optimization of frequency distribution of storm-water discharges for coastal ecosystem restoration

Optimization of Frequency Distribution of Storm-WaterDischarges for Coastal Ecosystem Restoration

Yongshan Wan, M.ASCE1; John W. Labadie, M.ASCE2; Kenneth D. Konyha3; and Thomas Conboy4

Abstract: Many coastal ecosystems have been adversely impacted by increased storm-water drainage due to expanding urbanization. Theecosystem of the St. Lucie Estuary �SLE�, located on the east coast of south Florida, has been greatly influenced by development of anintricate network of storm-water drainage canals in the tributary watershed. A suite of models dealing with watershed hydrology, reservoiroptimization, and estuary salinity and ecology are applied for optimal sizing and operation of storm-water reservoirs. The multipurposestorm-water control facilities provide for hydrologic restoration to predrained or natural hydrologic conditions for recovery of salinity-sensitive biota in the SLE, as well as supplemental irrigation water and pollution control through connected storm-water treatment areas.The optimization is challenging since the ecological goal is for mean monthly storm-water discharges to the SLE to coincide with thedesired natural frequency distribution, rather than simply attempting to control individual extreme events. The OPTI6 optimization modelapplies a genetic algorithm, coupled with a daily simulation model of the storm-water drainage network, to optimize the sizing and fuzzyoperating rules of reservoirs for controlling storm-water discharges to the SLE. Results indicate that the desired frequency distribution isclosely matched and the level of service for the supplemental irrigation demand is met under reduced storage requirements.

DOI: 10.1061/�ASCE�0733-9496�2006�132:5�320�

CE Database subject headings: Canals; Coastal environment; Detention reservoirs; Ecology; Estuaries; Fuzzy sets; Optimizationmodels; Stormwater management.

Introduction

Expanding urbanization along many coastal areas has adverselyimpacted shoreline and estuarine ecosystems through increasedstorm-water discharges and pollutant loadings. Along the eastcoast of south Florida, the ecosystem of the St. Lucie Estuary�SLE� has been greatly influenced by development since the early1900s of an elaborate network of storm-water drainage canals inthe watershed �Fig. 1�. These canals have drained many historicwetlands and allowed widespread agricultural and urban develop-ment. The current SLE watershed covers over 200,000 ha�500,000 acres�, of which about 50% is irrigated agriculturalland �primarily citrus�, 17% is rangeland, pasture and forest; 17%is urban, and only 16% remains as wetlands. These developments,in combination with emergency freshwater releases from LakeOkeechobee, have significantly altered the quantity, quality, tim-ing, and distribution of storm-water runoff into the SLE. Major

1Senior Supervising Engineer, Coastal Ecosystems Division, SouthFlorida Water Management District, 3301 Gun Club Rd., West PalmBeach, FL 33414. E-mail: [email protected]

2Professor, Dept. of Civil Engineering, Colorado State Univ., Ft.Collins, CO 80523-1372.

3Lead Engineer, Office of Modeling, South Florida WaterManagement District, 3301 Gun Club Rd., West Palm Beach, FL 33414.

4Staff Engineer, Coastal Ecosystems Division, South Florida WaterManagement District, 3301 Gun Club Rd., West Palm Beach, FL 33414.

Note. Discussion open until February 1, 2007. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on April 6, 2005; approved on November 14, 2005. Thispaper is part of the Journal of Water Resources Planning and Manage-ment, Vol. 132, No. 5, September 1, 2006. ©ASCE, ISSN 0733-9496/
2006/5-320–329/$25.00.
320 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT

effects of these manmade alterations are increased drainage, re-sulting in lower groundwater tables, and dramatic changes instorm-water runoff characteristics. Changes in basin hydrologyand water quality have significantly affected the overall salinityregime supporting the SLE ecosystem. Seagrasses and oysters,once abundant in the estuary, are virtually absent today �Haunertand Startzman 1985�.

Restoration of the SLE ecosystem is a major component of theComprehensive Everglades Restoration Plan �CERP� undertakenby the South Florida Water Management District �SFWMD�and the U.S. Army Corps of Engineers �USACE�. In support ofthis plan, the screening model SFSAM was developed by Watkinset al. �2004�. The proposed restoration plan includes a budget ofabout $1 billion for SLE restoration, centering on recapturing thehydrologic characteristics of the predrained or natural watershedusing storm-water detention reservoirs to aid in the recovery andprotection of salinity sensitive biota �USACE and SFWMD2004�. This endeavor, coupled with construction of adjacentstorm-water treatment areas, rehydration of drained wetlands, andremoval of muck sediment in the estuary, will improve the overallecosystem health of the SLE.

During the restoration plan formulation, a suite of modelsdealing with watershed hydrology, reservoir optimization, andestuarine salinity and ecology were employed for establishingthe hydrologic restoration targets and refining the alternatives�Wan et al. 2002�. Aquatic biologists have determined that biotain the SLE are more sensitive to the long-term frequency distri-bution of mean monthly inflows to the estuary, rather than toindividual extreme hydrologic events �Haunert and Konyha2001�. Optimization of the sizing of the storm-water detentionfacilities and development of optimal operating rules are par-
ticularly challenging since the objective is for mean monthly
© ASCE / SEPTEMBER/OCTOBER 2006

storm-water discharges to the SLE to match the desired natural orpredrained frequency distribution as closely as possible.

