INTRODUCTION
Reservoirs are structural measurements which help regulating and redistributing
watershed water resources temporal and spatial availability. Obtaining
optimal reservoir operation rules entails formulating it as a mathematical
program and optimizing it according to an objective function which represent
operator and/or some other user`s utility functions. Problem formulation,
while considering most of the operational considerations, may lead to
a linear or nonlinear mathematical representation. Solution of such problems
that includes discrete nonsmooth variables and functions are better handled
by evolutionary algorithms (Hota et al., 1999). Karamouz et
al. (2002) formulated an optimization model for Zayandehrud reservoir
operation rules by formulation and solved that by genetic algorithm. Another
important aspect of reservoir operation rules optimization would be uncertainty
of inflows to reservoir, which affects operational rules in long term
planning. Rules obtained from a long term planning will have more reliability
in comparison with rules obtained from a mid or short term planning. However,
in long term planning uncertainty of inflows affects optimality of rules.
Considering these uncertainties will improve reliability of operation
rules application. Most applied approach to uncertainty consideration
in reservoir operation planning is stochastic programming by Stochastic
Dynamic Programming (SDP), Stochastic Linear Programming (SLP) and chance
constrained formulation. Any of the approaches mentioned has limitations
such as computational burden or curse of dimensionality for SDP and SLP
approaches. Chanced constrained approach forces the problem to be solved
within a very small portion of feasible region, which eliminates possibility
of a better solution in excluded feasible region. Therefore, it is a very
conservative and noneconomic approach to uncertainty consideration. On
the other hand, Monte Carlo sampling approach to stochastic programming
will lead to a more reasonable method of solving a problem under uncertainty
from computational burden and precision point of view (Dembo, 1991). Scenario
based optimization as a Monte Carlo sampling approach to stochastic programming
has been used by many researchers in solving reservoir operation planning
(Dembo, 1991; Shapiro, 2007). Scenario Optimization as a scenario based
approach is much simpler and computationally lighter and faster approach
to stochastic programming. However, in all mentioned approaches indices
like expected value or variance of some objective function is optimized.
These indices do not represent uncertainty in a measurable manner. However,
an index like coefficient of variation shows uncertainty in a more meaningful
way. Formulation of reservoir operation problem considering such indices
will make a nonlinear discrete nonconvex problem which is difficult
to solve by gradient based algorithms. On the other hand, evolutionary
algorithms can cope with this problem which its combination with stochastic
programming will ensure high reliability of obtained operation rules.
In this research, reservoir operation rules optimization under uncertainty
of inflows in a long term horizon would be done by scenario optimization
and differential evolutionary algorithm. This type of problem formulation
and solving is applied to a real world case study, Zayandehrud reservoir
in Isfahan, Iran.
PROBLEM FORMULATION
Operational considerations of the reservoir, uncertainty of inflows to
the reservoir in future and performance criteria for reservoir operation
are important aspects that consideration of them makes the operation rules
reliable with least unfavorable effects. Considering these aspects will
formulate a nonlinear stochastic problem with noncontinuous nonconvex
objective function. Solving this problem will entail combination of two
efficient solution algorithms in a manner that their combination is still
effective. These two algorithms are Scenario optimization and Differential
evolutionary algorithm. In this research problem is formulated to be solved
in an optimizedsimulation algorithm. Reservoir operation is done according
to scenario optimization, say; releases are allocated as target releases
until upper and lower limit of storage values are met. Rule curves are
generated, simulated and optimized within a combined algorithm made up
of differential evolutionary algorithm and Scenario optimization. Main
equation of the system is:
where, S_{t} shows reservoir water volume, QI_{t} represents
the inflow to reservoir and Rl_{t} is the release decision
variable. This equation is corrected for evaporation losses if some conditions
are met:
In this nonequation
is the reservoir outflow rating curve and Target is the amount determined
in rule curves. If according to the release determined above S_{t+1}
is higher than maximum storage capacity then spillage exists and S_{t+1}
is set to maximum storage capacity. On the other hand, if S_{t+1}
is less than minimum storage capacity then S_{t+1} is set to minimum
storage capacity and release is set to:
Objective function of this problem consists of reliablity range to the
mean ratio in supplying water demands. Objective function of the reservoir
operation problem is shown below, which is minimized.
where, Relrange and Relmean stand for reliability range and mean, respectively.
The r is the importance factor.
It is clear that the problem formulation is nonlinear, discontinuous
and nonconvex though it represents a better uncertainty measurement index.
In the problem to be solved QI`s are indeterministic and have an uncertainty
band, which in optimum operation planning it should be dissipated as much
as possible in reservoir operation performance index defined earlier.
SCENARIO OPTIMIZATION
When problem conditions become stochastic, decision making needs more
considerations so that the failure probability of decisions applications
reaches the least possible value. While, decisions are deterministic anyway,
conditions are uncertain at present or in future. Scenario optimization
formulates this problem in a way that deterministic decisions with least
deviation from the optimum of each scenario are obtained. In mathematical
formulation a problem with some uncertain parameters could be expressed
as below (Dembo, 1991):
where, X`s, e_{j} and h_{j} (X) are deterministic decision
variable vector, right hand side and constraints, respectively. However,
b_{i}, g_{i} (X) and Z are uncertain parameters, constraints
and function, respectively. This problem is solved in a discretized deterministic
form,
is described by independent scenarios.
For each scenario S, a deterministic problem is formulated, which is called
subproblem:
Since none of the scenarios may happen in reality, optimum X will have
some slacks in satisfying uncertain constraint .
Therefore problem of finding optimum X when occurrence of scenarios is
uncertain and X in presence of uncertain parameters could be reformulated
as follow:
Another representation of the problem for X_{opt} could be:
The above problem which is solved after optimizing the subproblems is
called tracking model. In the above formulation, P_{s} is the
probability of each scenario occurance. By solving the tracking model
X_{opt} is determined, so that deviation form scenarios constraints
and the optimum objective function is minimized. Optmization problem defined
by Eq. (8) is solved after optmizating the subproblems
in Eq. (7). By this two stage solution method a stochastic
problem is shown in a single iteration for deterministic decision.
Differential Evolutionary Algorithm (DEA): Problem formulation
is not suitable for solution by gradientbased algorithms because of discontinuity
and nonlinearity of the problem. Therefore, an evolutionary algorithm
or dynamic programming approach must be selected to solve the problem.
From these two algorithms the one with least amount of computational burden
should be selected. Differential evolutionary algorithm has proved to
solve the complexity of this problem more efficiently than DP. This algorithm
has the least computational burden and robust in solving complex engineering
problems (Deb, 2001). Based on differential evolutionary algorithm, a
random population of feasible target releases is generated as follow:
Problem objective function, Eq. (4), is used for target
release chromosome assessment, mutation and crossover after their application
in reservoir operation simulation. Next generation is produced by mutation
and/or cross over while keeping the elites. Mutation occurs with probability
Mutprob as follows:
Δ is a random parameter in above equations. Crossover takes place
with probability (1Mutprob) as simple average:
Aside from mutation and cross over, elitism helps to keep the best of
each generations through evolution. Reliability is computed for total
demand supply as:

