Computer Simulation of Flood Operation in Multireservoir Systems

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Computer Simulation of Flood Operation in Multireservoir Systems Ewa Niewiadomska-Szynkiewicz SIMULATION 2004; 80; 101 DOI: 10.1177/0037549704042730 The online version of this article can be found at: http://sim.sagepub.com/cgi/content/abstract/80/2/101

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APPLICATIONS

Computer Simulation of Flood Operation in Multireservoir Systems Ewa Niewiadomska-Szynkiewicz Institute of Control and Computation Engineering Warsaw University of Technology ul. Nowowiejska 15/19, 00-665 Warsaw, Poland [email protected] In this article, a summary of experience with the development of methods for controlling multiplereservoir systems during flood periods, gained through computer simulation, is presented. Several decision mechanisms based on online forecasting and optimization are taken into consideration. Particular attention is focused on a two-level control structure with periodic coordination. The advantages of applying parallel global algorithms to determine the optimal decisions regarding outflows from the reservoirs are presented. Advantages of developing sophisticated control structures, as well as decision mechanisms with respect to the available quality of inflow forecasting and delays involved by data transmission and computation time, are considered. Finally, the results of simulations of online management in the case study of two reservoir systems located in the southern part of Poland are described and discussed. Keywords: Flood management, multireservoir systems, hierarchical control, simulation-based optimization 1. Introduction

by inundations. These can be related to the peak water levels as well as to the duration of flooding and to the current use of the floodplain. The types of flood damages and loss estimation models are discussed in literature [1, 7, 8]. In most cases, it is assumed that flood damage J can be expressed as a function of high flow levels Q downstream the dams, at the important cross sections [9]. This is basically due to the fact that the impact of other flow attributes on flood damages is very complicated and not easily identifiable. So, the following performance index can be considered: K J = vk · max[Qmax − Qlimit (1) k k , 0].

The main goal pursued by floodplain communities is to minimize flood losses. Increased protection of municipal, industrial, and agricultural developments can be achieved by employment of offline and online activities [1]. The control of water storage reservoirs is a basic online activity that can purposely change flow discharges at the important damage centers [1-6]. We have studied the development of various decision mechanisms for real-time flood operation in single- and multiple-reservoir systems for several years. As a result, we discuss and compare three control structures: decentralized, centralized, and hierarchical structures. The objective of this study is to present the summary of our experience in flood control, gained through computer simulation. The article is organized as follows: first, the control problem is described, and several control mechanisms are proposed. Next, the results of experiments performed for two river basin systems are presented and discussed.

k=1

and Qlimit denote the peak and In the above formula, Qmax k k the highest safe discharge (with respect to protection of banks and flood damages) at the kth damage center, respectively; vk denotes the weighting factor related to the flow at the kth cross section (different points have different importance); and K is the set of all damage centers. Another important objective of flood control is to prepare the reservoirs for the conservation control period after the flood to satisfy the requirements of different water users (hydropower production, water supply, navigation, etc.). It can be achieved through filling up the reservoirs to the desired capacity, xif (i = 1, . . . , m), by the end of the flood:

2. Description of the Problem 2.1 Objectives Let us consider a multireservoir system with m reservoirs located in parallel or serially (as depicted schematically in Fig. 1). Flood damage mitigation schemes are usually evaluated by means of multicriteria analysis. The main purpose of flood control is obviously to minimize damages caused SIMULATION, Vol. 80, Issue 2, February 2004 101-116 © 2004 The Society for Modeling and Simulation International

xi (tf ) = xif , | | | |

(2)

where xi is the storage capacity of the ith reservoir, and tf is the estimated control termination time.

DOI: 10.1177/0037549704042730 Downloaded from http://sim.sagepub.com at PENNSYLVANIA STATE UNIV on April 16, 2008 © 2004 Simulation Councils Inc.. All rights reserved. Not for commercial use or unauthorized distribution.

Niewiadomska-Szynkiewicz

Small Vistula

Dwory Smolice

Popedzynka Karsy

Szczucin

Bielany Zabno

Oswiecim Proszówki Skawa

Dunajec

Raba

Kroscienko

Nowy Sacz

Dunajec reservoir

Roznow

reservoir ROZNÓW

CZORSZTYN

Poprad Figure 1. Considered river systems: (a) Vistula river system and (b) Dunajec river system