The difficulties of applying conventional stochastic optimiza-tion methods to solve a problem where the goal is essentially tooptimize a probability distribution resulted in selection of a ge-netic algorithm �GA�, coupled with a daily hydrologic simulationmodel of the drainage network. The GA simultaneously optimizesthe sizing and operating rules of the storm-water detention reser-voirs for controlling storm-water discharges to the SLE. The flex-ible and adaptive operating rules comprise a fuzzy rule-basedsystem, with the GA optimizing the means of the fuzzy conse-quences of the rules. The objective of this paper is to describehow the GA-based optimization model is developed and appliedin the plan formulation process to optimally size and operate astorm-water detention system to achieve coastal ecosystemrestoration.

Reservoir Sizing and Rule OptimizationModel: OPTI6

Formulation

The optimization model OPTI6 was developed to determine theoptimal sizing and operating rules for detention reservoirs in theSLE watershed that �1� achieve the target long-term frequencydistribution of storm-water discharges to the estuary, �2� providesupplemental water supply at a specified reliability, and �3� mini-mize the required capacities of the detention reservoirs. Theweighting method of multiobjective optimization is applied toincorporating these three criteria

minimize �c=1

nc

wc�100Fc − 100Tc�2 + �i=1

nb

�wI�100Pi − 100��2

�1�

if Pi � �; 0 otherwise� + �i=1

nb

wS · si,max2

where Fc is the frequency distribution of mean monthly storm-

Fig. 1. Location map of St. Lucie Estuary and major drainage canalsin watershed. Canals draining basins are assigned same name asbasin. Basins C-25 and 1 discharge directly into Indian River Lagoon.

water releases to the SLE within discrete flow ranges c; Tc is the

JOURNAL OF WATER RESOURCES PLANNING

target probability of mean monthly storm-water runoff to the SLEwithin the discrete flow range represented by class c; Pi is theprobability of failing to meet the water supply requirements forirrigation associated with storage basin i in any year; � is theacceptable risk level for water supply, which is typically the 1-in-10-year drought in the SLE watershed; wc �c=1, . . . ,nc� areweighting factors providing a subjective rating of the relative im-portance of meeting each criterion for nc discrete flow frequencyclasses �the rationale for these weighting factors is detailed in asubsequent section�; wI is a weighting factor associated with vio-lating the risk target for irrigation water supply; wS is a weightingfactor associated with minimizing storage capacity requirementsof each detention reservoir/storm-water treatment area �STA�; nbis the number of storm-water reservoirs; and si,max is the maxi-mum storage capacity actually used in storage option i based onhydrologic simulation of the system.

Solution of Eq. �1� requires daily simulation of the drainagenetwork for calculation of the mean monthly probabilities Fc forall frequency classes c of storm-water releases to the SLE. Thedrainage network simulation assumes construction of offstreamreservoirs requiring pumping facilities for both diversion into andrelease from the basins, as depicted in Fig. 2. For this currentstudy, pumping costs were excluded from the objective function,with preeminence given to the three major criteria represented inEq. �1�. It is also assumed that a multicell STA is connected toeach detention reservoir for reducing loads of nutrients, pesti-cides, and other pollutants from storm-water runoff. The massbalance equation for the reservoirs with connected STA is

si,t+1 = sit + dit − rit + �rainit − evapit� · A�sit� − seepi · sit �2�

�for i=1, . . . ,nb; t=1, . . . ,nd�, where sit is storage in basin i at thebeginning of day t, combining both reservoir and STA storage�103 m3�; A�sit� is surface area of basin i as a function of storagesit�103 m2�; dit is pumped discharge diverted into basin i from theadjacent canal �103 m3/d�; rit is pumped release from the STA to

3 3

Fig. 2. Schematic of typical offstream reservoir with connectedstorm-water treatment area

the canal �10 m /d�; rainit and evapit are rainfall and evaporation

AND MANAGEMENT © ASCE / SEPTEMBER/OCTOBER 2006 / 321

rates, respectively, for basin i on day t �m/day�; seepi is the seep-age fraction per unit storage for basin i, and nd is the total numberof days in the simulation. The simulation assumes that a portionof the reservoir/STA seepage can return as lagged flow to thecanal and is added to the storm-water release to the SLE.

Additional constraints designed to maintain nonnegative flowsinclude

Iit + �j�Ji

qtransjit − dit � 0 �3�

Iit + �j�Ji

qtransjit − dit + rit − wsit − �k�Oi

qtransikt = qtryit � 0

�4�

�for i=1, . . . ,nb; t=1, . . . ,nd�, where Iit is unregulated storm-water inflow from basin i; qtransjit is the portion of flow originat-ing from basin j that is transferred to basin i; Ji is the set of basinstransferring flow into basin i; Oi is the set of basins receiving flowtransfers from basin i; wsit is irrigation water delivery from basini; and qtryit is storm-water release from basin i to the estuary.

The following bounds on the variables are imposed duringsolution of Eq. �1�:

0 � si,t+1 � si,max �5�

0 � dit � di,max �6�

0 � rit � ri,max �7�

�for i=1, . . . ,nb; t=1, . . . ,nd�.The primary decision variables in the optimization are the

scheduling of diversions dit pumped into the reservoir/STA anddischarges rit pumped out. However, the decision variables arereplaced by net inflow to the reservoir qit= �dit−rit� by preventingboth dit�0 and rit�0 on the same day t. Although allowingpumping both into and out of the reservoir/STAs within the sameday would likely increase the quantity of storm-water flows thatcould be treated in the STAs, it would also require increasedpumping costs. Although pumping costs are not directly mini-mized in this formulation, this assumption provides some degreeof pumping cost reduction.

