Development of Rainfall-Runoff Model Using System Dynamics (SD) Analysis

Transcription

1 th International Congress on Advances in Civil Engineering, 7-9 October 22 Middle East Technical University, Ankara, Turkey Development of Rainfall-Runoff Model Using System Dynamics (SD) Analysis Alireza B. Dariane, M. M. Javadianzadeh 2 Faculty, Department of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran,borhani@kntu.ac.ir 2 PhD Candidate, of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran, m.javadian@gmail.com Abstract In this paper, a basin Rainfall-Runoff model is developed using Tank model, based on System Dynamics (SD). The SD method is created based on feedback loops and is a tool for modeling and analyzing systems behavior. This model is designed in two views: main body and routing. The Routing view is described by three methods including Time-Area with unknown number of routing coefficients, the Time-Area procedure proposed by the USACE, and a triangular hydrograph. Furthermore, the Powell search method was used for calibrating the model. In final the results obtained by different methods are evaluated and compared. Keywords: Rainfall-Runoff model, System Dynamics analysis, Tank model, Triangular Transformation Function. Introduction Nowadays, water resources management and availability of the equipment needed for it are of great importance, taking into account the frequent droughts which break out in many parts of the world and Iran. The rainfallrunoff models are among the useful complicated models in terms of basin management. The basin models are usually designed through two different methods; using the transfer functions and phenomenological relationships. In the transfer functions, the development of experimental equations would be taken into account, based on the deal of access to the historical records. Therefore this method requires the longterm rainfall-runoff data. Moreover, if land use would be changed in the region, the aforementioned method would not be able to make a correct evaluation. In the second method, the watershed model would be set out based on the physical elements of each hydrological process. In such models, some assumed reservoirs may be envisaged in order to simulate groundwater storage and soil moisture. This type of models includes a variety of simple and complicated basin models. The parameters used in the phenomenological models can be measured by field gauging and such indications on the earth as soil conditions, topography and vegetation. Also these parameters are achievable through the calibration of the model. The phenomenological models are divided into two categories of physical and conceptual models. According to the physical models, the existing equations between the hydrological processes in the basin should be considered, whereas the conceptual models of rainfall-runoff have something to do with the hydrologists perception of the important elements in the basin. Presentation of simplified physical processes and summing up the spatial changes in the basin make such models less complicated than the physical models and this fact increases the popularity of these models. But it is worth noting that most of the parameters of the conceptual models do not have direct physical connotation and should be deduced from calibration (Lee et al., 25). Since the early 99s data-driven models were introduced for modelling the catchment basins. These models are also named black box models. The development of such models is based on the correlation between the output and input regardless of the hydrological processes which take place in the basin. Artificial neural network is one of the most practical tools in this modelling method (Julta, 26).

