Risk in integrated urban water systems: A demonstration using measures and assessment of rainwater tank use in households

Transcription

1 Risk in integrated urban water systems: A demonstration using measures and assessment of rainwater tank use in households Chi-Hsiang Wang and Jane Blackmore ewater Technical Report

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3 Risk in integrated urban water systems: A demonstration using measures and assessment of rainwater tank use in households C.-H. Wang and J.M. Blackmore CSIRO Sustainable Ecosystems September 9 ewater Cooperative Research Centre Technical Report

4 Contact for more information: Dr Jane Blackmore Urban Systems Program CSIRO Sustainable Ecosystems P.O. Box 3, North Ryde, NSW 67 Please cite this report as: C.-H. Wang and J. Blackmore, (). Risk in urban water systems: A demonstration using measures and assessment of rainwater tank use in households. ewater Technical Report, Canberra. ewater Cooperative Research Centre This report is copyright. It may be reproduced without permission for purposes of research, scientific advancement, academic discussion, record-keeping, free distribution, educational use or other public benefit, provided that any such reproduction acknowledges ewater CRC and the title and authors of the report. All commercial rights are reserved. Published online ISBN ewater CRC Innovation Centre, University of Canberra ACT 6, Australia Phone () Fax () info@ewatercrc.com.au Web ewater CRC is a cooperative joint venture whose work supports the ecologically and economically sustainable use of Australia s water and river systems. ewater CRC was established in 5 as a successor to the CRCs for Freshwater Ecology and Catchment Hydrology, under the Australian Government s Cooperative Research Centres Program.

5 Table of Contents Executive Summary...6 Introduction...7 Rainwater Tank Use in a Household Modelling of Roof Runoff and Rainwater Tank Storage Roof Runoff Module Tank Storage Module... 4 Modelling of Domestic Demand Toilet Flushing Garden Watering Modelling of Rainfall Risk Assessment Methods Partitioned Multi-objective Risk Method (PMRM)... 7 Example: Risk Assessment for Supply-Demand of Rainwater Tank Risk of Demand not Met Volumetric Reliability Optimal Tank Capacity Resilience of Rainwater Tank Use Concluding Comments... 7 Appendix A. Probability Distribution Functions... 8 Appendix A.. The Beta Distribution... 8 Appendix A.. The Exponential Distribution... 9 Appendix A.3. The Shifted Exponential Distribution... 9 Appendix A.4. The Pareto Distribution... 9 Appendix B. Non-stationary Poisson Process... 3 Appendix C. Modelling of Water Evaporation... 3 Appendix C.. Dry-Bulb Temperature... 3 Appendix C.. Wet-Bulb Temperature Appendix C.3. Algorithm for Generating Wet-Bulb Temperature Appendix C.4. Wind Speed Appendix C.5. Water Evaporation References... 38

6 Executive Summary Integrated urban water systems (IUWS) consider as a whole potable water supply, storm, and waste water discharges. An IUWS typically consists of a number of subsystems such as natural, social, technical, and economic systems, which interact dynamically with each other in a complex way over the temporal and spatial domains. In addition, the potential impacts of climate change, population growth, and technological advances on these subsystems are highly uncertain. All of these make prediction of performance and management of water use in an IUWS difficult. A tool that assists in assessing the impact of uncertainty and complex interaction, and subsequent treatment of the hazardous controls, is risk assessment and management. As part of a broader scope of study on risk assessment and management of integrated urban water systems (IUWS), this report demonstrates the risk assessment technique for rainwater tank use in households. The rainwater tank storage is assumed to be filled by roof runoff and, if instantaneous demand cannot be met, supplemented by tap water. Two types of household demands are supplied by the rainwater tank: toilet flushing and garden/lawn irrigation. The main events related to roof runoff are rainfall, splashing, and sometimes gutter overflow. Since rainfall events are stochastic in nature, historical data recorded by the Bureau of Meteorology are used to derive stochastic models for these stochastic processes. Similar to climate variables, demands of toilet flushing and garden irrigation are stochastic and, in addition, exhibit non-uniform diurnal patterns. Data from a water usage survey and measurements serve the basis for development of stochastic models for toilet flushing and garden watering. In practical application, low-probability, high-consequence events are often of major concern. In addition to the conventionally used unconditional expected risk, the partitioned multiobjective risk method, which divides the domain of possible consequences into a number of consequence ranges, is employed for risk analysis to obtain the conditional expected risks of a number of partitioned exceedance probability ranges. When quality observational data are available, the technique used in this report can be applied to any locations in Australia, and to a community of cluster or regional scales. Similarly, it can be expanded to consider full-blown integrated urban water systems composed of potable water supply, storm and waste water discharges. Resilience is regarded as a key property of sustainable development for IUWS. Since the undertakings of both risk management and resilience analysis are to enhance the sustainability of IUWS, studies on finding methodologies that integrate consideration of both risk and resilience as tools for enhancing sustainable IUWS are warranted.

