System dynamics modeling of lake water management under climate change

System dynamics method

The SD method applies systemic processing to simulate complex non-linear dynamics and feedback. Systemic processing resorts to various tools to simulate complex system behavior and performance²⁴. 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 employs the SD method to simulate storage change in Lake Urmia in one historical period (1957–2005) and two future periods (2021–2050 and 2051–2080). The lake’s water volume is the state variable, which is governed by inflows (precipitation, surface water inflows, and groundwater inflows) and outflows (evaporation, leakage, and surface water outflows). The lake’s mass balance equation is expressed as:

where S_t+1 , S_t, I_s, and O_s denote the lake’s storage at time t + 1, the lake’s storage at time t, the inflow rate to the lake at time s (units of volume/time), and the outflow rate from the lake at time s (units of volume/time), respectively.

The SD method employs the Euler and Runge Kutta methods for the solution of differential equations. The software STELLA, Vensim, Powersim, and Dynamo feature SD solvers²⁶. This work applies the widely-used Vensim software²⁷.

Climate change

The data sets needed for modeling Lake Urmia’s storage over the two future periods were generated after simulating the lake’s water balance during the historical period. HADCM3, a coupled atmosphere–ocean general circulation model’s (AOGCM) climate projections were used to generate precipitation and surface temperature projections over the future periods. The AOGCM data at coarse spatial scales were downscaled to the regional scale suitable for lake storage simulation. The commonly used downscaling methods are statistic and dynamic in nature^28,29. This works applies the delta-change downscaling method, in which monthly temperature and precipitation differences between the future and historical are calculated by²⁹:

where ∆T_t denotes the difference in long-term average temperatures simulated by HADCM3 for the future ((overline{T}_{GCM,fut,t})) and historical ((overline{T}_{GCM,hist,t})) periods in month t (°C); ∆P_t represents the difference in long-term average precipitations simulated by HADCM3 for the future ((overline{P}_{GCM,fut,t})) and historical ((overline{P}_{GCM,hist,t})) periods in month t (mm). Then, ∆T_t and ∆P_t are applied to project the future downscaled data as follows²⁹:

where T_obs,t, and P_obs,t denote respectively the observed temperature (°C) and precipitation (mm) in month t in the baseline period; and T_t and P_t are the downscaled temperature (°C) and precipitation (mm) in month t of the future period, respectively. Delta-change downscaling is a simple yet efficient option when it comes to spatial downscaling of climate change projections (e.g.^30,31,32). The gist of this method is to replicate the changing patterns that are projected by the atmospheric ocean general circulation models (AOGCMs) to generate the climate change patterns of hydro-climatic variables on a regional scale. As such, one would simply compute the relative changes in the long-term variations of the variable that is projected by the models within the baseline and future timeframes. These relative changing patterns would be applied to the historical data to project the impact of climate change on a local scale.

Rainfall-runoff modeling

The IHACRES (identification of unit hydrographs and component flows from rainfall, evapotranspiration and streamflow) model is herein applied to simulate runoff from precipitation. Ashofteh et al.³³ implemented the IHACRES model to investigate the effects of climate change on reservoir performance in agricultural water supply. Ashofteh et al.³⁴ evaluated the probability of flood occurrence in future periods with IHACRES.

The IHACRES model includes a non-linear loss module and a linear unit hydrograph module. The non-linear loss module converts the observed rainfall into the effective rainfall, after which the linear unit hydrograph module converts the effective rainfall into the simulated streamflow³⁵. Here, precipitation r_k in time step k is converted to effective precipitation u_k through the non-linear loss module employing a catchment wetness index s_k:

The effective precipitation is converted to the surface runoff in time step k with the linear unit hydrograph module. The parameters of this model can be set through a thorough grid numeric search and trial-and-error. Perhaps, one of the major 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 hydrologic simulation is to use data-driven models. Here, the multilayer perceptron (MLP), a variety of the artificial neural network (ANN) method, was also used to simulate runoff. This model consists of an inlet layer, one or several middle (hidden) layer(s), and an output layer. All of the neurons of a layer are connected to the ones in the next layer, forming a network with complete connections. The primary parameters in modeling the neural network of MLP 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 the third layers are respectively the system inputs and outputs. The middle layer consists of neurons that perform calculations on the inputs. Choosing the number of layers in a neural network is made by trial and error³⁷. From a hydrological simulation standpoint the main idea behind this model is to create a suitable artificial neural network that is capable of accurately converting a set of hydro-climatic variables such as precipitation and temperature as input data into streamflow values. It should be noted that, like most data-driven models, the process of opting for a proper neural network architecture (i.e., selecting the number of layers, number of neurons, and the forcing function) is, for the most part, a trial-and-error procedure.

One must objectively evaluate the performance of the hydrological models in order to opt for the setting of a suitable parameter. The root mean square error (RMSE), coefficient of determination (R²), and mean absolute error (MAE) are herein employed to assess the performance of the rainfall-runoff model. They are respectively calculated as follows:

where x_t , y_t, and N denote the simulated value in time step t; the observed value in time step t; and the number data values, respectively. Large errors have a disproportionately large effect on RMSE or MAE.

Performance criteria

Various quantitative measures can be used to assess the performance of water resources systems under different strategies. When it comes to water resources planning and management, perhaps, some of the most common performance criteria are the probability-based performance criteria (PBPC) (i.e., reliability, vulnerability, and resiliency)^31,38. In this context, reliability represents the probability of successful functioning of a system; resiliency measures the probability of successful functioning following a system failure; lastly, vulnerability is the severity of failure during an operation horizon^39,40. The basic idea behind a performance evaluation attribute is to provide a quantitative measure to describe and assess the performance of a system. In the context of water resources planning and management, these measures have proven time and again that they can be reliable options to evaluate a set of strategic management options objectively (see, e.g.^40,41,42,43, and⁴⁴, just to name a few).

Operating policy

Any water resources system requires something called the “rule curve,” which determines how water is allocated in a given situation⁴⁵. A common and effective rule curve when it comes to operation of water resource systems is the standard operation policy (SOP). SOP is a simple, and perhaps best-known real-time operation policy in water resources planning and management⁴⁶. The core principle here is to minimize the water shortage at the current time step with no conservation policy (e.g., hedging 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 reservoir^47,48. This rule curve is established as an attempt to balance various water demands including but not limited to flood control, hydropower, water supply, and recreation⁴⁹. A SOP operating system attempts to release water to meet a water demand at the current time, with no regard to the future. Thus, according to the SOP’s principle, the decision-makers, first allocate the available water to meet the demand of the stakeholder with the highest priority. After this first water demand is fully satisfied, the available water can be used for the next demand. Such an allocation process continues until no water is available.

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Source: Ecology - nature.com