At the beginning of each day, attempts are first made to satisfythe water supply requirements for irrigation wsit by removingwater in storage in the reservoir. If basin storage is insufficient tosatisfy the irrigation demand, then the remainder is supplied frombasin inflow and transbasin diversions. The irrigation demand isdelivered only if it can be fully satisfied for that day, with partialfulfillment of water supply not allowed. If water available in stor-age �at the beginning of the day� and inflow is insufficient, then awater supply failure is assumed to occur for that day. The remain-ing flow is available for diversion into the reservoir/STA

IF Iit + �j�Ji

qtransjit + sit � wsit

THEN

IF Iit + �j�Ji

qtransjit − wsit � 0

THEN qavailit = Iit + �j�J

qtransjit − wsit

i


ELSE qavailit = Iit + �j�Ji

qtransjit + sit − wsit

sit ← sit − Iit − �j�Ji

qtransjit + wsit

ELSE wsit = 0

wsfaili ← wsfaili + 1

qavailit = Iit + �j�Ji

qtransjit

where qavailit is the remaining canal flow available for diversionto the reservoir/STA; and wsfaili counts the number of days ofwater supply failure during each year of the simulation.

Transbasin diversions qtransjit are given a high priority in thesimulation and are assumed to occur up to the pumping capacityif flow is available �i.e., after irrigation demands are satisfied�.The importance of transbasin diversions is seen in Fig. 1, wherethe health of the estuary can be greatly enhanced if excess storm-water releases can be diverted from C-24 to the North Fork areavia Ten Mile Creek, or south from C-23 to C-44. This reducesstorm-water releases from the C-23/C-24 canals into the middleof the SLE and enhances the salinity balance.

Any desired transbasin transfer configuration can be specifiedin the simulation model, and transfers can be made to more thanone basin. Maximum transbasin diversion amounts can be speci-fied, along with additional restrictions if flooding would occur inthe receiving basin. Special rules regarding basins such as in C-44are required when levels in Lake Okeechobee dictate when flowscan move west to the Lake or east to the SLE. However, transfersfrom C-23 are not allowed to enter the lake. An option included inthe simulation logic allows for special circumstances, such as thereservoirs in C-23 and C-24 connected by an uncontrolled in-verted siphon. A reservoir rebalance routine can be invokedwhereby storage in the connected reservoirs is assumed to bebalanced at the end of each day �in proportion to their relativestorage capacities�.

Fuzzy Operating Rules

The optimal reservoir operating rules qi�Iit ,sit� represent feedbackpolicies whereby operators measure current day inflows and res-ervoir storage and then obtain reservoir operation guidelines fromthe rules based on those measurements as well as the time of year.The original operating rules aggregated the storage and inflowmeasurements together using various weighting factors, meaningit was possible that one day with high inflow and low storagemeasurements could result in the same operational guidelines as aday with low inflow and high storage conditions. To overcomethis problem, it was decided to develop fuzzy operating rules thatallow inflow and storage conditions to be distinguishable to theoperators and produce unique rules for all combinations of inflowand storage conditions on any day. Fuzzy rules also have theadvantage of not requiring an a priori mathematical structure forthe rules, such as linear decision rules.

The general structure of a fuzzy rule n is �Bárdossy and Duck-
stein 1995�

IF a1 is An1�a2 is An2� ¯ �aK is AnK

THEN Bn �8�

where Boolean logical operator � refers to AND, OR, or XOR.The IF portion of the rule contains the premises, and the THENpart is the consequence. Arguments in the rule premise are as-sumed to belong to fuzzy sets, and the consequence also belongsto a fuzzy set. In contrast to crisp sets, a fuzzy set assigns amembership value or degree of truth to elements of the set. Themembership values vary between 0 �indicating no truth to theassertion that the element is a member of the set� to 1 �indicatingcomplete confidence in the assertion�. For each set of facts pro-vided to the fuzzy rules �i.e., current measurements of inflows Iit

and storage sit in this case�, a degree of fulfillment �DOF��in �Iit ,sit� for basin i is calculated for the premises of each rule n.Although there are a number of ways of calculating DOF, themost popular is product inference

�in�Iit,sit� = �Ain1 AND Ain2� = �Ain1�Iit� · �Ain2

�sit� �9�

where �Aink�ak� is the membership value �between 0 and 1� of

argument ak �k=1: inflow Iit; k=2: storage sit� in fuzzy set Aink ofrule n for basin i. Since several rules can be simultaneously acti-vated in fuzzy rule-based systems for a given set of measure-ments, but at varying degrees of fulfillment, a method is neededfor combining the fuzzy consequences of each of the rules. Thenormed weighted sum combination method has the advantage ofproviding a convenient means of defuzzifying the fuzzy conse-quences �Bárdossy and Duckstein 1995�.

�Bi�x� =

�n=1

Ni

�in�Iit,sit� · �in · �Bin�x�

maxu

�n=1

Ni

�in�Iit,sit� · �in · �Bin�u�

�10�

where Ni is the total number of rules for basin i and �in is theinverse of the area under the membership function for the nthconsequence.

