OPTIMAL DESIGN OF URBAN STORMWATER DRAINAGE ......4.5.2. Sampling steps 39 4.5.3. Pairing 40 4.5.4....

OPTIMAL DESIGN OF URBAN STORMWATER DRAINAGE

SYSTEM UNDER UNCERTAINITY

Seneshaw A. Kebede MSc Thesis MWI SE 2014-18 April 2014

0 500 1000 1500 2000

Pa

ram

ete

rs

Number of model simulations

Sensitivity analysis for flow lost to flooding

P. imp CW

n N1 impN2 perv Rainfall

Dummy Flow lost to flooding

SWMM5 model

• Hydraulic simulation

Optimizer (NSGA-II)

• Generate new population

Objective function computation

• Flooding volume • Intervention cost

Read NSGA-II population and

update SWMM input file

Termination

Stop

Yes No

Start

New population

OPTIMAL DESIGN OF URBAN STORMWATER DRAINAGE

SYSTEM UNDER UNCERTAINITY

Master of Science Thesis by

Seneshaw A. Kebede

Supervisor Prof. D. Brdjanovic, PhD, MSc

Mentor S.D. Seyoum, PhD,MSc

Examination committee Prof. D. Brdjanovic, PhD, MSc (UNESCO-IHE)

S.D. Seyoum, PhD, MSc (UNESCO-IHE) B. Gersonius, PhD, MSc (UNESCO-IHE)

This research is done for the partial fulfilment of requirements for the Master of Science degree at the

UNESCO-IHE Institute for Water Education, Delft, the Netherlands

Delft April 2014

©2014by Seneshaw A. Kebede. All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without the prior permission of the author. Although the author and UNESCO-IHE Institute for Water Education have made every effort to ensure that the information in this thesis was correct at press time, the author and UNESCO-IHE do not assume and hereby disclaim any liability to any party for any loss, damage, or disruption caused by errors or omissions, whether such errors or omissions result from negligence, accident, or any other cause.

i

Abstract Stormwater drainage system is essential part of any urbanization and due to its high investment requirement attention must be given at the time of planning design and construction. The failure of urban storm water drainage systems results in wide range of property damage and it is a threat to public health. One of the reasons for failure of the urban stormwater drainage system results from neglecting uncertainties during the design or rehabilitation phase of the system.

The objective of this research is to produce optimum design of urban stormwater drainage systems considering uncertainties of parameters due to different drivers. An optimized urban drainage system which includes uncertainties of parameters would give a better approach which can maintain the level of functioning under different circumstances as a result of future changes. Due to time and computational power required to include uncertainties in to an optimization framework, certain parameters which represent uncertainties were selected from SWMM5 model parameters. The value of uncertain model input parameters for which urban drainage models are sensitive was considered in the design of urban stormwater drainage systems. This is because the value of these uncertain parameters greatly affects the performance of the system. And to select sensitive parameters sensitivity analysis was performed in this study. And different approaches were used to develop a technique that can consider sensitive parameters uncertainties in optimization problems.

Sensitivity analysis of urban storm water model parameters was done using model eFAST to select most sensitive parameters which represent uncertainty for further multi-objective optimization process. eFAST works by integrating SWMM5 with other functions which sample the parameters, set the frequency of variation and quantify the sensitivity indexes. Four outputs of urban storm water model (SWMM5) and six parameters were selected to see the sensitivity of the output for the input variability. From the result the parameters sensitivity shows different result for different type of output. One of the objectives in this research was to minimize flooding. Therefore parameters which were most sensitive to Flow lost to flooding were used to represent uncertainties in the proceeding optimization process. And from the sensitivity analysis result the sensitivity of a parameter mostly depends on the type of the output required rather than the model as a whole.

Therefore In this research optimized design of urban stormwater drainage system which includes uncertainties due to climate change, urbanization and variability of the land cover within a sub catchment was produced. And the Multi-objective optimization methodology produced was applied to a case study area Dhaka Bangladesh used in this study.

Once uncertain parameters are selected, multi objective optimization problem was formulated by integrating sampling techniques (LHS) to sample from uncertain parameters, storm water management model (SWMM5) and an optimizer NSGA-II . The objective functions of the optimization were minimizing flooding and minimizing intervention cost. Two methods were used to introduce uncertainties in to the optimization model. In the first method for each chromosome generated by the optimizer, the hydraulic model was simulated for all sets of the uncertain parameters sampled and average of the flooding volume over all of the samples was

ii

considered as one of the objective function. While in the second method modification was done to NSGA-II to make the search for optimum solution more efficient and to breed a population capable of surviving over multiple generations. Hence for each chromosome in a generation, the hydraulic model was simulated for one sample realization for the evaluation of the fitness of the chromosome and the output was used for objective value calculation. The optimization model was simulated for different number of samples and different standard deviations are taken to include the uncertainty bound of parameters considered.

From the result of the two methods to introduce uncertainties in to an optimization model optimum Pareto front were obtained. Optimization result in which uncertainty were not included show under design and was less costly. Optimization results which include uncertainties of parameters resulted in more costly design but better performance and can handle extra flow loads coming due to uncertainties. Comparative result of two of optimization methodologies which considers uncertainties of parameters show that method 2 shows wide range of distributions and was more costly design option than method 1. This is because the objective functions used to evaluate fitness of chromosomes in method 2 are from different sample realizations in a generation and this result in different hereditary characteristics of the chromosomes and modification was done to the optimizer. This helps the genetic algorithm to search for variety of fittest solutions for decision variables. In method 1 average of the objective functions over the entire sample realizations were used to evaluate the chromosome fitness in a generation. This leads to more similar characteristics of the chromosomes in a generation as compared to method 2 in the search procedure for optimum solution.

Keywords: Stormwater Drainage Systems, Multi-objective optimization, Uncertainty, Sensitivity Analysis, SWMM5, NSGA-II, LHS, Flood Volume, Intervention cost

iii

Acknowledgements First of all I would like to thank the Almighty God who has been with me by giving patience, health and blessing to successfully accomplish my work.

I would like to express my heartfelt gratitude to my supervisor Prof. Damir Brdjanovic for his constructive comment, fruit full advice and practical guidance in the progress of my work.

My special thanks to my mentor Dr. Solomon Seyoum who has been with me all the way through the thesis work without hesitation to help me in any way at any time. His continuous guidance, support and valuable comment let me pass all the difficult times I had in the period of my MSc study and without his support it would have been difficult to make it to this point. His wonderful experience and valuable guidance on the area of my study and programming language and coding helped me to successfully develop my model. I have benefited more from working with him and I could say I am lucky enough to work with him.

I would like to thank my mom Alemitu Tola whose continuous support, encouragement and prayers let me successfully achieve my career all the times. I am proud to have you mom! And I pray to God to keep you healthy and live longer.

Special thanks to my Wife Misaye Zewde, it is difficult to describe how lucky I am to have you in my life and God has given me more than I ever could have asked for. And your helpful support during my study made me stronger. I love you, and looking forward to our lifelong journey.

I am grateful to my uncle Bogale Tola, cousins Dr Kassa Hailu and Dr Negash Seyoum for being my role models to be successful in my careers.

I would like to thank my sister Million Bekele for being with me from the first date of my application to the UNESCO-IHE through all the times. And I will not forget the lucky moments I had you here in Netherlands and your effort to introduce me to my study and the Dutch environment.

Finally I would like to thank all my family, friends and all the IHE community and the time I had with you all will be memorable in my life.

v

Table of Contents

Abstract i

Acknowledgements iii

List of Figures vii

List of Tables ix

Abbreviations x

1. Introduction 1

1.1. Background 1

1.2. Problem statement 2

1.3. Research objectives 3

1.4. Research questions 3

1.5. Structure of the thesis 4

2. Literature Review 5

2.1. Urban drainage systems 5

2.2. Urban Hydrology 6

2.3. Urban hydraulics 7

2.4. Urban drainage design 9

2.5. Uncertainties on urban stormwater Drainage modelling 10 2.5.1. Urbanization 11

2.5.2. Climate change 12

2.5.3. Aging of pipe 13

2.5.4. Sub catchment width 13

2.5.5. Manning's roughness coefficient for the sub catchment 14 2.6. Urban Drainage Model 14

2.6.1. SWMM5 15

2.7. Sensitivity Analysis 17

2.7.1. Linear trends between input parameters and output. 17

2.7.2. Non Linear but monotonic trends between input and output parameters 18

2.7.3. Non Linear and non monotonic trends between input and output parameters 19

2.7.4. Extended Fourier Amplitude Sensitivity test – eFAST 20

2.8. Sampling techniques 22

2.8.1. Monte Carlo Sampling 22

2.8.2. Latin hypercube sampling 22

2.9. Optimization 23

2.9.1. Single-Objective Optimization 24

2.9.2. Multi-objective Optimization 24

2.9.3. Genetic Algorithm 24

2.9.4. Non Dominated Sorting Genetic Algorithm (NSGAII) 26

3. Case study Area Description 27

3.1. Dhaka 27

3.1.1. General 27

vi

3.1.2. Drainage systems of Dhaka city 27

3.1.3. Location of the study area 28

3.1.4. Drainage system within the study area 29

4. Research Methodology 31

4.1. General 31

4.2. Quantification of uncertain parameters 33

4.3. Hydraulic Model set up (SWMM5) 33

4.3.1. Description of sewer network in the study area 34

4.4. Sensitivity Analysis 35

4.4.1. Sensitivity Analysis (eFAST) Model 35

4.5. Optimization 38

4.5.1. Latin Hypercube Sampling 38

4.5.2. Sampling steps 39

4.5.3. Pairing 40

4.5.4. Objective functions and decision variables 40

4.5.5. Optimization process 41

5. Result and Discussion 47

5.1. Result 47

5.1.1. Simulation result of sensitivity analysis 47

5.1.2. Simulation result for Optimization without considering uncertainties of parameters 52

5.1.3. Simulation result for optimization considering uncertainties of parameters 53

5.1.4. Comparison of simulation result of optimization without considering uncertainties of parameters versus with uncertainties of parameters (Method1 &Method 2) 58

5.2. Discussion 60

6. Summary Conclusion and Recommendation 62

6.1. Summary 62

6.2. Conclusion 64

6.3. Recommendation 65

References 67

Appendices 70

Appendix A 70

vii

List of Figures Figure 2-1 IDF Curve for Cabudare city in Venezuela (CORDERO, 2012) ......................................... 6

Figure 2-2 Flow continuity theory for pipe flow ........................................................................................ 8

Figure 2-3 The influence of urbanization on runoff (Butler & Davies, 2004). .................................... 12

Figure 2-4 Expected change in design capacity of drainage systems in extreme event (Arisz & Burrell, 2006) .................................................................................................................... 13

Figure 2-5 SWMM5 model flow chart (Ketema, 2007) .......................................................................... 16 Figure 2-6 General schematic diagram of a sensitivity analysis model (Saltelli et al., 1999). ........ 21 Figure 2-7 eFAST sensitivity analysis performed on the Lotka-Volterra model (Marino et al.,

2008). ..................................................................................................................................... 22

Figure 2-8 Structure of GA Model ............................................................................................................ 25 Figure 3-1 Location map of Dhaka city and study area ( DMDP 1995).............................................. 29

Figure 3-2 Drainage system within the study area (Ahmed, 2008) ..................................................... 30 Figure 4 -1 General Methodology flow chart .......................................................................................... 32

Figure 4-2 the study area network on SWMM5 model ......................................................................... 34 Figure 4-3 eFAST sensitivity analysis model diagram(Simeone Marino et al., 2008). .................... 37 Figure 4-4 Intervals Used with a Latin Hypercube Sample of Size n = 5 in Terms of the PDF

and CDF for a Normal Random Variable (G. D. Wyss, Jorgensen, Kelly H, 1998) ......... 39 Figure 4-5 optimization methodology flow chart without considering uncertainties of

parameters (Z. Vojinovic et al., 2008) ............................................................................... 42

Figure 4-6 optimization methodology flow chart without considering uncertainties of parameters ............................................................................................................................ 43

Figure 4-7 optimization methodology due to uncertainties of parameters Method 1 (Santizo, 2012) ...................................................................................................................................... 44

Figure 4-8 optimization methodology due to uncertainties of parameters Method 2 ....................... 46 Figure 5-1 Result of sensitivity analysis of SWMM5 model parameters for peak flow model

output (case study: Dhaka Bangladesh) .......................................................................... 48 Figure 5-2 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5

model parameters for peak flow model output (case study: Dhaka Bangladesh)...... 48 Figure 5-3 Result of sensitivity analysis of SWMM5 model parameters for time to peak model

output (case study: Dhaka Bangladesh) .......................................................................... 49 Figure 5-4 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5

model parameters for time to peak model output (case study: Dhaka Bangladesh) .......................................................................................................................... 49

Figure 5-5 Result of sensitivity analysis of SWMM5 model parameters for flow lost to flooding model output (case study: Dhaka Bangladesh) .............................................................. 50

Figure 5-6 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5 model parameters for flow lost to flooding model output (case study: Dhaka Bangladesh) .......................................................................................................................... 50

Figure 5-7 Result of sensitivity analysis of SWMM5 model parameters for total lateral inflow model output (case study: Dhaka Bangladesh) .............................................................. 51

Figure 5-8 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5 model parameters for Total lateral inflow model output (case study: Dhaka Bangladesh) .......................................................................................................................... 52

Figure 5-9 Pareto front for optimization without considering uncertainties of parameters ....................... 53

Figure 5-10 Pareto front for optimization considering uncertainties of parameters (Method 1) ............... 54

Figure 5-11 comparison of Pareto optimal front for different standard deviations (Method 1) ......... 55 Figure 5-12 comparison of Pareto optimal front for different number of samples & the same

standard deviation (Method 1) .......................................................................................... 56

viii

Figure 5-13 comparison of Pareto optimal front for different standard deviations (Method 2) ....... 57 Figure 5-14 comparison of Pareto optimal front for different number of samples with the same

standard deviation (Method 2) ........................................................................................... 58

Figure 5-15 comparison of Pareto optimal front without uncertainties of parameters versus optimization considering uncertainties (Method1 and Method 2) ................................. 59

ix

List of Tables Table 2-1 Description of optimization terminologies (Walski et al., 2003) ......................................... 23

Table 4-1 uncertain parameters for this study and their quantified min and max values from literature (CK, 2001), (Sahlu, 2012 ), (Ahmed, 2008)..................................................... 33

Table 4-2 normalized value of uncertain variables ................................................................................ 37

Table 4-3 List of pipe diameters and unit costs. .................................................................................... 41

Table 4-4 Input parameters for NSGA-II ................................................................................................. 42

Table 5-1 value of objective functions for comparison of optimization without uncertainties versus optimization with uncertainties (Method 1 and Method 2) ................................ 59

Table A-1 Sub-catchment properties description of the study area .................................................... 70 Table A-2 conduit properties description of the study area .................................................................. 72 Table A-3 Junction node property description of the study area ......................................................... 74 Table A-4 Outfall node property description of the study area ............................................................ 76 Table A-5 100 Year rainfall data measured at 5 minutes for the study area ..................................... 76

Table A-6 Manning's roughness coefficient n for different pipes ........................................................ 77

x

Abbreviations SWMM Storm Water Management Model NSGA-II Non Dominated Sorting Genetic Algorithm LHS Latin Hypercube Sample eFAST extended Fourier Amplitude Sensitivity Test FAST Fourier Amplitude Sensitivity Test SOO Single Objective Optimization MOO Multiple Objective Optimization GA Genetic Algorithm 1D One Dimensional PDF Probability Density Function CDF Cumulative probability Density Function TR Return Period IDF Intensity Duration and Frequency SA Sensitivity Analysis Si First order sensitivity index STi Total order sensitivity index

IPCC International Panel of Climate Change CC Pearson's Correlation Coefficient PCC Partial Correlation Coefficient SRC Standard Regression Coefficient SRCC Spearman Rank Correlation Coefficient SRRC Standardized Rank Regression Coefficient

OPTIMAL DESIGN OF URBAN STORMWATER DRAINAGE SYSTEM UNDER UNCERTAINITY 1

1.1. Background

Due to human interaction with natural water cycle, rain water is diverted away from the natural system of drainage. This interaction leads to covering of the land with impermeable artificial surface and removal of trees for different human needs, which increases the amount of surface runoff by decreasing infiltration and therefore increasing the total volume of water to be conveyed. Therefore, Stormwater drainage systems are needed in developed urban areas to transport the surface run off to prevent the inconvenience (Butler & Davies, 2004).

Urban Stormwater conveyance systems design deals with the transport and disposal of stormwater everywhere in the back yards, streets, parking lots, and parks. Failure to convey Urban stormwater properly ranges from nuisance flooding yards, basements and roadway travel lanes, minor property damage, public health hazards, to wide spread destruction and loss of life. One of the reasons for failure of urban drainage systems is due to insufficient capacity of the drainage infrastructures arising from uncertainties of parameters which need to be considered during the design of these infrastructures (Durrans & Dietrich, 2007).

Urban stormwater drainage system needs to be designed properly because it is an essential part of any urbanization. Its proper operation and performance insures the safe discharge of run off to the required place. Urban drainage models are required to design these infrastructures for the specific demands of the planed urban area. Consistent maintenance methodology is required after construction to guarantee the proper functioning of the system and to facilitate the remedial works and optimize the allocation of resources(Z. Vojinovic et al., 2008).

The urban stormwater drainage infrastructure financing is high in terms of capital investment as well as maintenance cost. Therefore, optimized design for the sizing of drainage infrastructures should be considered. The purposes of optimizing a sewer network design in this research is to minimize the capital cost related to its construction and/or rehabilitation, and to minimize the flooding in the network using optimization algorithm provided that the design performance is fulfilled. The parameter which is directly related to the construction cost of the storm sewer system is the pipe diameter size. The scenarios considered and the run-off flow that the drainage system needs to transport determines the dimensions of the drainage infrastructures (MH Afshar et al., 2011).

CHAPTER 1

Introduction

Introduction 2

In addition to the conventional measures to reduce the cost of stormwater drainage systems and damage which occurs due to flooding, on site measures can also be used to reduce peak runoff flows. These measures are provision of infiltration ponds, rainwater collection and reuse systems, application of green roofs, construction of pervious pavements etc and can be applied to reduce the runoff quantity which has a positive impact in reducing the flood volume and the investment cost of storm water systems (Woods-Ballard et al., 2007).

Uncertainties of model input parameters due to different drivers are the challenges in the design and analysis procedures of urban stormwater drainage structures. These model input parameters might not be always known with a sufficient degree of certainty because of different drivers coming from climate changes, population growth, aging of infrastructures, error in measurements, or simply a lack of current techniques to measure them.

This research was intended to build optimized design of urban stormwater drainage systems by considering uncertainties of model input parameters due to different drivers. Considering uncertainties of model input parameters in the design insures the reliability of the system.

1.2. Problem statement

Conventionally storm sewer is designed to insure un-pressurized or just full condition so that all the sewer conduits should be designed to deliver a free-surface flow. Manning equation and Colebrook-White equation are used to show the relationship between pipe size and its capacity. The size of the sewer can be calculated based on these equations given a design flow (velocity or discharge) and pipe roughness. A steady flow approximation is mostly used to simplify the calculation. Each pipe is considered as a separate entity and in a sequence from upstream to downstream when designing a network. And the design is simply based on the idea of keeping pipe slopes as flat as possible without any concept of optimization which may lead to high cost due to bigger diameter of pipes. Optimized storm sewer layout and design aims at minimizing the network construction cost while a good system performance is ensured. Depending on the problem formulation, the problem can be considered as a single-objective or multiple-objective optimization problem whilst satisfying a set of constraints (Guo et al., 2008).