7 Introduction Safe and reliable urban water supply is one of the major challenges of the twenty-first century. The looming threat of climate change and the ever-expanding global population cast doubt on the sustainability of future supply. Projected climate change could result in more severe drought and, on the other hand, more intense storms and rainfall that cause increased flooding, pollution due to sewage overflow, and contamination of ground water. To deal effectively with these challenges, integrated urban water systems (IUWS) are increasingly being considered and introduced for water use planning and management. An IUWS integrates potable water, storm water, and waste water of different qualities, which typically consists of a number of subsystems, such as natural, social, technical, and economic systems. These subsystems interact dynamically with each other in a complex way over the temporal and spatial domains. In addition, the potential effects of climate change, population growth, and technological advances on these subsystems are highly uncertain. All of these make prediction of performance and management of water use in an IUWS difficult. A tool that assists in assessing the impact of uncertainty and complex interaction, and subsequent treatment of the hazardous controls, is risk assessment and management. Risk is often defined as a measure of the probability of an occurrence multiplied by the (usually adverse) consequence. For risk assessment, three questions are often asked (Kaplan and Garrick 98): What can go wrong? What is the likelihood that it would go wrong? What are the consequences? Answers to these questions help identify, measure, quantify, and evaluate risks and their consequences and impacts. Risk management builds on the risk assessment process by seeking answers further to three questions (Haimes 99): What can be done and what options are available? What are the associated tradeoffs in terms of all costs, benefits, and risks? What are the impacts of current management decisions on future options? Though risk management is distinguished from risk assessment for they address different sets of questions as stated above, risk management has sometimes been used to connote the entire process of risk assessment and management (AS/NZS 436 4). A guide that aims to assist the stakeholders and managers in understanding risks (Blackmore et al. 7) and a risk management framework (Yum et al. 7) that uses quantitative risk assessment as a part of decision-making process and risk mitigation planning have been developed by ewater for use in integrated urban water management. As a demonstration for risk analysis of IUWS, and for the sake of clarity but without loss of generality, this report conducts risk assessment for the use of rainwater tanks in households against the following measures: Failure to meet the demand planned to be supplied by the rainwater tank Relative risks of rainwater tanks of different sizes not meeting the demand The rainwater tank storage is assumed to be filled by rainfall runoff and, if instantaneous demand cannot be met, supplemented by tap water. Two types of household demands are supplied by the rainwater tank: toilet flushing and garden/lawn irrigation.