1

�in=�

−�

�

�Bin�x�dx �11�

The normed weighted sum combination method provides lessweight for fuzzy consequences that are more vague �i.e., vague-ness is correlated with the area under the membership function�.

It is now necessary to defuzzify the fuzzy combinations, withthe most popular method being mean defuzzification

qi�Iit,sit� =

�n=1

Ni

�in�Iit,sit� · q̄in

�n=1

Ni

�in�Iit,sit�

�12�

where q̄in is the mean of the fuzzy consequence of rule n andqi �Iit ,sit� is the defuzzified operating rule conditioned on thecurrent inflow Iit and storage measurements sit supplied as facts tothe fuzzy rule-based system. The advantage of using product in-ference for degree of fulfillment of rule premises, normedweighted sum combination of fuzzy consequences, and mean
defuzzification is that characterization of the structure of the

membership function of the fuzzy consequence �Bin�x� is not ex-

plicitly required, including evaluation of �in. Only the means ofthe fuzzy consequences q̄in for each basin i and rule n are needed,so they can be regarded as decision variables manipulated in theoptimization.

For this application, the fuzzy membership functions for thepremises are assumed to be structured as symmetric, triangularfuzzy numbers �Fig. 3�. For the types of premise arguments—basin storage and expected basin inflow for the current day—theRange for each type of argument is determined using the maxi-mum potential storage capacity for each basin and then calculat-ing the largest daily inflow that occurred during the simulationperiod. The desired number of arguments for each type of premise�i.e., storage or inflow� can be specified for each basin. The modelthen calculates the Support for each fuzzy number based on theRange and the specified number of arguments. A desired degreeof Overlap of the triangular fuzzy numbers �e.g., 0.25 representsan Overlap of 25% of the calculated Support of each fuzzy num-ber� can also be specified, where the recommended degree ofOverlap is usually a value between 0.20 and 0.333 �Bárdossy andDuckstein 1995�.

In effect, the specified number of arguments for the storageand inflow premises essentially divides these values into discreteclasses, although each class is governed by a continuous member-ship function as defined by the triangular fuzzy number. Forexample, if 6 argument classes are specified for storage and 7argument classes for inflow, then a total of 67=42 fuzzy rulesare generated by the model for a particular basin. Defining moreargument classes theoretically gives a more accurate representa-tion of the range of storage and inflow amounts that can occur ina basin, but also increases the number of rules and therefore thenumber of variables to be optimized. In this example, each of the42 rules for this basin produces a fuzzy consequence, the meansof which are the variables to be optimized. In this case, the meansof the fuzzy consequences q̄in �n=1, . . . ,Ni; i=1, . . . ,nb� are val-ues varying between −100 and 100, representing the percentageof total available flow actually diverted to the reservoir for thatparticular rule if q̄in�0, or the absolute value of the percentage ofstorage in the reservoir that is pumped out if q̄in0. Mean de-fuzzification then produces the overall operating rule qi�Iit ,sit�

Fig. 3. Triangular fuzzy numbers for premises �inflow or storage� forfuzzy rule-based system

whereby


IF qi�Iit,sit� � 0

THEN dit =qi�Iit,sit�

100· qavailit

AND rit = 0

ELSE IF qi�Iit,sit� � 0

THEN rit =abs�qi�Iit,sit��

100· sit

AND dit = 0

Rainfall in south Florida varies seasonally, with distinct wetand dry seasons. Wet season or summer rainfall results primarilyfrom convective and tropical storms, whereas frontal systemsgovern dry season or winter rainfall. To relate the operating rulesto seasonal influences, distinct rules qi

w�Iit ,sit� and qis�Iit ,sit�

are developed for each season by optimizing the means of thefuzzy consequences q̄in

w and q̄ins for the winter and summer sea-

sons, respectively.

Genetic Algorithm

Since solution of Eq. �1� requires execution of an imbedded dailysimulation model for producing the frequency distribution Fc forstorm-water discharges to the SLE, as well as probabilities Pi

of failing to satisfy irrigation demands for basin i, traditionalstochastic optimization algorithms are not well suited to solutionof this problem. A GA was therefore selected since it requiresno explicit analytical representation of the objective functionand constraint sets in the optimization. Gradient information isnot required, and discontinuities in the objective function havelittle effect since GAs are resistant to becoming trapped in localoptima.

Rooted in the mechanisms of natural selection in biology, GAswere first proposed by Holland �1975�, whose goals were to ex-plain the adaptive processes of natural systems and design com-puting systems embodying their important mechanisms. Withpublication of Goldberg �1989�, which completely covers GAconcepts, mathematical foundations, implementation, and appli-cations, researchers in a wide variety of fields have attempted toapply GAs. These have proven particularly attractive for solvingcomplex combinatorial problems quickly and reliably, as well asproviding easy interfacing to existing simulation models, andhave therefore become the evolutionary computation method ofchoice.

GAs differ from traditional optimization methods in that �1�they operate on a binary string coding of the variables �genotype�rather than the actual real-numbered values �phenotype� of thevariables; �2� rather than searching over sequential points in thesolution space, a GA generates an entire population of solutions ateach step �generation�; and �3� random processes play an impor-tant role in a GA. Real variables are coded into binary stringssuch as �1 0 0 1 1 0 1 0�, whose phenotype value is 154. Thelength of the string depends on the size and precision of the realnumber being coded. The biological analogy is that this representsa chromosome, with each bit representing a gene in the chromo-some with a particular locus or position in the string. For binary-
coded strings, the allele of each gene is the value 0 or 1.

