# Multiobjective Optimization Modeling Approach for Multipurpose Single Reservoir Operation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Kritsky and Menkel Method for Designing Reservoir Operating Storage

#### 2.1.1. Method for Calculating the Annual (Within-Year) Capacity Storage of the Reservoir

_{i}

_{+1}= S

_{i}+W

_{i}− R

_{i}− L

_{i}− Sp

_{i}

_{i}− ∑W

_{i}

_{i}

_{+1}and S

_{i}are the storages of the reservoir at the end and the beginning of period i in units of volume. Besides W

_{i}, R

_{i}, L

_{i}, Sp

_{i}and U

_{i}are, respectively, the inflow, net release, loss of water, spillways, and gross release from the reservoir during the time period i, all in units of volume. In the other hand, it’s possible to formulate Equation (7) as a fraction of (U) and (Wg) in the period of deficit, as shown in Equation (8). It’s possible because U = ∑U

_{i}and Wg = ∑W

_{i}for the selected reliability.

_{i}− f × Wg

#### 2.1.2. Method for Calculating the Multi-Year or Carryover Storage Capacity of the Reservoir

^{2}+ e × Cv × α + f × α

^{2}+ g × Cv

^{2}× α + h × Cv × α

^{2}+ i × α

^{3}+ j × Cv

^{2}× α

^{2}+ k × Cv × α

^{3}+ m × α

^{4}

**g**

^{10.07}; b = 0.2938 ×

**g**

^{5.734}; c = 0.1475 ×

**g**

^{3.255}; d = 0.09187 ×

**g**

^{4.047}; e = 0.3546 ×

**g**

^{4.0};

^{−8}×

**g**

^{(−25.92)}+ 0.07981; g = 0.1326 ×

**g**

^{7.615}; h = 0.3994 ×

**g**

^{6.332}; i = 0.5727 ×

**g**

^{8.03};

^{−5}×

**g**

^{(−13.09)}; k = 0.3003 ×

**g**

^{12.98}; m = 0.3397 ×

**g**

^{9.268}

#### 2.2. Formulation of the Optimization Mathematical Model

#### 2.2.1. Objective Functions

^{3}); Cu,i is the deficit unitary cost of the user u during the time period i, in ($/Mm

^{3}); Du,i is the water demand of the user u during the time period i, in (Mm

^{3}); and Ru,i is the water release received by the user u during the time period i, in (hm

^{3}). BEE is the annual benefit of the electrical energy generated in ($); Bc,i is the unitary benefit of electric power generated by the hydroelectric c during the time period i in ($/kW-h); Ec,i is the energy produced by the hydroelectric power plant c during the time period i in (kW-h); Nc,i is the power output generated by the hydroelectric power plant c during the time period i in (kW); Hi is the average gross head on the turbines of the hydroelectric power plant c during the time period i in (meters); Ri is the net release of the reservoir during the time period i in (Mm

^{3}); n is the efficiency of the hydroelectric power plant, which is dimensionless; K is a dimensionless proportionality parameter of units from Mm

^{3}/month to m

^{3}/seg, and its value is 0.3858; u represents the users; nu represent the total number of users; i represents the time period; c represents the hydroelectric power plant; nc represents the total of hydroelectric power plant; and T is the total time period.

#### 2.2.2. Constraints

#### 2.2.3. Constraints of the Reservoir

_{i}= 0.5 × θ

_{i}× (S

_{i}+ S

_{i}

_{−1})

_{i}

_{i}

_{i}is the net release of the reservoir during the time period i in (Mm

^{3}); L

_{i}is the water loss of the reservoir during the time period i in (Mm

^{3}); θ

_{i}is the coefficient of water loss per unit average volume stored in the reservoir during the time period i, which is dimensionless; A

_{i}and B

_{i}are auxiliary coefficients for each time period i, which are dimensionless, and are obtained by combining Equations (6) and (18). The remaining parameters and variables were previously defined. Finally, Equation (6) can be transformed in Equation (22), but considering that there are not spillways allowed during the total time period.

