. A fuzzy constrained programming model can be defined as follows:

(1)

where f is the objective, X is the decision variable, and A, B, and C are parameters; the symbol ‘’ denotes that the relationship between AX and B is fuzzy, and in order to obtain a deterministic solution, B is always considered as an interval parameter [B⁻, B⁺], with B⁻ and B⁺ representing the lower and upper bounds of fuzzy set B, respectively.

Solution scheme

In model (1), the objective function is assumed to change monotonously with the increase or decrease of the deterministic value in [B⁻, B⁺]. Hence, model (1) can be decomposed into two sub-models, models (2) and (3), to obtain two desirable objective values, i.e. f ⁻ and f ⁺, with f ⁺ as the most desirable objective value and f ⁻ as the least desirable objective value. As a result, the optimized objective value of model (1) will be located in [f ⁻, f ⁺].

(2)

(3)

To better communicate the fuzzy objective and constraints, model (1) can be transformed into the following deterministic model:

(4)

where λ is the decision variable and corresponds to the degree of satisfaction of the fuzzy objective or constraints, and the λ value ranges between 0 and 1. Values closer to 1 correspond to a solution with a higher possibility of satisfying the constraints/objective under more advantageous system conditions; conversely, values near 0 relate to a solution that has a low possibility of satisfying the constraints/objective under conservative conditions.

Constrained genetic algorithm

The GA proposed by Holland (1992) has been widely utilized to solve different optimization. It mimics processes in natural evolution and progresses to improve the fitness of solutions through iterations by implementing operators including selection, crossover, and mutation operators. Although the GAs are efficient in searching for globally optimal solutions, they still encounter many difficulties in dealing with complex constraints. Consequently, when GAs are applied to nonlinear constrained problems, the constraint-handling technique becomes critical (Subbaraj et al., 2011). Real-world search and optimization problems can be written as a nonlinear programming problem of the following kind:

(5)

where is the objective function, is the jth inequality constraints, and is the kth equality constraints. The ith variable varies in the range [,]. A solution that satisfies all the above equality and inequality constraints and the above variable bounds is called a feasible solution. The others are called infeasible solutions.

Generation of initial population

Real-coded method is used to code each chromosome as a vector of floating point numbers, with the same length as the vector of decision variables. The real coding representation is accurate and efficient because it is close to the real design search space. Here, a vector (p₁, p₂, p₃,…, p_n) represents a solution to the optimization problem. Initial individual is generated as:

(6)

where p_i,min and p_i,max are the lower and upper limits of p_i, and α_i is the uniformly distributed random number in the range 0–1.

The vector (p₁, p₂, p₃,…, p_n) is identified as an individual, and M individuals are combined to form the initial population. The fitness of each individual is then evaluated. The fitness is defined as a value that quantifies the optimality of a solution in such a way that any individual in the population can be compared and ranked against all the other individuals.

Tournament selection

The tournament selection method has several advantages (Goldberg and Deb, 1991). It is faster than the roulette wheel selection method (RWS) as a result of the computation of the average fitness required in the RWS. Moreover, the RWS is always noisy and suffers from scaling problem. In the case of tournament selection, sets (of fixed size called tournament size) of individuals are randomly chosen from the population, and the fitness of individuals from each set is compared with one another, and the individual with better fitness is taken to the mating pool. Consequently, tournament selection is easy to parallelize, and the scaling problem is eliminated by making the individuals compete with each other to gain entry to the mating pool. According to Blickle and Thiele (1995), the loss of diversity of the population increases with the increase in the tournament size. Hence, in order to keep the diversity loss to the minimum, the tournament size is chosen as two in this paper. This selection process is repeated once again in order to make the number of individuals in the mating pool equal to the population size. As a result, two copies of the best parents from the population will be present in the mating pool, whereas the worst parents will be eliminated (Subbaraj et al., 2011).

Generation of children solutions by simulated binary crossover

The SBX operator works with two parent solutions and creates two children solutions based on the single-point crossover operation of binary strings (Deb and Beyer, 2001). During this operation, common interval schemata between the parents are preserved in the children. The procedure of computing the children c₁ and c₂ from the parents x₁ and x₂ is described as follows.

First, a random number u between 0 and 1 is generated. Thereafter, from a polynomial probability distribution, parameter β_q is defined as:

(7)

where , and β is calculated using Equation (8):

(8)

where the parameter x is assumed to vary in [x_l, x_u], and the parameter η_c is the distribution index for SBX and can take any nonnegative value. A small value of η_c allows solutions far away from parents to be created as children solutions, and a large value restricts only near-parent solutions to be created as children solutions. The children solutions are then obtained as follows:

(9)

It is assumed here that x₁ < x₂. A simple modification to the above equation can be performed for x₁ > x₂. For handling multiple variables, each variable is chosen with a probability 0.5 and the above SBX operator is applied variable by variable. In all simulation results here, we have used η_c = 1.