An optimal design of urban storm sewer system requires a number of objectives to be met at a time by an optimization algorithm. For instance minimization of: 1) cost of the infrastructure; 2) total flooding volume across the network. Fulfilment of these conflicting objectives at a time is really a challenge of designing. To get a compromise solution of conflicting objectives application of optimization techniques is necessary and the optimization is handled by selecting multi-objective optimization problem.

Optimized urban drainage systems with the inclusion of uncertainties of parameters due to different drivers would give a better methodology which can maintain the level of functioning under different circumstances as a result of future changes. The value of uncertain model input parameters for which urban drainage models are sensitive need to be considered in the design of urban stormwater drainage systems because the value of these uncertain parameters greatly affects the performance of the system. Failure of these systems results in higher damage for example if the system is under designed, and an excessive precipitation occurs then flooding of the manholes occurs with the potential of damaging the performance of the structures. Therefore there is the need to develop a technique that can consider uncertainties in optimization problems.

Optimization techniques turn out to be essential in addressing the problem effectively and locating optimal solutions. During the design, the optimization technique is usually integrated with a sewer network hydraulic simulator, which evaluates the hydraulic performance of each potential solution.

The scope of this research is to develop an optimization methodology which considers uncertainties of parameters due to different drivers in the design or rehabilitation of urban stormwater infrastructures.


Including uncertainties in to an optimization model increases the reliability of the system due to the increased performance. Uncertainties are included in to an optimization model through parameters whose variability has effect on the output of a model and to identify these parameters sensitivity analysis was required. There is time and computational power required to include a large number of uncertainties in to an optimization model. Therefore prior to the optimization methodology, certain urban storm water model parameters were selected and sensitivity analysis is done to select most sensitive ones which represent uncertainties in the optimization process

Different kinds of decision support information tools are used in urban stormwater drainage systems for optimal design. In this study a one dimensional storm water management model SWMM5 was coupled with an optimizer and the model was used to minimise the two objective functions flooding volume and intervention cost of infrastructures and the minimum value of these objectives was used as information on further decision making process.

In a complex urban terrain to make a complete decision making tool, a risk map based on flood vulnerability and flood hazard of an area need to be produced. Vulnerability of an area is the degree of seriousness of the flooding which is based on the asset value of an area (commercial, residential, industrial, standard of living, agricultural etc.) and the probability of occurrence. Flood hazard is the physical damage that arises due to the occurrence of flooding that is the physical destruction the flooding might cause. To determine the flood hazard the one dimensional model used in this study need to be coupled with 2D model to produce flood inundation map of an area which will enable us to know the extent of the damage and this will be used as a decision making tool.

1.3. Research objectives

The overall objective of this research is to develop and test a methodology to find the optimum size of stormwater drainage systems given the layout, topography and boundaries considering the uncertainties of parameters due to different drivers. It involves determination of the size of the stormwater drainage system components required to convey a design flow. Optimum design of stormwater drainage infrastructure focuses primarily to prevent stormwater damage to property and to provide a storm water system that can be maintained economically.

The result of this research will support decision making for proper design and management of urban stormwater drainage systems under uncertain parameters. Additionally, the results can be used to determine how optimal must a sewer system be to face all possible scenarios coming from uncertain conditions.

The specific objectives of this research are:

� To perform sensitivity analysis of an urban drainage model to identify the most sensitive parameters to be considered in the optimization process.

� To analyze and adapt optimization tool for application in urban stormwater drainage systems which consider uncertainties.

� To couple the optimization tool with the stormwater management model, SWMM5. � To apply the developed method on a case study. � To analyse the effect of including uncertainties of parameters in the optimization process.

1.4. Research questions

The following research questions must be answered to meet the objective of the study 1. To which parameters urban drainage model is most sensitive?

Introduction 4

2. How is uncertainty included in the optimization model for the design of urban stormwater drainage system?

3. What is the effect of considering uncertain parameters in optimization of urban stormwater drainage systems?

4. What is the advantage of considering uncertainties in the optimization of urban stormwater drainage systems?

1.5. Structure of the thesis

This thesis contains six chapters. Chapter one provides the introduction which includes the background of the study, the problem statement, research objectives and research questions. In the second chapter literature review which contains a description of information related to area of the study are included. In this chapter the type of urban drainage systems, the hydrological flow of rainfall to generate runoff and hydraulic design of the drainage infrastructures to convey the runoff, uncertainties on urban stormwater drainage modelling and urban drainage model used in the study were described. This chapter also describes sensitivity analysis, the method used to select the most significant parameters whose variability has influence on the model outputs. Sampling techniques which is used to sample from range of uncertain parameters is also discussed in this chapter. Optimization frame work including uncertainties of parameters due to different drivers which was developed by integrating optimizer and urban drainage model was also stated in this chapter. Previous stormwater drainage optimization works were also reviewed in this chapter. In the third chapter the case study area used in the study where the developed optimization framework was applied was described. The fourth chapter describes the methodology used to achieve the objective of the research and this includes review and quantification of uncertain parameters, sensitivity analysis of uncertain parameters to select the most sensitive parameters by which uncertainties are introduced in to an optimization model, build optimization framework in which uncertainties are included in to an optimization model. The fifth chapter provides the result and discussion of the study which shows the findings of the study and discussion of the findings. And in the sixth chapter the summary conclusion and recommendation of the study were compiled.


2.1. Urban drainage systems

Due to the interaction of human activity with natural water cycle, water is abstracted for consumption from the environment, wastewater is returned to the environment after consumption and the natural characteristics of the land surface is changed due to the impervious cover by human interaction. Therefore, the amount of water which needs conveyance increases due to these activities and this creates a need to develop urban drainage system. Urban drainage systems are structures that are used to transport stormwater and sewage in urban areas. Based on the channel of flow these systems can be classified as piped systems and open channelled systems.

Also based on the type of water that flows, conventional drainage systems can be classified as:

� Separate sewer system: - urban drainage system in which wastewater and storm water flows in a separate conveyance systems.

� Combined sewer system: -A waste water collection system in which wastewater and storm water is transported through a single pipe system to the treatment plant.

In this study a stormwater drainage system is concerned. A storm water drainage system should be designed to collect and convey run-off generated within a catchment area during and after rainfall events, for safe discharge into a receiving watercourse or the sea. The magnitude of peak flows that have to be accommodated will depend primarily on the intensity of rainfall and the size, topography, soil type, configuration and land use of the catchment (Drainage Services Department, 2000).

Safe and efficient drainage of stormwater is important to maintain public health and safety, avoiding the potential impact of flooding on life and property and to protect the environment and to reduce cost of the drainage network pipes. Therefore to achieve this, it is necessary to assess the performance of the existing stormwater drainage system within the area, to upgrade the system and to design new systems when necessary to allow accommodation for future development.

CHAPTER 2

Literature Review

Literature Review 6

2.2. Urban Hydrology

Design rainfall

Rainfall inputs are the key drivers for stormwater drainage systems. These inputs may be single event which are associated with a return periods or probability of occurrence or a historical time series and the need for these types of rainfall inputs depends on the type of analysis required. Single event associated with probability of occurrence or the so called return period is commonly use one (CORDERO, 2012).

The return period can be defined as a function of the frequency:

�� = 1�(� ≥ )

(2.1)

Where TR is the return period, and P(X>x) is the probability that a given X event is exceeded in a given period of time.

A record of maximum annual rainfall is needed to calculate P(X=>x). The exceedence probability of xm where n is the total number of years and m is the rank of the values arranged in descending order is;

P(X ≥ x) = �� (2.2)

Intensity, Duration and Frequency relationship (IDF Curves)

IDF curves are source of information which tells about the intensity duration and frequency relationships and it shows a particular place where the rainfall data is collected. It shows the decrease in intensity of rainfall as the return period increases and it is built by frequency analysis. To produce IDF curve the maximum annual rainfall of specific duration is ranked by descending order and exceedence probability will be calculated. Then the intensity and the corresponding probability is integrated with probabilistic statistical distribution (CORDERO, 2012).It is a common practice to use existing IDF curves like those shown in Figure 2-1 for the design or rehabilitation of urban drainage network.

Figure 2-1 IDF Curve for Cabudare city in Venezuela (CORDERO, 2012)


Design Rainfall Duration

The peak flow in urban storm water drainage occurs when the rain that falls at a farthest distance from the outlet leaves the catchment. And this occurrence is more described by the duration it takes the rain drop to go from that point to the outlet and this period is called the concentration time. When designing urban drainage systems the design rainfall duration has to be at least equal to the concentration time. Concentration time is divided in to two parts these are: the time it takes the stormwater to travel over land surface (t overland) and the time it takes the stormwater to travel in the channel.

tc = toverland + tchannel Run-off Generation

The rain fall need to be transformed in to runoff after the design rainfall is obtained. When rain falls in a sub catchment, vegetation catches a certain amount before it reaches the surface and this is called interception. Based on the type of the surface cover, the rain fall follows different paths that is, part of it evaporates to the atmosphere and some infiltrates in to the ground and some will be stored in depressions and these loss of rain fall on the way to the drainage system is called abstraction. The water that contributes to the run off after all the above abstractions is called effective rain fall. To design stormwater drainage systems, runoff rates must be estimated, because these are the discharges for which the conveyance facilities must be designed (Durrans & Dietrich, 2007).

Prior to planning and design of urban storm water drainage system, it is the primary phase to estimate peak rate of runoff, runoff volume and flow time. A number of mathematical formulae have been used to estimate peak runoff, runoff quantity based on hydrological characteristics of catchment area. Among them rational method is the most popular hydrological flow computation methods. It considers rainfall intensity, dimensionless runoff coefficient and catchment area for peak flow calculation.

Q = C ∗ I ∗ A (2.3)

Where: Q = peak flow collected from catchment area and discharge to conveyance system (cms)

C = Runoff coefficient

I = Rainfall intensity. (Mm/hr)

A = Catchment area. (Ha)

Rainfall intensity and duration are assumed to be uniform over the entire area in rational method and storm duration must be equal to or greater than the time of concentration of the watershed. But the limitation of this assumption is the variability of the rainfall intensity in space in real cases. Even if the system has limitations it is the most implemented methods to compute peak overland flow.

2.3. Urban hydraulics

Once the flow is found based on the hydrological analysis stated earlier the next step is sizing of the storm sewer components based on hydraulic analysis. The appropriate sizing of urban storm sewer system components like, pipe channels and storages need to understand the hydraulic design concept. Hydraulics are also used for the analysis and modelling of existing systems in order to predict the relationship between flow-rate and depth for varying inflows and conditions.

Literature Review 8

Hydraulic design of a stormwater drainage system relies on the amount of stormwater collected from the catchment area. The collection system can be handled either by piped system or natural or artificial open channels. The most common types of flow in urban storm sewer system are a part full pipe flow in which a liquid flows in a pipe by gravity or pumped and an open channel flow in which a liquid flows in a channel by gravity with free surface at atmospheric pressure. The liquid only fills the pipe area when the flow-rate equals or exceeds the designed capacity, and the bed of the pipe slopes down in the direction of flow. Fundamental hydraulic principles underlying fluid flow include flow classification, conservation of mass, conservation of momentum and conservation of energy (Butler & Davies, 2004). Some of fundamental hydraulic theories used to analyze hydraulic computation are explained below:

Continuity of flow

In a section of pipe, the mass of the liquid entering the pipe must be equal to the mass leaving in time. Assuming that a liquid has a constant density, the discharge entering must be equal to the discharge leaving. This theory is known as continuity of flow (Butler and Davies, 2004).

A1v1 = A2v2 (2.4)

Q1 = Q2 (2.5)

Where: A1 and A2 are cross sectional area at point1 and 2 v1 and v2 are flow velocity (=Q/A) at point 1 and 2 Q1 and Q2 are flow rate (volume per unit time) at point 1 and 2 respectively

Figure 2-2 Flow continuity theory for pipe flow

Manning's equation is also one of the common formulae used in steady flow for closed conduits or open channel flows to compute flow rate or flow speed and it is shown below.

# = 1� $�%/'()/%

(2.6)

Where: Q is the flow rate in m3/s, A is cross-sectional area in m2


R, is the hydraulic radius in m; n is the manning roughness coefficient

S is the conduit slope for steady flow and kinematic wave flow routing, and for dynamic wave flow routing it is the friction slope (i.e. Head loss per unit length.

Hydraulic Flow Energy

A flowing liquid has three main types of energy, these are: pressure, velocity and potential and energy in hydraulic flow is expressed as head, energy per unit weight (common units, m).

The tree types of energy expressed as head are,

Pressure head, *+, Velocity head,

-.%, Potential head, z

Total head (H) is the sum of the three heads stated above, given by the Bernoulli equation:

H = 01ρ2 + 3%

22 + z (2.7)

And when a liquid flows in a pipe or a channel, some head (hL) is lost from the liquid. So, from the law of conservation of energy for water flowing between sections 1 and 2 in the full pipe, total energy at section 1 is equal to the total energy at section 2

H1 − hL = H (2.8)

OR

01ρ2 + 31%

22 + z1 − hL = 02ρ2 + 32%

22 + z2 (2.9)

Where: P= pressure (ρgh), v=velocity, ρ= Density of liquid g = gravitational acceleration, Z= elevation from datum. hL= head loss(difference between their total heads)

2.4. Urban drainage design

The design of urban storm drainage needs understanding of hydraulics in order to identify the proper size of components of the system such as ,pipes, channels shape, size of storage, etc. the analysis and modelling of existing system also needs hydraulics in order to predict the relationship between flow-rate and depth for varying inflows and conditions. The flow characteristics and cause of overflow in a network also needs hydraulics knowledge to take remedial action for flooding occurrence analysis in urban drainage design and the provision of efficient conveyance capacity for computed or anticipated storm volume or events require special attention (Jang et al., 2006).

Piped systems or artificial channels can be used to handle the urban stormwater collection systems. Gravity flow is the usual way of conveying stormwater except in areas which need pumping to enhance the flow. Hydraulic design of a system is essential so that the run off volume and peak flow of the stormwater

Literature Review 10

collected is conveyed properly from the surrounding catchment in order to reduce inconvenience, flood damage and public health risks.

There are varieties of flow conditions in urban drainage systems (Price et al., 2011)which range from:

• Free surface flow to surcharged flow (pressurized) • Steady to unsteady flow • Uniform to non-uniform flow (gradually or rapidly varying flow).

Open channel and conduit flows normally have a uniform cross section along its length and a uniform gradient. Because the dimensions of the cross section are typically one or two orders of magnitude less than the length of the conduit, unsteady free surface flows can be modelled using one dimensional flow equations.

Flood waves in conduits show certain distinctive characteristics (Price et al., 2011) including:

• Translation (the peak propagates downstream direction); • Attenuation (gravity forces tend to flatten the peak or disturbance along the Conduit or channel); • Distortion (change in shape of the wave profile).

2.5. Uncertainties on urban stormwater Drainage mod elling

A circumstance where we do not have precise knowledge to express a system into consideration is called uncertainty. When solving engineering problems that are uncertain, the method, data and modelling is based on assumptions or rough calculations which might be difficult to express in details. Uncertainty can be named as arbitrariness, vagueness, irregularity, changeability, variability (Van Gelder et al., 2001). It is necessary to identify the type, source and nature of uncertainty before trying to reduce it.

Usually when a stormwater drainage system is designed, the capacity of the system with respect to future population is taken in to consideration by calculating the future population growth for a certain design period. But the growth of a population varies continuously based on social and economic factors in a particular urbanized area. This variability leads to high uncertainty concerning the capacity of urban stormwater drainage systems to handle the flow of stormwater in a given area.

The integrity of the pipes inner surface is also affected through process due to aging of infrastructure and the time influence for deterioration has not been part of the models. Due to abrasion and corrosion taking place in the inner surface of pipes, the roughness of a pipe increases.

The other uncertainty is climatic effects on urban drainage. According to IPCC 2007 (Le Treut et al.), in the last 100 years there is a rise of global mean temperature by 7oc, the outcome of this change resulted in hydrological cycle changes and this in turn increases the rainfall intensity and resulted in regular precipitation. The vulnerability of urban drainage network increases due to the variations in precipitation and uncertainties due to these change need to be included in the design process to guarantee the proper functioning of the urban drainage infrastructures.

There are also other uncertainties of parameters due different drivers such as sub catchment width, and manning's roughness coefficient of the sub catchment that depend on the particular area to be considered.

Identification of the most significant input variables reduces the estimation of uncertainty and allows investigation in refining the knowledge about it. This helps to improve optimized design procedures of urban drainage network. In this study by identifying the most important input variables used in the estimation of the model output via a sensitivity analysis, most sensitive parameters are considered for


further optimized design of urban stormwater drainage system, and hence decrease their uncertainty and variability, and finally increase the confidence in the optimal design of stormwater drainage systems (King & Perera, 2010).

2.5.1. Urbanization Due to urbanization the land use system will be converted in to commercial, residential and industrial uses and this change results in hydrological impacts that diverts the rainfall from its natural system of drainage. As urbanization increases, the impervious portion of an area increases and an increase of impervious area due to urban development is related to hydrological impact and urban drainage is one of a significant part of this.

Increased urbanization results in covering of the land with impervious surface these surfaces are pavements, roofs, parks, parking lots and any other human-built structure that obstructs the penetration of the stormwater into the ground. Therefore, the water generated by rainfall or snow melt cannot infiltrate into the soil and is converted to run-off. And the runoff generated is then conveyed by gravity to enter the urban drainage network within a concentration time hence flooding of the area might occur due to excess run-off that has occurred due to the urban growth.

Hydrological changes associated with urbanization make the design and planning of urban infrastructures more difficult. The study of (Morgan et al., 2004) used census, climate (temperature and precipitation), land use and runoff data to consider changes in the hydrological cycle and studied the hydrological consequences of urban growth and expansion of impervious surface area between 1950 and 2000 in the Laurel Creek Watershed, Waterloo, Ontario. From the study, it is found that runoff volumes have noticeably increased due to urbanization.

The change in rainfall runoff event and the performance of drainage system is affected because of the amount of impervious area due to increasing population and new urban development's (Semadeni-Davies et al., 2008). The percentage of the area that is impervious, the soil type and the vegetation cover of the area determines the land use parameters used for the design of urban drainage infrastructures. If the surface runoff generated is greater than the designed capacity of the drainage system, the area will be flooded because of excess discharge, pollution of the environment will occur and maintenance cost of the drainage infrastructures increases. But flooding of the area will not occur and the runoff produced will completely drain in to into drainage systems if urban surface runoff is smaller than the designed capacity of the storm drainage system.

Urbanization results in:

• Increases the total volume of runoff. • High peak flows due to quick travel of runoff generated over the surface. Figure 2-3 shows this

phenomenon.

Literature Review

Figure 2-3 The influence of urbanization on runoff

2.5.2. Climate change Climate change affects the rain fall patterparameter for urban stormwater drainage systemsthe capacity of storm sewers and cause urban flooding. Uncertainty concerning rainfall may be as epistemic and random (Hutton et al., 2004

The magnitude of rainfall events and runoff flows is significantly affecclimate change and these values are the major inputs for designing u& Burrell, 2006). The value of precipitation is increasing in as the generalexperiments of the global warming are change, the designed flow in the drainage system varies with time and the system will be surcharged more often also flooding of the area will occur which in turn increase the maintenance cost of the infrastructure because frequent flooding results in degradation of the system. change on major and minor drainage syst

e influence of urbanization on runoff (Butler & Davies, 2004).

the rain fall patterns of a catchment and rain fall is the most important input water drainage systems. Excess precipitation during storm events may overtop

the capacity of storm sewers and cause urban flooding. Uncertainty concerning rainfall may be Hutton et al., 2004).

