Application of modified enhanced differential evolution algorithms for reservoir operation during floods: a case study

IS

Indira Sagar

DEA

Differential Evolution Algorithm

EDEA

Enhanced Differential Evolution Algorithm

MEDEA

Modified Enhanced Differential Evolution Algorithm

O_t

Outflow through spillway

t

Time steps in hours (t = 1,2,3, … T)

T

The time horizon for the problem

St

Initial storage at the beginning of time step t

TS

Target storage in the reservoir at the end of time horizon

MCM

Million cubic meter

Opt

Outflow through turbine

Weight assigned for the spillage

Vector of weights for achievement of target storage at the end of time horizon

Weight to meet the continuity constraint

exp

Exponential

bin

Binary

rand_t[0,1]

Uniformly distributed random numbers between 0 and 1 generated for each time step t.

D

Dimension of the problem

NP

Population size

F

Mutation factor

K

Scaling factor

CR

Crossover factor

G

Generation

G_max

Maximum number of iterations

Outflow at G generation within its feasible bounds

S_t+1

Final storage in the end of time step t + 1

O _max

Maximum outflow through spillway at time step t

O _min

Minimum outflow through spillway at time step t

Minimum permissible storage values capacity of reservoir (MCM) in tth hour

Maximum permissible storage values capacity of reservoir (MCM) in tth hour

It

Inflow to reservoir at time step t

1st random sample value of outflow through spillway at G generation

2nd random sample value of outflow through spillway at G generation

3rd random sample value of outflow through spillway at G generation

4th random sample value of outflow through spillway at G generation

5th random sample value of outflow through spillway at G generation

O_best

Best value from random sample of outflow through spillway

Worst value from random sample of outflow through spillway at G generation

Mutant vector of (G + 1)^th generation at time step t

Target vector of (G + 1)^th generation at time step t

FP

Target or parent vector

FC

Trial or child vector

SFP

Fitness parent function

SFC

Fitness child function

TS-1

1st Target Storage i.e. 2281.96 MCM

TS-2

2nd Target Storage i.e. 4810.59 MCM

TS-3

3rd Target Storage i.e. 6969.22 MCM

TS-4

4th Target Storage i.e. 9744.57 MCM

TS-5

5th Target Storage i.e. 10754.51 MCM

Reservoirs serve multiple purposes like hydropower generation, irrigation and municipal/industrial supplies along with flood control. Reservoirs also play a crucial role in flood reduction in a river. Operation of a reservoir is a tricky problem because of the uncertain inflows and varying demands. Maintenance of the storage for conservation and flood control are conflicting issues. Since the early 1970s, the reservoir operation problem has been solved using different Mathematical Programming (MP) techniques. The reviews published by Yeh (1985), Wurbs (1993), Labadie (2004), Rani & Moreira (2010) highlight the research progress in the development and applications of MP techniques for the optimization of reservoir operations. For reservoir operation during floods, various mathematical programming techniques such as Linear Programming (LP) (Windsor 1973), Nonlinear Programming (NLP) (Olcay & Larry 1990), Folded Dynamic Programming (Kumar et al. 2010), and Dynamic Programming (Mythili et al. 2013) can be cited as examples. There have been certain issues like function type, dimension of the problems and convergence to optimum value, etc., with the MP techniques despite having a strong mathematical foundation of MP techniques.

The Meta-Heuristic Algorithm (MHA) that was used to solve optimization problems have evolved in the last four decades with many advantages over the MP application to the optimization problems. MHAs are stochastic search algorithms used to obtain the optimum value of a given function. A detailed history and merit–demerits of MHAs can be found in Sorensen et al. (2018). The application of MHAs to water resources engineering is discussed by Reddy & Kumar (2020). A few examples of the applications of evolutionary algorithms to reservoir problems are Genetic Algorithms (GA) (Chang & Chen 1998; Jothiprakash & Shanthi 2006), Differential Evolution (DE) (Vasan & Raju 2004), Genetic Programming (GP) (Fallah-Mehdipour et al. 2012), Swarm Intelligence Algorithms such as Particle Swarm Optimization (PSO) (Moradi & Dariane 2009), Ant Colony Optimization (ACO) (Reddy & Kumar 2006), Honey Bee Mating Algorithm (HBMA) (Haddad et al. 2005), Bat Algorithm (BA) (Yang 2010; Ehteram et al. 2018), and Improved Bat Algorithm (IBA) (Ahmadianfar et al. 2016). Other algorithms include the Weed Optimization Algorithm (WOA) (Asgari et al. 2016), Harmony Search (HS) Algorithm (Bashiri-Atrabi et al. 2015), etc.