Parameter-based mutation

A polynomial probability distribution is used to create a solution c in the vicinity of a parent solution x under the mutation operator. The following procedure is used for a parameter x ∈ [x_l, x_u].

First, a random number u between 0 and 1 is generated, and then the parameter δ_q is calculated as follows:

(10)

where δ = min[(x − x_l), (x_u − x)]/(x_u − x_l). The parameter η_m is the distribution index for mutation and takes any nonnegative value.

After obtaining the value δ_q, the mutation child is given as:

(11)

In order to get slight perturbation and prevent the individuals from premature convergence, we set η_m = 100 + t, and the probability of mutation is changed as follows:

(12)

where t and t_max are the current generation number and the maximum number of generations allowed, respectively. Thus, in the initial generation, we mutate, on average, one variable (p_m = 1/n) with an expected 1% perturbation, and as generations proceed, we mutate more variables with lower expected perturbations.

Criteria of stop

The algorithm stops when the specified maximum number of generations is reached, or it terminates early depending on the unsuccessful generation of the algorithm. Two rules for stopping the progress of the algorithm are used: (i) the best solution is not changing for a specified interval of generations; (ii) the algorithm stops when Equation (13) is satisfied (Subbaraj et al., 2011):

(13)

where f_t is feasible solutions at the tth generation and f_t − 1 is feasible solutions at the (t − 1)th generation.

Penalty constraint handling strategy

To effectively implement the GA in solving a constrained optimization problem, we usually adopt the external penalty functions to convert a constrained optimization problem into an unconstrained one. There are two possible ways to construct the fitness function with penalty terms. Usually, a fitness function is constructed by adding penalty values associated with the infeasible solutions to the objective function. However, there is no straightforward way to get an effective penalty function, and a troublesome trial-and-error procedure must be undertaken to obtain a suitable set of penalty parameters and functions. In this study, we devoted a large amount of effort to reach a suitable penalty function, and the penalty form is used as follows (Deb, 1995):

(14)

CASE STUDY

The study area

The Yalong River is located in the Sichuan province, southwest China. The mainstream of Yalong River is 1571 km before joining the Yangtze River and drains an area of 136000 km². The mainstream has a natural drop of 3830 m, and gives an annual runoff of 60.20 billion cubic meters. The river has a theoretically hydropower reserve of 33.40 GW and a technically development capacity of 29.188 GW. This would be equal to an annual power generation of 151.636 TW∙h, accounting for roughly 5% of the total generation of the whole country.

A development scheme of 21 cascade reservoirs (or dams) has been planned along the mainstream, which have a total capacity of 29.17 GW and put the river in the fourth place in terms of installed capacity among China's 13 major hydropower bases. As illustrated in Figure 2, the Jinping-Ι and Jinping-П cascade projects are located in the midstream of the Yalong River. Their construction is expected to be completed in 2014. The Jinping-Ι reservoir is of yearly regulation, with the normal pool level and the dead level as 1880 and 1800 m, respectively. The Jinping-П is designed to cut the 150-km river bend by a group of power tunnels to use the natural drop created by the bend, and its dam is 7.5 km downstream of Jinping-Ι. The Jinping-П reservoir itself only has a capacity of daily regulation with the normal pool level and the dead level as 1646 and 1640 m, respectively, but when jointly operated with Jinping-Ι, it also has the capacity of yearly regulation.

When the Jinping-П reservoir operates, most of the storage water will be diverted by tunnels for generating hydropower. As a result, less water is maintained in the river bend, although there is a tributary Jiulong He with a mean annual runoff of 106.5 m³/s, which is located 36 km downstream of the Jinping-П dam. According to the investigation, the Jinping river bend provides habitats and spawning/incubation grounds for many important fish species of the subfamily Schizothoracinae, which are among the unique species found upstream of the Yangtze River. Hence, it is necessary to take into consideration the downstream ecological flow requirements when planning reservoir operations. Figure 3 shows the natural inflows at the Santan hydrological monitoring station located upstream of the Jinping-Ι reservoir during the hydrologic year from June of 2001 to May of 2002.

RESERVOIR OPERATION FORMULATION

Objective function

According to the planning and designing proposal, the two reservoirs are constructed mainly for hydropower generation. Therefore, hydroelectric power was used as the objective in this study:

(15)

where E is the total cascade annual hydropower outputs (kW∙h), K is the hydropower output coefficient, t is the number of time steps, Q is the discharge through turbines (m³/s), ∆H is the net water head (m), and T is the operation time (s). Superscripts 1 and 2 represent the Jinping-Ι and Jinping-П reservoirs, respectively. In this study, a monthly reservoir operation model was developed; thus, T is 1 year and t is a single month in the year.