2.2 Mathematical Models Each reservoir is described by the dynamic of a simple tank, with one input-inflow and one controlled outputoutflow from the reservoir. The volume of water stored in the reservoir is obviously constrained. The construction of spillways in the dam results in constraints on the outflows. Propagation of flood waves along the river reaches can be described in many ways at different levels of accuracy. The possible range varies from nonlinear partial differential equations, proposed in 1871 by Barre de Saint Venant, to simple (discrete in time and space) linear models such as Kalininin-Miliukov or Muskingum ones. The applicability of various simplifications of flow transformation descriptions is discussed in many papers (see, e.g., Moussa [10] and Napiórkowski [11]). The simple models play an important role in decision support systems for operational flood control and flood forecasting systems. One example is the Flood Forecasting System for the Grand River Conservation Authority, in which the Muskingum method is applied [12]. For practical application, the routing reach is often divided into a number of subsections. Each of those sections is then treated as one of a series of a few storage ele-

ments described by identical linear or nonlinear equations. Such technique was applied in the presented simulation study. The flood routing in the river channels, presented in Figure 1a, was described by 110 differential equations (11 sections, with each section formed by 10 nonlinear storage elements). A detailed description can be found in Niewiadomska- Szynkiewicz, Malinowski, and Karbowski [13]. In conclusion, flood control is generally complex in practical operations, especially when we are concerned with floods that occur as a result of massive rainfall and therefore have a violent character. Some characteristics of operational flood control that make it rather distinct from typical control problems are presented as follows: • complex dynamics of the problem, nonlinear and statedependent constraints on control variables (water releases), and a nonlinear flow transformation model; • limited knowledge regarding future inflows to the reservoirs and to the river reaches that are downstream to the reservoirs; • multiple objectives involving different combinations of flood control, benefits from hydropower generation, water supply for irrigation, municipal and industrial use, navigation, and so forth;

102 SIMULATION Volume 80, Number 2


SIMULATION OF FLOOD OPERATION IN MULTIRESERVOIR SYSTEMS

• multiple decision units and, at the same time, different individual objectives (see Fig. 2). These local goals are often in contradiction to the global objectives.

2.3 Operational Flood Management During the past decade, many control schemes that could be employed for operational flood control in simple and multiplereservoir systems have been reported [14]. In general, the available methods can be classified as follows: linear programming, nonlinear programming, dynamic programming, and machine learning methods. We can distinguish deterministic and stochastic approaches. The deterministic optimal release scheduling problem is formulated under the idealistic assumption that the future external inflows can be exactly predicted. Several authors have analyzed deterministic flood control problems in multireservoir systems and proposed linear and nonlinear numerical algorithms [6, 9, 15, 16]. A very simple but suboptimal method for operational control in a cascade system was described in Marien [17]. An extensive discussion of the optimal releases in a flood control problem with deterministic inflow forecasts can be found in Karbowski [18]. Flood control methods that use stochastic models of inflows (e.g., Markov chains, ARIMA models) are presented in Loaiciga and Marino [19] and Wasimi and Kitanidis [6]. Another possibility, which allows directly taking into account the stochastic nature of inflows, is to use the repetitive optimization based on multiple-inflow forecasts. Inflow forecasting is discussed in many papers (e.g., Krzysztofowicz [20]; Smith and Boyd [12]). The stochastic design requires considerable computing times. To overcome this problem, researchers have described different methods to decompose the original control problem into a number of small-scale problems. An aggregated stochastic dynamic programming model of multireservoir systems is proposed in Archibald, McKinnon, and Thomas [21]. The control task, which involves the decomposition and application of hierarchical techniques to solve the problem, is discussed in Niewiadomska-Szynkiewicz, Malinowski, and Karbowski [13]. The possibility of using machine learning methods of artificial neural networks and fuzzy systems for multipurpose reservoirs is discussed in Hasebe and Nagayama [22]. A fuzzy optimal model of real-time multireservoir operation for the flood system of the upper and middle reaches of the Yangtze River is presented in Cheng [23]. Results of applying fuzzy mathematical programming to the operation problem at the Green Reservoir in Kentucky can be found in Teegavarapu and Simonovic [24]. In recent years, parallel processing has provided a new impetus in systems engineering. Techniques based on original problem partitioning, such as hierarchical approaches, have become popular again [25]. Applications of hierar-

chical, multilevel methods to solve problems pertinent to water systems operation can be found in numerous works (see Adiguzel and Coskunoglu [26]; Findeisen et al. [27]; Haimes [2]). It should be noted that most of available literature is concerned with reservoir operation during conservation periods (resource allocation, hydropower production). In this study, a hierarchical scheme is proposed for flood control in multireservoir systems—namely, reservoirs located on a large river or on tributaries to such a river, which flows through different administrative regions of a country and even through different countries—when centralized control is not acceptable. In addition, there are some advantages of applying multilevel control structures: • Control decisions are made by the local operators based on their local rules and operational instructions from the center. Local operators can fully use their private information and experience. • The center is supplied with relevant data and monitors the system operation. Analysis of the situation is repetitively performed, and the outcome of this analysis involves deciding whether there is a need to issue new directions to the local operators. In any case, interventions of the central authority are much less frequent than in the case of centralized control. • It is natural to apply parallel techniques to simulate hierarchical control systems operation.

