The reservoirs that feed large hydropower plants should be managed in order to provide other uses for the water resources. Those uses include, for instance, flood control and avoidance, irrigation, navigability in the rivers, and other ones. This work presents an evolutionary multiobjective optimization approach for the study of multiple water usages in multiple interlinked reservoirs, including both power generation objectives and other objectives not related to energy generation. The classical evolutionary algorithm NSGAII is employed as the basic multiobjective optimization machinery, being modified in order to cope with specific problem features. The case studies, which include the analysis of a problem which involves an objective of navigability on the river, are tailored in order to illustrate the usefulness of the data generated by the proposed methodology for decisionmaking on the problem of operation planning of multiple reservoirs with multiple usages. It is shown that it is even possible to use the generated data in order to determine the cost of any new usage of the water, in terms of the opportunity cost that can be measured on the revenues related to electric energy sales.
The optimal use of water resources is increasingly being recognized as a strategic issue for the nations [
The optimal planning of the operation of single and multiple reservoir systems involves essentially the decision about how much electricity should be generated in each plant on each time period. This decision determines how much water is available for electricity generation in the next time periods, interacting with the effect of reservoir filling due to seasonal rainfall. Other usages for the water will rely on the availability of water, either in the reservoir (measured by the reservoir level) or in the river watercourse (measured by the water flow). For instance, the flood control is better if there is less water in the reservoir, allowing the impoundment of a large additional volume of water, while more water in the reservoir is better for irrigation purposes. For allowing navigability, there is a need for a minimum water flow in the watercourse, which imposes a minimum level of electricity generation in order to liberate such a flow.
The underlying optimization problem can be characterized as (i) a multiobjective optimization problem, since the possible alternative usages of the water are conflicting with each other and with the power generation. (ii) It is a dynamic optimization due to the interdependence of the time stages with any decision being constrained by the decisions taken in the former stages. (iii) The problem is nonlinear, mainly due to the nonlinear relationship between the water flow through the reservoirs and the water level in them. These nonlinear functions are also nonconvex, in general. The objective functions may also be defined as nonlinear functions. (iv) There are constraints related to the minimum and maximum admissible levels of water in the reservoirs and to the minimum and maximum water flow that can be sent to power generation.
Several methods, both deterministic and stochastic, have been used for tackling simplified instances of this general problem. Some of them are reviewed in [
In this work, a multiobjective genetic algorithm is employed to optimize the operation of a set of five existing Brazilian hydropower plants through the course of one year for a typical year and for a dryer than usual year. Two objectives related to energy generation are considered: the maximization of the minimal power generation along the year (which is related to the power delivery which can be ensured by the system on any time) and the maximization of the total energy generation along the year. Those objectives represent a typical tradeoff to be examined by decisionmakers that work under the viewpoint of the electricity system planning. Another objective, not related to energy generation, is also included: the navigability in the river downstream the reservoir. This objective is expressed as the need to maintain the water flow above a given minimum, which allows the operation of ships above a given draft. The inclusion of this objective exemplifies the employment of the proposed methodology for the generation of data for the decisionmaking process considering several stakeholders, in addition to the electric power system viewpoint.
The proposed algorithm is a variation of the NSGAII algorithm, which has been modified in order to include a new representation for the individuals, allowing a better constraint handling and new problemspecific mutation and crossover operators, which enhance the algorithm computational efficiency [
A hydroelectric central is shown schematically in Figure
Schematic representation of a hydroelectric central.
The difference between the water level in the reservoir and its level in the outflow channel is called the head. This difference determines the potential energy that can be transformed into electric power. The water level in the reservoir presents a nonlinear relationship with the stored volume, while the level in the outflow channel also depends nonlinearly on the sum of flow through the turbines and the spillway flow.
The following variables are involved in the model of a hydroelectric central:
In all cases,
The power generated by a hydropower plant
In this expression
There are both physical and operational constraints for the operation of hydroelectric power plants.
The conservation of mass constraint, over a time interval, imposes that the increase of stored water volume in each one of the reservoirs must be equal to the amount of water flowing into it (