The Differential Evolution Algorithm (DEA) is a genetic-simulating stochastic population-based meta-heuristic technique developed by Storn & Price (1997). Brest et al. (2006) explained DEA as a powerful evolutionary algorithm for global optimization in many real-life complex problems. Das & Suganthan (2011) have compared DEA with other evolutionary as well as flock behaviour algorithms indicating that DEA is the most simple and easy to implement among all its kinds. The number of parameters is the lowest and the implementation requires simple coding.

Since the DEA method is a stochastic method based on either a random number generated or binary generation, the convergence parameters are crucial thus selection of these parameters is the crux of the versatility of DEA. Storn & Price (2004) mentioned that the performance of DEA depends upon the selection of control variables, viz. the population size (NP), mutation factor (F), crossover factor (CR), etc. From the literature review, it is clear that there is no standard procedure to determine the parameters, despite mentions of some tools and techniques for the determination of parameters. Storn & Price (2004) suggested ten different mutation strategies to increase the efficiency of DEA. These strategies approach the solution in slightly different ways in the mutation operation. An application of ten strategies to reservoir planning considering optimal cropping patterns with an aim of maximizing net economic benefits was presented by Vasan & Raju (2004). Regulwar et al. (2010) applied these strategies for the operation of multiple objective reservoir systems. Adeyemo & Otieno (2010) applied all strategies for the maximization of hydropower in a reservoir system.

The DEA has certain limitations like poor search capability, less population diversity, slow convergence rate, etc. Many different procedures to circumvent the limitations have been proposed. One of which is the Enhanced Differential Evolution Algorithm (EDEA) proposed by Ahmadianfar et al. (2017). While the algorithm was successful in its application, it had some minor limitations thatwere overcome by a better algorithm known as the Modified Enhanced Differential Evolution Algorithm (MEDEA). The introduction of this algorithm is a major contribution to the present study.

In the present study, the main focus is on the application of DEA and its variants the EDEA and MEDEA to solve the problem of reservoir operation during flood and compare the results. All five strategies have been implemented. The main criteria for comparison were the level of the solution achieved from the viewpoint of optimization and the required number of iterations in achieving the objectives of the formulated problem.

For the present study, inflow data of the IS Reservoir for a flood sequence was acquired from the Narmada Control Authority (NCA). The independent sequences of 25-h flow data are chosen for the present study starting from 0000 h on 29th August to 0000 h on 30th August in 2020. This flood is one of the highest floods in the past decades.

The order of meeting the objectives in operating the IS Reservoir during a flood is as follows:

• (1)

  Maximize the releases through the turbine.

• (2)

  Spillage of water to be controlled to maintain the discharge within the safe limits for the downstream.

• (3)

  Bringing the reservoir to the level of target storage after the flood wave is over.

Keeping the priorities in mind the optimization problem is formulated. The optimization problem is discussed in the following paragraph.

The optimization problem discussed in the preceding paragraph represented by Equations (1)–(3) is solved using DEA and its variants the EDEA and MEDEA in the present study. The techniques are discussed in the following paragraphs.

The DEA model was adopted for the present study and a code is written on MATLAB. The following steps of optimization are discussed.

• (1)
  Initialization: For the present problem the adopted time horizon (T) is 25 h. Thus, an initial population of 25 spill values are generated (for time step t and generation G = 0) within the bounds. (Equation (4)).