_{i}+ Bi × S

_{i}

_{−1}+ W

_{i}− R

_{i}= 0

_{i}

_{−1}) − (S

_{i}

_{+1}) = 0

_{i}≤ Qde

_{i}

_{+1}is the reservoir storage at the end of the total time period in (Mm

^{3}), S

_{i}

_{−1}is the initial reservoir storage at the beginning of the month i in (Mm

^{3}), and Qde is the maximum gate release capacity of the reservoir in Mm

^{3}/month.

_{i}is obtained through a lineal correlation analysis between the historical observed storage and losses in the reservoir. It can be calculated by applying the continuity equation to the reservoir with information from the historical data, and the key would be to get the losses of the reservoir. Equation (19) establishes that the entire release of the reservoir is received by the users and turbinated through the hydroelectric power plant. Equation (22) establishes the reservoir water balance. Equation (23) is very hard for the algorithms, because the reservoir must be returned to its initial state at the end. This constraint implies that the parameter U of the reservoir depends exclusively on the inflow. Equation (24) guarantees that the release of the reservoir never exceeds the maximum release of the reservoir gate.

#### 2.2.4. Constraints of Kritsky and Menkel’s Method

_{T}+ L

_{T}

_{i}≤ Ds + Y

_{i}≥ (Y − Sa) + Ds

^{3}); hence: Y = max(S

_{i}− D

_{S}). This is a key variable for building the guide curves for long-term reservoir operation; R

_{T}is the annual net release of the reservoir in (Mm

^{3}); L

_{T}is the annual water loss of the reservoir in (Mm

^{3}); and Ds is the dead storage of the reservoir in (Mm

^{3}). The remaining parameters were previously defined.

_{T}) and the annual net release (R

_{T}) of the reservoir. Equation (27) is the lower bound, and guarantees that the variable (Y) always exceeds or equals the value of the within-year storage component of the reservoir (Sa), and ensures the multi-year storage. Equation (28) ensures that the sum of the components (Sa) and (Sh) is equal to the (Sh) of the reservoir. Equation (29) is an upper bound, and ensures that the variable Y is never greater than the operating storage volume of the reservoir. Equation (30) is an upper bound that is used to avoid spillways, because its limit is the volume corresponding to the safety volume of the reservoir. Equation (31) guarantees that the volume storage on the reservoir always exceeds the multi-year or carryover storage component; note that Sh ≈ (Y − Sa).

#### 2.2.5. Constraints of the Hydroelectric Power Plant

_{o}and a are the adjustment coefficients of the polynomial equation that represent the curve level versus the storage of the reservoir, and Ho is the elevation of the turbine axis (in meters).

#### 2.2.6. Constraints of the Users

_{MAX}u,i

_{MAX}u,i = ρ × Du,i

^{3}); Du,i is the water demand of the user u during the time period i (in Mm

^{3}); Def

_{MAX}u,i is the maximum allowable water deficit of user u during the time period i (in Mm

^{3}); ρ is the fraction of the user’s water demand that is accepted as deficit, which is dimensionless; and Ru,i is the water release received by the user u during the time period i (in Mm

^{3}). The remaining parameters and variables were previously defined.

#### 2.3. Control Equation for Generating Operational Guide Curves

_{min}≤ λ ≤ 1

_{min}is the minimum fraction of the reservoir operational storage. The remaining parameters were previously defined.

_{min}. The two model executions are explained as follows.

- ▪
- The monthly storages of the reservoir that make the guide curve corresponding to the Upper Line of Safe Release (ULSR).
- ▪
- The monthly distribution of the release of the reservoir, the annual net release (R
_{T}), the annual water loss (P_{T}), and the annual gross yield (U) of the reservoir. - ▪
- Key parameters for the reservoir management are obtained, such as: the degree of regulation (α); the multi-year or carryover storage component (Sh), the within-year storage component (Sa), the within-year relative capacity (βa), and the relative carryover storage capacity (βh) of the reservoir.
- ▪
- The monthly energy generation schedule, power, and benefit of the hydroelectric power plant.
- ▪
- In the case of the users, the monthly distribution and total values of the releases and water deficits are obtained.

