Introduction

Along with rising temperatures in recent decades, many regions across the world have experienced increases in frequency and intensity of prolonged severe droughts (Williams et al. 2015; Cook et al. 2015; Erfanian et al. 2017). Since the hydrological cycle and climate system are closely related, variables such as precipitation and run-off are directly affected by changes in the climate system (Goharian et al. 2020). This significantly impacts water resources management, particularly in regions where surface water is the primary source of water supply (Azizipour et al. 2022). In these regions, it is pivotal for water managers and decision-makers to implement a drought management plan in order to mitigate the adverse impacts of water scarcity (Azizipour and Afshar 2018).

In drought conditions, surface reservoirs may fail to deliver the planned yields, and in such conditions water managers would rather have a series of smaller supply shortages than one potentially catastrophic shortage (Shih and Revelle 1994). Conventional methods used to mitigate drought-related water deficits include water rationing and limited water releases from the reservoirs which save water by applying a hedging rule to prevent severe future shortages. In previous works, hedging has been introduced to reduce the risk and cost of possible future large shortages compared to the cost of more frequent small shortages (Draper and Lund 2004).

Different versions of hedging rules for reservoir operation have been proposed. Hashimoto et al. (1982) showed that the standard operating policy (SOP) is the best operating policy where the loss function on releases is linear. A continuous hedging rule was created by Shih and Revelle (1994) using a mixed-integer nonlinear model, followed by later development of a discrete hedging rule for reservoir operation under drought conditions (Shih and Revelle 1995). Tu et al. (2003) employed both a conventional rule curve and hedging rules and developed an optimization model for optimal operation of a multi-reservoir multi-purpose system. Draper and Lund (2004) analytically examined the relationship between beneficial water release and carryover storage value, suggesting the frequent optimality or near-optimality of two-point hedging policies for water supply operations. Shiau and Lee (2005) derived optimal hedging rules for simultaneously minimizing short- and long-term deficit characteristics for a water supply reservoir. A mixed-integer quadratic programming model was developed by Tu et al. (2008) to optimize hedging rules for a multi-reservoir system. Karamouz et al. (2012) employed hedging rules to develop a planning scheme for optimal operation of reservoirs under drought conditions to minimize the maximum water deficit. A hybrid optimization model was proposed by Taghian et al. (2014) to simultaneously optimize both conventional rule curve and hedging rules. Kumar and Srinivasan (2018) suggested a two-point hedging rule, considering different starting and ending water availability points, significantly improving reservoir performance compared to results obtained by the SOP-based two-point hedging rule. Recently, a simulation–optimization framework was implemented by Srinivasan and Kumar (2018) to find pareto-optimal solutions for long-term hedged operation of a single water supply reservoir.

Recently, an optimization model using a genetic algorithm was used to develop a new hedging model that responds earlier to drought periods and starts rationing earlier, resulting in better optimization of the modified shortage index (MSI) in the historical period than conventional rationing methods (Bashavard et al. 2022). A hedging-based policy proposed by Mostaghimzadeh et al. (2022) incorporates a forecast term to manage release decisions in two separate phases, utilizing an exterior optimization model to handle the trade-off between forecast uncertainty and future information.

Despite the steady progress in development and application of different forms of hedging rules, all use cases have been restricted to short-term reservoir operations under prevailing drought conditions. Hedging rules have previously only been applied to reservoir operations such that the social, economic, and environmental impacts of water scarcity in drought conditions are reduced. In this study, however, the concept of hedging is utilized to develop a sustainable lake restoration framework for application to Lake Urmia, Iran.

Significant growth of agricultural development in the Lake Urmia basin in the past few decades has led to substantial increases in agricultural water demand (Eamen and Dariane 2013) and reduced inflows of water to the lake. The increasing water demand was exacerbated with the observed changes in the region’s climate (Vaheddoost and Aksoy 2017), resulting in continuous decline of the lake’s levels and 88% decrease in its surface area within two decades (AghaKouchak et al. 2015). Sarindizaj and Zarghami (2019) addressed the issue in a recent study and developed a system dynamics model to assess the sustainability of proposed restoration plans. Bakhshianlamouki et al. (2020) also used a system dynamics model to quantify the impacts of restoration plans on the food energy water nexus in the Urmia basin.