The magnitude of rainfall events and runoff flows is significantly affected by uncertainties occurring due to climate change and these values are the major inputs for designing urban stormwater infrastructures

The value of precipitation is increasing in as the general circulation model (GCexperiments of the global warming are showing. When the rainfall intensity is changed due to climate

designed flow in the drainage system varies with time and the system will be surcharged more e area will occur which in turn increase the maintenance cost of the infrastructure

because frequent flooding results in degradation of the system. Figure 2-4 shows the drainage systems design.

12

ns of a catchment and rain fall is the most important input . Excess precipitation during storm events may overtop

the capacity of storm sewers and cause urban flooding. Uncertainty concerning rainfall may be interpreted

ted by uncertainties occurring due to rban stormwater infrastructures (Arisz

circulation model (GCM) the rainfall intensity is changed due to climate

designed flow in the drainage system varies with time and the system will be surcharged more e area will occur which in turn increase the maintenance cost of the infrastructure

shows the effect of climate

OPTIMAL DESIGN OF URBAN STORMWATER DRAINAGE SYSTEM UNDER UNCERTAINITY

Figure 2-4 Expected change in design capacity of drainage systems in extreme

The intensity of the problems in seand more periodic rainfall events. Not only surface runit rains with more intensity more water infiltrates the system

2.5.3. Aging of pipe The diameter of the pipe is reducedwall of the aged pipe and this increase the roughness of the pipe system will be reduced. The capacity and the roughness which comes due to aging of infrastructurespipes and accumulation of sediment creates variability in the internal sewer pipes geometry and highly affects the discharge depth relationship. significant parameter in the hydraulic model of drainage of the pipe need to be specified in order to solve the law of conservation energy equationsurban drainage systems. And in modelling of urban storm water systems, pipe roughness is among the most significant sources if uncertainty this is because aging changes the inner surface of the pipe and the characteristics of the pipe also changes. The changquality of water that is being transported and the corrosion of the pipe depends on the pipe material usedthe quality of water.

Generally the carrying capacity of the stormwater drainage system decreinfrastructures and this effect is significantly magnified because the service life of urban drainage infrastructures is measured in decades (50 yechanges over these years will be visible

2.5.4. Sub catchment widthSub catchment width is the characteristic width of the overland flow path for runoff generated. characteristic width of the sub catchment is estimated by maximum average overland flow


design capacity of drainage systems in extreme event (Arisz & Burrell, 2006

The intensity of the problems in sewer systems increases in consequence of climate change, with increased and more periodic rainfall events. Not only surface run-off is a problem but also infiltration to the pipes, as it rains with more intensity more water infiltrates the system (Berggren, 2008).

he diameter of the pipe is reduced by the development of corrosion by-product and particles on the inner increase the roughness of the pipe and consequently the performance of the

system will be reduced. The capacity and the head loss of the distribution system are largely affected by a due to aging of infrastructures (Williams & Hazen, 1911).

pipes and accumulation of sediment creates variability in the internal sewer pipes geometry and highly affects the discharge depth relationship. This phenomena shows hydraulic roughness of pipes is a significant parameter in the hydraulic model of drainage systems. Pipe roughness, diameter and the length of the pipe need to be specified in order to solve the law of conservation energy equations

ban drainage systems. And in modelling of urban storm water systems, pipe roughness is among the most significant sources if uncertainty this is because aging changes the inner surface of the pipe and the characteristics of the pipe also changes. The change in the characteristics of the pipe may be due to the quality of water that is being transported and the corrosion of the pipe depends on the pipe material used

Generally the carrying capacity of the stormwater drainage system decreases due to the aging of the infrastructures and this effect is significantly magnified because the service life of urban drainage infrastructures is measured in decades (50 years to 100 years) and the cumulative

will be visible.

width Sub catchment width is the characteristic width of the overland flow path for runoff generated.

width of the sub catchment is estimated by dividing the area of the sub catchment by the erage overland flow length. The flow path from the inlet to the farthest drainage point of the

13

Arisz & Burrell, 2006)

wer systems increases in consequence of climate change, with increased off is a problem but also infiltration to the pipes, as

and particles on the inner the performance of the

head loss of the distribution system are largely affected by a . Bio film formation in

pipes and accumulation of sediment creates variability in the internal sewer pipes geometry and highly shows hydraulic roughness of pipes is a

systems. Pipe roughness, diameter and the length of the pipe need to be specified in order to solve the law of conservation energy equations in modelling of

ban drainage systems. And in modelling of urban storm water systems, pipe roughness is among the most significant sources if uncertainty this is because aging changes the inner surface of the pipe and the

e in the characteristics of the pipe may be due to the quality of water that is being transported and the corrosion of the pipe depends on the pipe material used,

ases due to the aging of the infrastructures and this effect is significantly magnified because the service life of urban drainage

effect of hydrological

Sub catchment width is the characteristic width of the overland flow path for runoff generated. The the area of the sub catchment by the

length. The flow path from the inlet to the farthest drainage point of the


sub catchment is the maximum overland flow length. And average of the maximum flow paths of different possible options will be taken in to account. The path followed by the runoff should represent the slow flow over the previous portion of the sub catchment and the rapid flow over the impervious and adjustment should be made to the sub catchment width to produce good fits to measure the runoff hydrographs.

The time of concentration in urban drainage design is affected by the parameter catchment width in the hydrological portion of the storm water management model (SWMM).The parameter width defines the rectangular area of the sub catchment in urban drainage models. The width parameter is one of the main validation parameters used to regulate hydrograph shape, and to some extent hydrograph volume. For water shades with larger width the overland flow length will be shorter and this reduces the time of concentration and reducing the sub catchment width intern results in the increase of the length of the runoff flow and this increase the time it takes the water drops to the inlet of the drainage infrastructure. When the overland flow length is increased the runoff flow will be exposed to evaporation and infiltration in pervious surfaces in to the ground and this phenomenon reduces the runoff volume. Therefore Stormwater collection systems and some undefined watercourse are examples of how the overland flow length can be shortened by manipulating the sub catchment width(Huber et al., 1988).

2.5.5. Manning's roughness coefficient for the sub catchment

Manning's overland flow over impervious portion of the sub catchment (N1-imperv) and Manning's overland flow over pervious portion of the sub catchment (N2-perv) are the properties of the sub catchment which has a considerable contribution for the runoff generated from the urban settings and the value of these parameters mainly depend on the type of land cover of the area. And the value of these parameters is also variable due to considerable variability in the ground cover of the sub catchment. Therefore considering the uncertainty limits of these parameters in the design of the urban drainage systems ensures the reliability of the system and helps the safe discharge of the stormwater flows in an area.

2.6. Urban Drainage Model

Urban drainage models are used to asses' urban drainage performances and these models are mathematical models that describe a physical process and they contain different equations of algebraic or differential nature that can be solved analytically or numerically. These models are used to test the operational and structural alternatives of the urban drainage infrastructures and help to understand and predict the behaviour of stormwater drainage components (pipes, culverts manholes etc.)(Price et al., 2011).Design and analysis of urban drainage infrastructures using computer programs started in 1970s.the development of SWMM started ever since it was first discovered in the USA in 1971 (Rossman, 2010). Even if computer-based hydrograph method is developed in the UK in 1970s in the transport and road research laboratory; there has not been a standardized software package until early 1980s and it became improved with time afterwards(Butler & Davies, 2004).

Usually urban drainage systems and environment related data was collected by engineers to prepare reports on the performance of the system (Price et al., 2011). The increased value and quantity of the data collected and the requirement for storage resulted in the use of computers. Due to the complex nature of urban drainage and waste water systems control and management the use of modelling tools is required. The extent of models is also different it ranges from, physically-based methods, data driven models, hybrid models, to simple statistical regression models or empirical equations. For instance rational method is a simple model to convert rainfall into runoff that can be used to look at the likely effects of different rainfall intensities.

Urban drainage models are used for different purposes at different phases of urban stormwater drainage projects. It is used to determine the physical detail of the proposed system and to check the behaviour of the system when exposed to specific condition at the design stage. It is also used to analyse the existing


physical details to see the response of the model to specific conditions like depth discharge extent of surcharging and occurrence of flooding to check the importance of systems improvement and how to improve it. SWMM5 model is used for this study to analyse the performance of the existing urban stormwater drainage network and to identify the extent of flooding within the network and after that SWMM5 is coupled with genetic algorithm for further optimization of the drainage network.

2.6.1. SWMM5 SWMM5 is a one dimensional hydraulic model and it is used to simulate the water quality, hydraulic and hydrological analysis of urban drainage system by using single event or long term events (Kaushik, 2006). The hydrological section of SWMM5 is simulated by using the data from sub catchments and rain gages. The runoff generated for each sub catchments of the drainage basin is calculated after simulating the values of the losses due to evaporation and infiltration in to the ground. The discharges and water depths in the links and nodes of the drainage network is in the hydraulic section of SWMM5 and it can be simulated using input data of the nodes and links of the given drainage network.

SWMM5 model is distinct time simulation model which employs the law of conservation of mass, energy and momentum and it works on steady state, kinematic wave flow or dynamic wave flow routings. Precise simulation of back water effects, flow reversal and pressure flows are also simulated using SWMM5 Numerical equations are also solved using SWMM and it has ability to automatically adjust the time step as needed to maintain the numerical stability (Rossman, 2010). SWMM5 tracks the quantity and quality of runoff generated within each sub catchment, and the flow rate, flow depth, and quality of water in each pipe and channel during a simulation period and comprised of multiple time steps (Rossman, 2010).

Hydrologic processes that produce runoff from urban areas that are accounted by SWMM5 include:

• Time-varying rainfall • Evaporation of standing surface water • Snow accumulation and melting • Rainfall interception from depression storage • Infiltration of rainfall into unsaturated soil layer. Using either the Horton or Green-Ampt

equations for up to 100 sub-catchments. Horton method is adopted for this study. • Percolation of infiltrated water in to ground water layer. • Interflow between ground water and drainage system

• Nonlinear reservoir routing of overland flow.

Flexible set of hydraulic modelling potential is also included in SWMM5 model which is used to rout runoff and external inflow through the drainage system network of pipes, channels, storage units and diversion structures (Rossman, 2010).and these potential includes ability to:

• Handle networks of unlimited size. • Use a wide variety of standard closed and open conduit shapes as well as natural channels. • Model special elements such as storage or treatment units, flow divider, weirs, pumps, orifices • Account external flow and water quality inputs from surface runoff, ground water inflow, rainfall

dependent infiltration, dry weather sanitary flow

• It various flow regimes, such as backwater, surcharging, reverse flow and surface pond.

Figure 2-5 shows the simplified flow chart of SWMM5.


Figure 2-5 SWMM5 model flow chart (Ketema, 2007)

Steady state flow, kinematic wave flow and complete dynamic wave flow (St. Venant) routing equations can be solved using SWMM5 for precise simulation of back water, looped connections, surcharging, and pressure flow.

St. Venant equations: is a dynamic flow routing equation which is derived from continuity and momentum theory of flow and it deals with unsteady flow (depth of flow and velocity will vary with time or it is a gradually varied flow).

Continuity equation

7#7 + 7$

78 = 0 (2.10)

Momentum equation

7#78 + 7

7 :Q%$ ;+ 2$ 7<

7 − 2$((= − (>) = 0 (2.11)

local

Acceleration

term

Convective

Acceleration

term

Pressure

Force

term

Gravity

Force

term

Friction

Force

term

Inertia Term St. Venant Equation:


7?78 + ? 7?

7 + 2 7<7 − 2((= − (>) = 0 (2.12)

Where: V=flow velocity, t = flow time, Q= flow rate (discharge), A=flow area, x=distance along the canal, y = flow depth, g = acceleration due to gravity, So= bed slope of the cannel and Sf=frictional slope opposite to flow direction

SWMM5 is also available in free open source form, therefore it is easily accessible for use for free therefore, and the latest version SWMM5 is used for hydraulic simulation of storm water drainage network optimization design for this study.

2.7. Sensitivity Analysis

Sensitivity analysis identifies the critical input parameters of a computational model and it also computes the impact of input uncertainty on the outputs of a model. The partial deviation of the outputs functions with respect to input variations is examined when the model input parameters are known with little uncertainty. By altering input factors around a nominal value and performing multiple simulations, numerical computation of sensitivity measure can be performed. Considerable change of the model output may result due to small variation in a particular variable. Sensitivity analysis is carried out for each uncertain variable to measure the sensitivity of the model to the changes in the values of the uncertain variables. This was performed by varying one parameter with significant high frequency while the other parameters are varied with low frequency and observing how the output varies. This is because, the simulation result of the output obtained in a complex catchment modelling system of urban drainage models, the result is more affected by inter relationship between parameters values in a set of parameters than variation of individual parameters (Marino et al., 2008).

Once SWMM model is built sensitivity analysis of the uncertain parameters stated above was done in this study to check the most sensitive parameter. Then these parameters are ranked as per their sensitivity index results and the most sensitive parameters are used to represent uncertainty for further optimization process.

There are different types of sensitivity analysis techniques based on the type of mathematical models and the relationship between the input parameters and output value sand some of them are stated below.

2.7.1. Linear trends between input parameters and o utput. If the relationship between output and input follows a linear trend sensitivity analysis (SA) techniques that work better are Pearson's correlation coefficient (CC), Partial correlation coefficient (PCC), standardized regression coefficient (SRC) and others. And details of these techniques are stated below.

Pearson's correlation coefficient (CC) The quantitative estimation of linear correlation will be determined based on the result of simple correlation coefficient values of input and output values. Model parameters are reasonably ranked according to their contribution to the uncertainty by simple correlation coefficients which are derived from Monte Carlo simulation(Gardner et al., 1981). Pearson's product moment correlation coefficient is denoted by r and is defined as for the correlation between Xi and Y (Conover, 1980).

Kinematic wave

Diffusion wave

Dynamic wave


@ = ∑ (�B − �C)(DB − DC)EBF)[∑ (�B − �C% ∑ (DB − DC)%]EBF)EIF) )/% (2.13)

Where, r is Pearson's correlation coefficient, Xi is the ith input variable and Yi is the output for ith input variable

The degree of relationship between the input and output values are shown by the value of r which means larger values of r shows stronger relationship and negative value of r shows the output is inversely proportional to the input((IAEA), 1989).

Partial Correlation Coefficients (PCC) Strong correlation between input parameters may influence input output relationships and to account for correlation among input parameters partial correlation coefficient (PCC) is calculated (Gardner et al., 1981). For random variables X1 and X2 as input and the output variable Y, a partial correlation coefficient is.

@J)K/J% = @J)K − @J)J% ∗ @J%KL(1 − @%J)J%)(1 − @%J%K)

(2.14)

Where, the value @J)K/J% represents the partial correlation coefficient for X1 and Y while accounting for the affects of X2.

If no correlation between the parameters X1 and X2 exists and if the parameters are assumed to be independent,@%J)J% = 0. Therefore, @J)K/J% reduces to

@J)K/J% = @J)KL(1 − @%J%K)

(2.15)

where, again, X1 and X2 represent any two input variables and Y represents the output variable.

Standardized Regression Coefficients (SRC) When the input variables are independent in the model, SRC is used to quantify the influence of each input variable on the output. The relative magnitude of the regression coefficient is used for the sensitivity ranking. The influence of the parameter on the model as a whole is shown by the value of the coefficient. Regression coefficients provide a means of applying sensitivity rankings to input parameters and have been used for such in several investigations (Iman R.L. , 1980 ). Matrix techniques have been utilized for a model with many sensitive parameters to calculate the regression coefficients due to its complex regression equation (Krieger et al., 1977). The general form of a simple regression equation is,

Ŷ = NO + P bRZRT

(2.16)

where each Zk is a predictor variable and a function of (X1 ..... Xn) and each bk is a regression coefficient(Helton, 1986).

2.7.2. Non Linear but monotonic trends between inpu t and output parameters Sensitivity analysis techniques that are based on rank transformation are better choices when the relationship between output and input relationship is non linear but monotonic. These sensitivity analysis


(SA) techniques are Spearman correlation coefficient (SRCC), Standardized Rank Regression coefficient (SRRC), and others. To reduce the effect of nonlinear data rank transformation is used because one of the problems encountered in calculating test statistics, e.g. correlation coefficients, from raw data is that the data are not necessarily linear. And if the dependent variable is a monotonic function of the independent variables transformation of raw data in to rank works quite well (Iman, 1979). Examples showing raw data and ranked data of input and output matrix are stated below.

X =

UVVVWX1 N1 Y1X2 N2 Y2X3 N3 Y3X4 N4 Y4X5 N5 Y5]̂

^̂_ Y =

UVVVVW<1 = >(X1, N1, Y1) = 5.2<2 = >(X2, N2, Y2) = 7.9<3 = >(X3, N3, Y3) = 3.1<4 = >(X4, N4, Y4) = 6.4<5 = >(X5, N5, Y5) = 0.8]̂

^̂_̂

Ranking Ranking

XR =

UVVVW2 3 45 1 34 5 21 2 53 4 1]̂

^̂_ YR =

UVVVW35241]̂

^̂_

Spearman Rank Correlation Coefficient (SRCC) The degree of monotonicity between input and output values will be indicated by the rank correlation coefficient and will be used if the input and output relations are monotonic. Then the rank transformation of input and output values by replacing the values with their ranks will result in linear relationship. The rank correlation coefficient, or Spearman's p, can be calculated using the equation 2.13 for Pearson's r with the exception of operating on the rank transformed data (Iman, 1979).

Standardized Rank Regression Coefficients (SRRC) Standardized rank regression coefficient (SRRC) is calculated by using a similar method for Standardized regression coefficient (SRC) however Ranked data of input and output values will be used for the regression rather than using raw data. The greater the SRC or SRRC values, the more significant the influence of the variable is. The negative SRC or SRRC for inputs indicates that an increase in these variables will lead to a decrease in the corresponding output. 2.7.3. Non Linear and non monotonic trends between input and output parameters Methods based on breakdown of model output variance are the best choice when the relationship between input and output follows nonlinear and non-monotonic trend. Examples of these methods are the Fourier Amplitude Sensitivity Test FAST and its extended version eFAST (Marino et al., 2008).

In this study eFAST will be implemented for SA method because the relationship between urban drainage models (SWMM) input parameters and output values follows non linear and non monotonic trend and it is a model which requires a technique with high computational efficiency. eFAST has proven to be one of the most reliable methods to solve the problem among the variance-based techniques.

Raw Data

Ranked Data

Input Matrix Output matrix


2.7.4. Extended Fourier Amplitude Sensitivity test – eFAST (Saltelli et al., 1999) developed Extended Fourier Amplitude Sensitivity Test (eFAST) based on the existing Fourier Amplitude Sensitivity Test (FAST) developed by Cukier et al. (1973). The Fourier Amplitude Sensitivity Test (FAST) method is used for both sensitivity and uncertainty analysis and it estimates the expected value and variance of the output, and the contribution of individual inputs to the variance of the output. The FAST method is used when the interaction between input parameters is insignificant and in general, the classical FAST method is not efficient to use for high - order interaction terms (A. Saltelli & Bolado, 1998). However, the extended FAST method can address higher order interactions between the inputs.

eFAST is a variance decomposition method equivalent to analysis of variance (ANOVA). To introduce the concepts of ANOVA, let us denote the scalar variable of interest by Y and let's assume y is calculated by the model of the form.

Y = f(x) = f(X1, X2. . . Xp) (2.17)

Where, Y is a controlled by a vector of variable x through the function f. And ordinary differential equation or combinations of algebraic equations are used to generate the output. The objective of sensitivity analysis is to break down the variance of Y to the contribution arising from the parameters x and to measure the extent and significance of each of the components in x (Harris & Yu, 2012).