_{min}.

- ▪
- The monthly storage of the reservoir that makes the guide curve correspond to the Lower Line of Safe Release (LLSR). This guide curve touches the line corresponding to the Dead storage (Ds) of the reservoir at least once a month.

#### 2.4. Case Study: Carlos Manuel de Céspedes Reservoir (Santiago de Cuba)

^{2}. According to the studies carried out by the Delegación Provincial de Recursos Hidráulicos of Santiago de Cuba province, the average annual runoff to Contramaestre River basin has been estimated at 238 Mm

^{3}, of these, 152 Mm

^{3}(63.86%) occur in the wet season (May–October), and the remaining 86 Mm

^{3}(36.32%) occur in the dry season (November–April). The average annual rainfall in the basin is 1616 mm. This reservoir has a multi-year or carryover storage refill cycle, and is used in the following order of priority: agricultural irrigation and hydropower. Downstream of the reservoir around 20 agricultural companies are located that annually demand approximately 180 Mm

^{3}.

^{3}, an operating storage of 213 Mm

^{3}, and the dead storage is 30 hm

^{3}. There is a small hydroelectric power station (PCHE) with only one operational turbine, which has an installed capacity of 1500 kW. The turbine requires a minimum operating head of 7.70 m; which is guaranteed because the difference between the dead storage head and the turbine axis is 8 m. The activity of agricultural irrigation is carried out through direct intake from the Contramaestre River. Currently, the reservoir is operated through considering an operational rule that was obtained with a simulation model. The current operating rule establishes that during real-time operation, if the volume of the reservoir is within two guide curves, the save yield can be held. However, this rule does not consider the multipurpose use of the reservoir and the energy production.

^{3}, and the coefficient of variation (Cv) is 0.55. By using an inflow of this probability, a conservative scenario is applied because it refers to a medium—dry hydrological year. The maximum discharge of the Céspedes reservoir is 5 m

^{3}/s, and the reliability is 75%. It is common to use this reliability when the main user is irrigation.

^{3}. The annual water demand of 180 hm

^{3}for all of the users (20 agricultural companies) has been distributed uniformly at a constant rate of 15.0 Mm

^{3}/month. The fraction of the user’s water demand that is accepted as a deficit has been estimated at 20%, which means that the maximum monthly deficit of all the users is 3.0 Mm

^{3}. This value guarantees a volumetric reliability of 80%. The value of the energy tariff is 0.27 $/kW-h. The maximum benefit of the hydroelectric power station has been estimated as 291,600.00 $/month. This value considers that the turbine works at maximum power throughout the month.

## 3. Results and Discussion

^{−8}. The model run lasted 35 min in a Dual Core PC with 4 GB of RAM.

^{3}; this value could be interpreted as the safe yield of the reservoir for a given inflow time series. The annual water loss is 11.5 Mm

^{3}, and the annual gross release is 169 Mm

^{3}. The value of the annual gross release confirms the quality in the execution of the model, because it is equal to the annual inflow for a probability of 75%, thus implying that the releases of the reservoir are governed by the inflow.

^{3}, which the reservoir will not be able to satisfy.

^{3}kW-h of energy, with a total benefit of $1094.6 × 10

^{3}. This operational schedule is very useful for the Delegación Provincial de Recursos Hidráulicos of the Santiago de Cuba province. First, the installed turbine in the reservoir is too big for this system, because the maximum power proposed by the model is 498 kW, and the installed power is 1500 kW. So, this turbine can only work with about one third of its power.

_{min}= 0.40. This is the minimum fraction of the operating reservoir storage capacity for making the second execution of the model. After running the model with λ

_{min}, it is possible to get the storages, levels, and releases corresponding to the guide curve that represents the Lower Line of Safe Release (LLSR).

_{T}) and the curve of the (ULSR); the (SRZ) is located between the (ULSR) and the (LLSR), and the (RRZ) is located between the (LLSR) and the dead storage (Ds).

