Development of water re-allocation policy under uncertainty conditions in the inflow to reservoir and demands parameters: a case study of Karaj AmirKabir dam

The process of optimal operation of multipurpose reservoirs is accompanied by large dimensions of decision variables and the uncertainty of hydrological parameters and water demands. Therefore, in determining the optimal operation policies (OOPs), the decision making for water allocation is faced with problems and complexities. One of the effective approaches for sustainable management and optimal allocation from water resources is the multi-objective structural development based on the uncertainty of input parameters. The purpose of this study is to provide OOPs from Karaj AmirKabir multi-purpose reservoir with applying uncertainty in the inflow to reservoir and downstream water demand. The proposed approach has been investigated in two certain and uncertain models, and three objective functions of the system including maximizing hydropower generation, water supply demands, and flood control have been considered to formulate OOPs. Non-dominated sorting genetic algorithm-II (NSGA-II) was performed to optimize the three proposed objective functions and by applying multi-criteria decision-making (MCDM) methods, the best operation scenario was selected. In the uncertainty model, using the interval method and repeated implementation of the deterministic model for completely random scenarios that generated based on the variation interval of the uncertain parameters, the non-deterministic optimal allocation values were produced. Based on these optimal allocation values and the fitting of the standard probability distribution on it, the probability of occurrence of the deterministic allocation values was determined. Production of optimal probabilistic allocation policies can be very useful and efficient in providing real vision to managers to select appropriate policies in different conditions and rare hydrological events. The results obtained from the certain model shows that as a result of optimal allocation to demands, the fuzzy reliability, resiliency, and system stability indexes were improved to 67.81, 21.99, and 24.98 percentage, respectively. Also, in an uncertain model, applying changes of 48% and 22%, respectively, for the inflow and downstream demand has led to changes of 23%, 55%, and 18%, respectively, in the first, second, and third objective functions. The highest impact from uncertain conditions, has been related to the water supply demands with 55% of the range of variations So, the water supply demands, has a higher sensitivity and priority than other reservoir objective functions under uncertain conditions. Another important result extracted from this study is to determine the monthly probability of optimal allocations achievement. Accordingly, in the warm seasons and years in which the reservoir is facing drought, the occurrence probability of the optimal allocations decreases. Given the comprehensiveness of the proposed methodology, this approach is a very suitable tool for determining the optimal water allocations as probabilistic based on the scenarios desired by managers and reservoir operators.