The present study develops a stationary policy using a new version of hedging rule that accounts for time-dependent environmental demand to restore and sustain the lake water level under assumed climate change conditions and water allocation needs for the agricultural and municipal sectors. This study aims to address feasible scenarios for the lake restoration and its sustainability for various pre-set restoration target periods. To do so, the study focuses on minimizing water deficits for stakeholders in the basin while restoring lake level implementing a dynamic demand framework. In a novel dynamic approach, the lake water requirement for restoration is adaptively varied by applying a hedging rule for rationing its requirement over time. The long-term system operation policy is achieved by substituting all surface reservoirs in the basin with an equivalent large reservoir. To incorporate climate change scenarios, inflow to the surrogate reservoir is estimated using an ANN-based rainfall–runoff model developed for the basin. The system is simulated using the AnyLogic system dynamic platform and optimized using the built-in optimization tool (Laguna and Marti 2003). The surrogate reservoir is then operated with intent to dynamically restore lake water level while minimizing basin-wide water deficits. The model simultaneously manages water allocation to other water users in accordance with the lake restoration strategy. To simplify the modeling process, all reservoirs in the basin are replaced with an equivalent reservoir to allocate the available water between potential users. The outcome of the proposed model is a stationary policy for lake restoration accounting for dynamic variations in lake level and inflows to the reservoirs due to presumed changes in climate conditions. The performance of the proposed framework is assessed through applications to Lake Urmia under different restoration and operation scenarios.

Case study description

We conducted a case study of Lake Urmia, which is located in the northwestern corner of Iran (Fig. 1) and has a catchment area of about 52,000 km2. One of the largest permanent hypersaline lakes in the world and the largest lake in the Middle East (Hassanzadeh et al. 2012), extends as much as 140 km from north to south and 85 km from east to west during high water periods (Jalili et al. 2012). The range of elevation within the lake watershed fluctuates between 1280 and 3600. Also, the average temperature ranges from 7.6 to 20.7 degrees Celsius. The surface area of the lake was estimated to be as large as 6100 km2 in 1995. However, the shoreline has been severely receding since leading to shrinkage in the lake's surface area of approximately 88% (AghaKouchak et al. 2015).

Fig. 1
figure 1

Map of Lake Urmia and the most critical water projects in the basin (Hassanzadeh et al. 2012)

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The lake is endorheic or terminal; thus, the only way water leaves the lake is by evaporation. Therefore, its salinity is quite high, ranging from 217 to more than 300 g/l, approximately eight times that of seawater (Ahmadzadeh Kokya et al. 2011). This has led to limitations in aquatic biodiversity such that the lake does not support any fish or mollusk species other than phytoplankton (Abbaspour and Nazaridoust 2007).

Inflows to Lake Urmia primarily come from rivers, including 21 permanent and 39 periodic rivers, with the Zarrineh Rood being the most extensive river feeding into the lake. Rainfall directly over the lake also contributes to its water supply. The Lake Urmia basin experiences an average annual precipitation of 401 mm, with over 90% of the total precipitation concentrated between the months of October and May (WRMC 2021). During the last two decades, inflows to the lake have significantly decreased, mainly due to climate variability and change as well as diversion of water for upstream use. Since the lake’s watershed is a vital agriculture hub, and agriculture surrounding the lake mainly relies on surface water supplies, surface water is typically diverted for irrigation purposes. As a consequence, flow to the lake has decreased causing overall shrinkage of surface area and leading to additional social and environmental problems. Therefore, it is crucial to find a sustainable plan that ensures the minimum required flow is allocated to the lake.

Methods and tools

This study presents a framework for time- and storage-dependent restoration of lakes’ level considering climate change impacts. Figure 2 presents the flow of data and information through the proposed modeling approach. Storage dependability of the lake restoration program is considered by presenting a new perspective on the previously applied hedging rule to reservoir operation. Lake water levels are expected to change dynamically which requires varying environmental flow of water throughout the management period. Any restoration strategy may fail if this dynamic variability is disregarded. The combination of pre-defined agricultural and/or domestic demand and variable environmental demand in a hedging rule for reservoir operation introduces a new challenge. Developing a set of dynamic multi-hedging rules is intending as a potential solution to this problem.

Fig. 2
figure 2

Illustration of the proposed framework

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Climate change effects are considered by employing downscaled global climate model (GCM) outputs. These data are then used as input for rainfall–runoff modeling to simulate basin runoff under climate change conditions. A hydrologic simulation model is developed in the AnyLogic system dynamic platform to estimate basin response to future changes in climate.

Sections "Projections of climate change impacts" and "Lake’s hedging rule" introduce the components of the proposed framework, illustrated in Fig. 2, along with the specific details for demonstrating how it is applied to Lake Urmia.

Projections of climate change impacts

Climate impacts on water resources are commonly investigated through experiments with global climate models (GCMs), run with nominally 100–200 km horizontal resolution, and an array of hydrological models (Chen et al. 2007; Gyawali and Watkins 2013). The accuracy of the results is expected to improve using finer spatial and temporal resolution, especially for demonstrating characteristics and dynamics of local and regional systems. Several methods (referred to as downscaling methods) exist for extracting information from GCM output at spatial and temporal scales finer than their native resolution (Wilby et al. 2004). Downscaling methods are generally classified as statistical and dynamical. Due to the capability of working with limited data and computational efficiency of statistical downscaling, we employ SDSM (Wilby et al. 2002) to statistically downscale GCM outputs. SDSM uses a hybrid stochastic weather generator and regression-based method for downscaling.

