https://orcid.org/0000-0002-0956-4078
Abstract
Water scarcity throughout the world has led to major difficulties and complexities in managing water demands. These challenges gravitate towards the development of efficient methods for optimal reservoir operation. The present study aims to introduce a hybrid approach which integrates Invasive Weed Optimization (IWO) and Cuckoo Search Algorithm (CSA), with an objective to minimize the deficits for Indira Sagar Reservoir (ISR), India. To prevail over the limitations of the Weed Optimization Algorithm (WOA) and CSA, a critical comparison has been made in the study. The hybrid approach has improved the performance by 5 and 9% as compared to WOA and CSA, respectively. For the reservoir system, the Cv for 10 random runs was computed to be 0.0303 using the hybrid model, whereas for WOA and CSA, Cv was 0.22034 and 0.30698, respectively. Based on the performance measuring indices, results revealed that the hybrid model is more reliable and sustainable with the minimum error between release and demand. In addition, results reveal that the deficits have been reduced by 62% on average for the considered study period using the hybrid approach. Therefore, the results show that the proposed hybrid model has considerable potential to be used as an optimizer for complex reservoir operation problems.
HIGHLIGHTS
A hybrid approach (HIWCSA) has been applied to a reservoir system for deriving optimal operating policies.
The analysis of the study implies that the proposed method HIWCSA is performing better than the standard models.
Critical comparison and evaluation of the applied methods have been carried out on the basis of benchmark function and performance-measuring indices.
INTRODUCTION
Owing to the ceaseless growing demand for water and emerging climate change, recent years have driven researchers to search for judicious methods of water resources systems. This has led to water shortages which makes it vital to distribute the deficits optimally during the non-monsoon periods.
In the last decades, numerous researchers have applied various evolutionary algorithms and mathematical models to optimize reservoir operation systems. Various studies using mathematical models have been successfully reviewed by Jacovkis et al. (1989), Vedula & Mohan (1990), Kumar & Baliarsingh (2003), Arunkumar & Jothiprakash (2012) and Heydari et al. (2015). These studies used linear programming (LP), dynamic programming (DP), and non-linear programming (NLP) as the basic models for reservoir operation optimization which also had their own limitations such as LP could not be used for non-linear optimization problems, NLP needs large computational storage and time while DP has the limitations of the curse of dimensionality.
Keeping in view the above limitations and requirements, researchers adopted different heuristic and evolutionary algorithms like Fuzzy Logic (Esogbue & Liu 2006), Ant Colony Optimization (ACO) (Jalali et al. 2007), Artificial Neural Networks (ANNs) (Chaves & Chang 2008), Genetic Algorithm (GA) (Cheng et al. 2008), Particle Swarm Optimization (PSO) (Reddy & Nagesh Kumar 2007), Cuckoo Search Algorithm (CSA) (Yang & Deb 2009), and Weed Optimization Algorithm (WOA) (Mehrabian & Lucas 2006).
An extensive review of different evolutionary algorithms was carried out by Ahmad et al. (2014) and Rani & Moreira (2010), discussing various optimization modelling approaches. However, each of the above discussed algorithms exhibit different kinds of problems like premature convergence, unstable convergence rate, complex programming or getting trapped in local optima (Karami et al. 2019). Therefore, a need arose to look for new hybrid approaches to solve optimization problems. The hybrid approach aids in overcoming the deficiency of individual algorithms so that the algorithms complement each other giving better solutions to the problem. CSA and Invasive Weed Optimization (IWO) are both new heuristic algorithms for finding optimal solutions in the given search space.
CSA has been recently developed as one of the latest nature-inspired meta-heuristic algorithms and is proving to be potentially more efficient than other evolutionary algorithms. There have been only a few studies based on the Cuckoo Search (CS) model for reservoir operation. Yasar (2016) developed a CS model-based solution for the generation of optimal rule curves and it was revealed that the CS model improved the operation of the system and increased the energy production. Later, Rath et al. (2017) used the CSA to develop optimal crop planning strategies for maximizing net benefits. Furthermore, different variants of the CS model were also studied extensively by Salgotra et al. (2018). Various modified versions of the CS model were adopted for improving the exploration and exploitation properties of the model.
Another efficient optimization technique, WOA, have been recently encountered its applications in reservoir operation optimization. Asgari et al. (2016) introduced WOA for continuous and discrete time formulation for reservoir operation and compared it with classic LP, NLP methods, and GA. The results showed that WOA gave a superior performance to that of the other methods with faster convergence. Another application of WOA was presented by Azizipour et al. (2016). The authors applied a novel evolutionary algorithm named IWO for the reservoir operation of hydropower systems. The results showed that IWO performed more efficiently and effectively for single and multi-reservoir systems than PSO and GA. Later, Ehteram et al. (2018) introduced an improved weed algorithm for minimizing irrigation deficits for reservoir optimization and suggested that the model has the potential to solve complex problems related to water resources management.