The disadvantage of hierarchical methods is their complex structure, which results in sophisticated, often suboptimal, decision rules. The presented hierarchical decision mechanism (HDM) for the operational control of multireservoir systems during flood periods incorporates two decision levels: the upper level, with the supervisory control center, and the local level, formed by the operators of the reservoirs. HDM was compared with the centralized decision mechanism (CDM), an autonomous control of each reservoir, based on the local decision mechanism (LDM) and traditional rules (TR)—this system has been used for operational flood control in Poland so far. 2.4 Software Systems for Flood Simulation and Control In the previous sections, we discussed the techniques of the flood-related problem modelling. Many working models that are broadly used to train reservoir operators also assist them in water resources engineering or in real flood emergencies, such as the following: AquaDyn (www.technum.com), DELFT-FLS [28], FLOODSC (www.aessoft.com/hydros.html), MIKE 11 [29; see also www.dhisoftware.com/mike11], MODSIM [30; see also www.modsim.engr.colostate.edu], and RIVERWARE [31; see also cadswes.colorado.edu/riverware], among many others. These tools enable the user to investigate new problems and control rules as well as flood protection measures Volume 80, Number 2 SIMULATION 103



hydrological station m+1

....

hydrological station m+2

hydrological station m+k

side inflow forecast 2

side inflow forecast 1

side inflow forecast k

CONTROL CENTRE (central operator) inflow forecast 1

inflow forecast m

hydrological station 1

hydrological station 2

inflow forecast 1

reservoir capacity

inflow forecast 2

reservoir operator 1 reservoir outflow 1

reservoir operator 2

....

hydrological station m inflow forecast m

reservoir operator m

reservoir outflow 2

reservoir outflow m

FLOOD WAVE TRANSFORMATION (river basin) flows at the measurement stations

Figure 2. Control structure for multireservoir system operation during flood period

much more effectively than it was possible before. They are invaluable to achieve better understanding of the system operation, and they can indicate what elements of the flood control system should be improved first and what would be the rewards for such improvements. The professional engineering software systems provide extensible libraries of modelling algorithms and numerical solvers that may by used for a wide range of applications (hydrology, hydraulics, water management, water quality, irrigation, etc.). The advanced graphical facilities enable visual data checking and presentation of the information stored in the databases. The software packages are integrated, userfriendly tools for the detailed design, management, and operation of both simple and complex river and channel systems. Because of their flexibility, they provide a complete and effective design environment for water resources and water quality management, flood control, planning, and forecasting. The software environments are successfully used by water authorities and researchers. AquaDyn, offered by Hydrosoft Energie, Inc., is the hydrodynamic simulation package used by engineers and decision makers from about 15 countries around the world. DELFT-FLS [28], proposed by DELFT Hydraulics in the Netherlands, was developed to simulate the dynamic behavior of overland flow over initially dry land, as well as flooding and drying processes on every kind of geometry, including lowlands and mountain areas. FLOODSC is a user-interactive software system for watershed simulation. It offers tools for runoff hydrographs analysis and modelling. Pipe flow routing and

open-channel routing methods are provided. Special versions of this program are available in a few states in the United States. MIKE 11 is a general-purpose software for the simulation of hydrology hydraulics, water quality, and sediment transport in estuaries, rivers, irrigation systems, and other inland waters. It is broadly used in Europe as a training and forecasting tool. MODSIM-DSS is a generalized river basin decision support system and network flow model developed at Colorado State University. It is successfully used by the U.S. Bureau of Reclamation, several cities in Colorado (Ft. Collins, Greeley, Colorado Springs), and the Imperial Irrigation District. RIVERWARE, a general river and reservoir modelling environment, is applied and used to solve operational and planning problems by the Tennessee Valley Authority (Daily Scheduling Model) and the U.S. Bureau of Reclamation (Colorado River Simulation System). 3. Formulation of the Control Mechanisms 3.1 Traditional Rules (TR) The flood control rule for a retention reservoir can be a simple one. The basic so-called rigid release policy, described in detail in Malinowski and Niewiadomska-Szynkiewicz [32], assumes that the real release is equal to the admissible outflow ulimit , until the reservoir is filled. Then, the i real outflow ui must be equal to the inflow di . The admissible outflow ulimit is usually equal to the maximum outflow i that does not create substantial damage downstream of the dam.