If the reservoir is the first one on a river, the affluent flow is determined by nature. If not, the total volume coming from the upstream reservoir must be added to the incremental volume—that of water from other tributaries, added between the two reservoirs:
Here
The mass conservation establishes a direct relationship between the water outflow and the volume of water in the reservoir, as the affluent flow is known.
The volume of water in the reservoir is bounded below by the level of the forebay (intake channel) and above by the structural limit of the dam:
The operational parameters of the turbines also impose constraints to the problem, in terms of both a maximum power and a maximum volumetric flow:
If the volume variation in a given interval requires a flow through the turbine that exceeds the maximum allowed, it is assumed that some water was discharged through the spillway.
For ecological, sanitary, and economical reasons, a minimum volumetric flow to the river downstream of the reservoirs must also be ensured:
In this work the availability of the turbines is assumed. Therefore no spillage is allowed when the flow is smaller than the allowed maximum; that is, all the water passes through the turbines.
Finally, it is assumed here that every reservoir must be returned to its initial state at the end. Therefore,
This constraint can be hardcoded on the algorithm by excluding the volume at the end of the last interval from the decision variable set.
The optimization of the reservoir system usage is defined by the choice of the objective functions to be maximized or minimized. In this section, several formulations of different objective functions are presented.
Some possible objective functions related to energy generation are shown in this subsection. In all cases, the index
A usual expression [
The maximization of the total generated energy is represented by
This expression makes sense under the assumption that the system to be optimized has a generation capacity, that is, always less than the demand, that is, supposing that all energy that the plant can produce will be sold. In such a case, the maximization of
Another objective is the maximization of the minimum monthly power generation:
Under the viewpoint of an independent company of electric power generation which sells energy in a market, this objective corresponds to the maximization of the assured energy, which is the share of the total energy produced by the company that should be guaranteed to be furnished under any circumstance. This share is sold by a price that is defined in a long term contract which, in average, provides a better price for the energy, with less volatility. Under the viewpoint of the whole system, this objective corresponds to the minimization of the need of installation of backup power plants that would operate only in case of low availability of hydroelectric energy. Notice that this objective is different both from
After a share of the total energy produced by a company is compromised by a long term contract (the assured energy), the remaining energy is sold in a spot market. It is possible to maximize the profit that comes from the sale of that energy, by selling the energy in the moments in which the price is expected to be higher. This objective can be expressed as
Several works [
The control of water flow downstream a reservoir can contribute to avoid the occurrence of floods in urban or agricultural areas. Such an objective is sometimes expressed as [
A different formulation which takes into account only the positive violations of the threshold may be stated as
This formulation is less conservative, avoiding the floods while allowing the preservation of water in the reservoirs for other usages.
Several rivers have large seasonal variations of their water flows along the year, which makes them be navigable by large ships during part of the year only. The flow in those rivers can be regulated by the reservoirs, making them navigable throughout the year. A larger minimum water flow means that a larger ship is allowed to navigate the river, which means that the maximization of the minimum water flow leads to the maximization of the allowed ship’s draft:
The optimization of this expression leads to the maximization of the minimum flow downstream the reservoir
The problem of planning the usage of the water stored in hydroelectric plant reservoirs was shown to be related to several different objective functions, which may be conflicting with each other. Therefore, the suitable framework for performing such a planning is the multiobjective optimization. A brief explanation about the subject of multiobjective optimization is presented in this section.
A conventional singleobjective optimization problem is stated as the problem of finding the point, in a space of decision variables, in which an objective function reaches its minimum value. The multiobjective optimization problem, instead, searches for a set of points, the
Consider the minimization of a vector function
The points
Representation of the Paretoset, in the space
Finding the set
Although general purpose shelf routines devoted to evolutionary multiobjective optimization are usually quite userfriendly, it has been recognized recently that in several cases some problemspecific adaptations become necessary in order to achieve reasonable levels of algorithm performance [
This section describes the problemspecific evolutionary multiobjective optimization algorithm that was developed here. The computational tool employed in this work in order to estimate the Paretoset solutions for the problem of multiple reservoir operation is based on the classical NSGAII algorithm [
The basic structure of the NSGAII is presented in Algorithm
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(5) Evaluate