Turning to the issue of hybrid approaches, many researchers presented hybrid algorithms of different evolutionary algorithms. Khaddor et al. (2021) presented the effect of dam construction on flood management using the Gumbel law and the HEC-HMS rainfall–runoff process and the results showed better performance of the models. Sutopo et al. (2022) analyzed the effect of spillway width on outflow and flow elevation for probable maximum flood (PMF) using the Hershfield equation indicating that spillway crest width should be smaller for large storage volume. Mamidala & Sanampudi (2021) proposed a multi-document temporal summarization (MDTS) technique which generates a summary of related events from multiple documents and compared the performance with PSO, CS models.
Many current researchers reveal that better success rates can be attained in the context of convergence and precision by the combination of IWO and CSA models. Ho et al. (2015) presented a hybrid model combining harmony search and incremental dynamic programming for reservoir planning and optimization. Another contribution was made by Zhang et al. (2016), in which IWO and CS algorithms were combined with their respective features and compared with the basic IWO algorithm. The results showed that the hybrid approach could be successfully used as a fast and global optimization model. Later, Karami et al. (2019) introduced another hybrid approach combining a gravitational search algorithm and PSO for minimizing water supply deficiencies which in conclusion was considered a potential method for optimizing reservoir operation. Another hybrid approach by Lai et al. (2021) involved a whale optimization algorithm and levy flight distribution (LFWOA) for optimal reservoir operation. The authors found that LFWOA was superior to other meta-heuristic algorithms. Besides the optimal operation of dams, hybrid models have also been used in forecasting the discharge capacity of inflatable dams by Zheng et al. (2021). The authors used a hybrid model of PSO and GA and compared the results with other hybrid models based on statistical indicators. Furthermore, Hu et al. (2021) presented other soft computing and machine learning algorithms to determine the overflow capacity of a curved labyrinth. The authors used the Least-Square Support Vector Machine-Bat Algorithm (LSSVM-BA) and analyzed that the LSSVM-BA model signified the best prediction accuracy.
The present study focuses on optimizing a multi-objective reservoir operation problem based on a hybrid approach combining IWO and CSA, to maximize hydropower generation and minimize irrigation deficits. The novelty of the present study lies in the application of the hybrid approach, namely, the Hybrid Invasive Weed Cuckoo Search Algorithm (HIWCSA) for multi-objective reservoir operation. The hybrid algorithm has been tested on benchmark functions and the results are then compared with basic WOA and CSA models to evaluate the performance of the proposed hybrid model. The novel contribution of the present work is the integration of two meta-heuristic models, namely IWO and CSA to enhance the efficiency and capability of the models used in previous publications. The major contributions of the present study are:
Improved and faster convergence rate for the hybrid approach.
Attainment of improved precision and better function values as compared to the traditional algorithms.
In a previous publication (Trivedi & Shrivastava 2020), the standard PSO model was used with two enhancement models, namely EMMOPSO and TVEMMOPSO, considered a hybrid approach of different meta-heuristic models, although the study area and problem formulation are similar. In Trivedi & Shrivastava (2022), the study area considered was different and the parameters used in CSA were varied whereas in the present work, the comparison has been made with standard CSA with no parametric variation. Besides, in the previous work, only a single objective function was used and in the present work, multi-objective function has been used and therefore sensitivity analysis was done on all the parameters again to validate them based on different objective functions. The sensitivity analysis of the parameters of all the models has been discussed in the results section, in detail.
MATERIALS AND METHODS
Based on the literature review, the present study focuses on application of the hybrid approach to optimise multi-objective reservoir (ISR) and evaluating the efficiency of the HIWCSA model by comparing it with the standard WOA and CSA models. The models are discussed as follows:
Cuckoo Search Algorithm
CSA was first developed by Yang & Deb (2009) and is a meta-heuristic algorithm inspired by the cuckoo species. The algorithm uses important features of cuckoo species which evolve with the host bird species by laying an egg in the nest of the host bird. Each egg will represent a vector solution and each nest can have only one egg. High-quality eggs are carried over to the next generations. Available host nests are fixed and the host birds discover the cuckoo egg with some probability pa.
is the randomly generated solution by levy flight; 
is the randomly selected nest within the given range; 
is the step length; 
is the new best solution.
Eventually, the existing and new nest solutions are compared and the best nest is recorded and used for the next generations. The process of discovery and generation of new nests is repeated until maximum iterations are reached.