Introduction
In recent years, the aggravation of dry periods, especially in arid and semi-arid regions, has led to an increase in water shortages in water supply demands and social Extended author information available on the last page of the article problems (Malekpour and Tabari 2020). In order to manage surface water resources and develop them, reservoirs of dams are built as one of the effective structures in reducing the problem of water shortage and proper re-allocation of available water resources according to their temporal and spatial variations (Gu et al. 2017). Reservoir dams are usually used as one of the important engineering structures to meet various goals such as water supply demands of domestic, industry, agriculture, environment, flood control, hydroelectric energy production, ecological health, and shipping and they are called multi-purpose reservoirs. Considering the complexity governing the input and output components of reservoirs, diverse competitive goals, dynamic characteristics, nonlinear behavior and uncertainty of parameters, the management and operation of these structures requires a realistic and intelligent approach (Al-Jawad and Tanyimboh 2017;Bozorg-Haddad et al. 2021;Huang et al. 2022;Yang et al. 2022).
In the approaches that are presented for the dam operation, various goals are followed such as providing downstream water demands, flood control, hydroelectric energy production and the environmental demands of the downstream ecosystem of the reservoirs, and they are proposed as multi-objective optimization models (Ahmed and Mays 2012;Dong et al. 2021;Malekmohammadi et al. 2011;Ming et al. 2015;Solgi et al. 2016;Tian et al. 2019). In order to solve these multi-objective models, various methods such as weighting of objectives, use of objectives in constraints and meta-heuristics methods have been developed over the last three decades (Deb 2001;Fang et al. 2014;Karamouz et al. 2004;Liang et al. 1996;Xu 2020).
In reservoir operation models, due to the presence of competing and conflicting goals, it is not possible to provide a specific solution that satisfy all the goals. Based on this, a set of optimal solutions are presented as Pareto optimal solutions. In the problems of the dam operations, each optimal solution in this set of solutions is considered as an operation policy that has a different system performance according to the objectives of the decision-maker (Bozorg-Haddad et al. 2017;Tabari and Yazdi 2014;Tabari and Abyar 2021;Tabari et al. 2020;Tabari and Soltani 2012).
Using classic methods of solving multi-objective problems, only one solution of the pareto front can be determined. Therefore, to extract the optimal pareto front (optimal trade-off), it is necessary to repeatedly execute the structure of reservoir operation models to determine all possible optimal solutions located on the trade-off curve. This is not possible in many cases due to the large number of decision variables and the increase in computational costs as well as discontinuity in the goal space. In order to solve this problem in the last two decades, meta-heuristic methods such as Evolutionary Algorithm (EA), methods based on Swarm Intelligence (SI) and algorithms based on simulating physical and chemical laws have been widely used in solving multi-objective models for reservoir water resource planning (Fang and Popole 2020;KhazaiPoul et al. 2019;Mansouri et al. 2022;SeethaRam 2021;Sharifi et al. 2021;Tang et al. 2020;Tukimat and Harun 2019;Yang et al. 2020). In this regard, Yaghoubzadeh-Bavandpour et al. (2022) compared the performance of SI and EA algorithms for the optimal management and planning of water resources in the operation of reservoirs. These algorithms have advantages such as fast calculations, simple design and implementation, finding solutions near to global optimal solution, a lot of use, and superior generality. Also, these algorithms are able to overcome the limitations governing classical optimization methods in the operation of dams, such as a large number of decision variables, nonlinear relationships in the objective function and constraints, multiplicity of objectives, computational performance (Dahmani and Yebdri 2020;Wang et al. 2021).
Examining the literature review provided regarding to the application of multi-objective meta-heuristic algorithms in optimal operation models of reservoirs shows that finding an optimal operation policy (OOP) is very difficult due to different and conflicting objectives, physical limitations of the water resources system, and uncertain conditions governing it (Xu 2020) .
In the simulation of reservoirs, the water balance relationship is used and this relationship is affected by nondeterministic parameters such as precipitation, evaporation, inflow, downstream water demands, release rate, leakage, etc. So, if the uncertainty governing these input parameters is not properly addressed in the process of developing operation policies, the management and operation of the reservoir will be faced with errors and deviations from the actual conditions governing it in the development of optimal operation rules (Bozorg-Haddad et al. 2019;Rani and Moreira 2010;Tehrani et al. 2008).
Based on the classification done by Labadie (2004), the methods of applying uncertainty in optimization models are divided into two general categories: ISO (Implicit stochastic optimization) and ESO (Explicit stochastic optimization). In the ISO method, historical time series are used, and in the ESO method, probabilistic relationships are used to apply the uncertainty of the parameters. The results obtained from the comparison of these two methods show that the ISO are more accurate than the ESO (Celeste and Billib 2009). Other advantages of using the ISO method: less complexity, higher execution speed and greater simplicity in implementation (Jalilian et al. 2022).
The review of the studies conducted on the non-deterministic optimal models of reservoir shows that the researchers have mainly evaluated the uncertainty effects of parameters of inflow to the reservoir, evaporation and precipitation on the optimal amount of withdrawal from the reservoirs in the form of one or more objective functions (Banos et al. 2011;Jalilian et al. 2022;Lowe et al. 2009;Muronda et al. 2021;Zhao et al. 2011).
The most recent study conducted in this field is related to the research of Bozorg-Haddad et al. in 2021. In this research, by simultaneously applying the uncertainty associated with the parameters of the inflow to the reservoir and evaporation on the SOP model and without applying optimization algorithms and using the Monte Carlo method, the effects of uncertainty on the variables of the reservoir such as released water, the volume of the reservoir and the amount of spillway were investigated (Bozorg-Haddad et al. 2021).
The use of a multi-objective approach in reservoir management to minimize the possibility of flooding in the upstream and downstream areas of the dam under uncertain scenarios of inflow to the reservoir was carried out by Huang et al. in 2022. In this study, which deals with one aspect of reservoir operation (i.e., flood control), using AR-MOEA multi-objective algorithm (Adaptive Reference Multiobjective Evolutionary Algorithm), has simultaneously controlled the possibility of risk, vulnerability and reversibility of flooding to support the intelligent control of the operation of a reservoir under real conditions (Huang et al. 2022).
The evaluation results of previous studies show that paying attention to the main functions of a reservoir such as flood control, water supply demands, and energy production at the same time and based on fluctuations and uncertain variation of effective input parameters (such as inflow to the reservoir and downstream water demands) has not been given serious attention.
In these studies, has been emphasized on changing the optimization method (Ming et al. 2015;Zarei et al. 2019), focusing on the control details of a phenomenon (such as flood) (Chen et al. 2020;Huang et al. 2022), the use of different methods in the evaluation of the uncertainty of parameters (Ahmadi et al. 2010;Dai et al. 2018;Li et al. 2021;Muronda et al. 2021;Tabari 2015), and presenting different multi-criteria decision-making (MCDM) approaches to select the best optimal policies from the set of nondominate solutions produced by multi-objective metaheuristic algorithms (Malekmohammadi et al. 2011;Tabari and Abyar 2021;Yang et al. 2022). Therefore, combining the main functions of a dam along with the use of implicit methods of applying the uncertainty of the parameters and choosing the optimal reservoir operation policies based on the combination of efficient MCDM approaches is still necessary and providing improved approaches in research related to the management of reservoir water resources is needed to enrich previous studies.
In the development of reservoir operation approaches, the amount of available water based on the inflow and consumption downstream of the dams as state variables of the reservoir system and key components is very important in extracting water release policies from dams. The simultaneous application of the uncertain conditions of these two mentioned parameters in reservoir simulation relationships can play a significant role in producing relatively conservatively and intelligent operation policies. Therefore, in order to fill the research gap in this field, the present study has been done aims to develop an approach in order to solve multi-objective models of reservoir operation considering the uncertain conditions of the parameters of the inflow to the reservoir and the amount of water required downstream to provide optimal allocation values in a probabilistic manner. This structure can significantly help the decision-makers and managers of dam reservoirs to realistically evaluation of managerial decisions based on optimal operation probabilistic policies and its effects on the stability of the reservoir water resources system.
In this research, due to the robustness of the NSGA-II algorithm in solving multi-objective optimization problems, this method was considered to implement the developed three-objective management model under two deterministic and uncertainty conditions of input parameters. The goals considered in the proposed model are as follows: minimizing the amount of shortage of downstream water demands of the reservoir, controlling the floods entering the dam for proper placement inside the dam by maximizing the volume of flood storage, and increasing the amount of hydroelectric energy generation. Due to the simplicity of implementation of implicit methods of applying uncertainty, relatively high execution speed and appropriate accuracy, the interval method, which has received less attention in the operation of reservoirs, was considered in order to investigate the uncertainty of input parameters on the state of optimal policies of the reservoir.
In this method, using the historical time series of two uncertain parameters (the inflow into the reservoir and downstream water demand of the Karaj Dam, as one of Iran's strategic dams in providing the drinking demands of the capital city), and the permissible interval of their monthly variations, a significant number of completely random and improbable scenarios were generated from these two parameters. By implementing the developed management model for randomly generated scenarios, nondominate solutions with different degrees of intelligence in the optimal operation policies are extracted in the form of optimal trade-off curves between objectives.
In order to choose the solution with higher priority, five MCDM methods called Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Weighted Aggregate Sum Product Assessment (WASPAS), Complex Proportional Assessment (COPRAS), Compromising Programming (CP), and Modified-TOPSIS (M-TOPSIS) were used. According to the conducted surveys, these methods have been more applied in reservoir water resource management problems among different researchers Tabari et al. 2020. Finally, in order to evaluate the performance of OOPs in two deterministic and uncertainty models, fuzzy indicators for evaluating water resource systems that proposed by Safavi and Golmohammadi (2016) were used. These indicators show better the functional state of the water resources system in the limit values (threshold). Generally, the proposed new structure to supply the functional goals of the reservoirs can be presented as a new and effective perspective to adopt intelligent operation policies of the reservoirs under uncertain conditions. This approach can easily be implemented and used in multi-purpose dams located in other regions.

