2.1. Incorporating Ecosystem Health in Reservoir Operation Modeling
Various bio-indices such as species richness index, evenness index, species diversity index and dominance index have been proposed and used by ecologists to assess bio diversity and richness. These indices are mainly based on categorizing individuals of macroinvertebrates into different classes depending on their sensitivity to organic pollutions. Diversity indices have been preferred over other aforementioned indices for assessing ecosystem health of river systems by some researchers such as Wilsey and Stirling (2007), Gallardo et al. (2011) because they provide valuable information about rarity and commonness of species in a community. The most common and famous diversity indices are Shannon, Simpson, and Margalef diversity indices. Among the various components of the aquatic ecosystems including plants, birds, fish and benthic macroinvertebrates, certain benthic macroinvertebrates are sensitive to change, i.e., watershed degradation and water pollution, and can be monitored for their presence and relative abundance to give a measure of watershed and river ecosystem health. Therefore, macroinvertebrates diversity indices pave the way for one of the most efficient ways of biological assessments (Karr 1998). Being the basic components of the aquatic chains of rivers and ubiquitous in all aquatic ecosystems, limited mobility, long lifespan and species richness with varying sensitivity to pollution are the highlighted reasons for widely reported studies of benthic macroinvertebrares as biological monitoring techniques (Taylor and Baly 1997, Rosenberg et al. 1999, Lliopoul et al. 2003, Azrina et al. 2006).
In this study, maximizing Margalef Diversity Index has been defined as one of the objective functions of the reservoir operation optimization model. Correlation between bio-diversity indices and physicochemical properties of water has been report in different studies such as Yazdian et.al (2014). In this study, Genetic Programing (GP) has been used in order to define the mathematical relationship between Margalef diversity index value and water quality parameters. GP is able to identify relationships between dependent and independent variables even when there is no assumption about the form of the mathematical relation between the variables (Savic, et.al.,1999). Even though, the application of GP is not frequent in hydrology and ecology, but different studies have revealed that GP outcomes have very good conformity with conceptual model results (Sivapragasam et.al,2008; Yang,2010).
GP is an evolutionary algorithm which encodes a function as a tree composed of nodes and branches. GP optimization is based on the natural selection principles. The procedure of GP is similar to genetic algorithms in which selection, cross-over, and mutation operators are utilized to generate and improve solutions. The program evaluates the fitness of all initial solutions treated as “parent population” for the first generation. Then, based on the fitness values, two parents are selected to mate and reproduce. Considering predetermined size of population, resulting solutions known as “offspring” are produced based on cross-over and mutation procedures. Then, the new generations are treated as parent population for the next iterations. The process repeats until satisfying a stopping criterion. The procedure in GP is mainly guided by fitness. Determination of the fitness function to be adopted is an important aspect in GP since its performance largely depends upon how well this fitness function represents the objective or goal of the problem at hand.
2.2. Optimization Algorithm
Water resources management problems often have many decision variables and conflicting objective functions. Recent advancements in multi-objective evolutionary optimization algorithms have made them preferable over other optimization techniques for solving these types of problems. One of these optimization algorithms which have been introduced recently is Particle Swarm Optimization (PSO).
Compared to previously developed algorithms such as Non-dominated Sorting Genetic Algorithm (NSGA), few applications of Multi-objective Particle Swarm Optimization (MOPSO) in water engineering have been reported. Gill et al. (2006) used MOPSO for parameter estimation of Sacramento rainfall-runoff model. Reddy and Kumar (2007) suggested EM-MOPSO for optimization of a multi-propose reservoir operation which uses an external archive with a variable size. Baltar and Fontane (2008) applied MOPSO to solve three multi-objective problems: 1) finding the optimal solutions of the test functions for comparing their proposed algorithm with other Multi-objective evolutionary algorithms; 2) multi-purpose reservoir operation problem with four objectives; and 3) qualitative optimization problem with three objectives. Liu (2009) combined NSGA-II operators and a mutation operator in MOPSO and utilized it for calibrating NAM rainfall-runoff model. Azadnia and Zahraie (2010) utilized MOPSO for calibration of non-linear Muskingum routing model. Saadatpour and Afshar (2013) developed a pollution spill response management mode using MOPSO.
2.3. Particle Swarm Optimization Algorithm (PSO)
Particle Swarm Optimization was proposed by Kennedy and Eberhart (1995). PSO was inspired from bird flocking and their social behavior. Like other evolutionary algorithms PSO generates a random population of solutions which are called Particles in this algorithm. Each particle demonstrates a point in the decision space. The best position that has been visited by each particle is called personal best (pbest). Also, the best position of all particles is called global best (gbest). Each particle searches for the optimal point with respect of pbest and gbest using the following equation:
2.4. Multi-objective Particle Swarm Optimization Algorithm (MOPSO)
In multi-objective problems, multiple objectives should be optimized simultaneously rather than one. In such circumstances, there are more than one optimal solution which are called Pareto optimal. Raquel and Naval (2005) presented a multi-objective PSO algorithm namely MOPSO-CD which is also modified and used in this study for solving multi-objective reservoir operation optimization. This algorithm uses the Crowding Distance mechanism which was first introduced in the NSGA-II algorithm. An external archive is also used in MOPSO-CD to store non-dominated solutions. A mutation operator maintains the diversity of the non-dominated solutions. Using external archive, crowding distance and mutation operator enables the algorithm to find all feasible solutions and promotes a good distribution of solutions in the objective space. In each generation, non-dominated solutions in the population are stored in the external archive. In order to cover all feasible solutions with a uniform distribution, the population must move towards the regions with low density of nondominated solutions. Thus, the solutions in sparse regions must be selected as gbest. For this propose, the algorithm uses crowding distance. In each iteration, archive members are sorted based on their descending crowded distance value and one of the top members of the archive is selected as gbest. Whenever archive is full, one of the bottom members of the archive should be replaced by a new non-dominated solution. So, the population goes toward the regions with low density of solutions in the objective space.