Weed Optimization Algorithm
is the number of seeds generated; 
is the ith objective function value; 
and 
are the minimum and maximum values of the objective function, respectively.The production of seeds is explained as follows:
is the standard deviation of iteration i; 
is the maximum iteration; 
is the number of current iterations; 
is the standard deviation at the initial level; 
is the standard deviation at the final level; m is the modulus of non-linearity.The number of weeds that could survive is limited to Pm, i.e., maximum population. Plants with lower fitness go for the next iteration to produce seeds and others are eliminated. At this stage, unsuitable weeds are abandoned until optimal criteria are achieved.
Hybrid model (HIWCSA)
is the new updated population, randn() is the normally distributed random number; 
is the mutation factor.
is the step length of levy distribution; u and v are random numbers generated based on normal distribution; β is the scale factor.
Flowchart for HIWCSA.
Data analysis
The basic data used in the present study include monthly inflows, demand patterns, and details of the ISR. The monthly inflow data were collected from the Narmada Control Authority (NCA) office, Indore, MP. The data from 2009 to 2015 were acquired for experimenting with the models discussed in the work. The monthly demand data were generated from the cropping pattern and hydropower demand data acquired from the NCA Office, Indore, MP, the NVDA Office Indore, and the NHDC Office, Khandwa district.
Stepwise procedure for data processing:
With the given inflow and demand data, a continuity equation was applied for all the years to identify the deficit months, i.e., months in which demands were not met successfully.
Storage greater than the live storage was considered as spill for months to be analyzed.
The identified deficit months in step 1 were then considered as inputs for all the models discussed in the study to obtain optimal release policies.
The models were then run using MATLAB for all the years considered and all the developed models.
Based on the results, certain performance measuring indices were also obtained for critical comparison of all the models.
Flowchart for data processing and methodology.
CASE STUDY
Salient features of dam
| Characteristics | ISR |
|---|---|
| Type of dam | Gravity |
| Height (m) | 92 |
| Length (m) | 653 |
| Total capacity (MCM) | 12,200 |
| Spillway capacity (m3/s) | 83,400 |
| Full reservoir level (m) | 262.13 |
| Characteristics | ISR |
|---|---|
| Type of dam | Gravity |
| Height (m) | 92 |
| Length (m) | 653 |
| Total capacity (MCM) | 12,200 |
| Spillway capacity (m3/s) | 83,400 |
| Full reservoir level (m) | 262.13 |
Location map of ISR.
Problem formulation
is the total release in time t (MCM); 
is the total demand during time t (MCM); 
is the maximum energy produced (MKWH); 
is the coefficient of power production; 
is the hydropower release for the month t (MCM); 
is the head of hydropower plant (m).
represents the reservoir storage at time t; 
represents the monthly inflow at time t; 
represents the monthly release at time t.
represents the minimum storage at time t; 
represents the maximum storage at time t; and 
represents the reservoir storage at time t + 1; 
represents monthly demand at time t.
Considering the above-mentioned constraints, the mass conservation equation was used to analyze the deficit months annually using the monthly inflow and demand data for all the years. The deficit months were those in which demand could not be completely met. The analyzed data was then used to run the proposed algorithms, HIWCSA, WOA and CSA, in MATLAB 9.4.0 version to determine the optimal operational policies for the considered period which were compared to evaluate the efficiency of the hybrid approach to those of the standard algorithms.
Stepwise procedure
The basic decision variable is the amount of water released which represents the generation of random population in the hybrid model.
The reservoir storages have been determined based on the state continuity equation. The months in which demands were not satisfied were considered as the deficit months to be optimized by the proposed model.
The storage and release values are then compared with the constraint values.
The objective function is then determined and analyzed by the model for better solutions.
The population is then allowed to reproduce seeds within a given range based on the equation.
The produced seeds are then spread randomly with a varying standard deviation to introduce stochastic nature in the model based on Equation (6).
The solutions with low fitness values are excluded and updated for the next iteration.
The updated solutions are further modified by introducing the concept of CSA along with levy flight distribution based on Figure 4.
The model is then checked for end criteria either by a maximum number of iterations or the optimal solution obtained.
Level of reproduction for each plant with respect to fitness.
Model application in benchmark test functions
In order to validate and evaluate the efficiency of the developed model, HIWCSA, for reservoir operation problems, a set of basic benchmark functions were considered as discussed in the following. The performance of the proposed model based on those functions was then compared with WOA and CSA models. The mathematical functions used for testing the models are as follows:
Sphere function
where i is the dimension, i.e., number of variables.
Rastrigin function
Ackley function
[−30,30].Considering the benchmark functions discussed above, the models were tested on all the functions. For the optimization of the functions, the number of iterations used was 100 for two variables and all other parameters were the same as those used for the real time optimization models.