Since a GA is a heuristic search procedure, convergence tooptimal solutions cannot be guaranteed. However, practical expe-rience with GAs finds they are often able to locate global optimalsolutions to nonconvex, multimodal optimization problems whenother methods fail to do so �Michalewicz 1996�. GAs attempt tomaximize the genetic fitness of the individuals �i.e., variables� inthe population �i.e., solution set�, which corresponds to maximiz-ing �or minimizing� the objective function. They combine sur-vival of the fittest among binary alphabet representations with astructured yet randomized information exchange to form a searchalgorithm with some of the innovative flair of human search�Goldberg 1989�.

Michalewicz �1996� has shown that real-coded GAs outper-form binary-coded GAs for problems with continuous variables,although in some cases a binary-coded GA produced individualswith superior fitness. Real-coded GAs operate directly on thefloating point representation of the variables, with no distinctionbetween genotype and phenotype. An argument against real-coded GAs is poor representation in a finite population, whicheither forces larger population sizes or diminishes the effective-ness of statistical sampling. In this case, although the means ofthe fuzzy consequences q̄in

s and q̄inw of the operating rules are

continuous variables, only single decimal precision is requiredfor the optimal solution. Since high-precision solutions are un-necessary, a binary-coded GA is considered sufficient for thisapplication.

Holland’s �1975� GA is commonly called the simple geneticalgorithm �SGA� �Fig. 4�. Components of the SGA are a popula-tion of binary strings, stopping criterion, selection, genetic opera-tors �i.e., crossover and mutation�, and replacement. Using thegenetic operators, the algorithm creates the subsequent generationfrom the previous one until the stopping criterion is met, based ona number of generations such that the best solution of the popu-lation fails to improve. Each binary string represents a solutionfor the optimization problem.

To reduce the likelihood of genetic drift or premature conver-gence of the population to individuals with similar geneticstructure, niching methods based on fitness sharing have beendeveloped whereby the fitness of a population element is dimin-ished in proportion to the number of similar individuals in thepopulation, based on a sharing function that measures the degreeof similarity of the individuals �Sareni and Krähenbühl 1998�.

Fig. 5 shows the connection between the GA and the drainage

Fig. 4. Components of simple genetic algorithm

network simulation model, where the GA selects populations of


fuzzy consequences of the fuzzy rule-based system, which arethen evaluated through the daily simulation model. The dailysimulation evaluates mean monthly frequency distributions ofstorm-water discharges, water supply reliability, and requiredstorage capacity, which are returned to the GA for improvementin fitness �Eq. �1��.

Application of OPTI6 for Optimal Restoration PlanDevelopment

Hydrologic and Irrigation Demand Input

Application of OPTI6 requires daily watershed storm-water dis-charges into the SLE as well as irrigation demand on pumpagefrom the Floridian aquifer as input data. Hydrologic simulation ofcurrent watershed conditions was conducted using the HydrologicSimulation Program-FORTRAN �HSPF�, Version 12 �Bicknell etal. 2001�. Enhancements to HSPF for simulating the high watertable and wetland conditions prevalent in south Florida were in-corporated prior to model calibration and application �Aqua Terra1996�. The model was calibrated for drainage basins where long-term flow monitoring data are available at the discharge controlstructures.

The model was applied to six major drainage basins within thewatershed �Fig. 1�, which were further divided into subbasins,each segmented into six land-use types. These land-use catego-ries, including irrigated agriculture �primarily citrus�, nonirrigatedpasture, forest, wetland, and urban lands, are considered the mostimportant factors determining hydrologic response in the water-shed. The 1995 land-use coverage was used to represent currentdevelopment, with the resulting simulation under these conditionsreferred to as the 1995 base. Projected land use in 2050 consti-tutes the future development condition, with the associated simu-lation designated as the 2050 base. Substantial expansion of urbandevelopment is expected in 2050, but with irrigated agriculture

Fig. 5. OPTI6: Interaction of genetic algorithm for optimizing fuzzyoperating rules with drainage network simulation model

remaining at current levels along with decreases in forest and


pasture areas. Wetlands are not anticipated to decline in total acre-age. HSPF was applied to predicting how these changes in landuse could impact the hydrology of the basin.

An important aspect of the modeling process is to simulate theallocation of available water supplies to meet irrigation demands.The SLE watershed includes substantial irrigated acreage but lim-ited surface storage capacity. The major source of water supplyhas been the Floridian aquifer, an artesian aquifer with salinityconcentrations generally exceeding acceptable levels for directuse in agriculture. During extended periods of drought, these saltlevels can significantly reduce citrus yields. HSPF attempts tosatisfy the irrigation demands by drawing water from specifiedcanals. When the canals are dry, however, irrigation demands areunmet since groundwater cannot be mixed with better-quality sur-face water. The Agricultural Field Scale Irrigation RequirementsSimulation �AFSIRS� model �Smajstrla 1990� was applied to de-termining irrigation demands, water availability, and Floridianaquifer withdrawals within the SLE watershed. Daily storm-waterdischarges into the SLE and supplemental irrigation from the Flo-ridian aquifer were processed as input data for OPTI6 based onresults of HSPF and AFSIRS.