Constraints

The reservoirs' operation should follow the operation regulation rules and comply with many physical constraints, such as the reservoir water balance equation, the permissible bounds of water release and storage, and the final storage target.

(16)

(17)

(18)

(19)

(20)

(21)

where V, S, Q_in, Q_up, Q_loss, and N are reservoir storage (m³), discharge through other structures besides turbines (m³/s), inflow from upstream, water withdraw from the reservoir, water loss by evaporation and leakage, and hydropower output, respectively; V_d and V_m are the dead storage and the capacity of the reservoir; Q_max is the upper release limit through the turbines; N_f and N_i are the firm output and installed capacity of the generator; is the fuzzy ecological flow of downstream with the upper and lower limits of Q_emax and Q_emin, and the superscripts 1 and 2 represent the Jingping-Ι and Jinping-П reservoirs, respectively. Equation (16) is related to the water balance equation, and Equations (17)–(19) are related to the upper and lower bounds on release and storage, the release through the turbines, and the hydropower output, respectively. Equation (20) defines the relationship between hydropower outputs and discharges with net heads of hydroelectric power stations. The downstream ecological flow was assured by Equation (21).

RESULTS AND DISCUSSION

The fuzzy constrained reservoir operation model of the two cascade reservoirs was computed using the water flow data of the hydrologic year 2001 as inputs following the proposed method, as discussed in the above sections. First, the operation model was converted into two sub-models by letting be Q_emax and Q_emin. Note that both Q_emax and Q_emin have 12 variables, which means 12 ecological flow constraints of each reservoir in the operation model. Then, a population with 100 individuals, each of which was composed of 24 variables of water levels, was evolved to search for the optimal population and the best individual. Finally, the optimized total hydropower generation of two reservoirs and optimal downstream releases could be obtained. When the ecological flow is based on the minimum requirements, the optimized hydropower generation E1 is 4.146 × 10¹⁰ kW∙h. Whereas the ecological flow reaches the most desirable value in Equation (21), the optimized total generation E2 reaches 4.519 × 10¹⁰ kW∙h. Consequently, the final hydropower generation of reservoir operation would be in the range of 4.146 × 10¹⁰ to 4.519 × 10¹⁰ kW∙h under the ecological flow requirements. To achieve a compromise between generation and ecological flow, it needs to maximize the membership degree λ for the objective and constraints at the same time. After implementing the integrated deterministic reservoir operation model, the optimal hydropower generation E_opt is 4.379 × 10¹⁰ with an optimal membership degree of 0.62492. Thus, feasible discharges from the Jinping-П dam are released to maintain the downstream aquatic ecosystem with allowable loss of hydropower generation.

The computation processes of GA with the lower and upper bounds of the ecological flow requirements were further examined. As shown in Figure 4, the fitness values decrease dramatically during the evolutional processes of solutions. It is suggested that the computation method featured fast convergence, and the optimal solutions converged at the generation of less than 310. The convergence rate under the ecological flow constraints of Q_emin is faster than that under the ecological flow constraints of Q_emax, while the optimal solutions under the ecological flow constraints of Q_emin are reached until the iteration number reaches 305. In comparison with Q_emin, higher ecological demands downsized the solutions search space under the ecological flow constraints of Q_emax. However, there is still difficulty in searching for the optimal solutions in the local region. Generally, the entire evolution process is mainly dominated by the SBX operator. However, the function of mutation operator for preventing premature convergence is so significant that the optimal solutions must be achieved.

Water levels

Figure 5 represents the water levels of the Jinping-Ι and Jinping-П reservoirs over the hydrologic year. In order to achieve good utilization of hydropower on the condition that sufficient releases can be satisfied, the Jinping-Ι reservoir has 6 months to be implemented at the normal storage water level. While the Jinping-П reservoir mostly holds the dead water level during the year, the Jinping-П reservoir only has a capacity of daily regulation; consequently, the water level should be considered as the water level at the end of each month, and the daily changes of the water levels could be determined through short-term operation planning. Nonetheless, when operated jointly with the Jinping-Ι reservoir, the Jinping-П reservoir also has the capacity of yearly regulation. In the flood season, the Jinping-Ι reservoir stores water to elevate the water level, and in order to regulate the flood, abandoned water is directly released to the Jinping-П reservoir over the period from June to October. Especially, the abandoned water is as high as 4.67 × 10⁹ m³ in September with an average inflow rate of 4105.89 m³/s. In the dry season, the Jinping-Ι reservoir operates at the normal water level with decreasing hydropower generation because of low inflows from November to February. Then, the water level of the Jinping-Ι reservoir gradually decreases to ensure adequate water releases until the dead water level to guarantee the firm output of the Jinping-П generation station and to meet the necessary requirements of the downstream aquatic ecosystem of the Jinping-П reservoir.