Evaluation criteria
Certain performance measuring indices and statistical indices have been used in the present study to evaluate the performance of the proposed model (Srdjevic & Srdjevic 2017). The following indices were used:
Performance measuring indices
represents monthly release at time t, 
represents monthly demand at time t.
is the coefficient of variation of inflows.
Statistical indices
In the present study, two statistical indices were used for the performance evaluation of the models (Dang et al. 2020), namely, Mean Absolute Percentage Error (MAPE), and Transformed Root Mean Square Error (TRMSE).
is the demand and 
is the release, for ith duration.
are the release and demand values as per the expression 
, 
= 0.3, which scales down Q, i.e., release/demand.
RESULTS AND DISCUSSION
The hybrid model proposed in the present study is demonstrated through the ISR project using the HIWCSA model and compared with standard WOA and CSA models using MATLAB software. In the model, the input was the monthly inflow and demand for the deficit months to be optimized for ISR. The models were run to obtain outputs as the annual reduced deficits and monthly release pattern for years considered to be the deficit. In the years 2011, 2012, and 2013, there was sufficient rainfall and so they were not considered for optimization.
Sensitivity analysis and model parameters of all the algorithms
The optimization process of the algorithm is greatly affected by the initial random parameters of the respective algorithm. Therefore, it becomes essential to measure the parameter values accurately which is achieved by analysing the effect of a wide range of values of a particular parameter on the objective function value. In the present study, the objective is to minimize the deficit, so the parameter value which gives the minimum objective function value will be considered as the best value.
Sensitivity analysis of CSA parameters.
Sensitivity analysis of HIWCSA parameters.
To further validate the selected parameters, another sensitivity analysis was performed, presented in Tables 2–4. For the hybrid model, a different set of parameters has been considered in a combination of the initial and maximum population (Pi and Pm), the minimum and maximum number of seeds (NoSmin and NoSmax) and initial and final standard deviation (σi and σf). Random combinations of these parameters were run to analyze the results precisely. In Tables 2,3 and 4, the values in bold indicate the minimum objective function value for corresponding parameter of all the models. The minimum objective function value is 0.01 at Pi = 10 & Pm = 30 which indicates the best value for the objective function, as the objective is to minimise the deficits. Similarly, for the minimum and maximum number of seeds, the best objective function value is 0.01 at NoSmin = 0 and NoSmax = 5 combinations. The optimal number of iterations obtained is 500 with a mutation factor value of 0.5 giving the lowest value of the objective function. Finally, the minimum and maximum values of the standard deviation are taken as 0.01 and 2 with the lowest objective function value as 0.012. Other parameters 
and β were optimized at 0.01 and 1.5, respectively as shown in Table 2.
Sensitivity analysis for hybrid algorithm, HIWCSA
| Pi | Pm | OF | imax | OF | NoSmin | NoSmax | OF | σi | σf | OF | Sr | OF |  | OF | β | OF |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 10 | 0.062 | 100 | 1.33 | 0 | 1 | 0.674 | 0.01 | 1 | 0.708 | 0.1 | 0.017 | 0.01 | 0.011 | 0.5 | 0.02 |