To develop the SDSM model, three sets of data were used. Observed daily precipitation data from three weather stations, as well as NCEP/NCAR daily reanalysis data, spanning 1961–2001 were used in this study. These two sets of data were used for both calibration and validation procedures. The locations of the three weather stations are illustrated in Fig. 3.

Fig. 3
figure 3

Location of weather stations used for downscaling

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Runoff prediction using an artificial neural network model

There are two main steps required to assess the climate change impacts on hydrologic characteristics of a basin. First, we predict regional temperature and precipitation variables using the downscaling procedure explained in the previous section. Then, we forecast future runoff in the basin using the regional variables as input to the rainfall–runoff model.

The rainfall–runoff model plays a crucial role in studying climate change impacts on water resources management. Due to its importance, many conceptual hydrological models with differing degrees of complexity have been developed by hydrologists to capture potential complications associated with rainfall–runoff dynamics. Conceptual models usually involve several interconnected components such as infiltration, evapotranspiration, land surface characteristics, which are difficult to simulate, forcing many researchers to use distributed hydrological models, such as the Soil and Water Assessment Tool (SWAT). On the other hand, distributed hydrological models require several parameters specific to the catchment that are not easily observed, making it challenging to appropriately model rainfall–runoff for many basins (Huo et al. 2012). In contrary to conceptual and distributed hydrological models, artificial neural networks (ANNs) can work with limited data and still yield promising results. Therefore, this study utilizes an ANN for rainfall–runoff simulation. The developed ANN model uses both temperature and precipitation data which affects the rainfall–runoff modeling directly, by impacting evaporation, and indirectly, by acting as global determinants for the seasonal variability (Furundzic 1998).

Lake-level simulation model

As a modeling paradigm, system dynamics (SD) was first introduced by Forrester (1961) in an attempt to provide a better understanding of the dynamic behavior of complex systems. Since its introduction, SD has been widely used in various fields, including water resource management (Ahmad and Simonovic 2000; Li and Simonovic 2002; Bakhshianlamouki et al. 2020). Here, the lake levels are simulated, and hydrological performance is assessed, under various conditions using the AnyLogic system dynamic platform. The stock and flow diagram for the lake-level simulation is presented in Fig. 4. As illustrated in the figure, the lake’s volume is a stock variable that increases by rain volume and surface water inflow and decreases by evaporation volume.

Fig. 4
figure 4

Stock and flow diagram for lake-level simulation

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The mathematical relationship between variables in the stock-flow diagram is governed by the mass balance equation in the form of:

$$ {\text{LV}}\left( t \right) = \mathop \int \limits_{{t_{0} }}^{{t_{n} }} \left[ {I\left( t \right) + {\text{RV}}\left( t \right) - {\text{EV}}\left( t \right)} \right]{\text{d}}t + {\text{LV}}\left( {t_{0} } \right) $$
(1)

in which LV(t) [million cubic meters (MCM)] is equal to the lake volume at time t, I(t)[MCM] represents surface water inflow to the lake at time t, and RV(t) and EV(t) [MCM] are rain volume and evaporation volume on/from lake surface at time t, respectively, which are calculated as:

$$ {\text{RV}}\left( t \right) = {\text{RI}}\left( t \right) \times {\text{LA}}\left( t \right) \times C1 $$
(2)

and

$$ {\text{EV}}\left( t \right) = {\text{EI}}\left( t \right) \times {\text{LA}}\left( t \right) \times C1 $$
(3)

where RI(t) [mm] and EI(t) [mm] represent rain and evaporation intensity at time t, respectively, and C1 is a constant parameter for taking into account the variability of dimensions.

After the development of the stock-flow diagram (simulation model), the model is calibrated and validated by using historical data. The evaporation intensity, EI, in the mass balance equation is the only parameter that needs to be determined during the calibration procedure. The obtained coefficient in the calibration stage is then validated.

Lake’s hedging rule

Hedging rules are generally employed in the reservoir operation context, especially during drought periods when mitigation or avoidance of severe water shortage is of the utmost importance. A hedging rule dictates that the reservoir operator retains water for probable future dry periods at the cost of slight deficits in outflows for current demand. While hedging rules have been used extensively in reservoir management, this section explains why and how the concept of hedging rules can be extended and applied for lake restoration purposes.