2.5. Reservoir Operation Optimization Model
The following equations show the reservoir operation optimization model formulation:
\({M}{i}{n}{i}{m}{i}{z}{e}(\sum _{{m}=1}^{12}\sum _{{y}=1}^{10}{{L}{o}{s}{s}1}_{{m},{y}}, \sum _{y=1}^{10}\sum _{m=4}^{7}{{L}{o}{s}{s}2}_{{m},{y}})\)
3
\({{L}{o}{s}{s}1}_{{m},{y}}=\left\{\begin{array}{c}{{e}}^{\frac{1}{\left(.4+\frac{{{R}}_{{m},{y}}}{{{D}}_{{m},{y}}}\right)}}-{{e}}^{0.71} {{R}}_{{m},{y}}<{{D}}_{{m},{y}}\\ \\ 0 {{R}}_{{m},{y}}={{D}}_{{m},{y}}\\ \\ .08\times \left({{e}}^{.7\times \frac{{{R}}_{{m},{y}}}{{{D}}_{{m},{y}}}}-{{e}}^{.7}\right) {{R}}_{{m},{y}}>{{D}}_{{m},{y}} \end{array}\right.\)
4
\({{L}{o}{s}{s}2}_{{m},{y}}={{e}}^{\frac{1}{\frac{{M}{I}{}_{{m},{y}}}{3.5}+.4}}-{{e}}^{.71}\)
5
\({M}{{I}}_{{m},{y}}={10}^{\left(\frac{{{l}{o}{g}}_{10}{{T}{D}{S}}_{{m},{y}}}{\left({{R}}_{{m},{y}}+\left(\frac{{{T}}_{{m},{y}}}{{{l}{o}{g}}_{10}{{T}{D}{S}}_{{m},{y}}}\right)\right)}+0.2\right)}-1\)
6
Subject to:
\({{S}}_{{m}}={{S}}_{{m}-1}+{{I}}_{{m}}-{{R}}_{{m}}\)
7
\(0\le {{R}}_{{m},{y}}\le {{R}}_{{m}{a}{x}}\)
8
\({{S}}_{{m}{i}{n}}\le {{S}}_{{m},{y}}\le {{S}}_{{m}{a}{x}}\)−1
9
\({{S}}_{{T}}<{{S}}_{1}\)
10
Where, Rm,y, Dm,y, and MIm,y are reservoir water release, water demand, and Margalef diversity index in month m of year y, respectively. In Eq. (6) T is the river water temperature (°C) and TDS is total dissolved solids (mg/l). S and I are volume of water stored in the reservoir at the end of each month and the volume of inflow to the reservoir, respectively.
Objective functions in this model have been defined as damage functions that the goal is minimizing these two functions.
\({Losss1}_{m,y}\): If the required water of downstream is not supplied, the damage function is defined as function (4). In this function, it is assumed that the damages due to releasing water from system are function of discharge to demand ratio. This function is convex and exponential which the structure is composed of three parts. If the discharge is equal to downstream demand, then the damage will be zero. The function will increase exponentially in the case of inequality of discharge and demand, and when the discharge is less than demand, more damage is considered in the function than the case that the discharge is more than demand.
\({Losss2}_{m,y}:\)For providing ecosystem sustainability, optimization of biological index in downstream of dam reservoir has been selected as objective function. Regarding that maximum biological diversity index of Margalef is assumed 3.5 in different studies (Datta et al.,2010), a biological diversity ratio of 3.5 has been considered for calculation of ecologic damage. The shape of first and second objective functions have been considered similar for more coordination, and this shape of objective function for consideration of ecologic damage of the river is an innovation of this study. As in warm months that the water temperature is high and water flow discharge is low, environmental health of ecosystem is more vulnerable, hence, regarding long time inlet flow to reservoir, July to October months have been considered for minimizing ecological damage influenced by difference between diversity and its maximum.
Calculating ecosystem sustainability as second function requires outlet flow qualitative data from the reservoir per outlet flow and inlet flow quality in optimization period. For this purpose, for each particle in each iteration of evolutionary optimization model the qualitative simulation model has to be called. Among different quality simulation models (one dimensional and are multi-dimensional), WQRRS model has been utilized for its ability to be linked with optimization model and relatively low running time. WQRRS Model has been developed by U.S Army corps of engineers. A water quality simulation model is linked with the optimization model to produce the outlets release quality as well as the temporal and spatial variation of the concentration of water quality variables within the reservoir. The basic equation of water quality simulation model developed in this study is based on one dimensional advection-dispersion mass transport equation, which is numerically derived over space and time for each of the water quality constituents.