Case study
Considering the priorities of the IRAN ministry of energy in achieving the major goals of water resources management and achieving environmentally sustainable development in the strategic regions, the developed methodology was applied on Karaj Amirkabir dam (Karaj dam). Karaj dam in northwest of Tehran city is constructed on Karaj river in geographical coordinates of 508,100 m (east length) and 3,979,100 m (north width) (Fig. 1). It is the first multi-purpose dam in Iran. The power plant of this reservoir dam has been connected to the national electricity network for over 50 years and has a capacity of 90 megawatts (MW). Table 1 shows the characteristics of the dam reservoir and power plant. The lake of Karaj dam expands over 4000 hectares and is located in the Varian Gorge at a 23 km distance on Karaj-Chaloos road. Other general specifications of this dam are presented in Tabari (2017); Tabari and Sanayei (2018), and Tabari et al. (2020) research.
In this study, a 10-year (from April 2006 to March 2016) planning horizon has been considered for evaluation of the optimum management model of Karaj dam reservoir operation under deterministic and uncertainty conditions. To achieve the aim of this study, meteorological and hydrological data of the region including inflow to dam reservoir, evaporation, precipitation, downstream demands, and specifications of power plant and reservoir of Karaj dam are required. These data were collected from Iran Water Resources Management Company (IWRMC) and Tehran Regional Water Company. The time series of precipitation, evaporation, inflow to dam, and total demands are shown in Figs. 2 and 3.
Climatically, the studied area is influenced by the heights of Alborz, Chalus Valley and Karaj River, which has made this area cooler and more humid than the surrounding areas, including the Tehran city. According to the long-term climate data recorded for this region, the minimum, maximum, and annual average of temperature is -17°C, 42°C, and 15.1°C, respectively. July with an average of 35.2°C and January with an average of -2.5°C are considered to be the hottest and coldest months of the year, respectively. The average annual  precipitation of this area is 285.5 mm with a variation coefficient of 30.3% based on the Karaj synoptic station. The average total annual evaporation from class A pan is equal to 2429.6 mm. January and July with an average of 56.3 and 422.7 have the lowest and highest long-term evaporation, respectively. In the area of Karaj Dam, the precipitation compensates only less than a third of the evaporation in the region (Fig. 2). The highest amount of evaporation occurred in August 1991 with the amount of 388.17 mm. The amount of precipitation is also in a similar situation. In this way, there are months in which no rainfall has occurred and the highest amount of rain (213.7 mm) occurred in November 2010, and in the following years, its amount has decreased significantly.
Downstream demands of Karaj dam include domestic and agricultural demands. Drinking demands of dam is related to sum of the drinking water of Alborz Province and Tehran city.
The time series of inflow into dam reservoir follows a similar pattern every year, so that every year with the beginning of winter the inflow to the reservoir increases and the trend decreases until the end of spring (Fig. 3). In the summer and autumn, the inflow to the dam reservoir decreases. During the study period, the minimum and maximum inflow to the reservoir occurs in July and February, respectively.
In mathematical structure of developed water management model and in water continuity equation, it is necessary to determine the relationship between the Volume-Height-Area (VHA) of dam (Fig. 4). Based on Fig. 4, the area-volume and height-volume equation are as follows: where A t : reservoir area in month t (km 2 ), S t : volume of dam reservoir in month t (MCM), and H t : level of reservoir water surface in month t (m). In order to determine the power plant downstream water level, the stage-discharge curve of power plant was drawn. Based on this curve, the reservoir tail-water level (Tw t ) is calculated considering the amount of water release from dam reservoir (R t ) as follows: In Eq. (3), the minimum water height on the turbine of power plant is 1710.9 m and with increasing water release from dam reservoir, it increases linearly with coefficient

Methodology
In the developed approach in this study, due to the importance of applying non-deterministic conditions to multi-objective reservoir models and to evaluate the efficiency and comparability of uncertain operation policies, two deterministic and uncertain structures are developed in Sects. 3.1 and 3.2.