| 5 | 30 | 0.024 | 200 | 0.92 | 0 | 3 | 0.083 | 0.01 | 2 | 0.012 | 0.3 | 0.016 | 0.02 | 0.017 | 1.0 | 0.033 |
| 10 | 15 | 0.029 | 300 | 0.429 | 0 | 5 | 0.01 | 0.01 | 3 | 0.018 | 0.5 | 0.012 | 0.015 | 0.014 | 1.3 | 0.43 |
| 10 | 30 | 0.01 | 400 | 0.212 | 0 | 7 | 0.029 | 0.01 | 4 | 0.055 | 0.7 | 0.018 | 0.005 | 0.02 | 1.5 | 0.011 |
| 7 | 10 | 0.37 | 500 | 0.013 | 2 | 5 | 0.22 | 0.02 | 1 | 1.6 | 0.9 | 0.68 | 0.05 | 0.34 | 1.7 | 0.75 |
| 7 | 15 | 0.34 | 600 | 0.018 | 2 | 7 | 0.08 | 0.02 | 2 | 0.04 | 1.1 | 0.66 | 0.10 | 0.35 | 1.9 | 0.78 |
| 15 | 25 | 0.29 | 700 | 0.029 | 4 | 7 | 0.07 | 0.04 | 1 | 1.4 | 1.3 | 0.65 | 0.25 | 0.36 | 2.0 | 0.017 |
| 15 | 30 | 0.28 | 800 | 0.054 | 4 | 9 | 0.03 | 0.04 | 2 | 0.03 | 1.5 | 0.64 | 0.50 | 0.34 | 2.2 | 0.81 |
| Pi | Pm | OF | imax | OF | NoSmin | NoSmax | OF | σi | σf | OF | Sr | OF | | OF | β | OF |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 10 | 0.062 | 100 | 1.33 | 0 | 1 | 0.674 | 0.01 | 1 | 0.708 | 0.1 | 0.017 | 0.01 | 0.011 | 0.5 | 0.02 |
| 5 | 30 | 0.024 | 200 | 0.92 | 0 | 3 | 0.083 | 0.01 | 2 | 0.012 | 0.3 | 0.016 | 0.02 | 0.017 | 1.0 | 0.033 |
| 10 | 15 | 0.029 | 300 | 0.429 | 0 | 5 | 0.01 | 0.01 | 3 | 0.018 | 0.5 | 0.012 | 0.015 | 0.014 | 1.3 | 0.43 |
| 10 | 30 | 0.01 | 400 | 0.212 | 0 | 7 | 0.029 | 0.01 | 4 | 0.055 | 0.7 | 0.018 | 0.005 | 0.02 | 1.5 | 0.011 |
| 7 | 10 | 0.37 | 500 | 0.013 | 2 | 5 | 0.22 | 0.02 | 1 | 1.6 | 0.9 | 0.68 | 0.05 | 0.34 | 1.7 | 0.75 |
| 7 | 15 | 0.34 | 600 | 0.018 | 2 | 7 | 0.08 | 0.02 | 2 | 0.04 | 1.1 | 0.66 | 0.10 | 0.35 | 1.9 | 0.78 |
| 15 | 25 | 0.29 | 700 | 0.029 | 4 | 7 | 0.07 | 0.04 | 1 | 1.4 | 1.3 | 0.65 | 0.25 | 0.36 | 2.0 | 0.017 |
| 15 | 30 | 0.28 | 800 | 0.054 | 4 | 9 | 0.03 | 0.04 | 2 | 0.03 | 1.5 | 0.64 | 0.50 | 0.34 | 2.2 | 0.81 |
The values in bold indicate the minimum objective function value for corresponding parameter of all the models.
Sensitivity analysis for standard WOA
| Pi | Pm | OF | imax | OF | NoSmin | NoSmax | OF | σi | σf | OF |
|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 10 | 0.116 | 100 | 1.45 | 0 | 1 | 0.846 | 0.01 | 1 | 0.918 |
| 5 | 30 | 0.05 | 300 | 0.464 | 0 | 3 | 0.147 | 0.01 | 2 | 0.047 |
| 10 | 15 | 0.071 | 500 | 0.045 | 0 | 5 | 0.047 | 0.01 | 3 | 0.05 |
| 10 | 30 | 0.045 | 700 | 0.047 | 0 | 7 | 0.051 | 0.01 | 4 | 0.054 |
| Pi | Pm | OF | imax | OF | NoSmin | NoSmax | OF | σi | σf | OF |
|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 10 | 0.116 | 100 | 1.45 | 0 | 1 | 0.846 | 0.01 | 1 | 0.918 |
| 5 | 30 | 0.05 | 300 | 0.464 | 0 | 3 | 0.147 | 0.01 | 2 | 0.047 |
| 10 | 15 | 0.071 | 500 | 0.045 | 0 | 5 | 0.047 | 0.01 | 3 | 0.05 |
| 10 | 30 | 0.045 | 700 | 0.047 | 0 | 7 | 0.051 | 0.01 | 4 | 0.054 |
The values in bold indicate the minimum objective function value for corresponding parameter of all the models.
Sensitivity analysis for standard CSA
| iter | OF | pa | OF |  | OF | β | OF |
|---|---|---|---|---|---|---|---|
| 50 | 0.159 | 0.1 | 0.168 | 0.25 | 0.179 | 0.5 | 0.199 |
| 100 | 0.11 | 0.15 | 0.137 | 0.5 | 0.188 | 1.0 | 0.169 |
| 150 | 0.144 | 0.2 | 0.212 | 0.75 | 0.1 | 1.5 | 0.119 |
| 200 | 0.141 | 0.25 | 0.105 | 1 | 0.133 | 2.0 | 0.157 |
| iter | OF | pa | OF | | OF | β | OF |
|---|---|---|---|---|---|---|---|
| 50 | 0.159 | 0.1 | 0.168 | 0.25 | 0.179 | 0.5 | 0.199 |
| 100 | 0.11 | 0.15 | 0.137 | 0.5 | 0.188 | 1.0 | 0.169 |
| 150 | 0.144 | 0.2 | 0.212 | 0.75 | 0.1 | 1.5 | 0.119 |
| 200 | 0.141 | 0.25 | 0.105 | 1 | 0.133 | 2.0 | 0.157 |
The values in bold indicate the minimum objective function value for corresponding parameter of all the models.