2 ACE22 The model of Stanford basin was the first attempt to model the hydrological processes in a basin. Afterward several efforts were made to improve these models. What have been emphasized in these models are accuracy and simplicity, validity of the measured parameters and sensitivity of the results to any change in the parameters amount. The main disadvantage of these models is that they need a wide range of data some of which are sometimes impossible to be provided. Also, in most cases it is impossible to use these models in the regions out of the location of the model calibration. The cornerstone and the main processor of the basin models is a process of rainfall-runoff transformation. Tank model is one of the conceptual models which have been created for rainfall-runoff (Sugawara, 96). In this model, one can simulate the components of basin flow from earth surface to underground by means of some assumed tanks on the surface and underground. The flow can be modelled in a particular direction by paying attention to the linear and non-linear equations related to each tank. The tank model is premised on the existence of cascading tanks with various outlets which have been envisaged for the modelling of different factors affecting the rainfall-runoff process and the relationship between the discharge of each outlet and water height in the tank. The number of tanks varies with whether the model is permanent or temporary and whether or not the snow melt would be factored in. Sugawara envisaged four tanks in his proposed model. These tanks have been devised for simulation of surface discharge, intermediate discharge, sub-base discharge and base discharge. He believes that it is not correct to look upon the tank model as a black box, but in contrast it is possible to compare the process of the model function to what is happening in the nature. Li and Simonovic (22) managed to develop a model with 5 tanks, taking the snow melt process into account. In this paper, we try to use just two tanks according to the principal of simplification. The first tank is used to model the surface and subsurface runoffs as well as to measure the amount of moisture in the upper soil. By moisture in upper soil we mean the subsurface moisture which directly interferes with surface and underground percolation. The second tank is used to model the basic groundwater flow. To do so, the model has been created and calibrated in the system dynamics analysis. The method of system dynamics analysis is a way to embody the real world by showing the relationship between the factors of our desired system. This method was set out for the first time by Forster in 96 at the Massachusetts Institute of Technology (MIT) (Julta, 26), which is being applied widely in many fields of science such as biology, ecology and economy nowadays. Recently this method is catching on in water resources management(zhang et al., 27). Keyes and Palmer (993) used the aforementioned method in simulating drought studies. Fletcher (998) made used of this method as a decision analysis method in water scarcity management. Simonovic and Fahmy (999) used this method to make a long term evaluation of the water resources and analyze the policies practiced in the Nile basin in Egypt. Khan and his colleagues (27) modelled the interactions between a vast irrigated area and surface and ground water resources. Madani (27) tried to integrate the heterogeneous physical and social systems in the basin of Zayande Rood. Momeni and his colleagues (26) set out the policy of utilization of multipurpose tank and Sadeghi and his colleagues (26) accomplished the modelling of tank utilization in order to control floods by using system dynamic analysis. 2 The Model s Structure As mentioned above, a tank model composed of two tanks for surface and subsurface flows is taken into consideration. Development of the model considered in the present article is partitioned in two parts: ) creation of model and 2) its calibration. To create the model based on the structure schemed in Fig.; the relations required in the environment of system dynamics analysis are defined and provided. Analysis of the system dynamics is composed of two terms: system and dynamics. The system is defined as a set of the actions and reactions counteracting, compatible and having mechanisms relying on each other which altogether generally affect the system structure (Deaton & Winebrake, 2). The dynamics can be defined as a degree of any change per time unit. Accordingly, the systems which are changing along with time can be defined in an environment of the SD analysis. As the method acts based on the feedback loop principle; so it needs a data exchange among the system s different components, through which the system s general behaviour can be simulated, to search accordingly some more simple solutions for nonlinear systems (Li & Simonomic, 22). Simulation of the system along the time would assist us to better know the system and its borders, recognizing the key variables and defining mathematically the physical processes and systems variables and finally in planning the model structure. In the tank model; the process in which the rainfall is converted to runoff is such that, after the occurrence of rainfall ( ) firstly the overland flow on impermeable surfaces is calculated and the result is deducted from the primary rainfall. The impermeable surfaces is a set of surfaces like rocks, residential areas, rivers, lakes, etc., on 2

3 A. B. Dariane, M. M. Javadianzadeh which the rainfall has no enough time and directly transforms to runoff. If the remaining amount of rainfall is less than the infiltration potentiality then all the flow infiltrates into the soil; otherwise, some deal of the surface runoff (Q ) will flow in the basin. The flow rate infiltration is added to upper layer s moisture and it is considered the upper tank s inflow. To simulate the subsurface flow s behaviour and its deep infiltration into the lower layers, two outlets are worked out on this tank; one for subsurface flow and one for percolation. In the nature; the subsurface flow is a part of the upper surface moisture gradually joining the overland flow during its horizontal movement (Farahmand and Dariane, 2). Also; a part of the water, under the effect of the gravity force and in the shape of vertical flows, infiltrates into the lower layers, which is considered an inlet to the lower tank. The rate of the lower tank s outflow is calculated as the groundwater base flow. Thus, the spot runoff rate(q ) in each time interval equals the totality of flow rates resulted from rainfall on impermeable surface(q ), the overland runoff rate(q ), subsurface flow rate(q ) and groundwater flow rate(q ). In the next phase, using routing with three methods, that is, Time-Area method with unknown number of routing coeficient C, Time-Area method with the suggested method of HEC and triangular transformation function method; the basin s outflow is calculated. Figure. Schematic Diagram of the Tank Model To calculate the infiltration for this model; the Green Ampt non-linear method is used as follows: n = + () Sm s F K( s ) Where F is the infiltration rate, the unit for which is the same as the soil s hydraulic conductivity in the saturated state,. The soil s effective porosity and the primary moisture are represented by and, respectively. Other relations are shown in Fig., below. To define the model within the SD analysis environment; two views are considered. The first one includes the major body of the model. To define the model in the SD system, the system should be divided into the major components of SD system; and accordingly the model can be formed. These components are: stack, process, convertor, and relations among the parts. The stack acts as a reservoir to supply for another part of the system. The processes are the same as the system s current operations which determine the contents of the stacks along 3