8 In the remaining sections, the schematic of a single dwelling considered is first described, followed by the modelling of roof runoff collection and rainwater tank storage in time. Based on the results obtained from a water usage survey and measurement (Roberts 4 and 5), the stochastic models for toilet and gardening demands are developed. The climate data such as rainfall, temperature, and wind speed, collected by the Bureau of Meteorology are obtained as the basis for stochastic modelling of the quantity of runoff. The partitioned multi-objective risk method is then described and used for risk analysis of the hypothetical rainwater tank system. Though a rainwater tank system in a household is seemingly simple, the interaction of supply-demand in the temporal domain and the variability in climate condition as well as in water usage pattern make a closed-form solution impossible. Therefore, a Monte Carlo simulation technique is employed for computation of risk. The risk assessment technique demonstrated in this report can be applied to cluster as well as regional scales, and generalized to integrated urban water systems. These will be carried out in future work. Rainwater Tank Use in a Household The rainwater tank system considered in this report is assumed to be on a lot of a single dwelling, located in a suburb in Melbourne, of which the water supply is serviced by Yarra Valley Water. The layout of the dwelling is schematically shown in Fig.. The tank collects water from both the house roof and garage for use in garden irrigation and toilet flushing. The tank has a backup supply of tap water from the water service and spillage from the tank is directed to the storm water system. The rainwater is gravity fed to the tank and then pump fed to meet toilet and garden demands. The lot has a lawn area of m and a garden bed area of m, which corresponds to a property classified in Roberts (4) with a moderate size of lawn area and a small size of garden bed. For discussion in this report, the variables considered for risk assessment are classified as either factors or controls. A factor is defined as a variable that affects the performance of a system and is not adjustable via intervention, whereas a control is a variable that affects the performance of a system and that can be adjusted via intervention. Some factors relevant to rainwater tank systems are listed in Table and controls in Table.

9 To Storm Water Allotment = 45 m Shed Tank PUMP To Garden House To WC Roof area = m Fig. Schematic representation of single dwelling with a rainwater tank Table Factors affecting the performance of a rainwater tank Factor Domain Description/comments Climate Natural Includes precipitation, wind events, temperature, evaporation. Legislation/Regulation Political At the household level, regulation can be seen as a factor as it is outside the control of the individual householder. At larger scale, if water company risks of water usage reduction were being assessed, this could be a control. Water pricing Socio-economic Outside control of householder so is seen as a factor. Energy pricing Socio-economic As above. Table Control System monitoring (includes moisture sensor) Controls affecting the performance of a rainwater tank Description/comments Controls garden irrigation No water quality monitoring.

10 Raintank design (Potable water float valve, pump, mesh filter, tank and lid and overflow) Collection system design (includes roof, gutters and downpipes) Distribution system design (includes irrigation system, moisture sensor and indoor piping) System maintenance Soil structure Education Garden watering demand Toilet flushing frequency Not generally controlled by householder but by system manufacturer. Householder has control in selection of manufacturer. Not generally controlled by householder but by developer or plumber. Householder has control in selection of these. Not generally controlled by householder but by plumber. Householder has control in selection of tradesperson and their qualifications. Will be responsibility of householder. Householder could have an impact on soil structure in terms of house location/import of soils, addition of mulches, barks, nutrients. The householders level of education regarding rain tanks is under their control. Also linked to availability of information. The householder could chose low demand plants. The householder could reduce the number of toilet flushes. 3 Modelling of Roof Runoff and Rainwater Tank Storage Storage of rainwater in a rainwater tank is usually achieved by the collection of rainfall on a roof area connected by pipes to the tank. The process of collection, storage, and harvesting of rainwater can be represented by two modules (Mitchell et al. 8): a rainfall-runoff module for modelling runoff inflow from the roof catchment to the tank, and a storage module for modelling the variation over time of tank storage volume due to runoff inflow and rainwater harvesting. 3. Roof Runoff Module The quantity of rainfall runoff on a roof area depends on the intensity and duration of rainfall, evaporation, roof depression storage capacity that represents the amount of water retained by the roof, and some loss of runoff due to splashing, gutter overflow, etc. Roof retention of water may be dependent on the material and the slope of roof; however, Liaw and Tsai (4) found that the difference of retention between an inverted-v iron sheet roof and a level cement roof is negligible. Because the process of runoff is dynamic in nature, smaller time-step size can capture more accurately the variation of runoff in time. If one-minute time step size climate data are available, for example, then the rate of runoff at a given time step i, denoted by R i (mm/min), is in which F i and (mm), respectively; i be estimated by Ri Fi Si O i () O i are the rainfall intensity (mm/min) and roof depression storage capacity S is the rainwater retained in the roof store (mm) at time step i S can. i Si Si Fi Ri E i ()