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Brown, C.M.; Lund, J.R.; Cai, X.; Reed, P.M.; Zagona, E.A.; Ostfeld, A.; Hall, J.; Characklis, G.W.; Brekke, W.Y. The future of water resources systems analysis: Toward a scientific framework for sustainable water management. Water. Resour. Res.
**2015**, 51, 6110–6124. [Google Scholar] [CrossRef] - Marie, K.J. Modelo de Gestión Optima Multiobjetivo Para la Cuenca del río Kwanza. Ph.D. Thesis, Universidad Rey Juan Carlos, Madrid, Spain, 2012. [Google Scholar]
- Rani, D.; Moreira, M.M. Simulation–optimization modeling: A survey and potential application in reservoir systems operation. Water Resour. Manag.
**2010**, 24, 1107–1138. [Google Scholar] [CrossRef] - Singh, A. An overview of the optimization modelling applications. J. Hydrol.
**2012**, 466, 167–182. [Google Scholar] [CrossRef] - Reddy, M.J.; Kumar, D.N. Multiobjective Differential Evolution with Application to Reservoir System Optimization. J. Comput. Civ. Eng.
**2007**, 21, 136–146. [Google Scholar] [CrossRef] - Russell, S.O.; Campbell, P.F. Reservoir operating rules with fuzzy programming. J. Water Resour. Plan. Manag.
**1996**, 122, 165–170. [Google Scholar] [CrossRef] - Wurbs, R. Optimization of Multiple—Purpose Reservoir System operation: An Review of Modelling and Analysis Approaches; US Army Corps of Engineers; Institute for Water Resources Hydrologic Engineering Center (HEC): Davis, CA, USA, 1991; p. 34. [Google Scholar]
- Neelakantan, T.; Sasireka, K. Review of Hedging Rules Applied to Reservoir Operation. Int. J. Eng. Technol. (IJET)
**2015**, 7, 1571–1580. [Google Scholar] - Lerma, N.; Paredes-Arquiola, J.; Molina, J.L.; Andreu, J. Evolutionary network flowmodels for obtaining operation rules in multi-reservoir water systems. J. Hydroinform.
**2014**, 16, 33–49. [Google Scholar] [CrossRef] - Loucks, D.P.; Van Beek, E.; Stedinger, J.R.; Dijkman, J.P.; Villars, M.T. Water Resources Systems Planning and Management: An Introduction to Methods, Models and Applications; Springer Nature, Ed.; UNESCO-IHE: Paris, France, 2005; pp. 1–635. ISBN 9231039989. [Google Scholar]
- Lund, J.R.; Guzman, J. Derived operating rules for reservoirs in series or in parallel. J. Water Resour. Plan. Manag.
**1999**, 125, 143–153. [Google Scholar] [CrossRef] - Ding, Y.F.; Tang, D.S.; Meng, Z.Z. A New Functional Approach for Searching Optimal Reservoir Rule Curves. Adv. Mater. Res.
**2014**, 915, 1452–1455. [Google Scholar] [CrossRef] - Vogel, R.M.; Bolognese, R.A. Storage-Reliability-resilence-yields relations for over-year water supply systems. Water. Resour. Res.
**1995**, 31, 645–654. [Google Scholar] [CrossRef] - Votruba, L.; Broža, V. Water Management in Reservoirs; Elsevier: Prague, Česká Republika, 1989; pp. 1–439. ISBN 0080870244. [Google Scholar]
- Kritsky, S.; Menkel, M. Water Resource Calculations; Gidrometeoizdat: Leningrad, Russia, 1952. (In Russian) [Google Scholar]
- Martínez, J.B. Compendio de Temas sobre Diseño y Operación de Embalses, 1st ed.; Centro de Investigaciones Hidráulicas (CIH): La Habana, Cuba, 2001; pp. 1–365. [Google Scholar]
- Recio, I.A.; Martínez, J.B. Modelo para la operación de embalses simples utilizando las relaciones de Capacidad-Garantía–Entrega aplicadas a la teoría de Kritsky y Menkel. In Proceedings of the Convención Científica de Ingeniería y Arquitectura, La Habana, Cuba, 24–28 November 2014; Universidad Tecnológica de la Habana, CUJAE: Havana, Cuba, 2014. [Google Scholar]
- Recio, I.A.; Martínez, J.B. Sistema para operación de embalse simple implementado en el asistente matemático MATLAB. Rev. Ing. Hidrául. Ambient.
**2016**, 37, 28–42. [Google Scholar] - Draper, A.J.; Lund, J.R. Optimal hedging and carryover storage value. J. Water Resour. Plan. Manag.
**2004**, 130, 83–87. [Google Scholar] [CrossRef] - Recio, I.A.; Martínez, J.; Soto, L. Política de operación óptima de un sistema de embalses mediante modelos HEC-ResPRM y RK3. Rev. Ing. Hidrául. Ambient.
**2017**, 38, 44–58. [Google Scholar] - Yeh, W.W.G. Reservoir management and operations models: A State of-the-Art Review. Water Resour. Res.
**1985**, 21, 1797–1818. [Google Scholar] [CrossRef] - Labadie, J.W. Optimal Operation of Multireservoir Systems: State-of-the-Art Review. J. Water Resour. Plan. Manag.
**2004**, 130, 93–111. [Google Scholar] [CrossRef] - Marglin, S. Public Investment Criteria; MIT Press: Cambridge, MA, USA, 1967. [Google Scholar]
- Yeh, W.W.; Becker, G.L. Multiobjective analysis of multireservoir operations. Water Resour. Res.
**1982**, 18, 1326–1336. [Google Scholar] [CrossRef] - Goldberg, D.E.; Holland, J.H. Genetic algorithms and machine learning. Mach. Learn.
**1988**, 3, 95–99. [Google Scholar] [CrossRef] - Shiau, J.T. Optimization of reservoir hedging rules using multiobjective genetic algorithm. J. Water Resour. Plan. Manag.
**2009**, 135, 355–363. [Google Scholar] [CrossRef] - Oliveira, R.; Loucks, D.P. Operating rules for multireservoir systems. Water Resour. Res.
**1997**, 33, 839–852. [Google Scholar] [CrossRef] - Kumar, D.N.; Raju, K.S.; Ashok, B. Optimal Reservoir Operation for Irrigation of Multiple Crops Using Genetic Algorithms. J. Irrig. Drain. Eng.-ASCE
**2006**, 132, 1–8. [Google Scholar] - Jothiprakash, V.; Shanthi, G. Single reservoir operating policies using genetic algorithm. Water Resour. Manag.
**2006**, 20, 917–929. [Google Scholar] [CrossRef]