Many studies have reported the impact of climate change on arid and semi-arid regions, including Lake Urmia, in the form of decreasing precipitation and increasing temperature, as well as the magnitude and severity of droughts (Alizadeh-Choobari et al. 2016; Sanikhani et al. 2018; Abbasian et al. 2020; Schulz et al. 2020). This perspective necessitates the implementation of climate adaptation strategies and drought management approaches and justifies using hedging rules as a method for sustainable lake restoration. Considering constructed projects, under-construction projects, and planned projects in the Lake Urmia basin, the total volume of regulatory infrastructures in the basin is equal to 3869 MCM (Hassanzadeh et al. 2012), which is approximately equivalent to 60% of the average annual surface inflow to the basin. The proposed hedging rule intends to use the potential capacity of these infrastructures for lake restoration.

To apply the hedging rule concept to the lake, the sum of total capacity from all reservoirs in the watershed is aggregated up into one large equivalent reservoir. The approach of using a lumped reservoir has been used previously in various studies (e.g., Neelakantan and Pundarikanthan 2000; Tu et al. 2003). The equivalent reservoir, which is assumed to be located upstream of the lake, is operated with the primary objective of supplying water to meet downstream demand, including the lake demand. To fulfill the lake restoration purpose, a simulation–optimization model was developed next.

Simulation model

The complex and dynamic nature of the system is captured using a system dynamics simulation model with three main components: lake-level modeling, equivalent reservoir (ER) simulation modeling, and simulation of lake hedging rules. The former is previously described in Sect. "Lake-level simulation model."

The lake simulation model uses the simplest expression of mass conservation, namely the mass balance equation, for the equivalent reservoir. The mathematical form of the mass balance equation for the problem at hand is written as:

$$ V_{{{\text{ER}}}}^{t + 1} = V_{{{\text{ER}}}}^{t} + In_{{{\text{ER}}}}^{t} - R_{{{\text{ER}}}}^{t} - {\text{Evap}}_{{{\text{ER}}}}^{t} $$
(4)

where \(V_{{{\text{ER}}}}^{t}\) [MCM] is the volume of the equivalent reservoir, \(In_{{{\text{ER}}}}^{t}\) [MCM] is the inflow to the ER, \(R_{{{\text{ER}}}}^{t}\) [MCM] is water released downstream from the ER, and \({\text{Evap}}_{{{\text{ER}}}}^{t}\) [MCM] is evaporation from the ER, all in period t.

The third component of the simulation model is the lake’s hedging rule. In this study, we used a linear hedging strategy, proposed by Shih and Revelle (1994). According to this strategy, water release from the reservoir linearly increases in order to meet downstream water demand. In contrary to Standard Operating Policy (SOP), the slope of this line, K, is greater than 1 (K > 1), meaning that only a fraction of downstream demand is released from the reservoir when drought occurs or is anticipated. The level of storage plus projected inflow at which the rationing is triggered can be determined by:

$$ V_{1} = {\text{KD}} $$
(5)

where V1 [MCM] is the trigger volume; D is the downstream demand equal to the summation of lake, agricultural, and domestic demands. This hedging rule applies to a constant downstream demand. However, lake demand is a time-dependent variable and is positively related to water level. Thus, lake demand is a function of lake’s water level and dynamically derived via the following relationship:

$$ D_{L\left( i \right)} = {\text{TV}} - V_{\left( i \right)} + {\text{Evap}} $$
(6)

where Vi is the lake volume associated with water level i, DL(i) is lake demand at the level i (equivalent to the volume required to achieve the lake’s target level from level i), TV is the target volume of the lake (determined based on the volume required to reach the lake’s ecological level), and Evap is the evaporation from the lake’s surface.

The target volume of Lake Urmia corresponds to a water level of 1274.1 m above sea level (MASL), and the lake demand at each water level can be determined using Eq. (6). In this study, the range of water-level variability during restoration period was set to 1270 to 1275 MASL and lake demand was calculated for the water levels of 1270, 1271, 1272, 1273, 1274, and 1275 MASL. In the case of a lower (higher) water level than 1270 (1275), the lake’s demand is assumed to be the same as that of the lake’s water level of 1270 (1275) MASL.

Using six discrete water levels in Eq. (6) leads to six different values of constant water demand for the lake. Given that each hedging rule is only valid for a specific demand, six different values of K should be determined for six different hedging rules. Once K values are known, the trigger volumes V1 of individual hedging rules, can be calculated via Eq. (5).