11 in which E i is the rate of evaporation (mm/min). The rate of runoff (L/min) from a roof area of A (m ) is RA, i Ri A F L (3) where F L, the effective roof area loss factor, is the proportion of runoff lost due to splashing, gutter overflow, etc. 3. Tank Storage Module It is assumed that the rainwater tank is closed and leak-free; therefore, evaporation from the tank and losses because of leakage are neglected. In addition, though the natural processes of filling, spilling, and extracting water from a tank can happen in an arbitrary combination of orders, computer simulation of tank storage over time requires a pre-assumed order of these. Typically, extraction of water is assumed to occur either before or after spillage (Fewkes and Butler ; Liaw and Tsai 4). Fewkes and Butler () concluded that the assumption of water yield after spillage gives a conservative estimate of yield; in this report, yield after spillage is assumed in computation. The volume of water in tank before spillage, V B (L), at time step i is VB, i Vi R i (4) where V (L) is the volume of water stored in tank. The quantity of spillage, P (L), is P V C (5) i B, i where C is the tank capacity (L). The water yield, Y (L), subjected to demand D (L) at time step i is, therefore, Yi VB, i Pi D i (6) 4 Modelling of Domestic Demand Yarra Valley Water, a local water utility in Melbourne, conducted a two-stage project in 3 and 4 in order to enhance the understanding on residential water use. The first stage, conducted in the second half of 3, involved visiting and surveying 84 randomly selected households among the Yarra Valley Water customers on the appliance stock and water usage patterns of households (Roberts 4), which identified in detail all water using appliances, measured flow rates and flush volumes and ascertained usage behaviours for all water use. The second stage involved the selection of out of the 84 households surveyed in the first stage and installing meters and data loggers to collect water usage data at five-second intervals for two weeks in February 4 (summer) and two weeks in August 4 (winter). Per capita consumption estimates were derived in the second stage from the reported number of people in the households when the survey was undertaken. The water usage project conducted by Yarra Valley Water investigated both indoor and outdoor water usage of households. The types of indoor water use investigated include showers toilets clothes washers dishwashers evaporative air conditioners

12 tap use baths water leakage The types of outdoor water use include garden irrigation swimming pools outdoor spas car washing Among the surveyed water usage types, garden irrigation, pool use, and evaporative air conditioning generally occurred in summer only, while the usage patterns of all other water use types did not differ notably between summer and winter seasons. The daily water usage patterns in summer by Roberts (5) are shown in Fig.. In this report, garden watering and toilet flushing are considered for demand from rainwater tank storage. The occurrence rates of both will be deduced from that shown in Fig., as described in the following subsections. Fig. Percentages of hourly water use in summer days (from Roberts 5) 4. Toilet Flushing The domestic water usage monitoring program by Roberts (5) shows that toilet flushing has a clear diurnal pattern that occurs mostly from 8 am to am. The hourly flush rates per capita deduced from that survey are shown in Fig. 3. The flush rate of each occupant is 4. times per day (Roberts 5).

13 Hourly flush rate / person.4 Flush rate per person (half + full flushes) Fig. 3 Hour Hourly toilet flush rates 4 Roberts (4) gives the average volume of full and half flushes of some types of toilets, as 3 shown in Table 3, and the associated variability of flush volumes, as shown in Fig. 4. The 6/3- toilet i litre dual flush toilet is found to be the most used type (with 39.6% share of the surveyed households); therefore, in this report the 6/3-litre dual flush toilet is chosen for the hypothetical household Table 3 Measured flush volumes by toilet types (from Roberts 4) i