Similarly, sensitivity analysis values were obtained for standard WOA also with the same parametric values as those of the HIWCSA model, as shown in Table 3.
For CSA, the number of iterations was optimized at 100 and details of other parameters are also shown in Table 4.
Based on the sensitivity analysis performed above, Table 5 represents the parameter values of all the models. The common parameters of WOA and HIWCSA models are the same except for Sr, 
and β, which are added in the hybrid model.
Model parameters for different algorithms
| Model | Parameter | Value |
|---|---|---|
| CSA | Number of iterations, iter | 100 |
| Number of nests | 25 | |
| Discovery probability, pa | 0.25 | |
step size,  | 0.75 | |
| Scale parameter, β | 1.5 | |
| WOA | Pi, Initial population | 10 |
| Pm, Maximum population | 30 | |
| imax, Maximum number of iterations | 500 | |
| NoSmax, Maximum number of seeds | 5 | |
| NoSmin, Minimum number of seeds | 0 | |
| m, modulation of non-linearity | 3 | |
| σi, Initial standard deviation | 2 | |
| σf, Final standard deviation | 0.01 | |
| HIWCSA | Mutation factor, Sr | 0.5 |
step size,  | 0.01 | |
| Scale parameter, β | 1.5 |
| Model | Parameter | Value |
|---|---|---|
| CSA | Number of iterations, iter | 100 |
| Number of nests | 25 | |
| Discovery probability, pa | 0.25 | |
| step size, | 0.75 | |
| Scale parameter, β | 1.5 | |
| WOA | Pi, Initial population | 10 |
| Pm, Maximum population | 30 | |
| imax, Maximum number of iterations | 500 | |
| NoSmax, Maximum number of seeds | 5 | |
| NoSmin, Minimum number of seeds | 0 | |
| m, modulation of non-linearity | 3 | |
| σi, Initial standard deviation | 2 | |
| σf, Final standard deviation | 0.01 | |
| HIWCSA | Mutation factor, Sr | 0.5 |
| step size, | 0.01 | |
| Scale parameter, β | 1.5 |
Analysis of benchmark test functions for all the algorithms
To evaluate the efficacy of the hybrid model, all the algorithms were first tested on some mathematical test functions and then on the real case study. Sphere, Ackley and Rastrigin functions have been used in the present study to analyze the performance of the models. Table 6 shows the optimal function values and their statistical parameters for different mathematical functions for all the models. It can be concluded that the hybrid model has outperformed the standard WOA model as well as the CSA model significantly. The HIWCSA model has optimized the solutions in a better way by converging them to global optima, i.e., 0, precisely in Ackley and Rastrigin functions for almost all the runs. In the WOA model, there were only a few runs in which global optima was achieved while for other runs, the model has performed better than CSA.
Statistical measures of different functions from random runs of all the algorithms
| Test Function | Model | |||
|---|---|---|---|---|
| CSA | WOA | HIWCSA | ||
| Sphere | Function value | 3.81628 × 10-19 | 1.00 × 10-32 | 6.18 × 10-42 |
| 8.31628 × 10-19 | 3.63 × 10-32 | 8.92 × 10-43 | ||
| 4.41628 × 10-19 | 1.63 × 10-32 | 3.63 × 10-42 | ||
| 2.61628 × 10-19 | 1.28 × 10-31 | 1.63 × 10-42 | ||
| 1.21628 × 10-19 | 1.93 × 10-31 | 5.28 × 10-42 | ||
| Mean | 4.07628 × 10-19 | 7.67 × 10-42 | 3.52 × 10-42 | |
| SD | 2.66796 × 10-19 | 8.04 × 10-32 | 2.27 × 10-42 | |
| Ackley | Function value | 1.86282 × 10-5 | 3.50 × 10-15 | 0 |
