# Modeling the dynamics of lake water management systems under climate change

### System dynamic method

The SD method applies systemic processing to simulate complex nonlinear dynamics and feedbacks. System processing uses a variety of tools to simulate the behavior and performance of a complex system24. Systems evolve through states, which change with flows. An example of a state variable is water storage in the study of lakes. The SD method simulates changes in system states driven by flows and various feedbacks25.

This work uses the SD method to simulate the storage change in Lake Urmia during one historical period (1957–2005) and two future periods (2021–2050 and 2051–2080). Lake water volume is the state variable, which is governed by inflows (precipitation, surface water inflows and groundwater inflows) and outflows (evaporation, surface water leakages and outflows ). The lake mass balance equation is expressed as follows:

$$S_{t + 1} = intlimits_{t}^{t + 1} {[I_{s} – O_{s} ]ds + S_{t} }$$

(1)

or Syou+1 , SyouIsand Os designate the storage of the lake at the time you+ 1storage of the lake at the time you the flow entering the lake at the time s (units of volume/time), and the outflow from the lake at the times (units of volume/time) , respectively.

The SD method uses the methods of Euler and Runge Kutta for solving differential equations. STELLA, Vensim, Powersim and Dynamo software includes SD solvers26. This work applies the widely used Vensim software27.

### Climate change

The datasets needed to model the storage of Lake Urmia over the two future periods were generated after simulating the water balance of the lake in the historical period. HADCM3, an Atmosphere-Ocean Coupled General Circulation Model (AOGCM) was used to generate projections of precipitation and surface temperature over future periods. AOGCM data at coarse spatial scales were downscaled to the appropriate regional scale for lake storage simulation. Commonly used downscaling methods are statistical and dynamic in nature28.29. This work applies the delta change downscaling method, in which the monthly temperature and precipitation differences between the future and the historical are calculated by29:

$$Delta T_{t} = overline{T}_{GCM,fut,t} – overline{T}_{GCM,hist,t}$$

(2)

$$Delta P_{t} = overline{P}_{GCM,fut,t} – overline{P}_{GCM,hist,t}$$

(3)

or ∆Tyou denotes the difference in long-term average temperatures simulated by HADCM3 for the future ((overline{T}_{GCM,fut,t})) and history ((overline{T}_{GCM,hist,t})) periods in months you (°C); ∆Pyou represents the difference in long-term mean precipitation simulated by HADCM3 for the future ((overline{P}_{GCM,fut,t})) and history ((overline{P}_{GCM,hist,t})) periods in months you (mm). Then, ∆Tyou and ∆Pyou are applied to project future reduced data as follows29:

$$T_{t} = T_{obs,t} + , Delta T_{t}$$

(4)

$$P_{t} = P_{obs,t} { + }Delta P_{t}$$

(5)

or Job, t, and Pob, t denote respectively the temperature (°C) and the precipitation (mm) observed during the month you in the reference period; and Jyou and Pyou are the reduced temperature (°C) and the precipitation (mm) of the month you of the future period, respectively. Delta change downscaling is a simple but effective option when it comes to spatial downscaling of climate change projections (e.g.30,31,32). The essence of this method is to replicate the changing patterns that are projected by the Atmospheric Ocean General Circulation Models (AOGCM) to generate the climate change patterns of hydro-climatic variables at the regional scale. As such, one would simply calculate the relative changes in the long-term variations of the variable that are projected by the models into the baseline and future time frames. These relative change models would be applied to historical data to project the impact of climate change at the local scale.

### Rainfall-runoff modeling

The IHACRES model (identification of unit hydrographs and component fluxes from precipitation, evapotranspiration and discharge) is applied here to simulate precipitation runoff. Ashofteh et al.33 implemented the IHACRES model to study the effects of climate change on the performance of reservoirs in agricultural water supply. Ashofteh et al.34 assessed the probability of occurrence of floods in future periods with IHACRES.