14 Fig. 4 Variability of flush volumes (from Roberts 4) With a lack of detailed data, the probability distribution of toilet flush volumes are assumed to follow beta distributions to capture the minimum, maximum, and median values revealed in Fig. 4. The beta distribution is described in Appendix A.. The lower and upper bounds as well as distribution parameters for both full and half flushes are listed in Table 4. The ratio of the times of half flush to the times of full flush is 7:73 (Roberts 5). Occurrences of toilet flushing are assumed to be independent of each other and, therefore, can be assumed to follow a Poisson process. Furthermore, the flush rates in a day are not uniform, as shown in Fig. 3, occurrences of toilet flushing in time need to be modeled by a non-stationary Poisson process (Ross ), which is briefly described in Appendix B. Table 4 Probability distribution parameters for toilet flush volumes Flush type Lower bound (L) Upper bound (L) Mean (L) Standard deviation (L) Full flush Half flush 6.9.8

15 Garden watering rate. Probability density function.5.4 Full flush Half flush Fig. 5 Flush volume (L) Probability density functions for full and half flush volumes of 6/3 dual flush toilets 4. Garden Watering To simulate the events of garden watering, the probabilistic characteristics of the occurrence time, the duration, and the intensity of watering need to be examined and modeled. Since handheld hose was the most used watering method (Roberts 5), constituting 57% of households surveyed, it is assumed to be the watering method in this report. Similar to toilet flushing, occurrences of garden watering exhibits a clear diurnal pattern (Roberts 5). The watering rate of a household using handheld hose is 3. times per week. If the watering events are assumed to be independent of each other and considering the fact that the occurrence rates are not uniform over the day, as is deduced from Fig. and shown in Fig. 6., non-stationary Poisson process is needed to model the occurrences of garden watering. Roberts (4) states that, for handheld hose watering, the mean and standard deviation of watering duration are roughly the same, and both are about 37 minutes (Roberts 5). This implies that the duration of watering may be modeled by an exponential distribution, as described in Appendix A...5 Diurnal pattern of garden watering rate Fig. 6 Hour Hourly garden watering rates.4.3 i

16 Probability density function The average flow rate for a handheld hose was found to be 4.8 L/min. Clearly there are lower and upper bounds for the flow rate, beta distribution is used as its probability model with the following parameters: lower bound = 5 L/min, upper bound = 5 L/min, mean = 4.8 L/min, and standard deviation = 3 L/min. The probability density function is shown in Fig Fig. 7 Flow rate (L/min) Probability density function for handheld hose flow rate According to Duncan and Mitchell (8), in addition to seasonality, garden watering generally occurred only when the maximum temperature of the previous 4 hours exceeds 8 C. The threshold of 8 C for maximum daily temperature is therefore used for the occurrence of garden irrigation. 5 Modelling of Rainfall Climate parameters are the dominant factors for the quantity of rainfall runoff. Precipitation is clearly the most important one. Since the household location is assumed to be in suburban Melbourne, typically rainfall is the only source of precipitation. Loss of runoff due to evaporation is neglected in this study because it accounts for only about % of the rain tank yield; however, in locations of warm climate and of high wind, the effect of evaporation may become notable. Modelling of evaporation is presented in Appendix C. Probabilistic modelling of rainfall events is derived on the basis of climate data collected from 997 to 3 by the Bureau of Meteorology (BOM) at its Melbourne Regional Office weather station (Station No. 867). An example of recorded rainfall in a 4-hour period is shown in Fig. 8. To simulate the events of rainfall, therefore, three properties need to be considered: () inter-arrival time of rainfall events, () duration of an event, and (3) intensity of rainfall. Note in Fig. 8 that the rainfall intensity was recorded with a minimum of. mm/min and with increments in the multiple of. mm/min; i.e. with intensity value in the sequence of.,.4,.6, and so forth.