Restoration Target: Flow Distribution

To establish the target flow distribution as expressed in Eq. �1�, afavorable range of watershed storm-water discharges for salinity-sensitive biota in SLE, called the salinity envelope, was first es-tablished. Salinity in the upper part of the SLE is dominated bystorm-water discharges. Salinity modeling in the SLE indicatedthat once watershed inflows exceed 56.6 m3/s �2,000 ft3 / s� to85 m3/s �3,000 ft3 / s�, salinity in the upper SLE can be close tozero even during high-tide conditions �Hu 1999�. The combina-tion of biological understanding of the ecosystem and salinitymodeling in the SLE led to establishment of a salinity enveloperanging from 10 m3/s �350 ft3 / s� to 56.6 m3/s �2,000 ft3 / s� forjuvenile marine fish and shellfish, oysters, and submerged aquaticvegetation.

The next step in establishing the restoration target was to de-termine acceptable violations of this range that can occur and stillsustain the estuary ecosystem �Haunert and Konyha 2001�. It isassumed that the acceptable violations are confined by the tem-

Fig. 6. Flow distributions into St Lucie Estuary: 1995 baseconditions and predrained conditions

poral and spatial hydrologic variability of the predrained water-


shed. This concept is supported by other efforts undertaken torestore freshwater riverine ecosystems �Richter et al. 1997�.

The Natural Systems Model �NSM�, Version 4.5 �Van Zee1999�, was developed for simulating the predrained watershedhydrology. The NSM is a cell-based, 2D, coupled-surface andgroundwater model that simulates infiltration, evapotranspiration�ET�, and overland, river, and groundwater flow. The watershed isdiscretized into grid cells of 3.223.22 km �22 mi� in size.The primary spatial properties required to simulate the hydrologicprocesses in each cell include vegetation/landscape type, surfaceelevation, and aquifer properties. Rivers are represented as linearsegments of connected river cells, and cell vegetation is based onthe Natural Landscape Position coverage �Van Zee 1999�.

The dominant landscape/vegetation types under predrainedconditions were forested wetlands �mainly pine flatwoods� andwet prairie, occupying 59 and 35% of the predrained watershed,respectively. Since these landscape types have been identified inthe remaining natural Everglades, the model input parameters areimported directly from the Everglades NSM and the calibratedand verified South Florida Water Management Model �SFWMD1998� from which the NSM was derived. The NSM results were

Fig. 7. Recommended plan for S

further verified by comparing with long-term measured flow data


obtained from the Peace River basin located in the southwestcoast of Florida. The natural portion of the basin has landscape/vegetation types similar to the natural SLE watershed. The moni-toring data were normalized with the watershed area for compari-son with the NSM results.

According to NSM results, watershed development has in-creased average annual inflows into the estuary by about 45%,with a loss of 7.6 cm �3 in.� or 154 million m3 �125,000 acre-ft�of water storage. The increase in total flow is accompanied byextremely high flow events under the developed condition. Due tothe construction of drainage canals, only about 25% of the stormwater is drained to the estuary through the North Fork. In con-trast, flow to the middle of the estuary increased from 3 to 25%.Adverse salinity impacts caused by this hydrologic change sug-gest a need for flow diversion back to the North Fork to partiallyrestore historic flow patterns.

The monthly flow frequency distributions for the predrainedand developed watersheds as well as the Peace River basin arepresented in Fig. 6. The similarities of the distribution curve be-tween the NSM applied to SLE watershed and the monitoredPeace River basin support the NSM results. Compared with the

ie Estuary ecosystem restoration
t. Luc
1995 base condition, the predrained watershed had a substantially


lower probability of high flows. The NSM also showed that lowerflow rates entered the estuary during dry periods, suggesting thatthe estuary may not require flow augmentation during dry peri-ods, and consequently that irrigation demands do not directlycompete with environmental demands from the estuary. This ob-servation also supports the contention that hydrologic restorationshould focus on reducing the frequency of high flows to the SLE.Fig. 8 displays the restoration target monthly flow distribution asdefined by the NSM results with the emphasis placed on the high-flow ranges.

Reservoir Optimization Results

To achieve the targeted monthly flow frequency distribution, ca-pacities and fuzzy operating rules of the reservoir/STA were op-timized using the OPTI6 model. A multiobjective analysis wasconducted by varying the weighting factors in Eq. �1� until asuitable compromise solution was obtained between the threespecified criteria: �1� matching the frequency distribution ofstorm-water discharges to the SLE; �2� satisfying the minimumirrigation water supply reliability requirements; and �3� minimiz-ing the required storage capacities of the reservoirs. A GA driverprogram gafortran was obtained from Dr. David Carroll �CUAerospace, Inc., Champaign, Illinois; �http://cuaerospace.com/carroll�� as freeware for use in this study. Since Version 1.7 ofgafortran is written in FORTRAN 90, the program was convertedto ANSI C to maintain compatibility with other C/C�� softwareused in this study. The converted C code gaopt.c uses standard Clibraries and is readily compiled with the Gnu gcc compiler.

Extensive testing with gaopt.c confirmed that performance im-proved with the standard GA using a population size of 100�npopsiz�. Bounds on the fuzzy rule consequences are maintainedbetween −100 and +100, since the rules specify percent of avail-able flow or storage for diversion to or release from the reser-voirs. Discretization of the fuzzy rule consequences is requiredfor the binary-coded GA and is specified to maintain the singledecimal place precision of the variables. Tournament selectionwith a shuffling technique for choosing random pairs for matingis the only selection option available in the code. Niching wasalso invoked to enhance the diversity of successive generations.The recommended jump mutation rate of �1/npopsiz� was used inthe genotype, with a rate of 0.02 set for creep mutation, which is

Fig. 8. Comparison of mean monthly frequency distributions of SLEinflows between current distribution �1995 base�, target �NSMmodel�, and optimal plan

performed in the phenotype �i.e., the real value of the variable�.