Algorithm | α | βa | βh | βu | Sa (Mm^{3}) | Sh (Mm^{3}) | Su (Mm^{3}) | R_{T} (Mm^{3}) | U (Mm^{3}) | L_{T} (Mm^{3}) |
---|---|---|---|---|---|---|---|---|---|---|

GA | 0.71 | 0.236 | 0.5108 | 0.7474 | 87.06 | 145.58 | 213.0 | 157.5 | 169.0 | 11.5 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Recio Villa, I.; Martínez Rodríguez, J.B.; Molina, J.-L.; Pino Tarragó, J.C.
Multiobjective Optimization Modeling Approach for Multipurpose Single Reservoir Operation. *Water* **2018**, *10*, 427.
https://doi.org/10.3390/w10040427

**AMA Style**

Recio Villa I, Martínez Rodríguez JB, Molina J-L, Pino Tarragó JC.
Multiobjective Optimization Modeling Approach for Multipurpose Single Reservoir Operation. *Water*. 2018; 10(4):427.
https://doi.org/10.3390/w10040427

**Chicago/Turabian Style**

Recio Villa, Iosvany, José Bienvenido Martínez Rodríguez, José-Luis Molina, and Julio César Pino Tarragó.
2018. "Multiobjective Optimization Modeling Approach for Multipurpose Single Reservoir Operation" *Water* 10, no. 4: 427.
https://doi.org/10.3390/w10040427