| 6.56282 × 10-5 | 0 | 0 | ||
| 7.66282 × 10-5 | 0 | 0 | ||
| 7.96282 × 10-5 | 0 | 0 | ||
| 3.56282 × 10-5 | 0 | 0 | ||
| Mean | 5.52282 × 10-5 | 7.00 × 10-16 | 0 | |
| SD | 2.6857 × 10-5 | 1.40 × 10-15 | 0 | |
| Rastrigin | Function value | 1.71628 × 10-14 | 0 | 0 |
| 4.41628 × 10-13 | 0 | 0 | ||
| 2.91628 × 10-13 | 0 | 0 | ||
| 4.61628 × 10-13 | 0 | 0 | ||
| 3.11628 × 10-13 | 0 | 0 | ||
| Mean | 3.04735 × 10-13 | 0 | 0 | |
| SD | 1.58917 × 10-13 | 0 | 0 | |
| Test Function | Model | |||
|---|---|---|---|---|
| CSA | WOA | HIWCSA | ||
| Sphere | Function value | 3.81628 × 10-19 | 1.00 × 10-32 | 6.18 × 10-42 |
| 8.31628 × 10-19 | 3.63 × 10-32 | 8.92 × 10-43 | ||
| 4.41628 × 10-19 | 1.63 × 10-32 | 3.63 × 10-42 | ||
| 2.61628 × 10-19 | 1.28 × 10-31 | 1.63 × 10-42 | ||
| 1.21628 × 10-19 | 1.93 × 10-31 | 5.28 × 10-42 | ||
| Mean | 4.07628 × 10-19 | 7.67 × 10-42 | 3.52 × 10-42 | |
| SD | 2.66796 × 10-19 | 8.04 × 10-32 | 2.27 × 10-42 | |
| Ackley | Function value | 1.86282 × 10-5 | 3.50 × 10-15 | 0 |
| 6.56282 × 10-5 | 0 | 0 | ||
| 7.66282 × 10-5 | 0 | 0 | ||
| 7.96282 × 10-5 | 0 | 0 | ||
| 3.56282 × 10-5 | 0 | 0 | ||
| Mean | 5.52282 × 10-5 | 7.00 × 10-16 | 0 | |
| SD | 2.6857 × 10-5 | 1.40 × 10-15 | 0 | |
| Rastrigin | Function value | 1.71628 × 10-14 | 0 | 0 |
| 4.41628 × 10-13 | 0 | 0 | ||
| 2.91628 × 10-13 | 0 | 0 | ||
| 4.61628 × 10-13 | 0 | 0 | ||
| 3.11628 × 10-13 | 0 | 0 | ||
| Mean | 3.04735 × 10-13 | 0 | 0 | |
| SD | 1.58917 × 10-13 | 0 | 0 | |
Convergence curves for different mathematical functions for all the algorithms.
Analysis of random results of all the algorithms
Table 7 represents the results of ten random runs of all the models for the formulated problem. As can be seen in Table 7, the coefficient of variation for HIWCSA is lower than the other two models with a value of 0.0303 while for WOA and CSA the values are 0.22034 and 0.30698, respectively. Therefore, it can be concluded that HIWCSA is more reliable with a low Cv.
Ten random results for all the algorithms for ISR
| Run | CSA | WOA | HIWCSA |
|---|---|---|---|
| 1 | 0.194 | 0.061 | 0.032 |
| 2 | 0.111 | 0.061 | 0.033 |
| 3 | 0.126 | 0.081 | 0.016 |
| 4 | 0.157 | 0.07 | 0.023 |
| 5 | 0.223 | 0.078 | 0.02 |
| 6 | 0.215 | 0.045 | 0.029 |
| 7 | 0.223 | 0.045 | 0.03 |
| 8 | 0.299 | 0.045 | 0.038 |
| 9 | 0.299 | 0.045 | 0.034 |
| 10 | 0.301 | 0.058 | 0.009 |
| Best | 0.111 | 0.045 | 0.009 |
| Worst | 0.301 | 0.081 | 0.038 |
| Average | 0.215 | 0.059 | 0.264 |
| Standard deviation | 0.066 | 0.013 | 0.008 |
| Coefficient of variation | 0.30698 | 0.22034 | 0.0303 |
| Run | CSA | WOA | HIWCSA |
|---|---|---|---|
| 1 | 0.194 | 0.061 | 0.032 |
| 2 | 0.111 | 0.061 | 0.033 |
| 3 | 0.126 | 0.081 | 0.016 |
| 4 | 0.157 | 0.07 | 0.023 |
| 5 | 0.223 | 0.078 | 0.02 |
| 6 | 0.215 | 0.045 | 0.029 |
| 7 | 0.223 | 0.045 | 0.03 |
| 8 | 0.299 | 0.045 | 0.038 |
| 9 | 0.299 | 0.045 | 0.034 |
| 10 | 0.301 | 0.058 | 0.009 |
| Best | 0.111 | 0.045 | 0.009 |
| Worst | 0.301 | 0.081 | 0.038 |
| Average | 0.215 | 0.059 | 0.264 |
| Standard deviation | 0.066 | 0.013 | 0.008 |
| Coefficient of variation | 0.30698 | 0.22034 | 0.0303 |
Analysis of reduced deficits for ISR
Comparison of deficits for all the algorithms.
Analysis of release pattern for ISR
Monthly release pattern for 2009–2010.