Fig. 1: 
Stochastic differential evolutionary algorithm 
This index is computed for each chromosome representing target release
for reservoir operation within each generation over planning horizon.
Next generations are produced using mutation and crossover operators.
Δ used in mutation operator resembles stepsize in nonlinear programming
methods such as Newtonraphson (Coine et al., 1999; Deb, 2001).
Stochastic Differential Evolutionary Algorithm (SDEA): Each of
the algorithms reviewed earlier are robust in solving stochastic or nonlinear
nonconvex discontinuous problems. However, the problem to be solved here
entails use of both of them because of its nonlinearity and uncertainty.
Combining both algorithms in a way not to affect their robustness will
lead to another algorithm. This new algorithm makes use of scenarios within
differential evolutionary algorithm. Reliability range to average ratio
for each chromosome in the generation for all scenarios is computed and
used for making the next generation. Scenarios are made with different
methods which one of them is Indexed Sequential Modeling (ISM) (Labadie
et al., 1987). ISM is a very simple method to model uncertainty
when time series with long length is available and future trends are assumed
to be like past trends. It is simply selecting a length equal to the planning
horizon and shifting it according to the number of scenarios year by year
for example. Combined algorithm for problem solving is shown in Fig.
1.
Reservoir operation rules optimization is formulated and solved by the
combined algorithm SDEA.
CASE STUDY
Zayandehrud reservoir system at the upstream of Isfahan city in Iran
is selected to be formulated and solved by the modeling approach introduced
in this research. This reservoir plays an important role in the agriculture
and economics of the downstream subbasins. Reliability of supplying the
downstream demands would be of great economic and social importance to
the decision makers. Therefore, development of a long term reservoir operation
model which considers stochasticity of inflows will help to obtain the
most reliable reservoir operation rule. Reliability is taken as an objectiveto
be maximized. It is stochastic therefore another index is used to measure
stochasticity of reliability range to mean ratio. This index, that must
be minimized to ensure the dissipation of uncertainty in decision making,
makes the problem formulation nonlinear, discontinuous and nonconvex.
Therefore, Stochastic Differential Evolutionary Algorithm (SDEA) is used
to solve this problem. Application of SDEA in this case could be compared
to the actual operation for years 1975 to 1994. As said before inflows
scenarios to Zayandehrud reservoir are made up with ISM method that some
of them are shown in Fig. 2. Figure 2
and 3 show the scenarios considered according to ISM
method to represent stochasticity of reservoir inflows. Actual inflow
for years 1975 to 1994 do not mach any of scenarios and it is less than
all. It means model has no preknowledge about actual inflows, so, its
operation comparison to the actual operation becomes fair.
Figure 4 and 5, show evolution of objective
function and its components by generation for 25 chromosomes. Evolution of reliability
expected value (Fig. 5) has a jump in generation 2 which had
a range that led the objective function to 0.015 value. In fact, though in generation
2 mean has increased suddenly but range to mean ratio has a larger value in
comparison to next 3 to 17 generations. However, reliability mean has reached
1 and its range reach to 0.005 at optimum.
Model operation by SDEA shows meaningful improvements in reservoir operation
for validation years (Fig. 6). The SDEA needs relatively
small amount of generations and chromosomes to evolve which reduces the
computational burden. Besides, SDEA needs no ranking or fitness computation
which reduces the computational burden much.