After calculation of trigger volumes, several consequent rationing phases are ascertained, in which only αk fraction of downstream demand is met at the kth phase of rationing. As illustrated in Fig. 5, seven different rationing phases are considered, in which α ranges from 1 to 0.3, in this study. Water released from the reservoir at the kth rationing phase is allocated between downstream users based on:

$$ \alpha_{k} D = \alpha_{kD} D_{D} + \alpha_{kL} D_{L} + \alpha_{kA} D_{A} $$
(7)

where \(D\) represents the total combined downstream demand, \(D_{D}\), \(D_{L}\), \(D_{A}\) represent demand for domestic, lake, and agricultural purposes, and \(\alpha_{kD}\), \(\alpha_{kL}\), and \(\alpha_{kA}\) are the fractions of domestic, lake, and agricultural water demand at the kth phase of rationing, respectively. To reduce the complexity of the computation and its implementation, it is assumed that the relative priority between the lake restoration and agricultural demand can be defined by the authorities as \( = { }\alpha_{kA}\) /\(\alpha_{kL}\). With this assumption and through use of an appropriate simulation and optimization algorithm, one can feasibly determine the probability of attaining and sustaining each lake level during the simulation horizon.

Fig. 5
figure 5

Schematic representation of discrete hedging rule considered in this study

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It is important to note that the lake level at the end of each time step (i.e., t) is not the only selection criteria for model performance. Lake level at each time step does not necessarily address the lake condition over the entire planning horizon. Therefore, this paper employs a probabilistic approach by targeting the reliability of achieving and maintaining a satisfactory lake level during the entire operational horizon. The satisfactory lake level has been defined as 1274.1 MASL. To test and/or implement this approach, each lake-level restoration pattern would result in a specific duration curve associated with the restoration pattern addressed by its target level. The probability of maintaining any level is defined as:

$$ P_{{{\text{level}}}} = \frac{{i_{{{\text{level}}}} }}{T + 1} $$
(8)

in which ilevel is the number of time steps that lake level is equal or greater than target level, and T is the total number of observations.

Optimization model

Restoration of lake water levels and minimization of stakeholder deficits are the two main objectives of the optimization model. To derive the optimal values of trigger volumes for the hedging rules, the model proposed by Neelakantan and Pundarikanthan (1999) is extended. The mathematical representation of the optimization model is introduced by Eqs. (9) to (21).

$$ {\text{Min }}\sum \left( {D_{k} - R_{t + 1} } \right)^{2} \;\;\forall R_{t + 1} < D_{k} $$
(9)
$$ {\text{if}} S_{t} + I_{t + 1} - {\text{Eva}}_{t + 1} - T_{t + 1} > C \;\;\;R_{t + 1} = S_{t} + I_{t + 1} - {\text{Eva}}_{t + 1} - C $$
(10)
$$ {\text{if}} 0 \ge S_{t} + I_{t + 1} - {\text{Eva}}_{t + 1} - T_{t + 1} \ge C\;\;\;R_{t + 1} = T_{t + 1} $$
(11)
$$ {\text{if}} 0 \ge S_{t} + I_{t + 1} - {\text{Eva}}_{t + 1} - T_{t + 1} \;\;\;R_{t + 1} = S_{t} + I_{t + 1} - {\text{Eva}}_{t + 1} $$
(12)
$$ S_{t + 1} = S_{t} + I_{t + 1} - {\text{Eva}}_{t + 1} - R_{t + 1} $$
(13)
$$ {\text{if}} V_{1} \le S_{t} + I_{t + 1} \;\;\;T_{t + 1} = D_{k} $$
(14)
$$ {\text{if}} V_{1} \ge S_{t} + I_{t + 1} \ge V_{2^{\prime}} \;\;\;T_{t + 1} = \alpha_{1} \times D_{k} = 0.9D_{k} $$
(15)
$$ {\text{if}} V_{2} \ge S_{t} + I_{t + 1} \ge V_{3^{\prime}} \;\;\;T_{t + 1} = \alpha_{2} \times D_{k} = 0.8D_{k} $$
(16)
$$ {\text{if}} V_{3} \ge S_{t} + I_{t + 1} \ge V_{4^{\prime}} \;\;\;T_{t + 1} = \alpha_{3} \times D_{k} = 0.7D_{k} $$
(17)
$$ {\text{if}} V_{4} \ge S_{t} + I_{t + 1} \ge V_{5^{\prime}} \;\;\;T_{t + 1} = \alpha_{4} \times D_{k} = 0.6D_{k} $$
(18)
$$ {\text{if }} V_{5} \ge S_{t} + I_{t + 1} \ge V_{{6{^{\prime}}}} \;\;\;T_{t + 1} = \alpha_{5} \times D_{k} = 0.5D_{k} $$
(19)
$$ {\text{if}} V_{6} \ge S_{t} + I_{t + 1} \ge V_{7^{\prime}} \;\;\;T_{t + 1} = \alpha_{6} \times D_{k} = 0.4D_{k} $$
(20)
$$ {\text{if}} V_{7} \ge S_{t} + I_{t + 1} \;\;T_{t + 1} = \alpha_{7} \times D_{k} = 0.3D_{k} $$
(21)

where Dk is the total downstream demand at period t + 1, which is determined based on the current lake water level, and domestic and agricultural water demand. Rt+1, It+1, Evapt+1, and St+1 are water released from the reservoir, inflow to the reservoir, and evaporation from the reservoir and the reservoir storage, all at period t + 1, respectively. Tt+1 represents the volume determined to be released from the reservoir, and C is the reservoir’s capacity.