The next, more complete, semi-rigid rule works in the same way as the rigid one, as long as the inflow di does not exceed the admissible level ulimit . If, however, inflow i limit (d (t) > u ), then the outflow can be inexceeds ulimit i i i creased over the ulimit level: i , d i (t) ui (t) = max{ulimit i − (1 − p/100)(ximax − xi (t))/∆T }. The above formula is developed in such a way that the increased outflow should, during the time period from t to t + ∆T (e.g., ∆T = 24h), fill up the p% of the free capacity at time t (i.e., ximax − xi (t)), assuming that during this period, the average inflow will remain equal to d i (t). Thus, the above semi-rigid rule is based on the arbitrary parameter p ∈ [0, 100] and a short-term forecast of the future inflow d i (t), t ∈ [tl , tl + ∆T ], where tl denotes current time. The rigid and semi-rigid rules have been used in flood control in Poland so far. In this study, we will refer to them both as traditional rules (TR). 3.2 Local Decision Mechanism (LDM) In the case of LDM, similar to TR, the operation of each reservoir is independent of the others. The central dispatcher of the entire water system does not influence the decisions on the outflows from the reservoirs. The natural objective of the ith operator is to minimize the damage created by high water levels directly downstream of the reservoir and to have enough water in the reservoir after the flood event. Assume now that a reasonable inflow forecasting beyond the next ∆Ti hours (a long-term forecast) is possible at times tl , (l = 1, 2, . . . ) in the form of an inflow tl wave pattern d i (t), t ∈ [tl , tf ], where tf is the estimated control termination time. Hence, we can formulate the decision problem of the ith reservoir operator at time tl (3), under the constraints on the reservoir storage, releases, and assumption (2): (3) min max ui (t) . ui

t∈[tl ,tf ]

The solutions to the above optimization problem—the optimal release profiles uˆ i in the case of inactive constraints on the maximal release (a) and active constraints on the releases (b)—are presented in Figure 3. The optimization problem (3) is solved repetitively, at times tl , tl+1 , . . . , using current measurements and inflow forecasts, taking into account the constraints on storage (wimin ≤ wi (t) ≤ wimax , where w is storage of the ith reservoir) and releases (uimin (wi (t)) ≤ ui (t) ≤ uimax (wi (t))). We will refer to this rule as the local decision mechanism (LDM). 3.3 Centralized Decision Mechanism (CDM) In the case of CDM, the central operator has to design a multidimensional rule for establishing simultaneously the

releases from all reservoirs. One possibility of computing the releases is to formulate and solve the dynamic optimization problem, using the models of reservoirs, as well as the dynamic models of river reaches and forecasted inflows [15], if sufficient computing power is available. Another approach, which was adopted in the considered case study, is to calculate suboptimal outflows from the reservoirs under the assumption that, for the ith reservoir, release is related to a step function (switching between two constant values u1i and u2i ). In this case, the objective of an operator in charge of the entire water system is to determine, at times tc , tc+1 , . . . , release trajectories uˆ i (i = 1, . . . , m, where m is the number of reservoirs) minimizing the performance measure (1), satisfying the constraints on the reservoirs’ storage and releases, and corresponding to the optimal switching functions described by 1 2 uˆ i , uˆ i , and the time of switching Ti (see Fig. 4). By an appropriate choice of parameters u1i , u2i , and Ti , one can try to desynchronize the peaks of the flows on various rivers and prevent the culmination from different reservoirs from overlaping. The example of a calculated release profile, in the case of inactive constraints on the releases and minimal storage, is presented in Figure 4. It should be pointed out here that the central operator makes his or her decisions regarding releases from all reservoirs, so he or she performs complicated calculations based on measurements and forecasts of inflows in the entire considered river basin. In the case of the decentralized approach (LDM), each operator of the reservoir solves a less complicated problem—recalculating the release from his or her reservoir—so the operator can change his or her decisions more frequently, at times tl , tl+1 , . . . . It is obvious that ∆Tc ≥ ∆Tl , where ∆Tc = tc+1 −tc and ∆Tl = tl+1 −tl . 3.4 Hierarchical Decision Mechanism (HDM) Finally, in the case of the HDM, the local release rules are designed in such a way that a central authority (coordinator) may adjust them in the process of periodic coordination. The goal is to achieve the cooperation of the reservoir operators. The objective of the control center is to determine the optimal vector of coordinating parameters a, such that the global damage J , described in (1), is minimized. Calculated parameters are used to modify the local performance measures (3). It is assumed that the vector ai of coordinating parameters for the ith reservoir is related to the weighting function αi (t) described by two parameters, Ti and ci (see Fig. 5). Finally, the vector ai is given as ai = [Ti , ci ]. In particular, the ith reservoir problem can be defined by (4), under the constraints on the reservoir storage and releases: min max (ui (t) · αi (t)) . (4) ui

t∈[tl ,tf ]