Fig. 2: 
Scenarios of inflow to reservoir for 20 years planning
horizon 

Fig. 3: 
Total Inflow volume of each scenario for 20 years and
actual inflow 

Fig. 4: 
Evolution of objective function (reliability range to
mean ratio) for chromosome number 25 

Fig. 5: 
Mean reliability evolution for 25 chromosomes in each
generation (Reliability to supply 95% of demand or more) 

Fig. 6: 
Model vs. actual operation for years 1975 to 1994 (yearly
sums) 

Fig. 7: 
Reservoir operation obtained from model application
in Zayandehrud Reservoir 
This problem with 240 real
variables has reached optimum value at generation number 38 with 25 chromosomes.
Number of scenarios is 10, as shown in Fig. 3. Reservoir
operation according to SDEA solution of Zayandehrud reservoir operation
model is represented in Fig. 7. Demands include agricultural,
domestic and industrial, which domestic and industrial are 10 MCM per
month for all periods. Domestic and industrial demands are fully met by
model operation.
CONCLUSIONS
In this research stochastic programming is viewed from another point
of view by considering reliability range to mean ratio for supplying water
demands. It is more representative than expected value or standard deviation
in measuring uncertainty, as objective function to be minimized. Minimizing
the objective function tries to narrow the uncertainty of releases reliability
and sensitivity to uncertainty of inflows (inputs) as much as possible.
It will lead to a nonlinear, nonconvex and discontinuous mathematical
program that a suitable category of solution algorithms for it would be
evolutionary algorithms. A new hybrid algorithm based on differential
evolutionary algorithm and scenario optimization is set up and applied
in a real world case, Zayandehrud reservoir in Iran. Application of model
decisions shows substantial improvements in reservoir operation for a
test period of 20 years. This improvement is mostly related to the problem
formulation selected in this research which is much more representative
of uncertainty in a system operation. Optimization procedure tries to
make the decisions insensitive to problem input (inflows) uncertainty
as much as possible. This approach will increase reliability of obtained
decisions for application in real world cases.
ACKNOWLEDGMENTS
The technical contribution of Ms. Leyla Karimi and Mr. Yousef BazarganLari
is hereby acknowledged.