The objective function of the optimization model is defined by Eq. (9). The ER is operated according to Eqs. (10) to (12). Equation (12) describes the situation in which the projected inflow to the reservoir plus current storage is less than 30% of downstream demand in which case all available water is released to meet the demand.

In the hedging rule strategy, generally employed for reservoir operation purposes, using a monthly time step reduces usability of values of α < 0.6. However, in this study, an annual time step is considered, which is reasonable for lake restoration purposes and also enables the model to consider smaller values of α without any negative impacts on the applicability of the results.

The mass balance in the model is controlled through Eq. (13). Equations (14) to (21) determine the volume of released water from the reservoir under the rationing strategy. The constant values of \({\alpha }_{1}\) to \({\alpha }_{7}\) are set to steadily decrease from 0.9 to 0.3 meaning at any rationing phase, and depending on the projected inflow plus current reservoir storage, 30% to 100% of downstream demand is released from the reservoir.

The optimization model represented by Eqs. (9) to (21) is a nonlinear problem, which may be efficiently solved with an evolutionary algorithm. For this, the OptQuest engine of AnyLogic software is used.

Results and discussion

In this section, the effectiveness of the proposed hedging strategy for the dynamic restoration of Lake Urmia is illustrated. To assess the proposed hedging rules, we consider the 2011–2040 period divided into three separate decades with constant agricultural and domestic water demand in each decade. Climate change impact on precipitation and evaporation in the basin is considered as described in Sect. "Projections of climate change impacts," and an artificial neural network (see Sect. "Runoff prediction using an artificial neural network model") estimates surface runoff. The time series of the projected surface runoff in the Urmia basin, which is considered as the annual inflow to the equivalent reservoir, is illustrated in Fig. 6.

Fig. 6
figure 6

Time series of projected surface runoff in Urmia basin over 2011–2040

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Lake operating curves

The proposed simulation–optimization model is used to determine the optimal values of the slopes (K) for the hedging rules over three decades. The results of the proposed model are summarized in Table 1. The total demand, as defined by Eq. (7), exhibits a direct correlation with the water level of the lake under consideration. Consequently, a set of six distinct hedging rules is formulated for each decade, taking into account the varying water levels. By utilizing the determined K values for each hedging rule, the corresponding trigger volumes, denoted as V1, are specified. Additionally, the trigger volumes associated with different phases of rationing, namely V2 to V7, were computed using a similar approach.

Table 1 Hedging rules’ slopes (K) for the period of 2011–2040
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Once the trigger volumes for all distinct rationing phases have been determined, the operating curves of the lake can be derived to illustrate the relationship between water level and downstream water demand. Figure 7a, b, and c depicts the operating curves for the periods of 2011–2020, 2021–2030, and 2031–2040, respectively. It is important to highlight that the incorporation of water demand as a dynamic function of water level (as represented by Eq. 6) allows for the generation of comprehensive operating curves. These operating curves are generalized based on the lake's water level, presented along the horizontal axis, while the percentages displayed on the curves represent the proportion of the total downstream water demand to be fulfilled during each rationing phase.

Fig. 7
figure 7

Lake’s operating rules for the periods of 2011–2020 (a), 2021–2030 (b), and 2031–2040 (c)

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The operating rules presented in Figs. 7a–c could guide the lake’s restoration process in a few simple steps. First, the ER storage plus projected inflow and the lake’s level at any time would be used to select the proper rule curve from these figures. Next, the selected curve would determine the rationing severity (αk) and thus the total water released to downstream demand for that specific phase of the restoration process. Finally, the total combined water release is allocated between all downstream users based on Eq. (7). The allocation coefficients used in Eq. (7) depend on many factors and could be derived using different allocation approaches (i.e., optimization, game theory, etc.). Here, we set these allocation coefficients by assigning the highest allocation priority to domestic water supply and setting the relative priority of lake and agricultural water demand to one (β = 1; see Sect. 3.3.1 for more details on allocation coefficients). Table 2 presents the adopted water allocation coefficients at different rationing phases.