4 ACE22 the period of time. The convertors are variables that define the rhythm of system process; and finally establishing a mathematical link among the system s components, their interrelation is fixed (Razavi et al., 23). Here the tank model s two tanks are regarded as the tank variables. These tanks represent the soil moisture content and the groundwater. Based on the relations among the variables, Fig.2 is defined as the SD model s final structure within the environment of Vensim software. In this figure; the major view is shown under the title of rainfallrunoff. Since the model s output on the rainfall-runoff view is calculated as spot runoff; the basin routing of the spot runoff should be carried out at the next phase, that is, at the routing view (Fig. 3). Figure 2. Projection of the model tank within the environment of SD system (rainfall-runoff view). Figure 3. Routing view in SD model. 4

5 A. B. Dariane, M. M. Javadianzadeh The routing part has a significant role in calibrating the model. The routing method of this model is based on the Clark method. Using the attenuation and translation of the spot runoff hydrograph obtained from the model; the basin s output hydrograph can be obtained. In the Clark method; the Time-Area coefficient and the tank s definition at the basin s output are proposed as the two tools for attenuation and translation of the primary hydrograph, the process of which is shown in the Fig.4 (U.S. Army, 994). Figure 4. Clark Model. In the definition of the Time-Area method, which is derived from reasonable methods (Shaw, 994), it is tried to approximately determine the basin s parts having almost the same water flow s travel time toward the basin s output. The ratio of the area of each part to the basin s total area is called as runoff coefficient and shown as C; therefore: Ci A M i = Ci = A i=, (2) According to the important of this coefficient in the calibration process, it is tried to calculate it in three different methods. In the first method, the considered basin divided into n simultaneous parts of concentration. Therefore, n routing coefficients (C) are obtained. Having considered the Eq. (2); the resulted C coefficients are satisfied as the requirements for the calibration process. Having estimated the basin s coefficients and using the Eq. (3); the rate of the runoff under routing can be obtained. M QR = CQ i=,2,..., M t =,2,..., T (3) t i ( t +> i ) i= Where, Q is the spot runoff and the indices i and t are the counters of the coefficients, C, and time steps, respectively. In the second method suggested by USACE, the basin s coefficients are calculated using approximate relations. In this method, assuming an oval shape for the basin area and considering the number of the routing coefficients, the basin s coefficients are estimated: t.5 tc.44 ( ) fort tc 2 At IA= = A t.5 tc.44 ( ) fort tc 2 (4) Where, IA is the cumulative amount of C coefficients. To calculate the t; we should have the amount of the time steps of the basin s simultaneous lines as well as the concentration coefficient. Calibrating the model; the amounts of tc and number of C coefficient are estimated. Anyhow, the parameters of the model requiring calibration will be very fewer than those of the first method. Having the two mentioned parameters; the time steps are obtained; and hence using the Eq. (4) the amounts of IA and as a result the coefficients, C, can be estimated. 5

6 ACE22 tc t =, ti = ti + t, Ci = IAi IAi (5) Numberof C After calculation the coefficients, C, using the Eq. (3), the runoff under routing can be obtained. The third method uses the suggested transformation function. In this method, an equilateral triangular weighting function distributes the spot runoff at a certain time step among the subsequent time steps (Seibert, 25). The only unknown parameter in this method, which should be obtained through calibration of the model, is MaxBas, which is actually the basin s number of coefficients. Figure 5. Performance of the triangular transformation function on the basin output s hydrograph. The equation needed for this method is the following: i i 2 MaxBas 4 ( ) (6) 2 MaxBas 2 MaxBas i C = u du Solving this relation results in the coefficients, C, using which and the Eq. (3) the runoff under routing can be obtained. The sum of coefficients obtained through this method always equals. To estimate the model s parameters, in case of having the data of simultaneous storms and floods and considering each parameter s variation limit, the calibration of the model would be possible. Calibration of the model means estimating the model s parameter such that the model s outflows have the most correspondence with observed values. There are a lot of methods suggested for optimizing the model (Mortazavi and Dariane, 26); however because of the software s possibilities, the parameter estimation method applying the Powell Search Method is applied (Ventana Systems, 23). The Powell Search Method is one of the direct search methods, wherein the search is primarily started from one direction until the target function has reached a minimum, and then the search direction is changed and so continues to obtain the optimized point. Table. Parameters primary values in the calibration of model. Parameter Achp ks Ns fqf Fk agw sm gwl Qclark k C(i) Max Min. Table shows the primary values of the model in the calibration of model, where achp, Ks, ns, sm, gwl, k, Qclark, and C(i) represent flow component coefficient on impermeable surfaces, soil hydraulic conductivity in saturate state, soil effective porosity, soil primary moisture, groundwater primary level, primary height of water in Clark tank, and routing coefficients, respectively. 3 Results And Discussion 6