Input parameters are varied causing variation in the output of a model in eFAST and statistical notion of variance is used to compute this variation.

s% = j (yl − ȳ)%nlF)N − 1

(2.18)

Where N=sample size or total number of model simulations, yi=ith model output, ȳ=sample mean. The variance of the output is partitioned by the algorithm in the model to describe the fraction of the variances due to the variation in each input parameter. Different parameters are varied at different frequencies and the identity of parameters in the frequency of variation is encoded by eFAST to partition the variance. The strength of each parameter in the model output is then measured by Fourier analysis. And thus the measure of the models sensitivity to the parameter is determined by the strength of the parameters frequency propagation from input to the output through the computational model (Marino et al., 2008).

A sinusoidal function of certain frequency is defined by eFAST for each input parameter or search curve, x= f (j), j= 1, 2… NS and a value are assigned to x based on the sample number1 through the total number of samples per search curve, Ns. The distribution of the parameter values required (e.g. uniform, normal, etc.) Determines the choice of the sinusoidal curve and several criteria must be met by the frequency allocated to the parameters so that they can be distinguished during the Fourier analysis. The sinusoidal functions repeat the same samples because of their symmetric properties and hence re-sampling is implemented to avoid this incompetency (Saltelli et al., 1999). Random phase shift in to each sinusoidal function is introduced in to the eFAST algorithm by repeating it NR (the re-sampling size) times with different search curves. Therefore the total number of model simulations, N is given by.

N= NS x k x NR (2.19)

Where k is the number of parameters analyzed and the minimum recommended value for NS is 65.


eFAST is better than FAST because it calculates both the firsteach input parameter considered in a model. The variance of particular parameters frequency divided by the total variances (i.e. ratio of the output variance due to the ithe first order sensitivity, Si of a paravariation of a given parameter is represented by the first order sensitivity index

Where Vi is the partial output variance due to the i

eFAST calculates the total order sensitivity index, Ssummed sensitivity index of the whole corresponding seidentification frequencies. The remaining variance after involvement of the complementary set, Sremoved is STi (Marino et al., 2008

Or

Where V~i is the partial output variance due to all variables except the i

Figure 2-6 General schematic diagram of


eFAST is better than FAST because it calculates both the first-order sensitivity and totalut parameter considered in a model. The variance of particular parameters frequency divided by the

total variances (i.e. ratio of the output variance due to the ith variable to the variance due to all variables) is of a parameter. The fraction of the model output variance explained by input

variation of a given parameter is represented by the first order sensitivity index (Marino et al., 2008

Sl = VlV(Y) = V(E(Y Xl⁄ ))V(Y)

e partial output variance due to the ith input variable and V(Y) is the total output variance

total order sensitivity index, STi of a particular parameter first by calculating the summed sensitivity index of the whole corresponding set of parameters except i by using their identification frequencies. The remaining variance after involvement of the complementary set, S

Marino et al., 2008).

Stl = 1 − Sci Stl = 1 − V~i

V

Where V~i is the partial output variance due to all variables except the ith input variable.

ic diagram of a sensitivity analysis model (Saltelli et al., 1999).

21

order sensitivity and total-order sensitivity of ut parameter considered in a model. The variance of particular parameters frequency divided by the

variable to the variance due to all variables) is meter. The fraction of the model output variance explained by input

Marino et al., 2008).

(2.20)

input variable and V(Y) is the total output variance

a particular parameter first by calculating the t of parameters except i by using their

identification frequencies. The remaining variance after involvement of the complementary set, Sci is

(2.21)

(2.22)

input variable.


Figure 2-7 eFAST sensitivity analysis performed on the Lotka-Volterra model (Marino et al., 2008).

2.8. Sampling techniques

Sampling is the process by which values are randomly drawn from input probability distributions. Avery large number of model simulations will be required to reliably represent the distribution of uncertain parameters. But practically there is a time and computational power constraint to perform a large number of model simulations. Therefore sampling technique is used to cover the distribution of uncertain parameters. Latin Hypercube Sampling and Monte Carlo sampling are the most commonly used sampling techniques and details of these techniques will be sated below.

2.8.1. Monte Carlo Sampling Monte Carlo sampling refers to the traditional technique for using random numbers to sample from a probability distribution. Monte Carlo sampling techniques are entirely random that is, any given sample may fall anywhere within the range of the input distribution. Samples, of course, are more likely to be drawn in areas of the distribution which have higher probabilities of occurrence.

2.8.2. Latin hypercube sampling Latin Hypercube Sampling is a type of stratified Monte Carlo Sampling developed by McKay et al. (1979). Considering a variable Y that is function of other variables X1, X2... Xk it is used to investigate how the value of the X variables affects variable Y. The probability distribution of values of variables X is divided into 'n' non overlapping intervals based on equal probability. Then one sample from each non overlapping interval is selected randomly with regard to the probability density within the interval. 'n' Latin Hypercube sampled values from each input variable are paired randomly if the input variables are more than one.


The variables are considered to follow a normal distribution that allows for the definition of equal probability intervals. From the normal distribution of each variable, a cumulative distribution function is created. The values are then obtained from sampling of the normal distribution, which is equivalent to sampling the vertical axis of the cumulative distribution function and then acquiring the final sampled values from the horizontal axis. And this Sampling technique will be used in this study to sample from the range of uncertain parameters.

2.9. Optimization

Optimization is finding the best candidate solution or optimum solution from a set of alternatives using special mathematical techniques. The optimization may be done to maximize or minimize a certain function called objective function and it is done based on the target of optimization methodology required. And before formulation the optimization process, the decision variable and a constraint of the optimization process need to be defined so that the optimum solution is attained. A decision-making model that uses those programmed mathematical techniques is called an optimization model(Walski et al., 2003).

After defining the optimization problem and setting the objective functions and decision variables, the next step is continuing with the optimization algorithm which is used to minimize or maximize the objective function based on the objective of the optimization. And this is done by changing the decision variables and considering the constraints, until satisfactory solution will be attained.

Due to complex nature of urban drainage design optimization has gained importance and computational models used for these purpose has developed fast and their computational power has also increased. The objective of urban drainage infrastructures in optimization is minimizing the intervention cost and minimizing the flooding while maintaining a certain level of performance (Coelho, 2010).Many circumstances have to be met to achieve the optimum design due to the complex characteristics of urban drainage design and this kinds of problems are called multi-objective optimization problem.

A powerful tool which will be used for urban storm water management planning, design, upgrading operation and maintenance is obtained when optimization techniques are associated with storm water management model. Table 2-1 shows some basic terminologies encountered when we develop optimization tools.

Table 2-1 Description of optimization terminologies (Walski et al., 2003)

Terminologies Description Objective Function

A numerical function which is used to measure the fitness of chromosomes and it indicates the degree to which the objectives are achieved. For example, minimizing the costs and minimizing flooding volume.

Decision Variables

The parameters that can be changed to improve the performance of a system and values of which are to be determined. e.g. the diameter of pipe

Constraints The limits or restrictions that have to be considered in judging networks and solutions while the network operates. Constraints may be applied explicitly to decision variables (for example, pipe sizes are discrete) or implicitly to other system parameters (for example, net head loss around a loop must be zero)

Several techniques have been developed to solve optimization problems and "genetic algorithm" is one of most widely used one. The objective functions in optimization problems of urban drainage systems is non linear and the objective functions are not known analytically and they can have a number of minimum or maximum values, (local and global optimum).


2.9.1. Single-Objective Optimization When a mathematical formulation is used to find the best solution which satisfies maximum or minimum values of one objective function, it is called single objective optimization. Multi objective optimization problems can also be reduced to single objective by combining different objectives in to one or by using one objective function and letting the others as constraints and this lets all of the objective functions to be quantified as a single function. Consequently, simple combination of the objective functions in to one objective having the same units is involved to prioritize (D. Savic, 2002).

A number of objectives have been addressed by different optimization problems and Single objective optimization is a one of these tools that give information about the nature of the problem. But a set of alternatives which provide options for decision makers cannot be obtained from this approach.

Consider the following example with two objectives, minimize cost f(x) and maximize benefits f(y), in a pipe sizing problem. When f(y) is multiplied by -1, max f(y) is equivalent to min -f(y) = min f’(y). Therefore, the two objectives can be combined into single objective function being f = min f(x) + f’(y).

2.9.2. Multi-objective Optimization Multi objective optimization is a numerical tool developed to get compromised optimal solution of more than one conflicting objectives and is used to assess their results. The obtained tradeoffs give decision making easier and each solution in multi objective optimization is not dominated by other solution to give a single solution like single objective optimization. The objective functions in a multi objective optimization interact to give a compromised solution called Pareto optimal front based on the aim of the objective functions (D. Savic, 2002).

To get a compromise solution between two objective functions in a multi objective optimization, one objective must be worsened to make to improve the other objective and vice versa. It is better to use the outcome of multi objective optimization for decision making because the result gives wider set of alternatives. The outcome of multi objective optimization gives a Pareto front showing compromised result of the objective functions.

Optimization associated with urban drainage models gives better performance because it is a time consuming job trying a manual trial and error in urban drainage design/rehabilitation project and also it is difficult to get optimum solution in a huge drainage network with different pipe diameters and slopes in an area. a traditional design process combines a small number of network parameters (Vélez et al., 2007).

2.9.3. Genetic Algorithm John Holland (1975) introduced genetic algorithm (GA) for the first time and studied further with his staff for its development in the University of Michigan. Nevertheless in the past ten years considerable effort has been applied to genetic algorithms for further research and improvement. Genetic algorithm uses the process of selection and reproduction to solve problems for the evolution of individuals by using a numerical model. In a genetic algorithm in a group of population the survival of individuals is measured based on its fitness in the environment and definite rules of selection cross over and mutation are used to maintain a group of individuals that survive in the process. A selection criterion in a genetic algorithm is base on high fitness value among individuals and hence the chromosomes in a population is either eliminated or retained based on the fitness at the time of evaluation. Genetic algorithm combines two or more chromosomes by changing different variables to generate new offspring through a process of cross over and mutation. Arrays of binary data or genes represent the genetic information of individuals and therefore simple manoeuvring results in the operation of mutation and cross over (Murphy et al., 1993).

Genetic algorithm works with a group of population and dealing with a single solution is not the case in genetic algorithm. The value selected for decision variable represents an individual chromosome in a population and in each generation selection of the chromosomes is performed to produce the best fits for


the next generation. Based on the objective function and their fitness, the chromosomes may continue for many generations or being substituted by others in the process to choose the best fit (Liong et al., 1995). And the cycle continues initializing a random population after representing the genes and defining the fitness of the chromosomes. Mutation, crossover, and selection operators will be repeated to improve the required solution until termination criterion is reached. Structure of GA is shown on Figure 2-8.

Figure 2-8 Structure of GA Model

Application of Genetic Algorithm for optimization An improved genetic algorithm is built and applied for pipe network optimization by Murphy et al. (1993) and decision variables are represented with grey coding rather than binary coding. Traditional genetic algorithm is compared with improved ones in their result and the result shows improved genetic algorithm achieved better performance than the traditional optimization methods.

An improved genetic algorithm which is also called real coded algorithm is used by Vairavamoorthy and Ali (2000) for water distribution system. The project was aimed at minimizing the network capital cost by insuring the adequate pressure at every node within the network. The efficiency and the robustness of the method were checked to be best by testing the performance of various networks.

Abebe and Solomatine (1998) used a global optimization tool with various random search algorithms and network simulation model that can handle both static and dynamic loading conditions for water distribution pipe networks. From the study, promising solution was obtained by the genetic algorithm for water distribution network.


2.9.4. Non Dominated Sorting Genetic Algorithm (NSG AII) NSGAII is one of the most popular elitist multi-objective optimization algorithms that ensure the performance of a solution from generation to generation in which the fittest of all the solutions will survive to go for the next generation and it doesn't have hydraulic parameters to operate. The benefit of NSGA-II is its high speed in solving multi objective optimization algorithms in a selective manner (Deb, 2002).

The search procedure for simple genetic algorithm is different from NSGA-II because NSGA-II compares more than one objective and the non dominated method of ranking is used in NSGA-II to evaluate the fitness of the chromosomes within a generation. The fitness of all the objectives of the optimization is used to check the dominance of one solution. And crowding distance method will be used to further evaluate two chromosomes of the same rank of dominance and this leads to uniformly spread Pareto front with optimum solution (Deb, 2002).

Hydro dynamic simulation model used in this research is easily integrated with NSGA-II and this makes NSGA-II more important. And one of the objective functions in the integrated model is to minimize flooding in the network for design or rehabilitation of the drainage network so the biggest interest is to work only with the solutions that provide little or no flooding.

Review of Previous Stormwater Drainage Optimization Works To get less expensive design alternatives for urban stormwater drainage network dynamic programming was used by Li and Matthew (1990). Dynamic programming dominates the stormwater network optimization problems and decomposes a problem into a serious of simple sub problems that are solved with chronological orders by transferring information from one level to the next level of computation (MH Afshar et al., 2006).

Genetic algorithm and global optimizer was applied by for remedial work of least cost optimization of wastewater systems byZ.; Vojinovic et al. (2005). MOUSE is used for hydraulic simulation and optimization tool used was GLOBE which can use various random searching algorithms. And in comparison to the traditional approach, they obtained optimal solution and achieved economic benefits with this approach.

Liong et al. (1995) Linked stormwater management model (SWMM) with genetic algorithm for finding optimum value of catchment calibration of storm water parameters. And from the study peak flow prediction is found with high degree of accuracy due to the application of genetic algorithm.

An attempt to find a spreadsheet template to find the least cost design was made by Miles and Heaney (1988). And from the study the result it is shown that the spreadsheet method was able to considerably improve on the dynamic programming design and optimization problems previously solved by dynamic programming was solved by this method.

Fullerton et al. (2002) Concluded that genetic algorithm gives best way to find the global optimum by developing simplified model for stormwater flow optimization using commercial hydraulic simulator model called hydro works and genetic algorithm as an optimizer.


3.1. Dhaka

3.1.1. General Dhaka is the capital city of Bangladesh, and it is located in the geographic centre of the country. The city is surrounded by rives, Buriganga from south, Balu from east, Turag from west. Dhaka city covers a total area of 360 square kilometres. The elevation of Greater Dhaka lies between 2 to 13 m above mean sea level (msl). The elevation of the urbanized area lies at 6 to8 m above msl (Ahmed, 2008).

Dhaka is one of the densely populated cities in the world. The city drainage system is not able to manage the changing situation due to rapid urban growth, shortage of resources and lack of integrations in managing it. Due to high intensity rainfall events, flooding which resulted in high inundation occurred recently and the drainage system of the Dhaka city was unable to handle the flow. The frequent occurrence of these flooding resulted in environmental pollution and it resulted in a public health hazard for the city. Generally, environmental quality of the city is degrading due to uncontrolled rapid urbanization process and high rate of population growth. Flood in Dhaka City caused by local rainfall occurs in the built-up areas of the city several times a year on a various scale (Mark, 2002). Damage in the inland area of the area of the city occurred frequently in the city because of poor drainage capacities of existing infrastructures(Sahlu, 2012 ).

3.1.2. Drainage systems of Dhaka city The rapid growth of Dhaka city which is caused by highly growing population and unexpected migration to the city creates extra burden on the drainage system capacity of the city.Urban drainage system in Dhaka is a separate system in which waste water and stormwater of the city is handled in a different pipes and the storm water of the city is discharged to the surrounding rivers. Drainage Circle of Dhaka WASA is responsible for the maintenance and operation of the storm water system. Based on the drainage channel, the greater Dhaka city drainage area is divided in to 12 zones (JICA., 1991). The length of stormwater drainage pipes is 210 kilometres and the diameter ranges from 450 to 3000mm. The city has also box culverts and the length of these box culverts is about 7 kilometres and their size is 2.5 m X 3.4 m to 6 m X 4.1 m. Generally the city has 1100kilometers of underground and surface drains and 22 open channels of length 60 kilometres 10 to 30m wide (JICA., 1991).

During the monsoon period (May to October), the existing drainage system is unable to drain out the stormwater of Dhaka City. Average rainfall of 304 mm, 267 mm, 262 mm and 231 mm occurred in the

CHAPTER 3

Case study Area Description

Case study Area Description 28

years 2000, 2001, 2002 and 2003 which resulted in the overflows of the urban drainage system of the city (BMD, 2003). Currently urban stormwater drainage networks of Dhaka managed by WASA covers an area of 140 sq. km and the components of this drainage network are stated as follows (Ahmed, 2008).

• 185km long underground pipes of diameter 450 to 3000mm. • 22 open canals with total length of 65 km and width of 10 to 30 m. • 6.5 km of Box culverts with sizes ranging from 2.5 m x 3.4 m to 6 m x 4.1 m • Storm water-pumping stations, of capacity of 9.6m3/s and 10m3/s at Narinda and Kallyanpur

respectively

3.1.3. Location of the study area Segunbagicha catchment is an area shown on the figure below which has an area of 8.3sq.km. And the catchment covers a mixture of residential, commercial uses, government and non government offices, educational institutes and some small scale industries. Particularly, Motijheel, the commercial hub and centre of business activities of the city is located in this area. It has been selected for this study and the reasons for selecting Segunbagicha catchment for this particular study are (Ahmed, 2008).

• Segunbagicha is one of those areas which are frequently faced with flooding problem. • Some data was available for this area.

• This area comprises of some of the most important land uses of Dhaka city.


Figure 3-1 Location map of Dhaka city and study area ( DMDP 1995)

3.1.4. Drainage system within the study area The collected stormwater from each sub-catchment within the study area is drained by sewer pipes to the khal and finally it is drained to river system by pumps at the basin in front of the sluice gate. The storm water drainage system of Segunbagicha area is shown in the Figure 3-2.

Sub catchments 74 sub catchments which are supplied by a rainfall from one rain gage were used in the study area. Some of the main parameters used to represent or set the sub catchments were rain gage, outlet, total area, percentage of impervious, width, slope, curve length, infiltration method ( SCS runoff curve number),land

Case study Area Description 30

uses. The properties of the rain gage were described by the source and format of the precipitation data. In this study a 100 years return period rainfall data of 1 hour duration which was recorded at 5 minute time interval will be used.

Conduits and Nodes The pipe network contains a total of 88 conduits or links where by 75 circular pipes and 13 box culverts. The main parameters used to represent these conduits are, conduit length, cross sectional geometry, offset heights of the conduit above the inlet and outlet nodes and manning’s roughness. In addition, a total of 88 nodes, set based on their location (x, y coordinates), are used to join the links together. The nodes are represented through Invert elevation, initial depth and height to the ground surface.

Figure 3-2 Drainage system within the study area (Ahmed, 2008)


To answer the research question as well as to achieve the objective of the study the methodologies stated below were used.

4.1. General

An optimization frame work which considers the effect of uncertainties of parameters was developed for the design and rehabilitation of urban drainage systems. The range of possible parameter values representing future climate change, urbanization, aging of pipes, sub catchment width, Manning's roughness coefficient for overland flow over the impervious and pervious portion of the sub catchment(N1-imp and N2-perv), are compiled from the literature for the case study area.