Table 2 Water allocation coefficients
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The proposed operating rules for the lake are distinct from those used for reservoir operations due to use of an annual time step and incorporation of level of downstream demand met (values of αk). Generally, rationing with αK < 0.6 for monthly reservoir operations cannot be applied in practice, especially in a basin with dominant agricultural demand. In such a basin, supplying agricultural demand is of great importance during specific seasons, and therefore, it is not meaningful to use αK < 0.6. However, in this study, having an annual time step for the lake’s rule curves enables use of lower values of αK. In this case, the resulting water deficit can be adjusted between different months in a year or even decrease the area under cultivation for specific years.

Hedging rules assessment

To evaluate the effectiveness of the proposed hedging rules, a comprehensive analysis was conducted, considering five distinct and plausible scenarios. These scenarios were formulated by taking into account the current management practices within the basin as well as the perspectives proposed in this study, with the aim of demonstrating the operating rules' capability to facilitate the restoration of the lake level to its desired ecological condition. A summary of the specifications for each scenario is provided in Table 3, which serves to outline the key parameters and conditions associated with each scenario.

Table 3 Scenarios specifications
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As shown in the table, the first scenario represents the conditions in which domestic and agricultural demands are entirely met and excess water is diverted toward the lake. Lake demand is disregarded in this scenario, and therefore, the operating rule does not substantially affect the lake water level. Most of the previous studies in the basin (i.e., Abbaspour and Nazaridoust 2007; Environment, 2010) considered a static (constant) demand (3100 MCM per year) for the lake regardless of the current lake’s water level. The second and third scenarios investigate the consequence of this approach on lake water levels. Contrary to the static demand, this study suggests incorporating dynamic demand for the lake, which is investigated in the fourth and fifth scenarios. Both dynamic and static representations of lake demand were tested for two operation scenarios: the hedging method proposed in this study versus the SOP.

The performance of the various scenarios was evaluated by focusing on two primary objectives: the water level of the lake and its ability to meet the demands of stakeholders within the basin. The evaluation of the lake's water level was conducted by analyzing level duration curves, which provide a comprehensive depiction of the lake's water-level characteristics throughout the entire planning horizon. Furthermore, the assessment of water supply for stakeholders involved the utilization of performance indicators such as reliability, vulnerability, and resiliency, as proposed by Hashimoto et al. (1982). These indicators serve as valuable metrics to evaluate the effectiveness of the operating rules in meeting the water requirements of stakeholders, thereby providing insights into the system's overall performance and its ability to sustainably manage water resources.

While the assessment of the lake's physical and biological conditions is commonly based on the water level, relying solely on the lake level in any given year may lead to misleading evaluations of scenario performance. In periods of severe drought or excessive rainfall, the water level of the lake may fall below or exceed the target ecological level, respectively, which can result in an inaccurate representation of the average lake level over the entire planning horizon. To address this challenge, the occurrence probability of a water level exceeding the minimum ecological threshold for Lake Urmia, specifically set at 1274.1, is calculated during the restoration period. This calculation enables the development of lake-level duration curves (LDCs), which serve as a more comprehensive performance criterion for assessing the lake's condition throughout the planned restoration horizon. By considering the LDCs, a clearer understanding of the lake's behavior and its deviation from the ecological target level can be achieved, providing valuable insights into the effectiveness of the scenarios in restoring and managing the lake's water resources.

Figure 8 shows lake LDCs for the planned restoration period. The difference between curves shown in the figure is related to differences in ER operation rules and lake demand implemented in each scenario. As it is seen in the figure, lake demand significantly affects the LDCs. In scenario 1 where there is no demand considered for the lake, and only excess water is diverted to the lake, lower lake levels are observed with a higher exceedance probability. Using static demand for the lake in scenarios 2 and 3, leads to increased lake water levels, compared to scenario 1. The impact of using dynamic demand (scenarios 4 and 5) is apparent in that the lake experiences the highest water levels with higher occurrence probability.

Fig. 8
figure 8

Lake LDCs for the restoration period under different scenarios

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While Fig. 8 highlights the advantage of using dynamic demand for the lake, comparing the LDCs of scenarios 2 and 4 with those of scenarios 3 and 5 indicates that the ER operation rule has a negligible impact on the lake-level indicator. In other words, operating ER with either SOP or hedging rules leads to approximately the same LDC.

In practice, a successful implementation of any restoration plan is highly dependent on stakeholder satisfaction and cooperation. Ensuring a lower supply deficit distributed to stakeholders, namely agricultural and domestic water demand, facilitates adaptation of a proposed restoration plan. Therefore, it is crucial to evaluate the performance of different operation scenarios in their ability to meet water demand from stakeholders. Here, we calculated reliability, vulnerability, and resiliency, as performance criteria, based on domestic and agricultural water demands for five different scenarios (Table 4).