The IHACRES model includes a nonlinear loss module and a linear unit hydrograph module. The nonlinear loss module converts the observed precipitation into actual precipitation, after which the linear unit hydrograph module converts the actual precipitation into simulated discharge.35. Here the precipitation rk in time step k is converted into effective precipitation youk through the nonlinear loss modulus using a watershed moisture index sk:

$$u_{k} = , s_{k} times , r_{k}$$

(6)

Effective precipitation is converted to surface runoff at the time stepkwith the linear unit hydrograph module. The parameters of this model can be set through extensive numerical grid search and trial and error. Perhaps one of the main advantages of the IHACRES model over other commonly used rainfall-runoff models is its minimal input data requirement (i.e. air temperature and precipitation).31.35.

The other alternative for hydrological simulation is to use data-driven models. Here, the multi-layer perceptron (MLP), a variety of the artificial neural network (ANN) method, was also used to simulate the trickle. This model consists of an input layer, one or more intermediate (hidden) layers and an output layer. All neurons in one layer are connected to those in the next layer, forming a network with complete connections. The main modeling parameters of the MLP neural network are: (1) the number of neurons in each layer, (2) the number of layers in the network and (3) the forcing functions. A regular MLP neural network has three layers36. The first and third layers are respectively the inputs and outputs of the system. The middle layer is made up of neurons that perform calculations on the inputs. Choosing the number of layers in a neural network is done by trial and error37. From the perspective of hydrological simulation, the main idea behind this model is to create a suitable artificial neural network capable of accurately converting a set of hydroclimatic variables such as rainfall and temperature as input data into flow values. It should be noted that, like most data-driven models, the process of opting for an appropriate neural network architecture (i.e., selecting the number of layers, number of neurons, and function forcing) is, for the most part, a trial-and-error procedure.

It is necessary to objectively evaluate the performances of the hydrological models to opt for the adjustment of a suitable parameter. The root mean square error (RMSE), coefficient of determination (R2) and the mean absolute error ( MFA ) are used here to evaluate the performance of the rainfall-runoff model. They are respectively calculated as follows:

$$RMSE = sqrt {frac{{sumlimits_{t = 1}^{N} {(x_{t} – y_{t} )^{2} } }}{N}}$$

(7)

$$R^{2} = left( {frac{{sumnolimits_{t = 1}^{N} {(x_{t} – overline{x} ).(y_{t} – overline{y} )} }}{{sqrt {sumnolimits_{t = 1}^{N} {(x_{t} – overline{x} )^{2} } } .sqrt { sumnolimits_{t = 1}^{N} {(y_{t} – overline{y} )^{2} } } }}} right)^{2}$$

(8)

$$MAE = frac{{sumnolimits_{t = 1}^{N} {left| {x_{t} – y_{t} } right|} }}{N}$$

(9)

or Xyou , thereyou, and NOTdesignates the simulated value in time step you ; the observed value in time step t; and numeric data values, respectively. Significant errors have a disproportionate effect on RMSEWhereMFA.

### Performance criteria

Various quantitative measures can be used to assess the performance of water resource systems according to different strategies. In water resources planning and management, perhaps some of the most common performance criteria are probability-based performance criteria (PBPC) (i.e. reliability, vulnerability and resilience).31.38. In this context, reliability represents the probability of a system functioning correctly; resilience measures the probability of proper operation following a system failure; finally, vulnerability is the severity of failure over an operating horizon39.40. The basic idea behind a performance evaluation attribute is to provide a quantitative measure to describe and evaluate the performance of a system. In the context of water resources planning and management, these measures have proven time and time again that they can be reliable options for objectively evaluating a set of strategic management options (see, e.g.40,41,42,43and44Just to name a few).

### Operating policy

Any water resource system requires something called the “rule curve”, which determines how water is allocated in a given situation.45. A common and effective rule curve for operating water resource systems is the Standard Operating Policy (SOP). SOP is a simple real-time operating policy and perhaps the best known in water resources planning and management46. The basic principle here is to minimize water scarcity at the current time step with no conservation policy (eg, coverage rules) in place. The SOP, as a standard rule curve, determines how the operator acts to control a system at any given state of a tank47.48. This ruler curve is established as an attempt to balance various water demands including, but not limited to, flood control, hydroelectricity, water supply, and recreation.49. An SOP operating system attempts to release water to meet a water demand at the present time, without regard for the future. Thus, according to the principle of the SOP, the decision makers first allocate the available water to meet the demand of the player with the highest priority. Once this first water demand has been fully satisfied, the available water can be used for the next demand. Such an allocation process continues until no more water is available.

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