17 Probability density function Rainfall intensity (mm) Fig. 8 Time (hr) Example of recorded rainfall in a 4-hour period.5 Histogram of inter-arrival time of rainfall > min Data Model Fig. 9 Time (min) Histogram and fitted Pareto density function for inter-arrival time of rainfall From the rainfall recorded by the BOM, the histogram of inter-arrival time and the fitted Pareto distribution density function are shown in Fig. 9. The lower bound of the distribution is, with a median value of 7 minutes and a mean value of 87.3 minutes. The lower bound and the median are used to determine the distribution parameter as.553. The Pareto distribution is described in Appendix A.4. Note that the maximum inter-arrival time from the rainfall data is about 3.4 days; therefore, in this report, the maximum inter-arrival time is assumed to be 5 days. Examination of the rainfall data reveals that most rainfall events have duration, denoted as a random variable by D (and a realization D denoted by d), of either (with a probability of.9) or minutes (with a probability of.4), while durations longer than minutes may be approximately modeled by a Pareto distribution with distribution parameter D.43. That is,.9 when d minute; p d.4 when d minutes; D 3 D D D d.37 otherwise. (7)

18 Probability density function. The histogram and the fitted Pareto distribution for duration > minutes are plotted in Fig.. Histogram of rainfall duration > min Data Model Fig. Duration (minutes) Histogram and fitted distribution density function for rainfall duration > minutes The values of recorded rainfall intensity, assumed to be a random variable denoted by I (and a realization of I denoted by i) were found mostly to be either. (with a probability of.85) or.4 mm/min (with a probability of.8). The probability distribution of I.6 mm/min may be approximately modelled as a shifted exponential distribution described in Appendix A.3. Therefore, the following distribution density function is assumed for rainfall intensity:. i.5.85 when.5 i.3;.3 i.3 pi i.8 when.3 i.6;.3 I e I i.6.3 otherwise. The histogram and the fitted density function for intensity >.6 mm/min are plotted in Fig. (8)

19 Probability density function..8 Histogram of rainfall intensity >.6 mm/min Data Model Fig. Intensity (mm/min) Histogram and fitted density function for rainfall intensity >.6 mm/min 6 Risk Assessment Methods Risk is commonly defined as a measure of the probability and consequence of adverse effects (e.g. AS/NZS 436 4; HB 3 4). With this definition, the most common quantification of risk adopted by many disciplines is usually expressed as the expected consequence, an operation that multiplies each possible event by its probability of occurrence and sums all these products over the entire universe of events. This operation literally treats adverse events of high consequences and low probabilities of exceedance as the same in importance as the events of low consequences and high probabilities of exceedance. In the face of some unforeseen catastrophes such as dams bursting, levee overtopping, the so called extreme events, the consequence is typically immense and sudden. The impact, and the fear, of such events to the people affected are much higher than that of the low-consequence events occurring at a much higher frequency. Moreover, managers and decision-makers are most likely concerned with the risk associated with a specific case under consideration, rather than the likelihood of the average adverse outcomes that may result from all possible risk scenarios. In this sense, the expected value of consequence is not only inadequate, but also likely to lead to fallacious interpretations. As an example to demonstrate the inadequacy of computing risk to be the expected consequence, suppose that a major flood, with an occurrence probability of 6, would endanger human life and cause environmental damage on 6 acres of land, and a minor flooding event, with an occurrence probability of., would flood an area of acres. By the definition of expected consequence, the contribution of the two events to the expected risk would be the same; 6 6 i.e... This would distort the relative importance of these events and consequences as they are viewed and evaluated by the decision-maker. One approach that expresses not only the average risks but also captures the significance of catastrophic events is the partitioned multi-objective risk method (PMRM) (Haimes 4). In addition to the unconditional expected risk, the PMRM considers conditional expected risks of a number of partitioned exceedance probability ranges. It was developed for the purpose of solving multi-objective problems of a probabilistic nature.