  

  (4)

The and (in MCM). The term is set of 25 uniformly distributed random numbers between 0 and 1 generated for each time step t. From this vector of spill values the objective function is evaluated and stored in the form of the fitness parent function (SFP).

• (2)

  Mutation: From the string of the spill values the mutation operation was carried out. The mutant vector for (G + 1)th generation for different strategies was computed from the formulae presented in Table 2.

Table 2

Mutation for differential evolution strategies

Differential evolution strategies .	DEA .	Equation for mutation operation .
DE/rand/1/exp	DEA/1
DE/rand/2/exp	DEA/2
DE/best/1/exp	DEA/3
DE/best/2/exp	DEA/4
DE/rand to best/1/exp	DEA/5

Differential evolution strategies .	DEA .	Equation for mutation operation .
DE/rand/1/exp	DEA/1
DE/rand/2/exp	DEA/2
DE/best/1/exp	DEA/3
DE/best/2/exp	DEA/4
DE/rand to best/1/exp	DEA/5

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The obtained trial vector is used to evaluate the objective function for the selection of next generation vector.

• (4)
  Selection: Before invoking the selection process the objective function value is evaluated from the trial vector as the fitness child function SFC. For the selection of the final value of spills for the next generation the objective function values SFP and SFC are compared using the greedy criterion for the minimization problem as follows:

  

  (7)

The DEA strategies involve exponential as well as binomial functions for mutation. However, in the present study, only exponential functions for all the 5 strategies have been adopted and reported. This is because the binomial functions lead to premature convergence and get stuck in the local optimum. This was mentioned by Qin et al. (2009) and the experience gained in the present study.

The convergence of DEA harps on the efficiency of the mutation operation. The traditional DEA has a limitation of weak local search ability and slow convergence rate despite the fact that it is a simple and efficient evolutionary algorithm. (Fan & Lampinen 2003; Ahmadianfar et al. 2017). This limitation escalates when complex optimization problems are solved. To improve the performance of DEA Ahmadianfar et al. (2017) introduced an Enhanced Differential Evolution Algorithm (EDEA) with changes in the mutation operation of the traditional DEA. They mention that the improvement in mutation will only be possible if the exploration and exploitation path is invoked. In the DEA, the first and the second strategies (Table 2) select the random variables from the vector of the initial population. The approach is exploratory. The exploratory approach adopted by the other strategies use the best individuals with randomly selected variables. They combined the mutation strategy used by Fan & Lampinen (2003) and Mohamed & Sabry (2012) to generate a new mutation strategy using an exponential formula. In the present study, the mutation strategy of EDEA was further modified to enhance the efficiency of the process of mutation. For the modification placement of the variable O_(i,G) was changed. Further, the scaling factor K is introduced and used with F in the mutation operation of a new method named as Modified Enhanced Differential Algorithm (MEDEA). The difference between the two methods is shown in Table 3.

Table 3

Comparison of mutation operation of EDEA and MEDEA

EDEA (Ahmadianfar et al. 2017) .	MEDEA (Suggested in present study) .
else endif	⁠) else endif

EDEA (Ahmadianfar et al. 2017) .	MEDEA (Suggested in present study) .
else endif	⁠) else endif

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In Table 3, P_max and  P_min are two constants within the range between 0 and 1, respectively, is the current generation number and is the maximum number of generations for mutation operation of EDEA and MEDEA. are the tournaments best, better and worst five randomly selected vectors, respectively.

The dimensions, search space and global optimum values are shown in Table 4 and other parameters with the ranges are shown in Table 5. The computed results of the application of DEA, EDEA, and MEDEA on these functions are presented in Table 6. Due to the lowest average results and the standard deviation values, the MEDEA is quick in convergence and effective in determining the results in comparison to both other techniques.