SWMM5 model was built for the urban drainage systems of the area and sensitivity analysis of the parameters is done using eFAST (extended Fourier Amplitude Sensitivity Test). The eFAST method is used to estimate the expected value and variance of the output, and the contribution of individual inputs to the variance of the output. This method was used because significant interactions among inputs has effect on the model output on urban drainage model and eFAST handles such kind of interactions between input and output. eFAST was used to calculate the first order sensitivity index (Si) and total order sensitivity index (STi) which shows the sensitivity ranking of the parameters. And the first order sensitivity index Si is calculated by assigning a unique frequency to each input parameter, and thus the unique contribution of each parameter in determining the model output is calculated. And to calculate total order sensitivity indexes STi, the eFAST method varies i at a unique, high frequency, but all other parameters at low, not-necessarily unique frequencies. STi measures the combined first and all higher order effects of the ith input variable including all interaction effects involving the ith variable. Then the most sensitive parameters were selected from the above procedures for the proceeding optimization process.

Non Dominated Sorting Genetic Algorithm (NSGA-II) was used for the optimization process. The optimizer (NSGAII) and storm water management model (SWMM5) were linked with intermediate links. A code written with C++ is used to perform sampling, run the hydraulic model, calculate the objective function and carry out the optimization process .The outcome of the above procedures was the optimum solution of the urban drainage system pipe network. And the methodology was applied in the case study area Dhaka, Bangladesh where urban flooding problems are frequent occurrences. Figure 4 -1 below shows the general methodology flowchart that was used in this study. And, the details of the methodology are presented in the succeeding sections.

CHAPTER 4

Research Methodology

Research Methodology 32

Figure 4 -1 General Methodology flow chart

Q3. What is the effect of considering uncertain parameters on optimization of urban stormwater drainage systems?

Result and Discussion

Q1. To which parameters urban drainage model is most sensitive?

Q4. What is the advantage of considering uncertainties in the optimization of urban stormwater drainage systems?

Q2. How is uncertainty included in the optimization model for design of urban stormwater drainage system?

Optimization Methodology • With uncertainties • Without uncertainties

Build SWMM5 model

Quantification of uncertain parameters

Sensitivity Analysis

Literature review

Select most sensitive Parameters

Couple SWMM5 Model with Optimization Tool NSGA-II

Summary, Conclusion and Recommendation


4.2. Quantification of uncertain parameters

Uncertain parameters on stormwater drainage model used in this study are identified and their values are quantified from literature. To study the uncertainty in the model output that is generated due to uncertainty in the input parameters, sensitivity analysis is performed to quantify the variation in the model output and to allocate the quantified variation to different input parameters (Saltelli et al., 1999).

Input factors for urban drainage model consist of parameters and initial conditions for independent and dependent model variables. Input parameters in urban drainage model might not be always known with adequate degree of certainty due to different drivers coming from climate changes, population growth, aging of infrastructures, error in measurements, or simply a lack of current techniques to measure them (Marino et al., 2008). Table 4-1shows the types of uncertain parameter, uncertainty drivers and their quantified range of values.

Table 4-1 uncertain parameters for this study and their quantified min and max values from literature (CK, 2001), (Sahlu, 2012 ), (Ahmed, 2008)

Sub catchment properties

Uncertain Parameter Uncertainty Drivers Quantified min and max value of the parameters from literature

Rainfall data Climate change ±20% of the existing value Percent imperviousness (P.imp) Urbanization ±20% of the existing value Sub catchment width Error in measurement -15% to 41% of the existing value Manning's roughness over impervious portion of the sub catchment (N1-imperv)

Variability of land cover within the sub catchment

-40% to 65% of the existing value

Manning's roughness over pervious portion of the sub catchment (N2-perv)

Variability of land cover within the sub catchment

-30% to 20% of the existing value

Conduit properties

Uncertain Parameter Quantified min and max value of the parameters from literature

Manning n for conduit Aging of pipe -15% to 50% of the existing value

4.3. Hydraulic Model set up (SWMM5)

The amount and quality of runoff produced in storm water management model (SWMM5) is based on two parts and these are runoff and routing. In the runoff element the collection of different sub catchments that receive precipitation are found and the physical properties of these sub catchments is different and these properties are the basis of the amount and quality of runoff produced in the sub catchments and these properties are pervious and impervious fraction, infiltration calculation method, etc..And the transport of the runoff generated by means of conduits, open channels, storage units, pumps and flow regulators is handled by the routing element of storm water management model (Rossman, 2010).


4.3.1. Description of sewer network in the study ar ea

Sub catchments The case study area used in this study contains 74 sub catchments which are provided with a rainfall from one rain gauge. And the uncertain parameters selected from the properties of the sub catchment are rain fall data, percentage imperviousness, the width of the sub catchment, manning's roughness coefficient for overland flow for pervious and impervious portion of the sub catchment. The source of the precipitation data is described in the property of the rain gauge. 100 years return period rainfall data of 3 hours duration which was recorded at 5 minutes interval was used in this study.

Conduits and Nodes The pipe network in this study contains a total of 88 conduits where 75 are circular pipes and 13 are box culverts and only 75 circular pipes are used for the optimization process. Conduit length, cross sectional shape, offset heights of the conduit above the inlet and outlet nodes are parameters used to represent the conduits. And manning’s roughness coefficient is used as uncertain parameter for further sensitivity analysis procedures. The links are joined together by 88 nodes and initial depth, invert elevation and their height to the ground surface represents these nodes. SWMM5 uses the Manning equation to express the relationship between flow rate (Q), cross-sectional area (A), hydraulic radius (R), and slope (S) in all conduits.

Q = 1n AR%/'S)/%

(4.1)

Where n is the Manning roughness coefficient. The slope S is interpreted as either the conduit slope or the friction slope (i.e., head loss per unit length), depending on the flow routing method used.

In this study the sensitivity of SWMM5 model parameters selected from above on the output responses was evaluated and the parameters to which the model is most sensitive were used for proceeding optimization procedures.

Figure 4-2 the study area network on SWMM5 model


4.4. Sensitivity Analysis

Sensitivity analysis of model parameters studies the uncertainty and variability of input variables and their propagation to the model output through the computational model. Also the weighting of source of the uncertainty and variability which is found in the input variables of a model is measured by sensitivity analysis. Generally sensitivity analysis finds out the model input variables that bring considerable change to the model output. That means the sensitivity of the model output to the variation of the input variables is determined by sensitivity analysis (SA) (King & Perera, 2010).

Sensitivity analysis computes uncertainty of parameters in any type of complex model and identifies the critical inputs of a model and calculates the effect of input uncertainty on the outcome of a model. The partial derivation of the output function with respect to the input factors will be determined when input parameters are known with little uncertainty and sensitivity can be computed by performing multiple simulations by altering the input parameters around a nominal value. When the impact of input parameters on the output is based on values very close to the nominal value it is called local sensitivity analysis. But global sensitivity analysis is preferred to local sensitivity analysis when input factors are very uncertain. Monte Carlo simulations also called sampling based methods are used to implement global sensitivity analysis (Marino et al., 2008).

In this research sensitivity analysis of urban storm water model (SWMM5) parameters was performed. The interaction of SWMM5 model parameters is of high order and sensitivity analysis of this parameters require a model which can handle this interactions and of high computational efficiency. eFAST (extended Fourier Amplitude Sensitivity Test) has proven to be one of the variance based models which are capable of giving a solution to this kind of problems and was used in this study and detail description of this method was shown in section 2.7.4.

eFAST works by quantifying the variability of the input parameters variability and their propagation in to the output through a computational model by calculating the first order sensitivity index (Si) and total order sensitivity indexes (STi) .eFAST calculates the first order sensitivity (Si) of parameters by assigning a unique frequency to each input parameter, while the other parameters are constant and thus the unique contribution of each parameter in determining the model output is calculated. And to calculate the total order sensitivity indexes (STi) eFAST method varies i at a unique, high frequency, but all other parameters at low, not-necessarily unique frequencies. STi measures the combined first and all higher order effects of the ith input variable including all interaction effects involving the ith variable.

In general first order sensitivity index Si of a parameter is calculated to show the sensitivity of the ith input parameter when the higher order interaction between other parameters is not required. And total order sensitivity index STi is used to evaluate the combined first and higher order interaction between other input parameters and ith parameter has effect on the output values (Saltelli et al., 1999).

4.4.1. Sensitivity Analysis (eFAST) Model

In this research, the mathematical model used for the sensitivity analysis of uncertain parameters of storm water management model (SWMM5) is adapted from Global Uncertainty and Sensitivity Analysis in Systems Biology which is used for the study of the interactions between the components of a biological system, and modification is made to the model to outfit the objective of the study and applied to the case study area Dhaka Bangladesh (Marino et al., 2008).

For this study a sensitivity analysis frame work eFAST was used. It integrates storm water management model (SWMM5) for simulation of the drainage network with functions which sample the parameters, set


the frequency of variation of parameters in the simulation process, calculate the output and the sensitivity index of the parameters. The tool was developed on a Matlab environment which makes us of statistical functions in Statistical tool box of Matlab. Therefore the model takes parameters randomly from range of parameter values, run SWMM5 a specified number of model simulations by one parameter at a time with high frequency and the others at lower and registers the output on a Matlab work space and quantifies the magnitude of sensitivity of the output with respect to the input variation. This procedure continues until all the input parameters are evaluated for all the outputs. Diagrammatic representation of eFAST model and detail description of the functions implemented is shown in the Figure 4-3 below.

In this study, to perform sensitivity analysis of the SWMM5 model parameters and to identify the most sensitive ones, 6 parameters were selected (5 from the hydrology section of SWMM5 model and 1 parameter from the hydraulic section) and 1 dummy parameter which is used to evaluate the sensitivity index of other parameters was included. Since the parameters in storm water management model (SWMM5) are inter related on the generation of the output, the sensitivity analysis model (eFAST) works varying one parameter at a time with a very high frequency and the others at low frequency of variation and encode the variation in the output to calculate the total order sensitivity index (STi) by measuring the strength of frequency of propagation of parameters in to the output and the procedure was continued until the sensitivity index of all the parameters were calculated.

List of parameters selected for sensitivity analysis;

1. Percent imperviousness (P.imp.) 2. catchment width (cw) 3. Rainfall factor 4. Manning roughness for impervious part of sub catchment (N1 imp.) 5. Manning roughness for pervious part of sub catchment (N2 perv.) 6. Manning's roughness (n) for a pipe 7. Dummy parameter

In the eFAST model, a nonzero small sensitivity index was given to a parameter to which the model is completely independent. Therefore to show this, a Dummy parameter was introduced in the sensitivity analysis model and this parameter is not any part of a model equation and hence the model output is not affected by this parameter in any way and the model uses this parameter to check the significance of other parameters. Therefore parameters with total order sensitivity index less than or equal to the dummy parameter are considered non-significant to the model at all.

In the sensitivity analysis framework, four output of urban storm water model (SWMM5) were selected to evaluate the sensitivity analysis result. That means, to see if of a parameter is sensitive to a model output as a whole or if it depends on the type of output required and the selected outputs from the hydraulic model are stated below.

1. Peak Flow 2. Time to peak 3. Flow lost to flooding 4. Total lateral inflow

To start with the sampling procedure of the parameters by the model, the uncertainty ranges of the parameters were set to a normalized scale. Therefore, the existing values of the parameters on the case study area drainage network is assumed as mean values and its corresponding normalized value was set to 1 and set up the range of maximum and minimum parameter values given based on the range of values stated on the quantification of uncertain parameters in section on Table 4-1. Then these normalized values are registered on the function (Parameter_settings_EFAST.m) and then model eFAST performs the


required number of simulations for the input parameters and produces the output Y and calculates the sensitivity indexes.

Equation 2.19 was used to calculate the total model simulations in the sensitivity analysis and the value was

N= NS x k x NR=65*7*5=2275 model simulations

Table 4-2 normalized value of uncertain variables

Parameter Minimum Average (mean)

Maximum Standard deviation

Percent imperviousness (P.imp.) 0.8 1 1.2 0.2 catchment width (cw) 0.85 1 1.4 0.2 Rainfall factor 0.8 1 1.2 0.2 Manning roughness for impervious part of sub catchment (N1 imp.)

0.6 1 1.65 0.2

Manning roughness for pervious part of sub catchment (N2 perv.)

0.7 1 1.2 0.2

Manning's roughness (n) for a pipe 0.85 1 1.5 0.2 Dummy parameter 0.5 1 1.5 0.2

Diagrammatic representation of eFAST sensitivity analysis model and detail description of the functions that made up the model are described below

Figure 4-3 eFAST sensitivity analysis model diagram(Simeone Marino et al., 2008).

Description of the eFAST model functions shown in Figure 4-3 are given below,

Model_efast.m This is the major file of sensitivity analysis model which communicates with other files within a model it calculates the total sample size N=NS*k*NR by identifying the number of samples per search curve (NS) and number of re-sampling (NR) based on the number of parameters. It communicates with file

Siand S

ti

eFAST_sd.m

Swmm_efast.m

Parameterdist.m

Output: Y

SETFREQ.m

Parameter_settings_eFAST.m

Model_eFAST.m


(Parameter_settings_EFAST.m) to select the input parameters and also calls the function which sets the frequency of the parameters (SETFREQ.m) to select a frequency set for the corresponding group and it contacts (parameterdist.m) which sets the distribution of the parameters and finally (swmm_efast.m) which simulates the parameters and give the output is linked to model.efast.m and the output value of the process will be stored in the workspace of the matlab.

Parameter_settings_EFAST.m Minimum, maximum and baseline values as well as label of parameters and the duration of the model simulation and initial conditions for storm water management model SWMM5 and the model outputs label is defined by this function

SETFREQ.m In this file the frequency of the parameters will be selected and selecting frequencies of input parameters higher frequency will be set for a parameter at a time and lower frequencies will be set for others to identify the effect of other parameters interaction on a the output variation.

parameterdist.m Here the number of samples NS and type of distribution will be chosen for the sampling scheme.

swmm_efast.m: takes the parameters from Parameter_settings_EFAST.m and passes to swmm_efast.exe and swmm_efast.exe accepts the parameters and prepare SWMM5 input file and prepares the required output values and pass them back to swmm_efast.m and these values will be stored on matlab board.

efast_sd.m: This function is used to calculate Si, STi with respect to their particular distributions.

4.5. Optimization

4.5.1. Latin Hypercube Sampling Avery large number of model simulations will be required to reliably represent the distribution of uncertain parameters. But practically there is a time and computational power constraint to perform a large number of model simulations. Therefore sampling technique is used to cover the distribution of uncertain parameters.

Latin hypercube sampling is developed by McKay et al. (1979)and it is an approach which gives precise estimates and it is modified Monte Carlo sampling scheme. By Latin hypercube sampling, n different values are selected from each of the k variables X1...Xk from the range of non overlapping intervals on the basis of equal probability. One value from each interval is selected at random with respect to the probability density in the interval and the n values obtained from the variable X1 are paired with n values of X2, X3 etc. randomly. Hence (n × k) matrix of input parameters where the ith row includes specific values of each of the k input parameters to be used on the ith run of the SWMM5 model will be produced (G. D. Wyss & Jorgensen, 1998).

A Latin Hypercube Sampling technique is one of the sampling techniques and it will be used in this study to sample from the range of uncertain parameters. A Latin Hypercube Sampling technique has observations that are spread over the entire range of each input variable in a uniformly in a probabilistic sense. For sampling statistical distributions of uncertain parameters in this study, a Latin Hypercube Sampling scheme linked with the hydraulic model and the optimizer written on C++ will were used. Each uncertain variable was assumed to be distributed in a normal probability distribution during sampling technique. Due to the difficulty to find exact distributions which fit the data of each uncertain variable, normal distribution was assumed for each uncertain variable.


For this study the current value of uncertain parameters in the case study are assumed as a mean value and a standard deviation of 10% and 20% of the mean value will be used to cover the uncertainty bound of the parameters in the proceeding optimization process.

4.5.2. Sampling steps To generate a Latin hypercube sampling of size n=5 and input variables X, let us assume a normal distribution of X and the mean value be µ and a standard deviation of σ to determine the end points of the intervals. Figure 4-4 shows the probability density function (PDF) and cumulative probability density functions (CDF) curves and from the intervals it is shown that all of the 5 intervals have equal probability of 20% (G. D. Wyss, Jorgensen, Kelly H, 1998).

P (–∞ ≤ X1 ≤ A) = P (A ≤ X1 ≤ B) = P (B ≤ X1 ≤ C) = P(C ≤ X1 ≤ D) = P (D ≤ X1 ≤ ∞) = 0.2

Figure 4-4 Intervals Used with a Latin Hypercube Sample of Size n = 5 in Terms of the PDF and CDF for a Normal Random Variable (G. D. Wyss, Jorgensen, Kelly H, 1998)

To select n different values of LHS for each of the variables with k dimensions, we use the following steps (Santizo, 2012).

1. The range of each variables X1,...Xk will be divided in to non overlapping interval of equal probability

2. Each interval will be given a number 1,2,...n 3. Select a random number Um between 0 and 1 which is used to select the values randomly from

each interval. 4. Calculate a cumulative probability

�x = (1�)yx + (� − 1� ) (4.2)

Where: Pm is cumulative probability of ith variable Um is random number generated between 0 and 1 m is the ith variable and n is number of intervals generated

5. After finding the value of Pm find the corresponding value of standard normal (z-score) from inverse distribution table.


6. Finally the value of sample within the interval stated above can be found from the formula below �z{| = } ∗ ~ − �Y=@� + � (4.3)

Where: Xobs is the value of the sample observed within the interval, σ is standard deviation of the sample, z-score is the value corresponding to the cumulative distribution and µ is the mean of the sample.

4.5.3. Pairing To describe the general idea of pairing samples, let us write the value of each sample of X1 on a piece of paper and put the papers in to a box. Then write a value of each samples of X2 on other pieces of paper and put them in to a second box and repeat the procedure up to parameters Xk and at the end there would be k number of boxes each containing n pieces of paper with values written on them. Therefore to start the pairing procedure, draw one piece of paper from each box and this will give one Latin hypercube sample for the first SWMM5 model run. And repeat the procedure until all the samples are used and all the boxes are empty(G. D. Wyss, Jorgensen, Kelly H, 1998).

After pairing the samples, they will be ready to be input data for the hydraulic model (SWMM5), in the way that the first pair of uncertain variables will be used for the first model run and the running procedure will be repeated until NS columns of all paired uncertain variables are used.

4.5.4. Objective functions and decision variables Multi objective functions of conflicting objectives that can give compromise solutions are used in the optimization process of this study. The optimization algorithm is planned to minimize the two objective functions and these are intervention cost and flooding volume. The output of the optimization process will be a Pareto optimal front which is used as a tool for decision making.

The pipe diameter is the decision variable in the optimization process and the outcome of the optimization process results in minimizing the diameters of each of the pipes in the network provided that the objective functions are met.

Intervention cost Intervention cost components in the urban drainage cost are labour cost, overhead cost, material cost, earthwork cost required for unit length of pipe per diameters. Overhead cost and labour costs usually vary with association to many local factors, so it is difficult to quantify an accurate magnitude, so an average value of them is taken for this optimization model development. Earthwork cost includes excavation and backfill, which often dominate the whole project cost. Pipe cost is the expense, market price, of pipes it may vary with material type, size and mostly quantify per length. So an average actual cost that account all those cost components is taken as investment cost of the pipe.

The total cost (CT) of the network is calculated based on the cost per unit length associated with the diameter and the length of the pipe.

CT = P C(Dl) ∗ Lln

lF)

(4.4)

Where: n is the total number of pipes in the network, C (Di) is the cost per unit length of ith pipe with diameter Di and length Li. Available pipe diameters in the market and their cost per unit length are shown in the Table 4-3 below.


Table 4-3 List of pipe diameters and unit costs.