The uniform crossover method outperformed the single-pointcrossover method and was selected with the recommended cross-over probability of 0.5. Including elitism in the generational re-placement also produced superior results.

OPTI6 was applied to determining the optimal sizing andfuzzy operating rules for the detention reservoir/STAs in the SLEwatershed. In assigning penalty-weighing factors wc for monthlyflow ranges c in Eq. �1�, the highest penalties on deviation fromthe target frequencies were assigned to flows outside the favor-able range 10–56.6 m3/s �350–2000 ft3 / s�, particularly highflow frequencies above 56.6 m3/s �2,000 ft3 / s�. Flows in therange 10 m3/s �350 ft3 / s� were assigned the second-highestpenalty, with the less important intermediate ranges given lowpenalty weights. Penalty weights wI on irrigation water supplyfailure and wS for minimizing storage requirements were sys-tematically manipulated until the best compromise solution wasobtained.

During development of the optimal restoration plan, severalalternatives were tested with varying acreage of wetlands to berestored. The year 2050 was used to represent future watershedconditions, with projected land use in 2050 used for modelingwatershed hydrology with HSPF and irrigation demands and sup-plies with AFSIRS. As depicted in Fig. 7, the selected alternativeconsists of four off-line water storage reservoirs in the C-23,C-24, North Fork �NF�, and C-44 basins. Construction of thesefeatures includes water control structures, pumps, levees, and ca-nals and acquisition of approximately 4,937 ha �12,200 acres� ofland. These reservoirs, along with 36,422 ha �90,000 acres� ofrehydrated wetlands and flow diversion from C-23/C-24 to theNorth and South Forks, create a hydrologic regime similar to thenatural flow frequency distribution.

Fig. 8 shows that the optimal restoration plan provides an ex-cellent match to the NSM targets for the most important high flowfrequency classes, with the lower flow frequency classes of lessimportance displaying acceptable agreement. Significant im-provement over the current frequency distributions without thestorm-water control structures is clearly evident in Fig. 8, particu-larly for the critical frequency classes. Further evidence of thebenefits of the optimal restoration plan is seen in comparison withthe frequency distribution of flows in C-23 between the optimalplan and the current uncontrolled conditions. Although not givenhere, results show substantial reduction in storm-water dischargesto the middle SLE, which should provide significant ecologicalimprovement. Results also indicate that 20% of storm-water in-flows would be treated in the STAs for all the basins, resulting inreduced pollutant loadings to the SLE.

Fig. 9. Comparison of risk of water supply failure �base hydrologyand 2050 land use� between current �uncontrolled� and optimal plan

Fig. 9 highlights the multipurpose benefits of the detention


reservoirs where risks of failing to satisfy irrigation water supplyrequirements fall below the 10% risk target and provide signifi-cant improvement over the current uncontrolled conditions.Fig. 10 shows significant cost savings are achieved by OPTI6 inreducing sizing requirements of the detention reservoirs/STAs by45,400 103m3 �36,800 acre-ft�, or almost 30% of the original ca-pacity estimates. The actual footprint of the reservoir/STAs mayof course change during design and construction phases.

Sample defuzzified operating rules from OPTI6 for basin C-23are displayed graphically in Fig. 11. This display is for illustrationonly, since more accurate calculation of rules for real-time opera-tion would require extracting the fuzzy rules from OPTI6 into aseparate interactive program that would allow operators to enterthe basin name, current season, current day inflow, and currentday storage in the detention reservoir/STA. The fuzzy operatingrules calculate the optimal percent of divertible flow to the reser-voir, where divertible flow is the lower of available inflow, un-used capacity in the reservoir, and maximum pump capacity fordiversion. The rules also specify under what conditions releasesshould be made from the basins, which are represented as theoptimal percent of the smaller of current storage in the basin andavailable pump capacity.

Fig. 11 underscores the robustness of the fuzzy rule-based sys-tem in that it would be difficult to find a priori defined rules, suchas a piecewise linear structure or high-order polynomials, thatcould duplicate the flexibility of the fuzzy rules. The reason for

Fig. 10. Reduction in required storage capacity under optimal plan

Fig. 11. Sample operating rules for C-23 basin from fuzzy rule-basedsystem conditioned on current day storage volume and inflows�summer season�


the complexity of the fuzzy rule structure is related to the com-plexity of the objective function. Unlike other stochastic optimi-zation problems that attempt to simply minimize the expectedvalue of the objective, or utilize some other type of statisticalmeasure, the attempt here is to optimize the long-term frequencydistribution of storm-water discharges to the St. Lucie Estuary,along with other terms. Since the operating rules areconditional—that is, the optimal release policies are dependent oncurrent inflow and storage conditions �and season� in the basin—then the optimal release is related to the long-term “frequency” ofthose conditions occurring in the basin. This is further compli-cated by the fact that the operating rules being optimized areimpacting the frequency of storage conditions in the basin,whereas the inflows are assumed to be unregulated. In this study,constraining the rules to a more simplified structure resulted inpoorer performance in terms of satisfying the long-term ideal fre-quency distribution of storm-water discharges.