Similar to the CDM mechanism, the proposed parameters Volume 80, Number 2 SIMULATION 105



di u^ i

di u^ i di

di

ui

ui

tf t

tl

a)

tf t

tl

b)

Figure 3. The optimal release profiles with respect to the local decision mechanism (LDM). Active constraints: (a) maximal storage and (b) maximal storage and maximal release.

di u^ i ui1

di ui2

Ti*

tc

tf t

Figure 4. The optimal release profile with respect to the centralized decision mechanism (CDM) (inactive constraints on the releases)

ai(t) 1ci -

ai are used to prevent the culminations from different reservoirs from overlapping. The peak flow downstream of the ith reservoir can be accelerated to occur before time Ti , when ci > 1, or delayed when ci < 1. For ci = 1, local, independent control policy LDM is implemented since the performance (4) reduces to the one given by (3). Figure 6 presents the release trajectories, obtained in the case of inactive constraints on the release. The decision-making process in the case of HDM is as follows. In the beginning, the forecasts of inflows to all reservoirs and side inflows are calculated. Next, the central operator determines at intervals ∆TC hours—say, at time tc —the optimal values of the coordinating parameters ai (i = 1, . . . , m), minimizing the damage J created in the whole river basin and related to the given flood scenario. Each local operator determines the decision regarding the future release from his or her reservoir (i.e., until time tl + ∆TL ). This is done by solving, at times tl (at intervals ∆TL hours, where ∆TL ≤ ∆TC ), the optimization problem (4) for a given inflow forecast and coordinating parameters Ti and ci , taking into account constraints on storage and releases. It should be pointed out here that for the purpose of the control center optimization, both in CDM and HDM, the local-level (reservoirs) operation has to be simulated many times during the computations. The decision-making process of the central authority is presented in detail in Figure 7. 4. Case Study Results

tl

*

tf t

Ti

Figure 5. The weighting function αi (t)

The considered control schemes were adopted for the flood management of two reservoir systems located in the southern part of Poland:




di u^ i

di u^ i

ci>1

ci 0. Variant C : Decision and transmission legs, τD > 0, were taken into account while computing the releases. It was assumed at time tc that the optimal parameters a, calculated during the previous intervention of the central dispatcher (i.e., at time tc−1 ), were implemented until the time tc∗ = tc + τD . The storage volume of each reservoir and flow levels at the measurement points in the considered water system were predicted at time tc∗ , based on the available forecasts of the inflows. Next, the optimal variables of parameters for the horizon [tc∗ , tf ] were computed by the central operator. The most interesting simulation results obtained from the CFM and WFM forecasts of the 1970 flood, for the delays τD of 3, 6, 9, and 12 hours, respectively, are presented in Figures 10 through 12. The results can be compared with those that were obtained for delays equal to zero and presented in Figure 9 (variant M). In general, the available simulation results indicate that the hierarchical control HDM is more tolerant to time delays than the centralized one. Taking into account the delays during the decision-making process results in a greater reduction of the loss function (see Fig. 12). So, there is a need to address the problem of time delays in relation to the complexity of control mechanisms. The theoretical analysis can be performed only for the simplified systems (see Niewiadomska-Szynkiewicz Volume 80, Number 2 SIMULATION 109



Figure 9. Percent reduction of global damages (Vistula river system): (a) 1972 flood, CFM forecast; (b) 1972 flood, WFM forecast; (c) 1970 flood, CFM forecast; and (d) 1970 flood, WFM forecast

[33]). Computer simulation becomes the basic tool to solve this problem. HDM is more robust with respect to the communication breakdown between the control center and reservoir operators, too (see Fig. 13). The reduction of damages decreases only 1% to 3% (CFM) and 3% to 5% (WFM) but is still improved when compared with LDM. In the case of CDM, the value of the loss function increases rapidly, 5% to 6% (CFM). 4.2 Construction of a New Retention Reservoir The simulation is especially useful in the case of a new hydrological investment, such as the construction of a new retention reservoir. It seems necessary to • evaluate the capacity of the reservoir, taking into account water demands in the considered river basin; • evaluate the capacity of a mandatory reserve for flood emergencies; • analyze the influence of the planned reservoir on the flood situation in the considered water system.