Structure of developed deterministic multiobjective model
In developing of the deterministic model, the volume of the reservoir is controlled and managed in a way to simultaneously be realized the benefits of releasing water to meet the demands and produce hydroelectric power along with the desired and required capacity to control the floodwater in the reservoir. It should be noted that only paying attention to one of the mentioned goals can lead to failure in the operation of the system. For example, releasing a lot of water from the reservoir, despite significant energy production and short-term supply of downstream demands, will have side effects such as failure to water supply demands in long-term and reducing the volume of the reservoir below the desired level in terms of operation and stability of the reservoir. The volume of water storage in reservoirs can be divided into two parts: flood control section and the required height to generate hydroelectric energy. Determining the optimal value of each of these two reservoir storage sections requires the use of a suitable optimization algorithm. The executive approach of achieving to the monthly optimal allocation values is presented in Fig. 5. In this flowchart, the OOPs are produced based on the continuity relationship of reservoir, objective functions and constraints governing the reservoir, NSGA-II algorithm, and the MCDM methods. The details of different parts of this approach are presented below.
In this section, the mathematical structure of the objective functions and the constraints of the multi-objective deterministic model are presented.
First objective function: Minimizing the lack of energy production compared to the installation capacity of the power plant.
In this objective function, the minimization of the difference between the energy production of the power plant and its installation capacity in each month is considered. In other words, the closer the produced energy of the power plant is to its installed capacity, the electricity generation will be also increased. To calculate the production power, in addition to the amount of water released from the reservoir, the amount of water head affecting the turbines must also be defined , 2017.
where T: the total number of planning horizon (120 month); P t : power generation in month t MW ð Þ; g: acceleration of gravity (m/s 2 ); g: efficiency of the power plant; R t : The amount of allocated water to the downstream demands in the month t (decision variable) (MCM); PF: power plant functional coefficient (20%); Mul t : conversion factor of MCM to m 3 /s in the month t (this coefficient is equal to 2.59 in months with 31 days, 2.68 in months with 30 days and 2.5 in month with 29 days); H t and H tþ1 : reservoir water level at the beginning and end of month t (m), respectively; Tw t : reservoir tail-water level in month t (m); PPC: power plant capacity, and Loss t : penalty function (dimensionless).
Second objective function: Minimizing the difference between the downstream water demands of the dam reservoir and the allocated water ; Tabari and Soltani 2012) where D t : the amount of downstream demands in the month t (MCM).
Third objective function: Maximizing flood storage volume where S n = storage volume of reservoir at the normal level (MCM).

Definition loss function
There are different ways for applying constraints in optimization issues that each of them follows their own goal. One of the common methods for solving the constraintbased problems is to impose loss function in different ways such as summing, multiplying or combining these two in the objective function. The general form of applying these functions varies according to the type of variables and conditions of each system. In this study, this function was defined based on the constraints of proposed multi-objective optimization model and added to each of the objective functions. Proper selection and definition of this function, in addition to solving the problem quickly, increases the quality of the optimal solutions and ensures that the optimal point obtained does not exceed the constraints of the problem. A following loss functions is applied to penalize the infeasible solution in the NSGA-II algorithm: where S min : minimum of dam reservoir capacity (30 MCM). Other parameters have already been introduced.

Constraints
One of the most important equations in modeling the operation of a reservoir is the continuity equation based on the conservation of mass: where I t : inflow to the reservoir during month t (MCM); P t : precipitation on the surface of the reservoir during month t (m); Ev t : rate of evaporation from the surface of the dam reservoir in the month t (m); A t and A tþ1 = reservoir areas in the beginning and end of the month t (km 2 ), respectively. The overflow from the reservoir is taken into account as follows: where S max : maximum of dam reservoir capacity (206 MCM).
Constraints related to the amount of water allocated, reservoir volume and constraints related to the initial (S t¼1 ) and end (S T ) volumes of the reservoir are considered as follows: S min S t S max ; t ¼ 1; 2; . . .; T ð13Þ 3.2 Structure of developed uncertainty multiobjective model In the proposed uncertainty model, two parameters of the inflow to the reservoir and the downstream demands of the dam, which have significant effects on the development of the reservoir's operation policies, were considered as uncertain parameters. The factors causing uncertainty in the inflow to the reservoir are very varied, as an example, the following can be mentioned: the unrecorded data of the stream and floodway entering the reservoir, the interaction of groundwater flow with the reservoir lake, lack of access to complete data of inflow time series, instrumental and measurement errors, and the existence of uncertain behavioral variations. Also, the downstream water demand parameter is also faced with uncertainty due to the lack of accurate recording of allocations, the existence of illegal withdrawals and outside of the operation plan, unpredicted population increase, per capita consumption growth, and etc.
To apply the uncertainty of these parameters in optimization models, these parameters can be presented as fuzzy values, interval values or stochastic numbers with specific probability distributions (Bozorg-Haddad et al. 2021;Miao et al. 2014;Zhang et al. 2009). In this study, due to the unknown distribution information related to downstream water demand and the inflow to the reservoir, the interval method has been used to consider the uncertainty caused by these two parameters (Nemati et al. 2021;Tabari 2015). In this method, using 2000 random time series generated based on the interval of variations of two uncertain parameters considered (Figs. 6, 7), 2000 reservoir operation models with objective functions and constraints defined in Sect. 3.1 are created. By implementing the NSGA-II optimization algorithm on these models, the optimal trade-off curves between the objectives are produced.
The solutions presented on this curve are actually the optimal reservoir operation policies that have different functional in terms of the three goals of water supply demands, flood control, and energy production. For this purpose, the use of MCDM methods can be effective in choosing a scenario that establish a balance between goals in a desirable way. Considering the diversity of these methods and the presentation of different results from them, applying methods with greater acceptability and multiplicity of use and combining their results to extract the desirable scenario is considered as an efficient approach in this study. Explanations related to these decision-making methods are provided in Sect. 3.2.2. Also, in this study, the fuzzy indicators of reliability, resiliency, vulnerability, and stability were used to evaluate the performance of the obtained OOPs (the relevant details are provided in Sect. 3.2.3).  By extracting the best operating scenario for each of the 2000 produced models, can be determine the variation range of the decision variables (monthly release from the dam reservoir) as well as variations in the objective functions as a result of applying the uncertain conditions of the two input parameters. By producing a set of superior optimal scenarios as a result of applying various uncertainty conditions governing the water resources and demands of the studied system, it is possible to fit the monthly probability distribution on the optimal allocation values. By using this fitted distribution, it is possible to determine the probability values of deterministic allocations under different operating conditions. This approach can be very useful and efficient in providing real vision to operators for better planning of multipurpose reservoir systems. The implementation process of the developed approach is presented in Figs. 8 and 9.
In this section, the mathematical structure of parts of the objective function and constraints that change in an uncertainty model are presented.
where D AE t : the uncertain intervals of demand (MCM) (variable with uncertainty). The continuity equation in the deterministic model is defined by considering the uncertainty of the inflow as follows: where I AE t : the uncertain intervals of the inflow (variable with uncertainty).