Pipe Diameter r(m) Cost $/Length(m) Pipe diameter (m) Cost $/Length(m) 0.3 430.92 1.2 2659.96 0.45 802.44 1.22 2709.50 0.525 988.18 1.35 3031.48 0.6 1173.94 1.5 3402.98 0.7 1421.61 1.55 3526.82 0.75 1545.45 1.7 3898.32 0.825 1731.19 2.5 5879.69 0.9 1916.95 4.3 10337.77 1.05 2288.46 5.0 12071.47

5.5 13309.82 Flooding at the nodes SWMM5 calculates the flooding at the nodes and flooding at the nodes evaluates the performance level of decision variables created by NSGA-II and simulations are performed for chromosomes generated by optimizer NSGA-II. In the optimization process values of design with minimum flooding are given better fitness and will be chosen as candidate solutions and those with poor fitness are rejected in the process. As the optimization process goes on the value of flooding per design minimizes. The output of SWMM5 is read for the calculation of flooding which gives cubic meter per second and to obtain a total volume then this value is multiplied by the time step.

Floodingatnodes = t ∗ P(Sur.N)ln

lF)

(4.5)

Where: t is the time step used i is the number of nodes surcharged (Sur.N) i is the quantity of surcharge in cubic meters per second at the ith node.

4.5.5. Optimization process In this research, optimal design of urban stormwater drainage system was solved using optimization methodology that integrates SWMM5 drainage network simulator with NSGA-II, multi objective optimization method which is based on evolutionary algorithm method. To link an optimizer with hydraulic model a code is developed with C++ language. This code executes sampling of uncertain parameters used in an optimization process, simulate SWMM5 a number of samples times, convert random values given by the optimizer into pipe diameters, updates the SWMM5 with new pipe diameters and Ns number of samples and pass the objective function to the optimizer.

A SWMM5 template file was developed on the drainage network which contains active spaces where the randomly sampled uncertain parameters are changed and the proposed diameters of conduits will be replaced in each of the model simulation.

NSGA-II solves complex optimization problems and it is the most conventional algorithm used for multi-objective optimization problems. An optimum Pareto front displaying the trade-off existing between conflicting objectives will be the output of the algorithm. Intervention cost and flooding volume are the two objective functions considered for this research.

Searching for the potential optimal pipe sizes which assure both the minimum construction cost and minimum flooding volume are required through optimization process. The NSGAII (Non Dominated Sorting Genetic Algorithm) developed by Deb (2002) will be used when executing the optimization process in this study. To start its application NSGA-II is associated with different operators and parameters. NSGA-II parameters used for this study are described in Table 4-4.


Table 4-4 Input parameters for NSGA-II

Inputs Description Population Size 80 (Should be a multiple of 4) Number of Generations 40 (must be greater than 1) Number of objective functions 2 Number of real variables 75 Number of constraints 0 Min. value Real variables -1 Max. value Real variables 1 Probability of crossover 0.6 Probability of mutation 0.1 Value of distribution index for cross over 10 Value of distribution index for mutation 20 Plot option 1 (yes) X axis objective function 1 Y axis objective function 2

When started the genetic algorithm initialize with random population and it evaluates the objective functions to select the best set of chromosomes and it selects the best fit chromosomes by cross over and mutation. Then this process will be repeated a number of times to find a good solution satisfying the objective functions until the stated number of generations are reached. The general schematic diagram is shown Figure 4-5 below.

Figure 4-5 optimization methodology flow chart without considering uncertainties of parameters (Z. Vojinovic et al., 2008)

SWMM5 model • Hydraulic simulation

Optimizer (NSGA-II) • Generate new population

Objective function computation • Flooding volume • Intervention cost

Read NSGA-II population and update SWMM input file

Termination

Stop

Yes No

Start

New population


Two types of optimization methodologies are used in this study to show the effect of including uncertainties of urban stormwater parameters due to different drivers like to urbanization, climate change and variability of the land cover within a sub catchment. The methodologies applied are optimization of urban stormwater drainage without considering uncertainties of parameters and optimization of urban stormwater drainage due to uncertainties of parameters.

4.5.5.1 Optimization without considering uncertaint ies of parameters In this section, optimization methodology of urban drainage network which do not consider uncertainties of parameters is used. Optimized urban stormwater drainage design or rehabilitation method used in this study integrates SWMM5 drainage network simulator with optimizer NSGA-II, an evolutionary algorithm based multi-objective optimization method. SWMM5 is simulated for 100 years return period rainfall data of 1 hour duration and summation of flooding at the nodes is calculated and this will be used to calculate the objective function. Next investment cost function will be evaluated by calculating the construction cost of the chromosomes. Then these two objective functions are passed to the optimizer which generates new population of decision variables. NSGA-II performs selection, crossover and mutation. Then the decision is made when the criterion is met and a good Pareto front showing optimized value of intervention cost and flooding volume of the drainage network is the final outcome. The above procedure is shown with schematic diagram in Figure 4-6.

Figure 4-6 optimization methodology flow chart without considering uncertainties of parameters 4.5.5.2 Optimization considering uncertainties of parameters In this section a methodology for optimal design of urban storm water drainage systems due to uncertainties of parameters due to different drivers like to urbanization, climate change and variability of the land cover within a sub-catchment is considered and the application of uncertainties into an


optimization model is performed in two methods to show the effect of uncertainties of parameters and detail of the methodologies used are stated below.

Method 1 In this method objective functions are calculated by averaging specific samples realization of uncertain parameters in a generation. That means, SWMM is simulated Ns number of samples performed for each chromosomes in a generation, then flooding volume is averaged over number of LH samples and the average flood volume over the number of LH samples is used to calculate the objective function. Based on the fitness values of the objective functions, NSGA-II select the fittest chromosomes and generates new populations for decision variables. If the number of generations is reached a Pareto optimal front solution will be found, but when the generation are not reached, the new population will be generated and using NSGA-II and the cycle continues until the result meets objective functions. The procedure stated above is shown diagrammatically in the Figure 4-7.

Figure 4-7 optimization methodology due to uncertainties of parameters Method 1 (Santizo, 2012)

Method 2 In this method modification is done to NSGA-II to incorporate uncertainties of parameters in to the optimization model and uncertainties are included by randomly selecting one pair of sample from a group of Ns samples for each chromosome in a generation. And hence the fitness of the chromosomes in a


generation is evaluated by one sample realization of a group of Ns samples of uncertain parameters. The flooding volume is used to calculate the objective functions for each chromosome in the optimization process. That means SWMM is simulated one times for a randomly selected sample from Ns number of samples for each chromosome in a generation. This method aims at producing a fittest solution over a multiple generation which enables the calculation of objective values as the average of objective value over that chromosomes age. This methodology is adopted from D. A. Savic (2005) and are stated as follows.

1. Initialize NSGA-II and set the chromosome age to zero and calculate the objective function by evaluating the fitness of each chromosome by one of Ns number of LH samples. Then identify and save non dominated Pareto front.

2. Increase the age of the population by one and evaluate the fitness of each chromosome by one of Ns number of LH samples and calculate the objective function by averaging the present and past objective functions over the chromosome age.

i. Create children population and set the chromosome age to zero evaluate each chromosome by one of Ns number of LH samples and calculate the objective function.

ii. Combine the parent and children population and identify optimum Pareto front. 3. Increase the age of the population by one and evaluate the objective function of the whole of the

chromosomes identified in step 1 and create non dominated Pareto front then combine this value with the value obtained in step 2.ii provided that the dominant chromosome age is equal to or greater than the minimum determined age.

4. Repeat steps 2-3 until the number of generation is reached. Averaging of the objective functions in this method helps to obtain fit chromosomes as the searching procedure is proceeding. And the stated procedures are shown in Figure 4-8.


Figure 4-8 optimization methodology due to uncertainties of parameters Method 2


5.1. Result

5.1.1. Simulation result of sensitivity analysis Stormwater drainage infrastructures are recognized by complex hydraulic and hydrological features in an urban setting and urban drainage model is used to handle the complexities. Therefore, accurate determination of the input variables of urban models and their effect on model output is time consuming but important. Sensitivity analysis helps to quantify the effect of input variability and the propagation of this variability to the output through the computational model.

The objective of the sensitivity analysis in this study is to analyse the effect of uncertain parameters on the output of urban drainage model, quantify the sensitivity of these parameters in to first order sensitivity index (Si) and Total order sensitivity index (STi). This helps to rank the parameters as per their sensitivity indexes for further optimization process. Four outputs of urban storm water management model (SWMM5) were selected in the sensitivity analysis process to quantify the effect of variability of 7 parameters. With 65 numbers of sampling schemes and 5 number of re-sampling, the total number of model simulations used in the process was 2275 for each output. The outputs of the model for which the parameters sensitivity analysed are

1. Peak Flow 2. Time to peak 3. Flow lost to flooding 4. Total lateral inflow

The output and the parameters from the simulation result of eFAST model were plotted on one graph for each of the outputs sated above. As it can be seen from the sensitivity analysis curves Figure 5-1, Figure 5-3, Figure 5-5, and Figure 5-7 for each output, the frequency of the output show higher values for most sensitive parameters and lower values for less sensitive parameters and description of result for each output is given below.

5.1.1.1. Sensitivity analysis result for peak flow

Peak flow is the maximum run off attained from an area and this happens when all parts of the watershed contributes to the flow. From the total order sensitivity index result in Figure 5-2, the peak flow of the study area shows sensitivity index of 45% for N1-impervous, 21%for P. imp, 15% for n, 10% for rainfall factor, 6% for N2-pervous, 3% for catchment width, and 2% for the dummy parameter. Therefore from the result it is clearly seen that peak flow is mostly influenced by the pervious portion of the sub catchment. Hence, Manning roughness for impervious part of sub catchment (N1 imp.) and Percent imperviousness (P.imp.)

CHAPTER 5

Result and Discussion

Result and Discussion 48

are the most sensitive parameters. The effect of Manning's roughness (n) for a pipe and Rainfall factor are lower and are next sensitive parameters. While Manning roughness for pervious part of sub catchment (N2 perv.) is least sensitive parameter and the significance of catchment width (cw) is very low.

Figure 5-1 Result of sensitivity analysis of SWMM5 model parameters for peak flow model output (case study: Dhaka Bangladesh)

Figure 5-2 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5 model parameters for peak flow model output (case study: Dhaka Bangladesh)

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5.1.1.2. Sensitivity analysis result for time to pe ak The time to peak is the time it will take the maximum flow to reach the catchment outlet. From Figure 5-4 the total order sensitivity index result for time to peak is, 30% for n, 19% for N1-imp, 15% for N2-perv, 11% for cw, 10% for rainfall factor, 7% for P.imp, and 7% for the dummy parameter. From the result we can see the roughness coefficient of conduits and the roughness coefficients of sub catchment over impervious and pervious portion are most sensitive to the output and will have inverse relationship. The width of the sub catchment and rain fall values are less sensitive to output for the study area and percent imperviousness is not significant for this output.

Figure 5-3 Result of sensitivity analysis of SWMM5 model parameters for time to peak model output (case study: Dhaka Bangladesh)

Figure 5-4 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5 model parameters for time to peak model output (case study: Dhaka Bangladesh)

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5.1.1.3. Sensitivity analysis result for flow lost to flooding

Total order sensitivity index results as shown in Figure 5-6 For flow lost to flooding with respect to the input parameters is 41% for N1 imp, 20% for P.imp, 12% for N2 pervious, 11% for rainfall factor, 8% for cw, 6% for n and 3% for dummy parameter. And from the result, flow lost to flooding has shown higher sensitivity to manning's roughness for the impervious part of the sub catchment, percent imperviousness, manning's roughness for the pervious part of the sub catchment and rainfall factors. This output was less sensitive to catchment width and manning's roughness for conduits.

Figure 5-5 Result of sensitivity analysis of SWMM5 model parameters for flow lost to flooding model output (case study: Dhaka Bangladesh)

Figure 5-6 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5 model parameters for flow lost to flooding model output (case study: Dhaka Bangladesh)

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P. imp CW n N1 imp N2 perv Rainfall Dummy Flow lost to flooding

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index of flow lost to flooding


5.1.1.4. Sensitivity analysis result for total late ral inflow

Figure 5-8 shows the sensitivity analysis of the output total lateral inflow and the output response to parameter variations. First order and total order sensitivity analysis results of all parameters and their ranks are shown in this figure. From the total order sensitivity analysis result of parameters for the output total lateral inflow, it can be seen that the sensitivity index is 30%for N1 imp, 15% for n, 13% rainfall factor, 13% N2 pervious, 12% P.imp, 9% for cw and 7% for dummy parameter. For total lateral inflow to the drainage system in the study area the sensitivity analysis result is as follows. Manning's roughness for the impervious portion of the sub catchment, Manning's roughness coefficient for conduits, Rainfall factors and Manning's roughness for the pervious portion of the sub catchment were more sensitive parameters in the sensitivity analysis. Percent imperviousness and sub catchment width were less sensitive to this output.

Figure 5-7 Result of sensitivity analysis of SWMM5 model parameters for total lateral inflow model output (case study: Dhaka Bangladesh)

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Sensitivity analysis for total lateral inflow

P. Imp CW n N1 imp N2 perv Rainfall Dummy Total Lateral Inflow


Figure 5-8 First order sensitivity index (Si) and total order sensitivity index (Sti) for SWMM5 model parameters for Total lateral inflow model output (case study: Dhaka Bangladesh)

Most sensitive parameters for the output flow lost to flooding were used for the next optimization process because one of the objective functions is minimizing flooding at the nodes. Therefore from Figure 5-6, the total sensitivity index (STi) ranking of the parameters is N1 imp., P.imp, N2 perv., Rainfall, catchment width, manning's roughness coefficient for pipe (n) and dummy parameters respectively.

5.1.2. Simulation result for Optimization without c onsidering uncertainties of parameters

In this approach uncertainties of parameters due to different drivers (urbanization, climate change and variability of the land cover within a sub catchment) are not considered in the optimization process. Therefore, the value of the storm water management model parameters Manning roughness for impervious part of sub catchment (N1 imp.), Percent imperviousness (P imp.), Manning roughness for pervious part of sub catchment (N2 perv.) and Rainfall factor used in the optimization model are certainly known values in this case. The optimization was done to attain the objective functions in the optimum design/rehabilitation of urban drainage network of the case study area. One of the objectives of this study is to show the effect of considering uncertainties of parameters. This approach is done to show the comparison between optimization of urban drainage network considering uncertainties of parameters versus without considering uncertainties of parameters.

Several model simulations were performed by changing the NSGA-II parameters (number of populations and number of generations) until a good Pareto front showing optimum value of intervention cost and flooding volume is obtained .The number of population and the number of generations used in this study to produce the optimum Pareto curve is 80 and 40 respectively. From the result, the range of optimized values for intervention cost is $22.6 million-$52.3 million and the range of values for flooding volume is 673m3-35,400m3. The optimum Pareto front curve for this approach is shown in Figure 5-9.

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Figure 5-9 Pareto front for optimization without considering uncertainties of parameters 5.1.3. Simulation result for optimization consideri ng uncertainties of parameters

The purpose of this approach is to produce optimal design of urban storm water drainage system which considers uncertainties of model parameters. Therefore to represent the uncertainties of the storm water management model parameters, Manning roughness for impervious part of sub catchment (N1 imp.), Percent imperviousness (P imp.), Manning roughness for pervious part of sub catchment (N2 perv.) and Rainfall factor were considered uncertain parameters in the optimization model. The optimization model frameworks for this approach were described in the methodology section 4.5.5.2 of this study. The uncertainty drivers to these parameters are urbanization, climate change and variability of the land cover within a sub catchment. To cover the uncertainty bound of these parameters, the existing value of the parameters was set as the mean value and a standard deviation of 10 % and 20% from the mean value were used in the optimization model. Two methods were used to introduce uncertainties of parameters in to the optimization model and multiple runs were performed for each method. Avery large number of model simulations will be required to reliably represent the distribution of uncertain parameters. But practically there is a time and computational power constraint to perform a large number of model simulations. Therefore, sampling is proved to be efficient method to introduce uncertainties in to an optimization model. Population of 80 and generation of 40 were used for Latin Hypercube samples of 5, 10 and 20. But for Method 2 due to the modification done to the optimization process, additional parameter called minimum chromosome age of 30 was used. The result in two of the methods aims at minimizing the objective functions, intervention cost and flooding volume and it is stated as follows.

5.1.3.1 Result of method 1

In this method at every generation in the optimization process hydraulic model is simulated N number of LH samples to evaluate the fitness of chromosomes coming from the optimizer NSGA-II. The objective

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function is calculated by the average of the objective functions over a number of LH samples. Multiple model simulations were performed to test the performance of the methodology. 5, 10 and 20 LH samples with standard deviation 10% and 20% of the mean values were used to include the uncertainty bound of the parameters. The Figure 5-10 below shows the optimum Pareto fronts curve for LH sample of 20 and standard deviation of 20% from the mean value.

Figure 5-10 Pareto front for optimization considering uncertainties of parameters (Method 1)

From the result of optimum Pareto fronts for all of the model simulations in this method, the range of intervention cost varies from $24.5 million to $57.7 million and the flooding volume varies from 1620 m3to 38,700m3. From the Figure 5-10, 3 points were selected to describe the Pareto curve obtained. Value at point 1 shows maximum flood volume and minimum intervention cost. The value at point 2 shows compromised value of flood volume and intervention cost because it is taken from the central part of the Pareto curve. And value at point 3 shows minimum flood volume and maximum intervention cost. Point 1 is preferred when considering minimising flooding and point 3 is preferred when considering minimising the intervention cost. The decision maker will choose the appropriate value based on the economic constraints and the required level of performance of the drainage network.

To see the effect of using different standard deviations for the distribution of uncertain parameters, Comparison was made in the Figure 5-11 below.

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Figure 5-11 comparison of Pareto optimal front for different standard deviations (Method 1)

From Figure 5-11 above, the optimum Pareto front curve obtained from optimization framework with uncertainty range of 20% standard deviation shows a shift upwards. This shows it is more costly than the curve with uncertainty range of 10% standard deviation from the mean. The difference in the Pareto optimal front curves obtained for 10 number of Latin hypercube samples is more noticeable than it is for 5 number of Latin Hypercube samples. Therefore, increasing standard deviation in a distribution of uncertain parameters shows significant change.

Comparison was also made in this method to show the effect of using different number of Latin hypercube samples in the optimization process while the standard deviation is the same. Figure 5-12 below shows these comparisons.

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Figure 5-12 comparison of Pareto optimal front for different number of samples & the same standard deviation (Method 1)

As can be seen from Figure 5-12, the optimum Pareto curves for 5, 10 and 20 LH samples for standard deviation of 20% more or less overlapped. The differences are not considerable when compared to the Pareto curve of same samples with different standard deviations. But as the number of samples increase, the shape and the distribution of the Pareto optimal front gets better.

5.1.3.2 Result of method 2 NSGA-II is modified to produce this method and different approach than method 1 is used to introduce uncertainties in to the optimization model. In this method the fitness of the chromosomes in a generation is evaluated by one sample realization of uncertain parameters of N LH samples. The objective function is calculated from the outputs of hydraulic model over the sample realization. The purpose of this method is to produce the fittest solution over a multiple generation which is used to calculate the objective value as the average of the new and previous over that chromosome age. Minimum chromosome age of 30 is used for this approach. The minimum age the chromosome stayed is one of the selecting criteria and the other NSGA-II parameters used for this method are the same as that of method 1. In this method multiple simulations were performed for 5, 10 and 20 Latin Hypercube samples. Standard deviation range of 10% and 20% from the mean value of uncertain parameters was considered to cover the uncertainty bound and the optimum Pareto fronts was obtained.