The simulation study was performed for the planned ´ reservoir Swinna Por¸eba on the Skawa River in the Vistula

river system. The main goal of this reservoir is flood protection of one of the oldest cities in Poland—Cracow. The experiments were performed for two historical floods and three variants: T : Only the Tresna reservoir was considered. ´ ´ T-SPs : Two reservoirs, Tresna and Swinna Por¸eba, were considered, and the assumed flood reserve capacity ´ in Swinna Por¸eba was 25 × 106 m3 . ´ T-SPb : Two reservoirs were considered, but the flood re´ serve capacity in Swinna Por¸eba was increased to 6 3 130 × 10 m . The results for two big historical floods from 1970 and 1972, as well as for various management types and mandatory reserves, are depicted in Figure 14. The bar diagrams present the percent reductions of the flow culminations in Cracow, which were equal to 2945 m3 /s in 1970 and 2308 m3 /s in 1972 for the uncontrolled floods. The simulation ´ study indicates that the new reservoir, Swinna Por¸eba, may play an important role in the flood protection of Cracow, but the results strongly depend on the designed mandatory reservoir capacity and the type of management.




Figure 10. Percent reduction of global damages for the 1970 flood: CFM inflow forecasts and different delays: (a) CDM and (b) HDM

Figure 11. Percent reduction of global damages for the 1970 flood: WFM inflow forecasts and different delays: (a) CDM and (b) HDM

4.3 Parallel Global Optimization for Optimal Flood Control 4.3.1 Optimization Methods The optimization problem of the control center consists of determining a vector of parameters a that minimizes the flood damage J , through which the control center influences releases from the reservoirs. For the purpose of the control center optimization, both in CDM and HDM, the local level (reservoirs) has to be simulated many times during the computations. The simulation of the operation of the complete control structures is then a complex task, requiring both advanced computer tools and the ability to create realistic conditions of the simulation experiment. The important problem considered in the case study was to develop the optimization algorithm to solve the task of the central operator. The measurement function J , de-

scribed in (1), is nondifferentiable and nonconvex. However, the dimension of the considered optimization problem is not high—in considered applications of four to six decision variables, it is quite complicated and time-consuming. The presented controllers realize the optimizer-simulator scheme that is illustrated in Figure 7 (i.e., in every optimization step, a value of the performance index J is calculated based on the simulation of the whole considered river system operation). The optimization is realized as follows: after assuming certain values of parameters a, simulation of the reservoir operation and flow transformation in the whole river basin until the predicted end of the flood tf is performed. Then, the value of the flood damage J , related to the given vector a, is computed. So, in every optimization step, the release trajectories from all reservoirs due to rule (4), taking into account current forecasts of inflows, are calculated. Next, flows at all cross sections in the whole

Volume 80, Number 2 SIMULATION 111



% 14 12 10 8 6 4 2 0

0

1

2

3

4

5

6

7

8

9

10

11

12

delay [ h] HDM (variant B)

HDM (variant C)

CDM (variant B)

CDM (variant C)

Figure 12. Increase of the flood damages with respect to values obtained for delays equal to zero

Figure 13. Average percent reduction of global damages with respect to the uncontrolled flood (case of communication breakdown)

river basin are computed, applying the nonlinear flow transformation model described in section 2.2. Due the growing complexity of systems taken into consideration, as well as the possibilities of modern computers, we can observe increasing interest in the development of the algorithms that are designed to search for

the global minimum (maximum). They are broadly used in modelling [34], reservoir management [35], and flood control [36]. Since the discussed optimization problem involves cumbersome calculations (e.g., numerical simulations), it seems reasonable to apply nongradient global optimization methods and parallel supercomputers to solve




Figure 14. Percent reduction of the flow culmination in Cracow (Bielany station): (a) 1970 flood, CFM forecast; (b) 1970 flood, WFM forecast; (c) 1972 flood, CFM forecast; and (d) 1972 flood, WFM forecast

it. Two heuristic, global algorithms—controlled random search (two versions: CRS2 and CRS3) [37] and evolutionary strategy (ES) [38], in sequential and parallel versions— were applied to solve the central dispatcher optimization problem. The comparison was made with the standard Nelder-Mead (NM) nonlinear simplex method [39]. CRS and ES methods are briefly described in the appendix. The ES and CRS methods, although quite simple, are not only efficient but also easily adaptable to the parallel environment. The parallel versions of the algorithms were developed and tested. The goal was to improve the accuracy of the solution, not to speed up the algorithms. In the parallel versions, several independent instances (threads) of the considered algorithms were executed, each on a separate processor. The final solution was the best one calculated by all processors. The numerical experiments were carried out on the Cray Superserver 6400 with the help of the Parallel Support Library (PSL). 4.3.2 Numerical Results Many calculations were performed for each optimization method. The question was how the parallel, global algo-

rithms influence the optimization results and thus influence the issues of the operation of a multireservoir system during a flood. The numerical experiments were performed for the Vistula river basin system (Fig. 1a). Only big floods were considered1 : two historical, which occurred in 1970 and 1972, and one hypothetical scenario.2 The most interesting numerical results are collected in Tables 1, 2, and 3. The tables present the best (i.e., the lowest) and the worst (i.e., the highest) values of the performance index (1) related to flood losses in the system, obtained during 10 runs of each optimization algorithm, and the percent reduction of the performance index with respect to the Nelder-Mead algorithm. The results obtained by the sequential version of the Nelder-Mead, ES, CRS2, and CRS3 methods are compared in Table 1. Tables 2 and 3 present the results of parallel versions of the CRS2 and CRS3 for a different number of processors (four and eight). 1. In the case of smaller floods, damages are small, and coordination of the local-level operation does not bring significant improvement. 2. The hypothetical scenario is the set of inflows, which was generated as a combination of several historical flood events that occurred in the Vistula river basin. It represents a typical flood with two culminations.