NSGA-II optimization algorithm
A review of the initial study on solving multi-objective optimization problems shows that these problems were initially solved by using the weighted sum of objective functions or converting them to a single-objective problem based on the e-constrained method (Malekmohammadi et al. 2011). To solve the problems governing these methods, the Non-dominated Sorting Genetic Algorithm (NSGA) was presented by Srinivas and Deb (1994). In this algorithm, GA can find widely different Pareto-optimal solutions using diversity-preserving mechanisms. This algorithm has the following drawbacks: (1) the computational complexity of this algorithm is of degree m 9 N, where m is the number of objective functions and N is the population size. In other words, with the increase in population size and sorting for each population, the computational cost of NSGA will increase dramatically.
(2) The elitism method is not used to achieve the solution. This method guarantees the preservation of the good solutions of the previous generation when creating the new generation after applying the operators of the genetic algorithm. As a result, the possibility of converging to the optimal solution is faster and the search efficiency is improved. To improve the capabilities of the NSGA algorithm, Deb et al. (2002) presented a modified version of the NSGA, called NSGA-II. This algorithm uses binary tournament selection, elitism nondominated sorting, and crowding distance operator in order to extract optimal trade-off between objective functions. To get acquainted with the details of this method, refer to the Deb et al. (2002). In order to execute the proposed deterministic and uncertainty model using the NSGA-II algorithm, the code of this algorithm was prepared and implemented in the MATLAB2018b (version 9.5) environment.

Ranking the optimal solutions of NSGA-II based on MCDM methods
Extraction of the most practical optimal solution among solutions located on the optimal trade-off is an important Fig. 9 Generation of reservoir operation probabilistic policies using the structure of the developed uncertainty multi-objective model and challenging issue. Unlike single-objective metaheuristic algorithms, which have an optimal solution near to the global optimal solution, the output of multi-objective optimization algorithms is a set of optimal solutions located on the optimal trade-off curve. In this study, each of these solutions, which examine different aspects of the objective's functions, is considered as a scenario for the operation of the dam reservoir. The use of decision-making methods can be very helpful in selecting the disputed scenarios in order to ranking and selection the solution in a way that satisfies all of the objectives relatively (Banihabib et al. 2017

Performance evaluation criteria of developed models
To evaluate the performance of the deterministic and uncertainty multi-objective management model, the reliability, resiliency and vulnerability fuzzy criteria were considered. These fuzzy criteria are more accurate in assessing the operation status of water resources systems than their non-fuzzy state and have been developed by Safavi and Golmohammadi (2016). By summing up these criteria with the sustainability index, we can evaluate the water resources system in terms of sustainability of water resources and water supply demands. In order to develop relationships related to performance criteria based on fuzzy theory, first of all, a utility function should be defined. This function expresses the concept that the closer the water supply is to the demand, the desirability and satisfaction of system performance will increase. Therefore, the parameters of this function must be defined in the form of S (El-Baroudy and Simonovic 2004).
In this study, based on the opinion of Tehran regional water company operation experts, the bell-shaped function was chosen as the representative of the fuzzy membership function of system desirability. Based on this membership function and by replacing it in the classical relations presented by Hashimoto et al. (1982)  Fuzzy reliability index: where x t is the percentage of water supply demand and l x t ð Þ, the level of system desirability in the tth time step (obtained based on Fig. 10). Fuzzy resiliency index: Fuzzy vulnerability index: Fuzzy sustainability index:

Results and discussion
Based on the developed deterministic and non-deterministic models and its implementation using the NSGA-II algorithm, the best operation scenarios were extracted based on MCDM methods. In this section, in order to better show the efficiency of the developed approach, the results of two models are first presented separately, and then, using the water resources system evaluation indicators defined in Sect. 3.2.3, the performance and capabilities of the multi-objective uncertainty operation model is investigated.