Ecological flows

The downstream releases from the Jinping-П reservoir, Q_down, should be equal to or greater than the ecological flow demands as required by Equation (21). Figure 6 depicts the changes of Q_down during the year, and we can find that the downstream releases are mostly between the bounds of Q_emax and Q_emin, and the minimum ecological flow is satisfied over the year. In September, the inflow rate is so high (4105.89 m³/s) that the two hydroelectric power stations are run at their maximum hydropower outputs. To regulate the flood, the downstream release from the Jinping-П reservoir is greater than the most desirable ecological flow requirement in September. We also compared the downstream releases with the ecological flows proposed by the habitat method (Li et al., 2010) as shown in Figure 6. The optimal releases mostly guarantee the requirements of downstream habitat, although there is a lack of water release in March, April, and May as a result of special requirements of the target fish species chosen by the habitat method; it also demonstrates the feasibility and applicability of the proposed method in this study.

Ecological demands for water flows vary during the year. One the one hand, the organisms living in the stream and inhabiting the river bank have adapted to the hydrologic regime, and a specific or solid life pattern for each organism has been shaped. In this study, simulated downstream releases accord well with the pattern of the natural hydrologic regime. Hence, it can be supposed that the organisms' perception for hydrological change is not disturbed, and as a result, such damages as habitat loss would not happen although there may be influences on the adaption of specific fish or animals. On the other hand, special hydrological conditions, e.g. peak flow, may be ecologically significant for specific species. The extreme discharge as shown in Figure 6, which corresponds to the peak flow in the natural runoff in September, plays an important part in meeting special requirements of organisms, such as the spawning/incubation requirements of many fish species. Thereafter, the proposed downstream releases satisfy the necessary ecological flow and are beneficial to the protection of downstream ecological integrity.

Tradeoff between hydropower generation and ecological flows

In order to achieve a satisfactory scheme accepted by both the project decision makers and the environmentalists, the tradeoff between the objective of hydropower generation and the constraints of the ecological flow would be further discussed. Figure 7 demonstrates the optimal hydropower generation under different ecological flow demands. The dashed line in the figure denotes the changes of optimal hydropower generation with the increasing membership degree of ecological flow, and the solid line, which represents the membership degree of the operation system benefit, has an extreme point. According to the ‘max–min’ principle in decision theory, the vertex is the optimal point, and thus, a compromise is reached between optimal hydropower generation and fuzzy ecological flow constraints. At the vertex, a best system performance is achieved and the optimal operating rules are obtained. To maintain the downstream ecological flows, the hydropower generation is reduced in order to guarantee adequate water release, whereas the ecological flow requirements are uncertain, as well as the hydropower generation objective. The concept of membership function is utilized to maximize the hydropower generation and ecological flows at the same time. In terms of decision theory, the conflict between the hydropower generation objective and the ecological flow demands is then resolved, and thus, we can make the most satisfactory decisions.

CONCLUSION

A fuzzy constrained nonlinear programming model has been developed for implementing reservoir operation under uncertainty. The model incorporates fuzzy ecological flows as constraints and improves the computation efficiency by using self-adaptive real-coded GA with SBX. In its solution process, the fuzzy constrained operation model is transformed into two deterministic sub-models, which correspond to the lower and upper bounds of the desired objective. This transformation is based on an interactive algorithm. To solve the complex nonlinear programming problem, adaptive GA is employed in search of optimal solutions with advantages of fast iteration and stable convergence. Thus, the proposed methodology not only provides a framework for implementing eco-friendly reservoir operation but also considers uncertain information, which is very important in ecological studies. As a result, we obtain the optimal result and operating rules to assist the reservoir management. The proposed model and solution framework could be very effective by considering fuzzy parameters or similar uncertainty information in the real world, and can suitably reach a compromise between the objective function and fuzzy constraints.

The applied case study showed that the optimal solutions for an eco-friendly reservoir operation could be obtained through self-adaptive GA with SBX, where an optimal output was achieved and the ecological flow demands were met. The adaptive GA was enhanced in its global searching ability and provided an adequate, effective, and robust way for searching reasonable reservoir operating hydrographs in an eco-friendly manner. In addition, uncertain ecological flow demands were expressed and optimized during the year, and the downstream water releases changed with the natural flow regime and were supposed to provide good performance in protecting downstream ecosystem. The developed model also proposed a framework for implementing an eco-friendly reservoir operation that was useful and easy to realize.

However, there are still limits when using the proposed method to solve fuzzy constrained nonlinear programming problems because we assumed that the objective value changes monotonously with the change of fuzzy parameters in constraints. Moreover, it will be also more difficult to obtain the optimal solutions when there are two or more uncertain parameters in the model. Consequently, we need more complex uncertainty analysis methods or techniques to deal with fuzzy parameters or uncertainty variables in the real world in future studies.