Volume 80, Number 2 SIMULATION 113



Table 1. Optimization results of different methods for historical and hypothetical data Optimization Method NM CRS2 CRS3 ES

Hypothetical Flood Best Resulta Worst Result 1587 1574 / 0.8% 1574/ 0.8% 1515 / 4.5%

1587 1587 1583 1572

Historical Flood 1970 Best Resulta Worst Result 1989 1892 / 4.9% 1877 / 5.6% 1773 /10.9%

Historical Flood 1972 Best Resulta Worst Result

1989 1907 1892 1792

773 765 / 1.0% 764 / 1.2% 722 /10.8%

773 768 770 770

a. Reduction of criterion with respect to the Nelder-Mead method.

Table 2. Optimization results for historical data concerning the flood in 1970 Optimization Method

Processors Number

Best Resulta

Worst Result

Time (s)

CRS2

1

1892 / 4.9%

1907 / 4.1%

12.2

4

1862 / 6.4%

1879 / 5.5%

13.5

CRS3

8

1837 / 7.7%

1888 / 5.4%

14.2

1

1877 / 5.6%

1892 / 4.9%

12.2

4

1861 / 6.5%

1874 / 5.8%

13.7

8

1810 / 9.0%

1889 / 5.0%

15.3


Table 3. Optimization results for historical data concerning the flood in 1972 Optimization Method

Processors Number

Best Resulta

Worst Result

Time (s)

CRS2

1

765 / 1.0%

768 / 0.7%

11.5

4

744 / 3.8%

757 / 2.1%

14.2

8

743 / 4%

754 / 2.5%

15.1

1

764 / 1.2%

770 / 0.4%

12.1

4

751 / 2.7%

763 / 1.3%

14.0

8

735 / 5.0%

753 / 2.6%

16.0

CRS3


The available numerical results indicate that the global optimization algorithms enable improvement in relation to the standard Nelder-Mead simplex method. In most cases, the best results were obtained by ES, but the time required to compute a solution was longer than in the CRS methods and can involve decision delays. The CRS3 method provided better results with respect to CRS2. However, the reduction of the computation cost with respect to the CRS2 method was not very big. The number of processors influences the effectiveness of the optimization method: the more processors, the better the results.3 In general, the global algorithms and their parallel implementation can improve the efficiency of the control systems. 3. ES and CRS are nondeterministic methods, so in every execution of the algorithms, we obtain different results. Because of this, the following situation sometimes may occur: the more processors, the worse the results (see Table 2).

5. Summary and Conclusions In general, the available simulation results indicate that the hierarchical control structure provides for satisfactory cooperation of reservoirs and allows for a considerable decrease of the flood damage. It was demonstrated that hierarchical control with periodic coordination can be an alternative approach to an online flood operation in such water systems where the centralized control is difficult to adopt. The conclusion to be drawn is that in all cases, a thorough computer analysis and simulation is required to develop and verify a control scheme prior to actual implementation. In recent years, it has become apparent that in systems engineering, theoretical and experimental research has been supplemented by computer simulation. Computer simulation allows the assessment of system




behavior, under realistic conditions, when real-life experiments are very difficult to organize and perform since they are always costly and, in many cases, just impossible at the preimplementation phase. Because of the complexity of the considered simulations, the direction that should bring benefits is parallel computing, in which the whole task is partitioned between several processors. In the presented case study, not only optimization but also simulation can be realized in parallel. The whole system consists of several subsystems operating autonomously, except to interact with other subsystems (see Fig. 2). It is natural to model such a complex system as a set of calculation processes that can be handled by distributed processors. A parallel simulation better reflects the structure of the system and allows the reduction of computation time.

operates through a simple cycle of stages: initialization (random selection of an initial population P ), evaluation (a performance function evaluation at each individual of P ), selection (selection of a population for genetic manipulation), and recombination and mutation (production of new chromosomes). Several schemes of evolutionary strategies can be selected (see Michalewicz [38]). In the presented case study, the multimember (µ + λ)-ES variant was considered. In this scheme, λ offspring individuals are created from µ parents by means of the recombination and mutation. The best µ individuals out of parents and offspring are selected to form the next population P . The algorithm is terminated when the improvement in the last few populations is less than ( denotes assumed small value). 7. Acknowledgments