Deterministic model result
The developed deterministic model structure with 120 decision variables was performed using NSGA-II algorithm in the MATLAB2018b programming environment for the period of 2006-2016. In the NSGA-II, the proper selection of the size of the population of chromosomes is of great importance in reducing the execution time and the speed of achieving a solution near to the global optimum. For this purpose, a sensitivity analysis was performed on the initial population of chromosomes and the optimal trade-off curve for populations between 50 and 350 was evaluated in terms of convergence status and diversity of solutions on the optimal pareto curve. In this evaluation, all the parameters of the NSGA-II algorithm were considered the same for different populations (Table 2) and the number of objective function evaluations considered for them was equal to 1000. It should be noted that the values presented in Table 2 were obtained by trial and error. By implementing the developed multi-objective model using the NSGA-II on a computer with 16 gigabyte RAM and CPU Corei7-9700k, the optimal trade-off curve corresponding to each population size was extracted. To calculate the convergence criterion of the trade-off curves obtained by NSGA-II, the index presented by Chen et al. (2007) was used. Based on the values of the aggregation distance for each solution, the convergence criterion can be presented as Eq. (10) (Tabari and Abyar 2021): where d e m is the limit values on the converged trade-off curve, d i is the amount of aggregation distance for each solution on the trade-off curve, d is the average values of the solutions aggregation distance, and n is the number of solutions located on the converged trade-off curve.
The ideal value for this criterion is zero. Therefore, an algorithm with a small Spread value means that it has been capable to find a set of non-dominate solutions with good density. Based on this index and by calculating its value for populations of 50-350 (Fig. 11), can be found that the most appropriate population size for this study is 250.
By selecting the population size equal to 250 and running the developed deterministic model using the NSGA-II for 10,000 iterations, the trade-off curve near to global optimal solution was obtained. By ranking the optimal solutions based on the approach of using MCDM methods, the superior solution was selected. In the used decisionmaking methods, objective functions were considered as decision criteria and each of the solutions located on the optimal pareto curve were considered as alternatives. By aggregating the ranking results related to five decisionmaking methods using the BAM method, the solution shown in Fig. 12 (point A) was selected as the most suitable operation scenario for allocation to the determined goals. This point is actually chromosome number 18 from among the considered 250 chromosomes.
After selecting the best scenario, the values of the objective functions were calculated and compared with the current allocation. As shown in Fig. 13, the optimized allocation is proportional to the water demand. In fact, under optimal conditions, the release perfectly matches the demands of less than 42 MCM And its surplus is stored for allocation in other months. The release of reservoir in the   warm months is outside the capacity of the dam reservoir system, which needs to be supplied from other water resources such as groundwater resources. In optimal allocation, the annual adjustable water volume of reservoir is 310 MCM per year and the average monthly allocation is 25.85 MCM. Another point in optimal allocation is that optimal release is less affected by dry and wet years. According to the SPI drought index, the years 2008 and 2013 were evaluated as dry and very dry. The optimal allocation time series obtained from the deterministic operation model shows that in order to achieve long-term operation of the reservoir, the drought factor has not been a limitation for allocating and the demand has been regularly supplied. Therefore, it can be found that the consumption optimization plays an effective role in the optimal allocation of water resources.
The storage volume of reservoir according to Fig. 14 in optimal mode has more values compared to the current allocation of the system, indicating a higher volume of storage and, consequently, higher hydropower generation. The average reservoir storage volume in the optimal mode is 142.7 MCM (94% of dam reservoir volume), in the standard operation policy is 78 MCM (51% of dam reservoir volume) and in the existing operating mode is 110.2 MCM (72% of dam reservoir volume). Filling with 94% of the reservoir storage volume evaluated desirable. In the years 2008 and 2013, the reservoir storage volume due to drought has been significantly reduced, while it has been less effective in optimal allocation.
The amount of hydropower generation is also compared in Fig. 15. The average monthly hydropower generation in the optimal mode is 52.2 MW, and in available mode it is 47.1 MW, which indicates a relative increase in energy production under optimal operation policy of the dam reservoir. The average annual hydropower generation are 627 MW and 565 MW in optimal and current allocation, respectively. Because of drought in the years 2008 and 2013, the reduction in hydropower generation has been tangible both in optimal allocation and in the current allocation mode (A and B sections shown in Fig. 15). However, the reduction in hydropower generation had much less effect on the optimal allocation. Also, due to the third objective function for maximizing the flood storage volume, in some months the amount of hydropower generation under optimal condition has been lower than the current operation, which justifies the shortage of energy production. In other years, energy production follows of a specific regularity.
In order to comprehensively evaluate the exploitation policies produced, fuzzy indicators (presented in Fig. 15 Evaluation of optimal operation policy performance in hydropower generation  Table 3, by implementing optimal water allocation policies, the performance of the system water supply demands based on the fuzzy reliability index has increased from 22.84 to 67.81%. In fact, the number of months that the system has successfully met the downstream demands has been optimally greater than the actual operating conditions of the system. Accordingly, the severity of system failures has been reduced by 10.55% (from 100 to 89.55%). The investigation of the fuzzy resilience index has also indicated an increase in the number of favorable periods of the system from 18.27 to 21.99%. The comparison of the overall performance of the developed model using the system stability index shows a significant increase in the stability of the Karaj dam reservoir operation system over a period of 10 years.