Conclusions

The St. Lucie Estuary �SLE� ecosystem restoration plan has beendeveloped based on the integration of a suite of models to simu-late watershed hydrology, reservoir optimization, estuary salinity,and ecology. The Natural System Model �NSM� was crucial forestablishing the hydrologic restoration targets and justifying flowtransfers between basins. The OPTI6 model coupled a geneticalgorithm with a daily drainage network simulation model usingHSPF inflow calculations for optimal sizing and operation of thereservoir/STAs in the SLE watershed. Robust operating rules ob-tained through application of a fuzzy rule-based system were thekey to providing optimal solutions that achieve target meanmonthly frequency distributions for storm-water inflows to theSLE for restoration of the estuarine ecosystem.

In addition, the multipurpose benefits of the detention reser-voirs are clearly evident by maximizing use of the attachedstorage-treatment areas �STAs� for pollutant load reductions, aswell as maintaining desirable risk targets for supplemental irriga-tion water supply from storm-water. Significant cost reductionswere also achieved under the optimal plan through reduction oftotal sizing requirements for the detention basins by over 30%from the initial estimates. Results indicate that the optimal resto-ration plan has the potential to restore and protect the mesohalineecosystem in the SLE.

Since these results provide only an indication of the potentialfor improving SLE conditions, future work will focus on obtain-ing verification results of the performance of the operating rulesusing an independent data set. Utilization of a short 2- to 3-yearverification period would fail to verify the long-term performanceof the operating rules as to producing correct mean monthly fre-quency distributions for storm-water discharges to the estuary.Selecting a longer verification period from the historical data setnot included in the calibration runs would likewise reduce theportion of the historical data set used for developing the operatingrules and therefore raise questions as to whether the operatingrules properly reflect the long-term statistical characteristics ofbasin inflows. As the historical dataset is augmented by both realand synthetically generated data, future work will focus on ob-taining verification results.

References

Aqua Terra Consultants. �1996�. “Modifications to HSPF for high water
table and wetlands conditions in South Florida.” Rep., submitted to

South Florida Water Management District, West Palm Beach, Fla.Bárdossy, A., and Duckstein, L. �1995�. Fuzzy rule-based modeling with

applications to geophysical, biological, and engineering systems,CRC Press, Boca Raton, Fla.

Bicknell, B., Imhoff, J., Kittle, J., Jobes, T., and Donigan, A. �2001�.Hydrologic simulation program-FORTRAN, version 12, user’sManual, National Exposure Research Laboratory, Office of Researchand Development, U.S. Environmental Protection Agency, Athens,Ga.

Goldberg, D. �1989�. Genetic algorithms in search, optimization and ma-chine learning, Addison-Wesley, Reading, Mass.

Haunert, D., and Konyha, K. �2001�. “Establishing St. Lucie Estuarywatershed inflow targets to enhance mesohaline biota.” Appendix E.,Indian River Lagoon—South Feasibility Study, South Florida WaterManagement District, West Palm Beach, Fla.

Haunert, D., and Startzman, J. �1985�. “Short-term effects of a freshwaterdischarge on the biota of St. Lucie Estuary, Florida.” Technical Pub-lication 85-1, South Florida Water Management District, West PalmBeach, Fla.

Holland, J. �1975�. Adaptation in natural and artificial systems, Univer-sity of Michigan Press, Ann Arbor, Mich.

Hu, G. G. �1999�. “Two-dimensional hydrodynamic model of St. LucieEstuary.” Proc., Environmental Engineering (1999), G. C. Schafran,ed., ASCE, Reston, Va., 434–443.


Michalewicz, Z. �1996�. Genetic algorithms�data structures�evolutionprograms, Springer, Berlin.

Richter, B., Baumgartner, J., Wigington, R., and Braun, D. �1997�. “Howmuch water does a river need?” Freshwater Biol., 37, 231–249.

Sareni, B., and Krähenbühl, L. �1998�. “Fitness sharing and nichingmethods revisited.” IEEE Trans. Evol. Comput., 2�3�, 97–106.

Smajstrla, A. G. �1990�. Agricultural field scale irrigation requirementssimulation (AFSIRS) model, Version 5.5 Technical manual, Univ. ofFlorida, Gainesville, Fla.

South Florida Water Management District �SFWMD�. �1998�. A primer tothe South Florida water management model (version 3.5), SouthFlorida Water Management District, West Palm Beach, Fla.

U.S. Army Corps of Engineers �USACE� and South Florida Water Man-agement District �SFWMD�. �2004�. “Central and Southern FloridaProject: Indian River Lagoon—South.” Final Integrated ProjectImplementation Rep. and Environmental Impact Statement, Jackson-ville, Fla.

Van Zee, R. J. �1999�. Natural system model documentation, version 4.5,South Florida Water Management District, West Palm Beach, Fla.

Wan, Y., Konyha, K., and Sculley, S. �2002�. “An integrated modelingapproach for coastal ecosystems restoration.” Proc., 2nd Inter-AgencyHydrologic Modeling Conf., Las Vegas, Nev., 13.

Watkins, Jr., D. W., Kirby, K. W., and Punnett, R. E. �2004�. “Water forthe Everglades: Application of South Florida Systems Analysis
Model.” J. Water Resour. Plan. Manage., 130�5�, 359–366.

optimization of frequency distribution of storm-water discharges for coastal ecosystem restoration

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