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Details about the
The nonstandard components of the algorithm which are proposed here are described next.
The authors proposed, in a previous work [
First, a modified lower bound to the volume of the reservoir is imposed at the end of each interval, ensuring that the reservoir can be replenished at the end interval with the available upstream affluent flow and discounting the minimum downstream volume:
A modification to the upper bound constraint was also introduced in order to avoid cases in which the affluent flow minus the minimum downstream flow becomes insufficient to allow for the desired increase in volume:
The decision variables were also changed from the monthly volume of water stored in the reservoir to the fractions of the monthly allowable volumes:
Within those limits, the volumetric flow through the turbines is always greater than the minimum, though it can still lead to a calculated power generation which is greater than the maximum allowed or it can be itself greater than the maximum allowed volumetric flow. To deal with those cases, it is assumed that some water was spilled, and the true flow through the turbine is one that satisfies both conditions (
The crossover adopted in the proposed algorithm is described in Algorithm
(1) Inputs:
(2) Output:
(3)
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In Algorithm
The mutation employed in the algorithm is performed on each variable, independently of the individual, as shown in Algorithm
(1) Inputs:
(2) Output:
(3)
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In Algorithm
The proposed constrainthandling procedure (the unconstrainedencoding formulation) constitutes the main modification in the conventional NSGAII which leads to a much enhanced algorithm performance, in terms of the solution set quality. In order to assess the importance of that procedure, a preliminary study of its effect was performed on a singlereservoir system, considering a series of comparisons of the proposed procedure with (i) a naive approach, in which the conventional constrainthandling procedure of NSGAII is employed [
This reducedrange constraint management is defined as follows. It can be observed that most of the infeasible individuals generated in the naive approach are related to the need for replenishing the reservoir, if at the end of any interval its volume becomes significantly lower than the initial one. If the affluent flow becomes insufficient, the conservation of mass leads to an infeasible flow through the turbines. To deal with this situation, instead of imposing a single lower bound to the volume of the reservoir, a lower bound can be imposed at the end of each interval:
The lower bound volume at the end of a period becomes equal to the upstream affluent water minus the minimum downstream volume.
The experiments were run using data for a typical year (from May, 1976 to April, 1977), with initial and final volumes of the reservoir set to 95 percent of the maximum. The multiobjective optimization problem that was solved considered only the Nova Ponte hydropower plant in the Araguari River. The problem was formulated as follows:
The crossover and mutation probabilities were studied in preliminary experiments and finally set at 85% and 35%, respectively, in order to improve the exploration of the domain. This mutation probability is for each individual, since a Gaussian operator is employed.
Only some few experiments that were run with the naive formulation yielded a population with feasible individuals, even when the initial population was handtailored to include only nearfeasible individuals. The Pareto fronts for 8 of those runs, with 400 individuals and 5000 generations, are shown in Figure
Efficient sets for 8 runs of the basic formulation, with 400 individuals and 5000 generations.
The proposed approach and the reducedrange constraint formulation were able to find feasible solutions in all runs. For those algorithms, 21 experiments were run with 400 individuals and 2500 generations. The reducedrange formulation was able to generate a fully populated Pareto front, but with large variation in solution quality between runs, as can be seen in Figure
Efficient sets for 21 runs of the reducedrange formulation, with 400 individuals and 2500 generations.
Every run of the new proposed formulation yielded a well spread Pareto front, as shown in Figure
Efficient sets for 21 runs of the unconstrainedencoding formulation, with 400 individuals and 2500 generations.
Efficient sets for every run of the reducedrange and unconstrainedencoding formulations, with the combined Pareto fronts.
In this section, the results of two case studies are presented. The first one deals with power generation objectives only, and the second one includes an objective that is not related to power generation, the river navigability.
The case studies consider a small subset of the Brazilian electric generation system. This subsystem is composed of five hydroelectric power plants, in the Paranaíba river basin, on the southeastern region of Brazil, with a total installed power of 4,765 MW (see Figure
Schematic representation of the subsystem composed of five power plants.
Three of the considered power plants are installed on affluents of the Paranaíba River: two on the Araguari River and the Nova Ponte dam, hereafter referred to as
The climate in the region features two very well defined seasons: a dry winter, from April to September, and a rainy summer, from October to March. This determines the volumetric affluent flow to the reservoirs, for which data is available from 1931. The power generated by each plant is given by (
Energy production function polynomial coefficients.
Coefficient  Power plant  