Uncertainty model result
In the implementation of the developed uncertainty model, two parameters of the inflow to dam reservoir and the downstream water demand were assumed as uncertainty parameters. By implementing the developed uncertainty model for each set of generated random values of the uncertainty parameters, the 2000 optimal water allocation scenario was obtained over a period of 120 months. In order to better analyze the results of the non-deterministic model, the optimal allocation values in the deterministic model along with the variation interval of the optimal allocation values were drawn in the format of Fig. 16. According to this figure, the deterministic optimal allocation values are between the minimum and maximum allocation values associated with the uncertainty model. These operating intervals help managers to choose the best policy in different hydrological conditions. These significant variation in the optimal allocation values indicate the high sensitivity of reservoir operation models to the uncertainty of the inflow and downstream water demands.
Evaluation of the uncertain behavior of optimal allocation time series shows that in the best conditions (highest monthly inflow and lowest downstream demand), the operation system of the Karaj dam reservoir will not be able to meet all demands. Due to the significant deviation of the deterministic optimal values from their non-deterministic interval, it is important to pay attention to the accurate estimation of two uncertain parameters and the hydrological condition governing the modeled period in extracting operation rules near to real conditions. Also, the range of variation of the dam storage volume and the hydropower generation based on generated scenarios are shown in Figs. 17 and 18, respectively. The storage volume of reservoir, like the certain model, is lower in the months of November to March due to the aim of keeping the reservoir empty for controlling the spring floods and consequently the hydropower generation decreased in these months. Another noteworthy point is that if the dam reservoir is faced with dry and very dry   of the uncertainty interval obtained for it, and it is necessary to apply more uncertainty in these periods. Investigating the amount of hydropower generated in the deterministic model compared to the uncertainty model shows that the amount of hydroelectric energy produced in the deterministic condition in most months is equal to the maximum power of the system (Fig. 18). This shows that the energy production policy in the deterministic optimal model is based on the maximum use of system power.

Behavioral analysis of the defined objective functions under uncertainty condition
Based on the deterministic model of dam reservoir operation, it can be found that the dam operation manager can only have an operation scenario in executing the certain model, but the values of the objective function models of the certain model change with any unexpected changes occurring in the inflow to reservoir or in the downstream water demands. For this purpose, understanding the maximum and minimum range of objective functions will helps in order to better management and decision making on sustainable operation of dam reservoirs. Therefore, the range of variations in the objective functions under uncertainty condition are presented in Table 4. On average, applying changes of 48% and 22%, respectively, for the inflow and downstream demand has led to changes of 23%, 55%, and 18%, respectively, in the first, second, and third objective functions. It should be noted that the range of variations applied to the inflow has been determined based on the 10-year hydrological statistics of Karaj dam. Obviously, with the increase of the investigated years, the variation range of the output parameters of the uncertainty model will increase. The obtained results show that the uncertain parameters had different effects on the objective functions. The highest impact from uncertain conditions, has been related to the water supply demands with 55% of the range of variations. In fact, water supply demands in stochastic planning have a higher sensitivity and priority compared to other goals. Energy production and flood control are in the second and third priorities, respectively, for the development of uncertain policies. Therefore, it is necessary in the optimal allocation to demands, uncertainty parameters are properly collected and estimated to minimize the deviation of the allocated water from the optimal values.
To evaluate the performance of proposed uncertainty model, the variations in each of the fuzzy performance criteria are calculated (Table 5). For this purpose, each of the 2000 operational scenarios has been analyzed and the minimum and maximum value of the fuzzy indexes were extracted. Based on these limit values, it is possible to estimate the variations in the performance of the Karaj dam reservoir operation system under the implementation of deterministic operation policies.
According to Table 5 can be found that in case of deviation of the input parameters of the dam reservoir water resources system from the measured values, the reliability, resiliency, vulnerability, and sustainability criteria of the system can experience changes of up to 20%, 14%, 10%, and 15%, respectively, compared to the certain conditions. Therefore, the possibility of planning to deal with adverse conditions in the dam operation system will be provided. The vulnerability criterion of the uncertainty model provides the user with a 10% interval, that is, the severity of system failures is 10% less and more than the optimal deterministic model. Finally, the system sustainability index can have up to 15% variations. This assessment expresses the level of expectations from the dam reservoir operation system.

Probabilistic investigation of allocations in a deterministic and uncertain model
Providing the probability of occurrence associated with each optimal allocation can provide a better view of the dam reservoir operation plan to the operating managers. In this research, the standard normal probability distribution  has been used in order to investigate the probabilities of optimal allocations and obtain the confidence level of the allocation. Based on the developed uncertainty methodology and considering the results of the optimal allocation values associated with the 2000 times implementation of the proposed management model, the monthly probability distribution related to the optimal allocation was extracted.
Accordingly, the cumulative probability distribution function of each month was used to determine the probability of allocations of the deterministic model. As an example, the cumulative distribution function of August 2006 is presented in Fig. 19. According to the figure, the probability of achieving the optimal allocation of 37.36 MCM (based on deterministic model) for this month is equal to 39%.
Similarly, to assessment the optimal allocations obtained from the deterministic model, the probability of allocations was extracted. According to Fig. 20, the July, August, September and October have the lowest probability of achieving the optimal allocation. Therefore, it is necessary to take management actions to use other water resources (such as groundwater) to meet water demands in these months.
The average monthly allocations and their probability are shown in Fig. 21. Generally, in the hot seasons of the year, despite the increase in the amount of water allocated to the demands, due to the uncertainty in the inflow to dam reservoir and water demands, the probability of achieving to optimal values decreases below 50%. Based on this, it can be found that the optimal allocation values obtained from deterministic models cannot have high certainty in meeting the demands and it is necessary to be considered the effects of parameters that have high uncertainty in the process of extracting optimal operating policies. In order to better display the effects of uncertainty applied to the input parameters of the uncertainty multi-objective model, comparatively, the average uncertainty included on the input parameters and the average uncertainty associated with the optimal allocation amount, which is a key and influential parameter in the operation policy of multipurpose reservoirs, were drawn in Fig. 22. The findings of the study showed that the months that are associated with more allocation (May-September) and are mainly concentrated in the hot months with high evaporation, the amount of unauthorized withdrawals from the upstream of the dam reservoir that supplies the inflow increases, and therefore the level of uncertainty in allocation policies to demands will increase. Then, in generating uncertain conditions related to the optimal allocation, the uncertainty related to the inflow to reservoir plays a more important role.