The comparisons of optimum Pareto fronts obtained from 5 and 10 number of LH samples and different standard deviations of 10% and 20% of the mean for this method are shown in Figure 5-13. The optimum Pareto front curve which is produced by optimization frame work which considers uncertainty bound of 20% standard deviation from the mean value shows more expensive options than that of 10% standard deviation. The upward shift of the optimum Pareto front curve in this method also shows considerable change when the number of samples were increased. That means 10 Latin Hypercube samples show considerable variations than 5 Latin Hypercube samples.

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Figure 5-13 comparison of Pareto optimal front for different standard deviations (Method 2)

For the same standard deviation for the distribution of uncertain parameters in this method, the effect of different number of samples was compared in Figure 5-14 below.

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Figure 5-14 comparison of Pareto optimal front for different number of samples with the same standard deviation (Method 2)

From Figure 5-14 above, it is observed that for LH samples 5 and 10 the Pareto optimal curves show more or less similar values which shows the variation in flooding volume and intervention cost is relatively low. A slight upward shift of the curve is observed for 20 LH samples, which shows relatively expensive solutions in this case.

5.1.4. Comparison of simulation result of optimizat ion without considering uncertainties of parameters versus with uncertainti es of parameters (Method1 &Method 2)

In the optimization of urban drainage network which do not consider uncertainties, the uncertainty of model input data was not taken in to consideration. That means, the parameter values were considered certain and the optimization framework stated in section 4.5.5.1was used. This approach was developed for comparison since one of the objectives of this study is to see the effect of uncertainties in the optimization of urban stormwater drainage systems. In the other approach, uncertainties are included in to the optimization model by two methods (Method 1 and Method 2) as stated in section 4.5.5.2. Figure 5-15 shows comparison of the optimization of urban drainage network without considering uncertainties of parameters with that of optimization which considers uncertainties of parameters (Method 1 and Method 2). An optimum Pareto front curve result for LH samples of 20 and a standard deviation of 20% of the mean value were used for the comparisons for Methods 1 and 2.

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Figure 5-15 comparison of Pareto optimal front without uncertainties of parameters versus optimization considering uncertainties (Method1 and Method 2)

As it can be seen from the Figure 5-15 the Pareto front curves from the results of an optimization due to uncertainties (Method 1 and Method 2) shift upwards indicating more expensive option than optimization without uncertainties. To describe the deference, consider a value on line A of the optimum Pareto front curve of optimization which does not consider uncertainties. This point has a flooding volume of 35,400 m3and intervention coast of 22.6 million$. To maintain this intervention cost in the optimization of the drainage network which considers uncertainties of parameters the flooding volume will be increased due to uncertainties of parameters considered. And when two of the methods (Method 1 and Method 2) of optimization due to uncertainties of parameters are compared Method 2 show more expensive options than Method 1.

Three random values are selected from the Pareto optimal front curve along the lines A, B, C as shown on the Table 5-1 below to show the comparisons of the design options of all the optimization methodologies.

Table 5-1 value of objective functions for comparison of optimization without uncertainties versus optimization with uncertainties (Method 1 and Method 2)

Optimization without uncertainties Optimization with uncertainties (method 1)

Optimization with uncertainties (method 2)

point Intervention cost (millions $)

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A 22.6 35,400 24.5 38,700 25.2 39,500 B 30.5 14,300 33.2 17,600 34.1 19,800 C 44.6 1,740 45.8 4,770 46.4 5,870

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5.2. Discussion

Sensitivity analysis result of the storm water management model parameters of the study area were analysed for different model outputs. The first output was Peak Flow and it shows more sensitivity to the parameters related to the imperviousness of the sub catchment, roughness coefficients of the conduits and rain fall data (N1 imp, P.imp, n, Rainfall), whose alteration in the model highly propagates to peak flow. The peak flow generated in the study area is more dependent on these parameters which implies that when there is big concern on the amount of peak flow in this area these parameters need to be taken in to consideration. Time to peak was the second output and it was more sensitive to the parameters which are related to the roughness coefficients of the conduit and sub catchment which impedes the velocity of flow as well as the width of the sub catchment which decides the time it takes the flow to reach the outlet (n, N1 imp, N2 perv, cw). These parameters uncertainty need to be considered when time to peak is considered output. The third model output in this study was Total lateral inflow and it was more sensitive to manning's roughness coefficients of the impervious sub catchment and conduit, the rainfall data and manning's roughness for pervious sub catchment (N1 imp, n, Rainfall factor, N2 perv). These are parameters related to the generation of runoff and inflow in to the drainage systems. The fourth output considered in this study, Flow lost to flooding, was more sensitive to manning's roughness for impervious part of the sub catchment, percent imperviousness, manning's roughness for pervious part of the sub catchment and rainfall data (N1 imp, P.imp, N2 perv, rainfall data). This indicates the flooding of the drainage network of the study area is more affected if these parameters are altered in the model. From the sensitivity analysis model result, it can be concluded that the sensitivity of a parameter depends on the type of output required than the general model as a whole. From the above procedure 4 parameters (N1 imp, P.imp, N2 perv and rainfall data) which are more sensitive to flow lost to flooding were used to introduce uncertainties in to the proceeding optimization model because one objective of optimization in this study was minimizing flooding.

For the optimization of urban stormwater drainage design or rehabilitation under uncertainties of parameters, two methods were used to introduce uncertainties of parameters and optimum Pareto fronts with compromised solutions of two conflicting objectives were obtained. That means the higher the intervention cost the lower the flooding volume and the higher the flooding volume the lower the intervention cost. Uncertainties of parameters due to different drivers introduced extra flow. Therefore larger diameter of the drainage network is required to carry the flow loads which in turn increases the intervention cost of drainage network. Therefore the Pareto optimal front obtained helps the decision makers to choose the appropriate solution based on their priorities.

From Figure 5-11 for optimization of drainage network considering uncertainties of parameters(Method 1), as the range of standard deviation from the mean value to cover the uncertainty bound increases from 10% to 20% of the mean value the Pareto optimal front curve shifts upwards to the right. This sows an expensive option which results in increase of the flooding volume and intervention cost. The increment in values is more noticeable when the number of Latin Hypercube samples increase from 5 to 10. In the comparison made for Method 2 of the optimization in Figure 5-13 for standard deviation of 10% and 20% of the mean value this method also shows shift upward of the Pareto optimal curves for standard deviation of 20%.The reason for shift in the Pareto optimal curve is, as the range of standard deviation widens from the mean value the value of uncertain parameters used in the optimization process increases and this increase in value increases the corresponding intervention cost and flooding volume.

Form the result in Figure 5-12 the larger the number of LH samples used for the same standard deviation, the better the distribution of solutions and the shape of the Pareto front curve gets smoother. This is because when large number of samples used within a given standard deviation bound in the optimization process, it covers large span of uncertainty sets. In the comparison made for Method 2 in Figure 5-14 for the same case, the Pareto optimal curve for 20 Latin Hypercube samples the Pareto optimal curve shows upward shift and wider distribution of solutions which could be due to the modification done to the search procedure in the optimization methodology.


When comparing optimization of urban storm water drainage systems without considering uncertainties of parameters with optimization which consider uncertainties of parameters (Method 1 and Method 2) in Figure 5-15, the optimal Pareto curve shifts upwards to the right. This shows increase in intervention cost and flooding volume of the drainage network. This is because of the design option made in the optimization process to accommodate the increased run off flow due to uncertainties of parameters in the methods stated. But for optimization without uncertainties of parameters, uncertainties due to climate change, urbanization and variability of the land cover within a sub catchment are not considered and non dominated Pareto front is obtained for certain value of parameters. On the same figure comparisons of optimization methodologies (Method1 and Method 2) which include uncertainties was also shown and the optimization Pareto curve result of Method 2 shows an upward shift than Method 1. This shows more expensive options of Method 2 because in Method 1 uncertainties of parameters are included in to the optimization model through multiple realizations of uncertainty sets and average of the hydraulic model output over the samples was used in the objective value calculation. The averaging effect in Method 1 has the probability of making the characteristics of the chromosomes heredity more similar and this limits the selecting criteria of the optimizer. While in Method 2 one sample realization of uncertainty sets will be used to evaluate the fitness of each of the chromosomes in a generation. In Method 2, due to the modification done to the optimizing algorithm NSGA-II, the searching procedure for optimal solution is more efficient even though less number of samples was used in the fitness evaluation of chromosomes. Additional selecting criteria of parameter called minimum chromosome age was added to the optimizer in method 2 where fittest solution with age greater than or equal to this age will be selected in the process. This lets the objective function calculation of the chromosomes as the average of past objective function over that chromosomes age. Therefore uncertainties need to be considered during the design stage of urban drainage design or rehabilitation so that additional flows handled and the proper functioning of the infrastructure is ensured.

The computational time required for the optimization process of Method 1 is quite longer than Method 2 and it will be time demanding to use it for complex multi objective optimization problems which require many model simulations. In this study the number of populations and number of generations used were 80 and 40 respectively. Method 1 uses all sample realizations for fitness evaluation of chromosomes in a generation and average of the values are used for objective calculation. Therefore if we use 20 Latin Hypercube samples, 1600 hydraulic model simulations are required in a generation for Method 1. While in Method 2 one sample realizations are required for fitness evaluation of chromosomes in a generation which implies 80 hydraulic model simulations are required in a generation for 20 Latin Hypercube samples.

Summary Conclusion and Recommendation 62

6.1. Summary

Human interaction such as clearing of the natural vegetation and covering of the surface with artificial impermeable surface in the day to day activity diverts the water cycle from the drainage system it was naturally following. Consequently, the infiltration capacity of the surface decreases and the runoff generated results in increased volume of water in an area. Urban stormwater drainage system is used to transport and safely discharge the excess volume of water created in an area from the point of generation to the disposal area. The inability to discharge stormwater of an urban area results in flooding which causes widespread of damage to properties, disruption of traffic, public health hazard to a larger magnitude of destruction and loss of life.

The design of urban stormwater infrastructures need to be properly done because it is the essential part of any urbanization. The finance required for this infrastructure is considerably high in terms of capital investment cost as well as maintenance cost. Uncertainties need to be considered in the design stage because if neglected under design of the infrastructure might result in failure to convey the extra flows. Uncertainties occur in circumstances in which parameters of urban drainage model are not known with sufficient degree of certainty and this might occur due to different drivers like climate change, population growth, aging of infrastructures or lack of current techniques to measure them. Due to time and computational power required to include uncertainties in to an optimization of drainage network, most sensitive uncertain parameters was considered. The sensitivity of these parameters was quantified by sensitivity analysis model.

This research focused on optimum design of urban stormwater drainage infrastructures which considers uncertainties of parameters due to climate change, urbanization and variability of land cover within the sub catchment.

The following methodology was followed to consider uncertainties in to an optimization model frame work in this study. First 6 uncertain parameters were selected from urban stormwater model (SWMM5) parameters and the uncertainty bound of these parameters quantified from literature. Secondly sensitivity analysis of uncertain parameters was conducted using a sensitivity analysis model eFAST (extended Fourier Amplitude Sensitivity Test). Thirdly most sensitive parameters which were useful for the objective of an optimization process were selected to represent uncertainties. Fourthly the selected uncertain parameters were introduced in to an optimization model frame work developed. Two methods (Method 1 and Method2) were used to introduce uncertainties in to the optimization model. Fifthly the results obtained from two of the methods were compared with each other and with optimization result which do not

CHAPTER 6

Summary Conclusion and Recommendation


consider uncertainties to see the effect of considering uncertainties in to the optimization model. Finally conclusion was drawn based on the finding of the study and recommendations were made. Description of models used above is described below.

Once uncertainty bound of the selected parameters is compiled from literature, storm water management model (SWMM5) of the study area was built and sensitivity analysis of these parameters was conducted using model eFAST (extended Fourier Amplitude Sensitivity Test).eFAST was used to quantify the effect of input parameters variability on the output values. It works by integrating storm water management model SWMM5 for simulation of drainage network with other functions which sample the parameters, set the frequency of variation of the parameters in the simulation process, and calculates the sensitivity index of the parameters. In the sensitivity analysis process, Percent imperviousness (P.imp.), catchment width (cw), Rainfall factor, Manning roughness for impervious part of sub catchment (N1 imp.), Manning roughness for pervious part of sub catchment (N2 perv.), Manning's roughness (n) for a pipe, were parameters varied. The parameters sensitivity was quantified for the outputs, Peak Flow, Time to peak, Flow lost to flooding, Total lateral inflow. eFAST quantifies the sensitivity of the parameters and calculates the first order sensitivity index (Si) and the total order sensitivity indexes (STi) of each of the parameters. The model uses a parameter whose variation do not have any effect on the model outputs called Dummy parameter. Dummy parameter is used to evaluate the sensitivity of a parameter to the model output. That means a parameter whose sensitivity index is less than or equal to the dummy parameter is considered insignificant to the model output.

From the sensitivity analysis result of the study area, peak flow has shown higher sensitivity index for parameters showing imperviousness of the sub catchment manning's and rainfall (N1 imp, P.imp, n, rainfall).Time to peak has shown more sensitivity for manning's roughness coefficients of conduits and the sub catchment as well as catchment width (n, N1 imp, N2 perv, cw) which affects the peaking time. Total lateral inflow was sensitive to manning's roughness coefficients of the sub catchment and conduit as well as the rainfall data (N1 imp, n, Rainfall, N2 perv) and this is related to the generation of runoff. Flow lost to flooding became more sensitive to manning's roughness for impervious part of the sub catchment, percent imperviousness, and manning's roughness for pervious part of the sub catchment and rainfall (N1 imp, P.imp, N2 perv., rainfall) .From the sensitivity analysis model result, it can be concluded that the sensitivity of a parameter depends on the type of output required than the general model as a whole. From the above procedure 4 parameters (N1 imp, P.imp, N2 perv and rainfall) which are more sensitive to flow lost to flooding were used to introduce uncertainties in to the proceeding optimization model because one objective of optimization in this study was minimizing flooding.

After selecting sensitive parameters which are used to include uncertainties, an optimization framework which includes SWMM5 for hydraulic simulation, LH sampling to sample from a range of uncertain parameters and an optimizer NSGA-II was produced. Two methods (Method 1 and Method 2) were used to introduce uncertainties in to an optimization model. Method 1 includes uncertain parameters as multiple sample realization of uncertainty sets in a generation. This means, all samples generated were used to evaluate each chromosome in a generation and average of the SWMM5 output over all of the samples was used to calculate the objective value. The procedure is repeated until number of generation is reached and optimum Pareto front showing compromised solution of the objective functions was obtained. In Method 2 modification was done to NSGA-II to incorporate uncertainties in to an optimization model. In this method, one pair of uncertain parameters sets from Ns group of samples was randomly selected to evaluate each chromosome in a generation and the SWMM5 model output was used to calculate the objective value. In this method, modification was done to the optimizer to perform efficient searching of optimum solution which is capable of surviving over a number of generations. This allows the calculation of the objective value of chromosomes as the average of the present and past objective function over that chromosome age. Even if small number of samples were used in this method for chromosome evaluation, the efficiency of the search is higher and the search procedure in this method utilizes better population at each generation.

Summary Conclusion and Recommendation 64

In Method 1 and Method 2 for the optimization process, the existing value of the parameters was taken as a mean value and a standard deviation of 10% and 20% of the mean was considered to include uncertainty bound of the parameters. Population of 80 and generation of 40 were taken as the parameters of the optimizer for both methods and additional parameter called minimum chromosome age of 30 was used for (Method 2). 5, 10, 20 LH samples of the uncertain parameters were taken to simulate SWMM5 model in the optimization process. The result of the two approaches shows the value of the optimum Pareto front obtained for the standard deviation of 20% of the mean has shown more expensive design option than the 10%. This was because bigger standard deviation widens the uncertainty bound of parameters. Comparisons were also made between the result of an optimization without uncertainties of parameters and 2 of the methods which includes uncertainties of parameters and two of the methods. Method 1 and Method 2 have shown higher capacity of handling extra flow due to uncertainties considered and also they have shown more expensive design options. Optimization result due to uncertainties of parameters of Method 1 and Method 2 were also compared and the result shows optimization result of Method 2 shows a wide distribution of design options and was more expensive than Method 1.This was due to the averaging effect of sample realizations in Method 1 and modification of the search procedure implemented in Method 2 in the optimization process.

6.2. Conclusion

The main objective of this study was to develop and test a methodology to find optimum size of stormwatr drainage system considering uncertainties of parameters due to different drivers such as climate change, urbanization and variability of land cover within the sub catchment. A sensitivity analysis model framework eFAST which incorporates SWMM5 model with other functions was used to select sensitive uncertain parameters that were used in the optimization process. The optimization framework includes SWMM5 model for hydraulic simulation, Latin Hypercube sampling techniques to sample from the distribution of uncertain parameters and a genetic algorithm based optimizer (NSGA-II) for the optimization process.

From the sensitivity analysis result it is shown that one parameter has shown different sensitivity index values for different model outputs. This shows a parameter which shows higher sensitivity to one output might be less sensitive to the other output. From the sensitivity analysis result, it can be concluded that parameters sensitivity analysis need to be performed based on the type of output required for the purpose of the study.

To introduce all values of uncertain parameters in to an optimization model from the statistical distribution of uncertain parameter values, a very large number of model simulations are required to reliably represent the distribution of uncertain parameters. But practically it is time demanding and computational power of the device used is also need to be considered. To manage this sampling of the parameter values was proven to be efficient to represent the distribution of uncertainty bounds.

In this study two methods were used to introduce uncertainties in to an optimization framework. Method 1 introduces uncertainties through multiple sample realization of uncertainty sets in a generation. And Method 2 uses one sample realization of uncertainty sets in a generation and modification was made on the optimizer in this case. For both of the approaches for a large number of Latin Hypercube samples used in the optimization process, the optimum Pareto curve gets better distribution of solutions and the curve gets smoother. From this it can be concluded that the number of Latin Hypercube samples used need to be increased for optimum design of urban drainage network which considers uncertainties of parameters.

Comparison of optimization result for different standard deviations for the distribution of uncertain parameters in an optimization which considered uncertainties in Method 1 and Method 2 was done in this study. The result of the optimum solution for increased standard deviation has shown expensive solutions which lead to increased flooding volume and intervention cost of infrastructures. The standard deviation for


statistical distribution of uncertain parameters is the measure of the values of deviation from the mean and it is determined by the data collected to represent the uncertain parameter. Therefore the data collection process needs to be given attention for parameters concerned so that better distribution is obtained and deviation of the parameter values from the mean value is known.

In this study, the result of optimization of drainage network which do not consider uncertainties was compared to the result which considers uncertainties (Method 1 and Method 2). From the result, the optimization which considers uncertainties show higher capacity of the drainage system which increases the flooding volume and intervention cost of the infrastructures due to extra flows coming from uncertainties. From this a conclusion can be drawn that uncertainties need to be included in to an optimization model to improve the reliability of the system.

6.3. Recommendation

The following recommendations were made based on the results of this study so that expansion is made for future researches related to this topic:

In this study a 1D storm water management model (SWMM5) was used for hydraulic simulation of optimization model to calculate one of the objective functions flooding volume. This gives the total volume of flooding in the drainage network. It is difficult to know the spot where the flooding occurred and the depth of flooding cannot be known from this approach although it is one of the decision making tools in urban drainage optimization designs. To clearly know the flooding depth in the drainage network to calculate the damage as a result of flooding, using 1D/2D coupled model it is recommended.

The number of population and number of generation as well as the number of Latin Hypercube samples used in the optimization frame work in this study is limited due to time limitation. It is recommended to use higher values to obtain improved solutions.