6. Appendix 6.1 Description of Optimization Methods Controlled Random Search Methods. In principle, controlled random search methods (CRS2 and CRS3) [37] were designed as a combination of a local optimization algorithm and a global search procedure. The CRS2 algorithm starts from the creation of the set P of points, selected randomly from the domain. Then, the best xL (i.e., that of the minimal value of the performance index) and the worst xH (i.e., that of the maximal value of the performance index) points are determined, and a simplex in n-space is formed with the best point xL and n points (x2 , . . . , xn+1 ) randomly chosen from P . Afterwards, the centroid xG of points xL , x2 , . . . , xn is determined. The next trial point xQ is calculated, xQ = 2xG − xn+1 . Then, if the point xQ is admissible and better (i.e., f (xQ ) ≤ f (xH )), it replaces the worst point xH in the set P . Otherwise, a new simplex is formed randomly and so on. The CRS3 algorithm is a combination of the CRS2 procedure and the local optimization procedure based on Nelder-Mead simplex method [39]. The local algorithm is triggered automatically. After completing the local search, the global search is continued. The CRS3 method tends to speed the convergence of the algorithm with respect to CRS2. The algorithm is terminated when f (xH )−f (xL ) < ( denotes assumed small value). Evolutionary Strategies. Evolutionary strategies (ESs) [38] emulate biological evolutionary theories to solve optimization problems. The ESs comprise a set of individual elements (the population P ) and a set of biologically inspired operators defined over the population. According to evolutionary theories, the most suited elements in a population are likely to survive and reproduce. In computing terms, an evolutionary strategy maps a problem onto a set of real-value vectors x ∈ Rn (chromosomes), with each vector representing a potential solution. The ESs manipulate the most promising individuals in their search for an improved solution. A general evolutionary strategy algorithm

The work reported in this article was supported by the Polish Committee of Scientific Research (KBN), grant No. 7 T11A 022 20. 8. References [1] Thampapillai, D. J., and W. F. Musgrave. 1985. Flood damage mitigation: A review of structural and nonstructural measures and alternative decision frameworks. Water Resources Research 21 (4): 411-24. [2] Haimes,Y.Y. 1978. Hierarchical analyses of water resources systems. New York: McGraw-Hill. [3] Marien, J. L., I. M., Damazio, and F. S. Costa. 1994. Building flood control rule curves for multipurpose multireservoir systems using controllability. Water Resources Research 30 (4):1135-44. [4] Plate, E. J. 2002. Flood risk and flood management. Journal of Hydrology 267:2-11. [5] Unver, O. I., and L. W. Mays. 1990. Model for real-time optimal flood control operation of a reservoir system. Water Resources Management 4:21-46. [6] Wasimi, S. A., and P. K. Kitanidis. 1983. Real-time forecasting and daily operation of a multireservoir system during flood by linear quadratic Gaussian control. Water Resources Research 19 (6): 1511-21. [7] Dutta, D., S. Herath, and K. Musiake. 2003. A mathematical model for flood loss estimation. Journal of Hydrology 277:24-49. [8] Wind, H. G., T. M. Nierop, C. J. Blois, and J. L. de Kok. 1999.Analysis of flood damages from the 1993 and 1995 Meuse floods. Water Resources Research 35 (11): 3459-65. [9] Windsor, J. S. 1973. Optimization models for the operation of flood control systems. Water Resources Research 9 (5): 1219-26. [10] Moussa, R. 1996. Analytical Hyami solution for the diffusive wave flood routing problem with lateral inflow. Hydrological Processes 10:1209-27. [11] Napiórkowski, J. J. 1992. Linear theory of open channel flow. In Advances in theoretical hydrology: A tribute to James Dooge, edited by J. P. O’Kane, 3-15. New York: Elsevier. [12] Smith, A. F., and D. Boyd. 1998. Flood forecasting on the Grand River. Paper presented at the Coping with Floods: A Civics Perspective workshop, Polish Environmental Partnership Foundation, May, Cracow, Poland. [13] Niewiadomska-Szynkiewicz, E., K. Malinowski, and A. Karbowski. 1996. Predictive methods for real time control of flood operation of a multireservoir system: Methodology and comparative study. Water Resources Research 32 (9): 2885-95.

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Ewa Niewiadomska-Szynkiewicz is an assistant professor at the Institute of Control and Computation Engineering, Warsaw University of Technology, Warsaw, Poland.



Computer Simulation of Flood Operation in Multireservoir Systems

Computer Simulation of Flood Operation in Multireservoir Systems

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