Monthly release pattern for 2010–2011.
Monthly release pattern for 2014–2015.
Analysis of results based on performance measuring indices
For the performance evaluation of the models, certain performance and statistical measuring indices have been used in the present study for the reservoir operation of ISR. Table 8 represents different indices used for all the algorithms. As can be seen in Table 8, the volumetric reliability index is high for HIWCSA at 63% than the other two models, WOA and CSA at 59 and 54%, respectively, signifying that HIWCSA meets the demands in a better way than other models. As the vulnerability index signifies the failure of an event, therefore it can be seen that the HIWCSA model produced a low value at 37% which is 4 and 9% lower than WOA and CSA models, respectively. This indicates that for the HIWCSA model, the intensity of failure was less than that for the other two models, WOA and CSA, and the reservoir system would face more deficit.
Performance-measuring indices for all the algorithms for ISR
| Model | Reliability (%) | Vulnerability (%) | Resilience | Sustainability index | Shortage index | MAPE (%) | TRMSE |
|---|---|---|---|---|---|---|---|
| CSA | 54 | 46 | 0.028 | 0.008 | 4.06 | 47 | 7.1 |
| WOA | 59 | 41 | 0.058 | 0.02 | 3.65 | 42 | 6.2 |
| HIWCSA | 63 | 37 | 0.081 | 0.032 | 3.3 | 38 | 5.6 |
| Model | Reliability (%) | Vulnerability (%) | Resilience | Sustainability index | Shortage index | MAPE (%) | TRMSE |
|---|---|---|---|---|---|---|---|
| CSA | 54 | 46 | 0.028 | 0.008 | 4.06 | 47 | 7.1 |
| WOA | 59 | 41 | 0.058 | 0.02 | 3.65 | 42 | 6.2 |
| HIWCSA | 63 | 37 | 0.081 | 0.032 | 3.3 | 38 | 5.6 |
Turning to the resilience index, HIWCSA shows a higher value of 0.081 than WOA and CSA models, indicating that the HIWCSA model has a higher probability of recovering from failure and met the demands more frequently than the other two models. Similarly, the hybrid model performs better with regard to the sustainability index and shortage index with values of 0.032 and 3.3, respectively. These results show that the hybrid model is more sustainable and has the lowest value of shortage index as compared to WOA and CSA models.
The MAPE has been used to validate the algorithms used in the study for accuracy. A lower error value signifies higher accuracy, as can be seen in Table 8, indicating that better accuracy is obtained in operating the reservoir using the hybrid model. Furthermore, HIWCSA shows a lower value for MAPE and TRMSE indices in Table 8 as compared to WOA and CSA models, at 38% and 5.6, respectively, signifying minimum error for HIWCSA between release and demand.
Analysis of results based on convergence curves
Convergence curves for all the algorithms for ISR.
CONCLUSIONS
The present study focused on operating a multi-reservoir system, i.e., ISR for deriving optimal operational policies using a novel hybrid approach, the HIWCSA model, and a critical comparison has been made with the standard WOA and CSA models based on the results obtained.
The hybrid model was first tested on benchmark functions and proved to converge in a better way, achieving the global optima. Then, the models were run on the real-time reservoir operation problem and it can be concluded that the hybrid model improved the overall performance of the reservoir system as compared to the other two models with better convergence and lower function values.
The main purpose of the study was to minimize the deficits annually for the considered years. The results showed that the deficits were reduced by 59, 66, and 62% for the years 2009–2010, 2010–2011 and 2014–2015, respectively, using HIWCSA compared to the other two models. For performance evaluation of the models, certain indices were also estimated which showed that the HIWCSA model produced better results than the other two models in terms of reliability, vulnerability, resilience, shortage index, sustainability index, MAPE, and TRMSE.
The limitations in the present study could be overcome by using a large amount of data sets regarding all the variables used in the study. It is suggested that a more critical comparison of the proposed hybrid approach with other meta-heuristic models and/or hybrid models could improve the optimization study with reference to the convergence rate and precision of the algorithms. It can hence be concluded that the proposed hybrid approach (HIWCSA) has significant potential to optimize a range of complex reservoir systems. Thus, the future direction of the study could be forecasting inflows with growing demands for optimal reservoir operation.
CONSENT TO PUBLISH
The authors have given consent and approval for manuscript submission
AUTHORS CONTRIBUTIONS
All authors contributed to the study's conception and design. Methodology development, application and data analysis was done by M.T. Data collection and conceptualization was done by R.K.S. Draft preparation and review were done by M.T.
FUNDING
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
DATA AVAILABILITY STATEMENT
Data cannot be made publicly available; readers should contact the corresponding author for details.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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