Conclusion remarks
The conservation of the downstream ecosystem of the dam reservoirs and its water supply demands will be fulfilled as a result of the correct implementation of its operation policies. Considering the complexity of the reservoir system of dams from the point of view of the uncertainties in the input parameters, applying approaches that use simulation, optimization and decision-making tools to produce operation rules under real conditions is necessary and deserves attention. In this research, to achieve sustainable management of water resources and hydropower from the Karaj dam reservoir, a robust stochastic multi-objective reservoir operation framework was developed by considering the uncertainty in the parameters of the inflow to the dam reservoir and downstream water demands. In proposed management model, three objectives of downstream water supply demands, controlling the volume of floods entering the dam reservoir and increasing the amount of hydropower generation were considered. Due to the simplicity of implementation of implicit methods of applying uncertainty, relatively high execution speed and appropriate accuracy, the interval method, which has received less attention in the operation of reservoirs, was chosen in order to investigate the uncertainty of input parameters on the state of optimal policies of the reservoir. Based on this method, a significant number of completely random and improbable scenarios (2000 times) were generated using the monthly variations of input parameters during the planning horizon.
By implementing the proposed multi-objective management model for each of the generated scenarios using the NSGA-II algorithm, a set of noninferior solutions are produced in the form of pareto curves. In order to select the compromise solutions with the best rank in terms of the defined objective functions among the solutions located on the optimal trade-off curve, five decision-making methods named TOPSIS, WASPAS, COPRAS, CP, and M-TOPSIS were applied. Due to the divergence of results from these decision-making methods and to benefit from their capabilities, the Borda method was used to overcome the variety of prioritizations made by five decision-making methods and to aggregate their results. On this basis, for each generated scenario, an optimal operation policy is obtained using the three-objective reservoir operation model. Finally, by fitting the probability distribution to the set of optimal operation policies, can be determined the probability of optimal allocation under different operation scenarios for each of the months. These optimal values of the decision variables (monthly release from the dam reservoir) which are associated with the probability of its realization due to the uncertain conditions of the input parameters, an intelligent mechanism to flood control, water supply demands, and produce hydroelectric power provides to the dam managers. In fact, the implementation of operation policies based on the proposed approach can lead to a significant increase in benefits of reservoir operations as a result of reducing the risk caused by applying uncertainties related to input parameters. The results obtained from this study can be summarized as follows: 1. The results of the deterministic model show that the optimal operation policies have led to a significant increase in the fuzzy index of system stability and its value has reached to 24.98. Under these conditions, the average volume of the dam reservoir has been 94% full and the implementation of optimal operation policies has been able to increase the average annual energy production by 627 MW (more than 10% increase compared to the current situation). 2. Based on the results of the uncertain model can be found that on average, applying changes of 48% and 22%, respectively, for the inflow and downstream demand has led to changes of 23%, 55% and 18% in the first to third objective functions. Accordingly, the greatest effect of uncertainty has been on the water supply of downstream demands with 55% of the range of variations that needs to be considered in the development of optimal operation policies. 3. Due to the significant deviation of the deterministic optimal values from their non-deterministic interval, it is important to pay attention to the accurate estimation of two uncertain parameters (inflow and downstream demand) and the hydrological condition governing the modeled period in extracting operation rules near to real conditions. 4. Presentation of deterministic allocation values as operating rules, due to the uncertain nature of the parameters governing the operation of the dam reservoir, it increases the possibility of making errors in the correct estimation of allocation values to the demands. So, in this study, the probability of occurrence of each of the optimal allocation values obtained from the deterministic model was determined. The findings of the present study show that the probability of achieving optimal allocation values in summer season is less than 50% and planning to operation from other water resources such as groundwater resources is necessary. 5. In months with more allocation and high evaporation, the amount of unauthorized withdrawals from the upstream of the dam increases, and therefore the level of uncertainty in allocation policies will increase. So, in generating uncertain conditions related to the optimal allocation, the uncertainty related to the inflow to reservoir plays a more important role in the hot months.
The developed approach provides a non-deterministic modeling method that based on which can be improved the level of intelligence of the OOPs of the dam reservoir system for the set goals. In this methodology, the uncertainties of the proposed structure are simulated in a desirable way due to the complexity of the uncertain parameters. It is suggested for future studies to be consider the uncertainty of other parameters such as precipitation, evaporation, environmental demands, etc. simultaneously in an uncertain structure. Also, by using other methods of applying uncertainty, the performance of other uncertainty approaches be evaluated in order to choose the appropriate method in the dam reservoir system modeling.
Funding The authors have not disclosed any funding.
Data availability Enquiries about data availability should be directed to the authors.