Due to time constraint in this study, in the sensitivity analysis model extended Fourier Amplitude Sensitivity Test (eFAST) the number of samples per search curve (NS) used was 65 which is the minimum value suggested by eFAST. And in further studies it is recommended to use higher values to increase the number of model simulations and to check the consistency of the result and compare the ranking of the sensitivity indexes for each parameter in the process. The decision variable used in this study was only pipe diameter and there are other drainage infrastructure components like storage volume which have contribution to the investment cost of the infrastructure as well as effect in minimizing flooding volume. Therefore it is recommended that this structure is optimized in further studies.


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Simeone Marino, Ian B. Hogue, Christian J. Ray, & Kirschner, D. E. (2008). A Methodology For Performing Global Uncertainty and Sensitivity Analysis in Systems Biology, Journal of Theoretical Biology. Retrieved from Uncertainty and sensitivity functions and implementation (matlab functions for eFAST)

website: http://malthus.micro.med.umich.edu/lab/usadata/ Vairavamoorthy, K., & Ali, M. (2000). Optimal design of water distribution systems using genetic algorithms.

Computer‐Aided Civil and Infrastructure Engineering, 15(5), 374-382. Van Gelder, P., Roos, A., & Vrijling, H. (2001). Risk based design of civil structures: Delft University of

Technology, Faculty of Civil Engineering and Geosciences. Vélez, C. A., Lobbrecht, A., Price, R., Mynett, A. E., Popescu, I., Galvis, A., & Restrepo, I. (2007).

OPTIMIZATION OF URBAN WASTEWATER SYSTEMS USING MODEL BASED DESIGN AND CONTROL. CASE STUDY OF CALI, COLOMBIA. Saneamiento básico y ambiental en america latina, 1, 358-370.

Vojinovic, Z., Solomatine, D., & Price, R. K. (2005). Dynamic least-cost Optimization of waste water systems remedial works requirements.

Vojinovic, Z., T, A. S., & Barreto, W. J. (2008). Optimising Sewer System Rehabilitation Strategies between Flooding, Overflow Emissions and Investment Costs Walski, T. M., Chase, D. V., Savic, D. A., Grayman, W. M., Beckwith, S., & Koelle, E. (2003). Advanced water

distribution modeling and management: Haestad press. Williams, G. S., & Hazen, A. (1911). Hydraulic Tables: The Elements of Gagings and the Friction of Water

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SAND98-0210, Sandia National Laboratories, Albuquerque, NM.

Appendices 70

Appendices

Appendix A

Table A-1 Sub-catchment properties description of the study area

Name Rain gage

Outlet

Node

Total

Area

Pcnt.

Imperv Width Slope N-Imperv N-Perv

101 Gage1 19 2.71 19.5 113.00 1 0.05 0.4

102 Gage1 23 2.71 19.5 113.00 1 0.05 0.4

103 Gage1 20 2.71 19.5 113.00 1 0.05 0.4

104 Gage1 24 2.71 19.5 113.00 1 0.05 0.4

105 Gage1 25 8.36 32.5 146.67 1 0.05 0.4

106 Gage1 26 8.36 32.5 146.67 1 0.05 0.4

107 Gage1 27 4.18 25 154.82 1 0.05 0.4

108 Gage1 16 4.18 25 154.82 1 0.05 0.4

109 Gage1 9 3.25 35 120.17 1 0.05 0.4

110 Gage1 29 6.49 35 127.24 1 0.05 0.4

111 Gage1 30 6.49 35 127.24 1 0.05 0.4

112 Gage1 31 3.25 35 120.17 1 0.05 0.4

113 Gage1 86 4.09 39 194.69 1 0.05 0.4

114 Gage1 28 4.09 39 194.69 1 0.05 0.4

115 Gage1 18 4.09 39 194.69 1 0.05 0.4

116 Gage1 22 4.09 39 194.69 1 0.05 0.4

117 Gage1 35 3.39 21 161.38 1 0.05 0.4

118 Gage1 65 3.39 21 161.38 1 0.05 0.4

119 Gage1 12 3.39 21 161.38 1 0.05 0.4

120 Gage1 33 3.63 21 93.00 1 0.05 0.4

121 Gage1 21 3.63 21 92.97 1 0.05 0.4

122 Gage1 36 9.61 65 152.60 1 0.05 0.4

123 Gage1 37 9.61 65 152.60 1 0.05 0.4

124 Gage1 38 9.61 65 152.60 1 0.05 0.4

125 Gage1 39 9.61 65 152.60 1 0.05 0.4

126 Gage1 41 10.45 6 158.33 1.5 0.05 0.4

127 Gage1 42 10.45 6 158.33 1.5 0.05 0.4

128 Gage1 43 10.45 7.5 158.33 1 0.05 0.4

129 Gage1 44 10.45 7.5 158.33 1 0.05 0.4

130 Gage1 45 10.45 7.5 158.33 1 0.05 0.4

131 Gage1 46 10.45 9 158.33 1 0.05 0.4

132 Gage1 47 10.45 9 158.33 1 0.05 0.4

133 Gage1 48 4.18 48.75 99.52 1 0.05 0.4


134 Gage1 14 4.18 32.5 99.52 1 0.05 0.4

135 Gage1 49 4.18 48.75 99.52 1 0.05 0.4

136 Gage1 50 4.18 48.75 99.52 1 0.05 0.4

137 Gage1 51 4.18 48.75 99.52 1 0.05 0.4

138 Gage1 52 4.18 48.75 99.52 1 0.05 0.4

139 Gage1 53 4.18 48.75 99.52 1 0.05 0.4

140 Gage1 54 4.18 48.75 99.52 1 0.05 0.4

141 Gage1 55 4.18 48.75 99.52 1 0.05 0.4

142 Gage1 56 4.18 48.75 99.52 1 0.05 0.4

143 Gage1 58 3.40 70 94.33 1.5 0.05 0.4

144 Gage1 59 3.40 70 94.33 1.5 0.05 0.4

145 Gage1 60 2.48 70 118.19 1 0.05 0.4

146 Gage1 78 2.48 70 118.19 1 0.05 0.4

147 Gage1 63 6.54 48.75 128.28 1 0.05 0.4

148 Gage1 64 6.54 22.5 128.28 1 0.05 0.4

149 Gage1 66 5.97 25 124.31 1 0.05 0.4

150 Gage1 67 5.97 25 124.31 1 0.05 0.4

151 Gage1 68 5.97 25 124.31 1 0.05 0.4

152 Gage1 69 5.97 25 124.31 1 0.05 0.4

153 Gage1 70 5.97 25 124.31 1 0.05 0.4

154 Gage1 71 5.97 25 124.31 1 0.05 0.4

155 Gage1 72 5.97 25 124.31 1 0.05 0.4

156 Gage1 79 1.57 18 65.33 1 0.05 0.4

157 Gage1 80 1.57 18 65.33 1 0.05 0.4

158 Gage1 81 3.54 19.5 168.52 1 0.05 0.4

159 Gage1 32 3.54 19.5 168.52 1 0.05 0.4

160 Gage1 21 3.54 19.5 168.52 1 0.05 0.4

161 Gage1 83 4.44 60 105.74 1 0.05 0.4

162 Gage1 84 4.44 60 105.74 1 0.05 0.4

163 Gage1 85 4.44 60 105.74 1 0.05 0.4

164 Gage1 82 5.62 60 170.21 1 0.05 0.4

165 Gage1 57 5.62 60 170.21 1 0.05 0.4

166 Gage1 40 9.61 19.5 152.60 1 0.05 0.4

167 Gage1 61 6.54 19.5 128.28 1 0.05 0.4

168 Gage1 62 6.54 19.5 128.28 1 0.05 0.4

169 Gage1 73 2.51 4.5 76.00 1 0.05 0.4

170 Gage1 74 2.51 9 76.00 1 0.05 0.4

171 Gage1 75 2.51 9 76.00 1 0.05 0.4

172 Gage1 76 2.51 9 76.00 1 0.05 0.4

173 Gage1 77 2.51 12 76.00 1 0.05 0.4

174 Gage1 87 54.86 3 373.21 1 0.05 0.4

Appendices 72

Table A-2 conduit properties description of the study area

Name

Inlet

Node

Outlet

Node

Length

Manning

N

Inlet

Offset

Outlet

Offset Shape Diam.

1 1 5 133.4 0.02 0 0 CIRCULAR 2.5

17 18 19 64.983 0.02 0 0 CIRCULAR 0.75

18 19 20 262.42 0.02 0 0 CIRCULAR 0.9

19 20 21 315.93 0.02 0 0 CIRCULAR 0.9

20 21 32 68.092 0.02 0 0 CIRCULAR 0.9

21 22 23 76.288 0.02 0 0 CIRCULAR 0.6

22 23 24 257.95 0.02 0 0 CIRCULAR 0.9

23 24 25 742.58 0.02 0 0 CIRCULAR 1.2

24 25 26 187.69 0.02 0 0 CIRCULAR 1.35

25 26 27 322.04 0.02 0 0 CIRCULAR 1.5

26 27 16 18.919 0.02 0 0 CIRCULAR 1.5

27 28 29 119.65 0.02 0.45 0.6 CIRCULAR 0.6

28 29 30 207.91 0.02 0.6 0.6 CIRCULAR 0.6

29 30 31 54.47 0.02 0.6 0.6 CIRCULAR 0.6

30 31 9 29.543 0.02 0 0 CIRCULAR 0.6

31 32 33 134.09 0.02 0 0 CIRCULAR 1.7

32 33 34 121.27 0.02 0 0 CIRCULAR 1.7

33 34 35 163.82 0.02 0 0 CIRCULAR 1.7

34 35 11 13.394 0.02 0 0 CIRCULAR 1.7

35 36 37 209.33 0.02 0 0 CIRCULAR 0.6

36 37 38 124.11 0.02 0 0 CIRCULAR 0.75

37 38 39 246.36 0.02 0 0 CIRCULAR 0.9

38 39 40 189.41 0.02 0 0 CIRCULAR 1.35

40 41 42 84.194 0.02 0 0 CIRCULAR 0.75

41 42 43 321.24 0.02 0 0 CIRCULAR 0.9

42 43 44 230.22 0.02 0 0 CIRCULAR 1.05

43 44 45 136.42 0.02 0 0 CIRCULAR 1.2

44 45 46 76.289 0.02 0 0 CIRCULAR 1.35

45 46 47 24.894 0.02 0 0 CIRCULAR 1.5

46 48 51 49.86 0.02 0 0 CIRCULAR 0.6

47 49 51 59.302 0.02 0 0 CIRCULAR 0.45

48 50 53 115.31 0.02 0 0 CIRCULAR 0.6

49 51 52 113.75 0.02 0 0 CIRCULAR 0.6

50 52 56 100.43 0.02 0 0 CIRCULAR 1.22

51 53 52 59.048 0.02 0 0 CIRCULAR 0.75

52 54 53 149.02 0.02 0 0 CIRCULAR 0.75

53 55 54 87.611 0.02 0 0 CIRCULAR 0.45

54 56 14 149.24 0.02 0 0 CIRCULAR 1.22


55 57 58 213.98 0.02 0.45 0.6 CIRCULAR 0.45

56 58 59 289.39 0.02 0.6 0.65 CIRCULAR 0.525

57 59 60 256.3 0.02 0.65 0.75 CIRCULAR 0.6

58 60 78 13.357 0.02 0.7 0.75 CIRCULAR 0.7

59 61 62 74.468 0.02 0 0 CIRCULAR 0.6

60 62 63 86.157 0.02 0 0 CIRCULAR 0.75

61 63 64 84.262 0.02 0 0 CIRCULAR 0.75

62 64 65 129 0.02 0 0 CIRCULAR 0.9

63 65 11 19.061 0.02 0 0 CIRCULAR 0.9

64 66 67 290.77 0.02 0 0 CIRCULAR 0.75

65 67 68 321.18 0.02 0 0 CIRCULAR 0.9

66 68 69 187.15 0.02 0 0 CIRCULAR 1.05

67 69 70 233.1 0.02 0 0 CIRCULAR 1.2

68 70 71 206 0.02 0 0 CIRCULAR 1.55

69 71 72 194.08 0.02 0 0 CIRCULAR 1.7

70 73 74 50.743 0.02 0 0 CIRCULAR 0.6

71 74 75 48.805 0.02 0 0 CIRCULAR 0.9

72 75 76 262.39 0.02 0 0 CIRCULAR 1.2

73 76 77 88.669 0.02 0 0 CIRCULAR 1.35

74 77 3 74.608 0.02 0 0 CIRCULAR 1.55

75 78 79 157.78 0.02 0.7 0.7 CIRCULAR 0.7

76 79 80 46.653 0.02 0.7 0.7 CIRCULAR 0.7

77 80 81 61.462 0.02 0.7 0.9 CIRCULAR 0.7

78 81 32 28.021 0.02 0 0 CIRCULAR 1.7

79 82 83 134.35 0.02 0.3 0.3 CIRCULAR 0.3

80 83 84 131.76 0.02 0.3 0.95 CIRCULAR 0.3

81 84 85 276.92 0.02 0.95 0.45 CIRCULAR 0.45

82 85 86 112.14 0.02 0.45 0.45 CIRCULAR 0.45

83 86 28 47.174 0.02 0.45 0.45 CIRCULAR 0.45

84 87 88 79.976 0.02 0 0 CIRCULAR 1.35

85 88 89 198.08 0.02 0 0 CIRCULAR 1.55

86 89 90 106.32 0.02 0 0 CIRCULAR 1.7

87 90 3 40.322 0.02 0 0 CIRCULAR 1.7

16 72 5 174.61 0.02 0 0 CIRCULAR 1.7

89 47 1 252.49 0.02 0 0 CIRCULAR 1.55

5 6 2 227.71 0.02 0 0 CIRCULAR 5.5

88 4 5 288.79 0.02 0 0 CIRCULAR 4.3

Appendices 74

Table A-3 Junction node property description of the study area

Name

Invert

Elev. Max.Depth

Init.

Depth

Surcharge

Depth

Ponded

Area

1 0 6.63 0 0 4.91

3 1.35 5.49 0 0 19.63

4 0.65 6.66 0 0 23.76

5 -0.14 6.59 0 0 28.27

6 -0.73 7.38 0 0 33.18

7 1.33 6.07 0 0 19.63

8 1.28 5.48 0 0 19.63

9 1.25 6.00 0 0 19.63

10 1.13 5.62 0 0 19.63

11 1 6.48 0 0 19.63

12 0.98 6.27 0 0 19.63

13 0.9 6.14 0 0 19.63

14 0.86 5.69 0 0 19.63

15 0.8 6.22 0 0 19.63

16 0.7 6.11 0 0 23.76

18 5 2.11 0 0 4.91

19 4.57 2.62 0 0 4.91

20 4.15 3.47 0 0 4.91

21 3.74 3.61 0 0 4.91

22 5.89 1.22 0 0 4.91

23 5.58 1.67 0 0 4.91

24 5.08 2.56 0 0 4.91

25 4.5 2.48 0 0 4.91

26 4.35 2.34 0 0 4.91

27 4.09 2.74 0 0 4.91

28 3.92 3.22 0.45 0 4.91

29 3.86 2.78 0.6 0 4.91

30 3.78 3.16 0.6 0 4.91

31 3.83 3.37 0 0 4.91

32 2.23 5.28 0 0 4.91

33 2.13 4.99 0 0 4.91

34 2.04 5.40 0 0 4.91

35 1.97 4.92 0 0 4.91

36 5.05 1.60 0 0 4.91

37 4.69 2.67 0 0 4.91

38 4.22 2.96 0 0 4.91

39 3.83 3.13 0 0 4.91

40 3.58 3.47 0 0 4.91


41 3.36 3.54 0 0 4.91

42 3.16 5.66 0 0 4.91

43 2.91 5.49 0 0 4.91

44 2.64 4.02 0 0 4.91

45 2.15 5.96 0 0 4.91

46 1.73 6.52 0 0 4.91

47 1.62 6.82 0 0 4.91

48 4.32 2.83 0 0 4.91

49 4 3.52 0 0 4.91

50 3.98 3.59 0 0 4.91

51 3.4 4.02 0 0 4.91

52 2.86 4.62 0 0 4.91

53 3.65 3.98 0 0 4.91

54 4 2.82 0 0 4.91

55 4.38 3.02 0 0 4.91

56 1.89 4.96 0 0 4.91

57 4.28 3.55 0.45 0 4.91

58 3.85 4.15 0.6 0 4.91

59 3.45 3.79 0.65 0 4.91

60 3.3 4.44 0.75 0 4.91

61 3.8 3.28 0 0 4.91

62 3.47 3.73 0 0 4.91

63 3.21 4.13 0 0 4.91

64 2.89 4.59 0 0 4.91

65 2.8 4.69 0 0 4.91

66 5.05 2.35 0 0 4.91

67 4.45 2.86 0 0 4.91

68 3.85 3.05 0 0 4.91

69 3.45 3.43 0 0 4.91

70 2.9 4.25 0 0 4.91

71 2.51 4.61 0 0 4.91

72 2.35 4.48 0 0 4.91

73 4.94 2.12 0 0 4.91

74 4.67 2.38 0 0 4.91

75 4.28 2.76 0 0 4.91

76 3.86 3.17 0 0 4.91

77 3 4.01 0 0 4.91

78 3.13 4.61 0.7 0 4.91

79 2.97 4.75 0.7 0 4.91

80 2.95 4.49 0.7 0 4.91

81 2.94 4.70 0 0 4.91

82 4.95 2.54 0.3 0 4.91

Appendices 76

83 4.81 2.74 0.3 0 4.91

84 4.39 3.02 0.6 0 4.91

85 4.14 2.35 0.45 0 4.91

86 4.25 3.07 0.45 0 4.91

87 4.16 2.91 0 0 4.91

88 3.86 3.21 0 0 4.91

89 3.55 3.45 0 0 4.91

90 3.5 3.51 0 0 4.91

Table A-4 Outfall node property description of the study area

Name Invert Elev. Outfall Type

2 -1.66 FREE

Table A-5 100 Year rainfall data measured at 5 minutes for the study area

Name Date Time Value

Gage1 9/16/1996 15:15 0

Gage1 9/16/1996 15:20 6.32

Gage1 9/16/1996 15:25 7.7

Gage1 9/16/1996 15:30 9.6

Gage1 9/16/1996 15:35 12.29

Gage1 9/16/1996 15:40 16.32

Gage1 9/16/1996 15:45 22.75

Gage1 9/16/1996 15:50 19.13

Gage1 9/16/1996 15:55 14.09

Gage1 9/16/1996 16:00 10.82

Gage1 9/16/1996 16:05 8.57

Gage1 9/16/1996 16:10 6.96

Gage1 9/16/1996 16:15 5.76

Gage1 9/16/1996 16:20 0

Gage1 9/16/1996 16:25 0

Gage1 9/16/1996 16:30 0

Gage1 9/16/1996 16:35 0

Gage1 9/16/1996 16:40 0

Gage1 9/16/1996 16:45 0

Gage1 9/16/1996 16:50 0

Gage1 9/16/1996 16:55 0

Gage1 9/16/1996 17:00 0

Gage1 9/16/1996 17:05 0

Gage1 9/16/1996 17:10 0


Table A-6 Manning's roughness coefficient n for different pipes

Type of Pipe Min. Max

Glass, brass or copper 0.009 0.013

Smooth cement surface 0.01 0.013

Wood-stave 0.01 0.013

Vitrified sewer pipe 0.01 0.017

Cast-Iron 0.011 0.015

Concrete, precast 0.011 0.015

Cement mortar surfaces 0.011 0.015

Common-clay drainage tile 0.011 0.017

Wrought Iron 0.012 0.017

Brick with cement mortar 0.012 0.017

Riveted-steel 0.017 0.02

Cement rubble surfaces 0.017 0.03

Corrugated metal storm drain 0.02 0.024

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