1  2  3  4  5  

7.261600  5.774166  4.555294  7.985410  5.244398 

6.84174  18.1900  34.9788  5.7208  3.1731 

−7.87711  0.0  196.0  5.0  2.0 

9.11797  0.0  0.0  0.0  0.0 

−6.49561  0.0  0.0  0.0  0.0 

1.95365  0.0  0.0  0.0  0.0 
The operational parameters and constraints for each hydropower plant, listed in Table
Operational parameters and constraints for the five hydropower plants and their reservoirs.
Parameter  Power plant  

1  2  3  4  5  
Minimum volume ( 
2,412  974  470  4,669  4,573 
Maximum volume ( 
12,792  1,120  1,500  17,725  17,027 
Maximum power (MW)  510  408  375  1,192  2,280 
Minimum flow ( 
125  080  070  170  190 
Maximum flow ( 
510  675  570  1,048  3,222 
The first case study was performed using as objectives the expressions (
Those objectives are conflicting, as can be inferred from a simple reasoning: the situation in which the reservoir has a higher level of water generates more power than a situation of lower water level for the same volume of water passing through the turbines due to the difference of gravitational potential energy in the water. Therefore, given a certain water inflow regime in the reservoir, the best policy for reaching a maximum total generated energy would be to avoid the generation when the reservoir is with low level, in order to make the level as high as possible in the future, in this way leading to the extraction of more power in that future for the same total water volume. This policy would lead to some periods of very small power generation. On the other hand, a policy which tried to avoid the moments of low power generation would spend the water more uniformly and consequently would not be able to take so much profit of the power peaks that would be obtained in the moments in which the reservoir would be almost full, because those moments would occur less frequently under such a policy.
The optimization was performed with the modified NSGAII algorithm as described previously. As indicated by previous experience, the crossover and mutation probabilities were initially set at 85% and 35%, respectively, in order to improve the exploration of the domain. The mutation probability is per individual, for a Gaussian mutation operator. This value becomes similar to the mutation probability per individual for a simulated binary crossover, employed in reference [
The experiments were run with data for both a typical year (from May, 1976 to April, 1977), with initial and final volumes of all the reservoirs set to 95 percent of the maximum and a dryer than usual year (from May, 2000 to April, 2001), with initial and final volumes of the reservoirs set to 85 percent of the maximum.
The efficient solutions sets that resulted from preliminary experiments with the 197677 data were widely spread. Extending the number of generations to 50,000 reduced this problem somewhat, as can be observed in Figure
Results for the 197677 case. Efficient sets from three sets of 30 experiments, with 400 individuals and 50,000 generations with mutation probabilities of 35% in black, 50% in blue, and 65% in green. In red, the efficient set for a single run initialized with the combined efficient set, with 400 individuals and 25000 generations.
Shown in Figure
Results for the 200001 case. In black, the efficient sets for 30 experiments, with 400 individuals and 50,000 generations. In red, the efficient set for a single run initialized with the combined efficient set, with 400 individuals and 25000 generations.
There was a clear difference in the results from the algorithm in the two cases. An explanation was found in the analysis of the decision variables for the overall most efficient solutions. In the 197677 case, though those solutions originated from four different experiments throughout the solution set the fractions of the allowable volumes were nearly constant for the two largest power plants, Emborcação and Itumbiara (see Figure
Results for the 197677 case. Boxplot of the decision variables (
In the 200001 case, the spreads were significant for every power plant, except Itumbiara (which, being the furthest downstream, receives a steadier inflow of water) (see Figure
Results for the 200001 case. Boxplot of the decision variables (
The second case study considered the problem of guaranteeing the navigability in the segment of the Araguari River between Nova Ponte dam and Miranda Dam, according to objective function (
An adaptation of the
Paretoset estimates for 10 runs.
As expected, the minimum power that will be generated is much less affected than the mean power when the navigability objective is considered. However, even the minimum power objective has some conflict with the navigability objective because a given navigability index requires a given water flow, while a given minimum power can be obtained with different turbined water flows, provided that the reservoir level has different values—therefore making those objectives different.
Another visualization of the same results can be obtained considering the prices for assured energy and variable energy. In order to exemplify this, an example was built considering the price of 120/MWh for the assured energy and 80/MW for the variable energy. Figure
Total revenue versus assured energy for different navigability conditions. A zoom view of the region in which the maximum values occur is presented in the bottom.
Figure
Point of maximum revenue versus assured energy for different navigability conditions.
This work presented an evolutionary multiobjective optimization approach for the study of multiple water usages in multiple interlinked reservoirs, including both power generation objectives and other objectives not related to energy generation.
The classical algorithm NSGAII was employed as the basic multiobjective optimization machinery. This algorithm was modified in order to cope with specific problem features. The main modification, which caused the major enhancement in the algorithm performance, was the new encoding scheme. This encoding procedure allowed the implicit handling of most of the constraints involved in the problem, in this way removing an important computational bottleneck.
The case studies, which included the analysis of a problem involving energy generation objectives only and of another problem which involved also an objective of navigability on the river, were tailored in order to illustrate the usefulness of the data generated by the proposed methodology for decisionmaking on the problem of operation planning of multiple reservoirs with multiple usages. It was shown that it is even possible to use the generated data in order to determine the cost of any new usage of the water in terms of the opportunity cost that can be measured on the revenues related to electric energy sales.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the support provided by the Brazilian agencies CAPES, CNPq, and FAPEMIG. The essential collaboration of their colleague Oriane M. Neto