Table 4

Dimensions, search spaces, global optimum values, and acceptance level of test functions

Test function .	Function .	Dimension .	Search space .	Global optimum .
Ackley	f1	25		0
Sphere	f2	25	[−5.12,5.12]D	0

Test function .	Function .	Dimension .	Search space .	Global optimum .
Ackley	f1	25		0
Sphere	f2	25	[−5.12,5.12]D	0

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Though the MEDEA technique performs better, DEA and EDEA are both used to determine the optimal operation of the reservoir in the study. The models were run for all five strategies with DEA, EDEA, and MEDEA for five target storage values. Each of the problems was solved with a sufficient number of iterations. The major issue was to choose a proper set of parameters for all three models. Storn & Price (1997) initially suggested the population size (NP) between 5*D and 10*D (where D is the dimension of the problem), values of Mutation Factor (F) to be in the range of 0–2 and the Crossover Factor (CR) in the range between 0 and 1. Storn & Price (2008) while discussing the application of ten strategies recommend revised values of NP as 10D, and reduced the range of F and CR between 0.5 and 1.0. Vasan & Raju (2004) have mentioned the range of CR between 0.4 and 1.0 and that of F between 0.4 and 1.0 at an increment of 0.5. Regulwar et al. (2010) adopted CR values from 0.7 to 0.95 at a discrete increment of 0.05 and F values ranging from 0.2 to 0.9 with a discrete increment of 0.05. All these values were not found to be suitable in the present study thus a trial and error procedure was adopted. For the present problem, the selection of control parameters for DEA, EDEA, and MEDEA is shown in Table 5.

The optimization results of fitness functions of all strategies of DEA, EDEA, and MEDEA for the reservoir system are calculated as shown in Table 8. The standard deviation of the fitness function of MEDEA is lower than EDEA and DEA. It is also recommended that a low standard deviation qualifies the convergence ability, stability and precision of MEDEA. It is also clear that the worst solution of MEDEA (equal to 4.13 × 10¹⁰) was better than the best solution of EDEA (equal to 8.83 × 10¹⁰).

Table 9 indicates a comparison of the achievement of different Target Storage values with various strategies of DEA, EDEA and MEDEA for the year 2020. According to the percentage deviation of target storage the MEDEA attains lowest values in comparison to the strategies of EDEA and DEA. In the hierarchy MEDEA performed the best, EDEA performed better than DEA.

The present study demonstrates an application of three variants of DEA, viz. the DEA, EDEA, and MEDEA to derive short-term operation policies for a single reservoir during a flood event. The EDEA technique was proposed by Ahmadianfar et al.(2017), and the MEDEA technique is a major contribution of the present study. The techniques were first tested on benchmark optimization functions, namely the Ackley and Sphere functions, to establish their suitability. The main objective of the present study was to solve multiple objective optimization problems of reservoir operation during a flood event. A case study of Indira Sagar Reservoir in India was considered for a single flood event for the year 2020. The problem had three major objectives in the following order: to maximize the hydropower generation by passing maximum discharge through turbines, to minimize the spills and attainment of stipulated target storage level at the end of the time horizon. The values of five target storage were specified by the state government. Weighing technique was used to solve the multi-objective problem. The weights were heuristically chosen to ascertain the fulfilment of the objectives according to the hierarchy. All the models were solved for five strategies. The choice of the optimization parameters was difficult and a tricky issue. Trial and error procedure has been adopted to finalize the parameters for all the algorithms. The study results indicate that the objective of hydropower maximization was fulfilled in all the algorithms at all the target storage values. The flood releases were also within the limits of the spillway capacity. The target storage values achieved were compared in the form of percentage deviations at the final iteration of all three algorithms with all the strategies. In this regard, the MEDEA technique performed better than EDEA. The DEA did not perform up to the mark in all the strategies. It is established that in all the strategies the MEDEA performed better since the solutions are stable and achieved fast convergence of optimization in comparison with DEA and EDEA. It can be recommended as a successful, quick converging, fast computing, and stable algorithm for solving large-scale reservoir operation problems.

The authors would like to thank Narmada Control Authority (NCA), Indore for the data and other details of the ISP. The authors are thankful to The Director, SGSITS, Indore and the Principal, IPS Academy Institute of Engineering and Science, Indore for the motivation and encouragement they provided.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.