7 A. B. Dariane, M. M. Javadianzadeh To implement and calibrate the model, the data related to 5 storms and floods simultaneously at the basin known as Kesilian located in Mazandaran province of Iran with an area of 67.4 Km 2 was taken as the input data of the model. For Calibration and validation of the model; the second to fifth storms are set as parallel with each other and the model s parameters are so calculated to calibrate the four storms together. Then the results of the calibration of this total storm were used to calculate the runoff resulting from the first storm and the accuracy of the system was accordingly examined. In Table 2 the dates of occurrence of all storms and their durations are shown. Table 2. Date of occurrence of storms and their durations. st Storm (Validation) 2 nd Storm 3 rd Storm 4 th Storm 5 th Storm Date of Occurrence Oct Dec 9 98 Oct May 982 Nov Duration of storm The Tables 3 to 5 comparatively shows the results of the model with three different methods of routing for the st to 5 th storms, respectively. Here, three methods of system dynamics analysis are introduced as the system dynamics analysis using Time-Area routing with unknown number of coefficients, system dynamics analysis using Time-Area routing of HEC, and triangular transformation function, abbreviated as SD-C, SD-HCE, SD- Triangle, respectively. To calculate the accuracy rate of estimation in each method, the parameters of the error index was calculated and compared with each other in the Tables 4 to 6. To compare the results, the total sum of square standard errors and the Nash-Sutcliff coefficient, peak time correlation factor, and the peak value were used. The standard error is obtained from the equation below. The values that are closer to zero indicate the best performance of the model: T ( ) 2 Q t obs Q = calc SE = (7) N The Nash-Sutcliff criterion is obtained from the Eq. (8), which can vary from to minus infinity. The index closer to indicates a model of more efficiency. R T t t Q calc Q t= obs NS = T t Q t obs Q = obs ( ) ( ) (8) Another equation to consider for evaluation of each model s results is the correlation factor. This factor indicates the correlation degree existing between the values observed and those calculated by model. Where Q obs, Q calc,,and are observed flow rate, calculated flow rate, average calculated flow rate and average observed flow rate, respectively. One of the important criterions to evaluate how efficient are the rainfall-runoff models is the accurate estimation on the value and the occurrence time of the peak flow rate, which has been considered in the present article. Table 3. Parameters estimated in the calibration for total storm. SD-C SD-HEC SD-Triangle achp.59.5 ks Ns fqf.8 fk.2 Agw sm e-6 e-6 5.e- gwl...42 Qclark k t c - - No. C or Maxbas

8 ACE22 Table 4. Comparison of the three models efficiencies SD-C SD-HEC SD-Triangle Storm(Verify) Storm2 Storm3 Storm4 Storm5 Nash-Sutcliff R Peakobs/Peaksim Tpobs/Tpsim SE Nash-Sutcliff R Peakobs/Peaksim Tpobs/Tpsim SE Nash-Sutcliff R Peakobs/Peaksim Tpobs/Tpsim SE Table 3 shows the values obtained through each method of calibration for the general storm. Table 4 compares the results obtained in each method. As it is observed for the st storm, which is considered validation, the routing method SD-C, having the least error and the most correlation between the data calculated and those observed, presents the most suitable answer; although, with the two methods of SD-HEC and SD-Triangle, the estimation of the peak point has done more properly. Q (m 3 /s) Rain (Cm) Rain Qout Obs Time (hour) Figure.6. Runoff hydrograph obtained for 5 storms through SD-C method Fig.6 shows the model s estimation for the quintuple storms using the SD-C method. In the present study, as mentioned before, the 2 nd to 5 th storms are included in calibrating the model and the st storm is used for validation. The obtained results show that the performances of the SD-HEC and SD-Triangle methods are weaker than the two methods of SD-C; the main reason of which is the simplification made for estimations of the basin s coefficient for the last two methods. It should be noted that the calibration cost is reduced with applying simplification; so that, the Time-Area routing method with unknown number of coefficients, C, requires calibrating 3 parameters; but SD-Triangle and SD-HEC require calibrating only and 2 parameters of model, respectively, naturally having less accuracy in results. To select a suitable method; the calibration cost and its accuracy should be taken into consideration. In complex systems having several sub-basins; the triangular transformation function can show its more real values. 8