Table 4 Performance criteria for ER in meeting agricultural and drinking water demands
Full size table

As shown in the table, the values of resiliency in scenarios 1, 4, and 5 are equal to 0, and scenarios 2 and 3 yield small values for resiliency. This emphasizes the critical situation in the basin in which the system faces serious challenges in being able to meet rapidly growing demands. In a 30-year restoration period, the summation of annual agricultural and domestic water demands is estimated to be 8300 MCM, while the potential of long-term surface water in the basin is approximated as 6600 MCM. This shows that the water demand in the basin is increasingly growing to an extent beyond the basin capacity to supply that volume of water.

Table 4 also indicates that the reliability of meeting demand under scenarios 4 and 5 is equal to 0. The reliability of meeting demand under scenario 3 is slightly higher than scenarios 4 and 5, while in scenario 1, the reliability reached 80%. These results demonstrate that the system fails to fully meet the agricultural and domestic demand in the basin for all periods considered in this study. However, as expected, eliminating the lake water demand (scenario 1) resulted in achieving higher reliability, while considering static (scenarios 2 and 3) or dynamic (scenarios 4 and 5) demand for the lake decreases the system’s capability to supply domestic and agricultural demands, and consequently the reliability is decreased.

While resiliency and reliability metrics indicate a similar performance for scenarios 2 to 5, the vulnerability metric reveals distinct performance improvements for scenarios that use the hedging rule. As shown in Table 4, the highest vulnerability of the system is associated with scenarios 2 and 4 in which the SOP is used to operate the ER. The lower vulnerability in scenarios 3 and 5 denotes that using the proposed hedging rule for operating the ER can substantially reduce the severity of failure in meeting agricultural and domestic water demands. For the static demand scenarios, the vulnerability of scenario 2 (using SOP) is 150% greater than that of scenario 3 (using hedging rule). Comparing scenarios 4 and 5, which consider the dynamic demand for the lake, reveals that using hedging rules instead of SOP for operating the ER improves the system’s vulnerability by 50%.

The final assessment of scenarios requires considering both lake level and performance criteria for the ER simultaneously. As illustrated in Fig. 8, scenarios 4 and 5 yield approximately the same LDCs outperforming other scenarios in lake-level restoration over the 30 years planning horizon. Regarding the ER performance criteria, scenarios 4 and 5 resulted in a value of zero for both reliability and resiliency. However, the vulnerability of scenario 4 is equal to 2263 MCM, while this is reduced to 1141 MCM in scenario 5, meaning that the average intensity of recurrent supply deficits in scenario 5 is approximately 50% less than that of scenario 4 which is achieved by spreading the total deficit volume more evenly across the restoration period. Overall, scenario 5 produces superior results regarding both lake-level restoration and the ER’s performance criteria.

Conclusion

In this study, we have presented a comprehensive framework for the dynamic restoration of lakes, taking into account the impacts of climate change on water resources. While hedging rules are traditionally employed in the context of reservoir operations, our approach focuses on their application in lake restoration. To determine the optimal hedging rules for the lake, we developed a simulation–optimization model. Utilizing a system dynamics model, we simulated the lake's basin, considering its dynamic nature and inherent complexities.

To manage surface water for lake restoration, we harnessed the potential of water storage infrastructures within the basin by consolidating all surface reservoirs into a single equivalent reservoir. This consolidated reservoir was operated in a manner that ensured the achievement of the lake restoration targets while also considering the diverse objectives of various stakeholders. Furthermore, this study introduced the concept of dynamic demand, which extends beyond static or constant demand considerations, for the lake restoration process. By incorporating the dynamic nature of water demand, we sought to enhance the accuracy and relevance of our approach.

The proposed framework was implemented in the Lake Urmia basin, which faces significant challenges due to climate change and increasing water demand. Five distinct scenarios were developed, each reflecting the existing water resource management practices in the basin. We employed two key performance criteria to assess the effectiveness of these scenarios.

Firstly, we utilized the lake's level duration curves (LDCs) as a performance criterion, enabling us to analyze the lake's water-level profile over the planning horizon. This evaluation provided valuable insights into the ability of each scenario to restore the lake to its desired water level.

Secondly, we evaluated the ability of each scenario to meet the water demands of stakeholders by calculating reservoir performance indices, namely reliability, resiliency, and vulnerability, for the consolidated equivalent reservoir. These indices allowed us to gauge the performance and robustness of each scenario in addressing stakeholder water demands.

The results of our analysis demonstrated that, among all the scenarios considered, the proposed method stood out as the only approach capable of successfully restoring the lake to the target water level. Moreover, it exhibited the lowest severity of water deficits during the restoration period, indicating its high practicality and feasibility for implementation.