9 A. B. Dariane, M. M. Javadianzadeh 4 Conclusion In this paper, the system dynamics analysis method applied in creation of rainfall-runoff model as well as the efficiencies of different routing methods were studied. The results show that each of the three routing methods shed light on calibration and validation. For the hydrographs having complicated forms, the results show that the method of HEC is inefficient in the process of calibration and validation. The major factor of this method s weak performance is the simplification of basin calibration. It is worth noting that with calibration simplification leads to considerable reduction in the related cost; but on the other hand, it also reduces the results accuracy. For decision on selection of a suitable method; the calibration costs and it accuracy should be taken into consideration. In complex systems having several sub-basins; the triangular transformation function can show its more real values. In such conditions; complex models because of too many calculations may give in far weaker results. There is also a notable point that the triangle transformation function method of routing has the least number of parameters for calibration; nevertheless it gave suitable answers. References Farahmand, Z. and Dariane, Alireza B. (2). Comparing calibration of a precipitation-runoff model using bee mating algorithm. The st National Congress on Applied Researches of Iranian Water Resources, Kermanshah, Iran. Fletcher, E. J. (998). The use of system dynamics as a decision support tool for the management of surface water resource. First lnt. Conf New Information Technologies for Decision Making in Civil Engineering, Monteral, Canada, Julta, A. S. (26). Hydtological Modeling of Reconstructed Watersheds Using a System Dynamics Approch. A Thesis Submitted University Saskatchewan, Saskatoon, Canada. Keyes, A. M. and Palmer, P. N. (993). The role of object-oriented simulation models in the drought preparedness studies. Proc., 2th Annu. Int. Con$, Water Resources Plan. and Manage. ASCE, Seattle, Washington, Khan, S., Yufeng, L. and Ahmad, A. (27). Analysing complex behavior of hydrological systems through a system dynamics approach. Environmental Modeling & Software, doi:.6/j.envsoft Lee, H., McIntyre, N., Wheater, H. and Young, A. (25). Selection of conceptual models for regionalization of the rainfall-runoff relationship. Journal of Hydrology, 32, Li, L. and Simonovic, P., (22), System dynamics model for predicting floods from snowmelt in North American prairie watersheds. Hydrological Processes, Madani, K. (27), Water Transfer and watershed development: A system dynamics approach. World Environmetal and Water Resources Congress, (doi.6/4927 (243) 55). Momeni, E., Tajrishi, M. and Abrishamchi, A. (26). Modeling of Application of Multi-Purpose Pool Using System Dynamics Method. Water & Wastewater Magazine, No.57, pp Mortazavi, M. and Dariane, Alireza B. (26). Comparative Study on Methods of Algorithm of Ants and Genetics in Tank Model Calibration. The 7 th Intl. Conference on Civil Engineering, Tarbiat Modares University, Theran, Iran. Phein, H. N. (997). A hybrid model for daily flow forecasting. Water SA, Vol 23, No 3, Powell, M. J. D. (27). A view of Algorithm for Optimization witout Derivatives. Presented in University Hong Kong Razavi, M. and Moshrefi, R. (23). Dynamic Modeling for Environmental Systems. Dayton Michael L., Wayne Brick James J., Institute of Scientific Publication of Sharif University of Technology. Sadeghi, N., Tajrishi, M. and Abrishamchi, A. (24). Modeling for Pool Application to Control Floodwater Using Dynamic System Analysis Method. The st National Congress on Civil Engineering NCCE, Theran, Iran. Seiber, J. (25). HBV light ver 2. User s Manual. Stockholm University. Shaw, E. M. (994). Hydrology in practice. 3rd edn. London: Chapman & Hall. Simonovic, S. P. and Fahmy, H. (999). A new modeling approach for water resources policy analysis. J. Water Resources Research, 35(), Simonovic, S. P., (22). World water dynamics: Global modeling of water resources. Journal of Environmental Management, 66, Singh, V. P. (995). Computer models of watershed hydrology. Water Resources Publications. Sugawara, M. (979). Automatic Calibration of the Tank Model. Hydrological Science- Bulletin-des Sciences Hydrologiques, 24, 3, 9. US Army Corps of Engineers (994). Flood-Runoff analysis (EM-2-47). 3. 9

10 ACE22 US Army Corps of Engineers (2). Hydrological Modeling System (HEC-HMS). Technical Reference Manual, Hydrologic Engineering Center. Ventana Systems (23). Ventana Simulation Environment. Zhang, X. H., Zhang, H. W., Chen, B., Chen, G. Q. and Zhao, X. H. (27). Water resources planning based on complex system dynamics :A case study of Tianjin city. Communication Nonlinear Science and Numerical Simulation, doi:.6/jcnsns