20 6. Partitioned Multi-objective Risk Method (PMRM) Conventionally risk is computed as the unconditional expectation of consequences. In contrast, the PMRM divides the domain of possible consequences into a number of consequence ranges, and computes conditional expectation of consequences for each of the consequence ranges. A conditional expectation of consequences is defined as the expected value of consequence given the occurrence of events within a specified consequence range. The values of conditional expectations are therefore dependent on where the probability axis is partitioned. If, for example, a decision-maker is concerned about the once-in-a-thousand-year events, the portioning should emphasise and reflect the risk related to that level of exceedance probabilities. By partitioning the consequence axis (and the corresponding probability axis) into a number of sub-ranges, the PMRM generates a number of conditional expected-value functions (termed risk functions), one for each range, which represent the risk given that the consequences fall within specific ranges of the probability of exceedance (Haimes 4). Fig. shows an example of partitioning the probability axis into three ranges and mapping these partitions onto the consequence axis. Fig. Mapping of the probability partitioning onto the consequence axis Let and be two exceedance probabilities partitioning the domain of consequence X into three ranges. The consequences that correspond to and are x and x, respectively, as shown in Fig.. Consequences less than x are considered to be of low severity occurring at high frequencies, and consequences greater than x are of high severity occurring at low frequencies. The consequences in between x and x may be considered of moderate severity. For each of the three ranges, the risk function, computed as the conditional expected consequence, provides a measure of risk associated with the range. Let r L x denote the risk function for the low-consequence, high-probability events and function of consequence X. Then x xp x dx px x the probability density X rl x (9) Similarly, the risk function, r M is x, for the moderate-consequence, moderate-probability events

21 M x xp x X x dx r x () and the risk function, r H x, for the high-consequence, low-probability events is xp x X x dx r x () H In contrast to the conditional risk functions, the conventional risk r E unconditional expected consequence, is x, expressed as the r x xp x dx () E X 7 Example: Risk Assessment for Supply-Demand of Rainwater Tank As an example of risk assessment for the use of rainwater tank in a household, consider a single dwelling with a rainwater tank installed, as shown in Fig.. The household, located in Yarra Valley of Melbourne with an allotment of 45 m, has four occupants (two adults and two children). It has a roof area of m for collecting rainwater to the rainwater tank. Garden irrigation and toilet flushing use water pumped from the tank that is topped with tap water when it is short in supply of rainwater. Two 6/3-litre dual flush toilets were installed in the household. A garden area of about 4 m is irrigated. Hand held hose is used for garden watering. Supply of rainwater and demands of gardening and toilet flushing are modeled according to the methods described in the previous sections. Monte-Carlo simulation is used for supplydemand analysis over one year, which is assumed to be in 3, the year at which the water usage survey conducted by Yarra Valley Water was carried out (Roberts 4). Note that in Melbourne the Stage drought restrictions were introduced in November and Stage restrictions in August 3. To investigate the effect of rainwater tank size, a range of tank sizes are used in computation:, 3, 4, 5, 6, 9, 3,5, 8,, and,5 L. For each tank size, a sample of size is simulated. The simulation time step size is one minute as the climate data of one-minute step size are available from the Australian Bureau of Meteorology. The initial tank storage level is assumed to be uniformly distributed between and the tank capacity. The roof area for rainwater collection is m. Following Mitchell et al. (8) the roof depression storage is assumed to be. mm and the effective roof area loss factor, accounting for the loss of runoff due to splashing and gutter overflow, is 5% of the roof runoff. For 6/3 dual flush toilets, the probability of full flush when flushing occurs is.73, and that of half flush is.7 (Roberts 5). As stated in Section 4., the flush volumes of both full and half flushes follow beta distributions with distribution parameters listed in Table 5, in which q and r are the probability distribution parameters, as described in Appendix A.. Table 5 Statistics and probability distribution parameters for flush volumes of 6/3 dual flush toilets Flush Type Minimum Maximum Mean St. Dev. q r (L) (L) (L) (L) Full Half

22 Exceedance probability Since a hand held hose is used for garden and lawn irrigation, the watering duration is assumed to be exponentially distributed with a mean duration of 37 minutes. The corresponding flow rate is assumed to follow a beta distribution with minimum and maximum flow rates of 5 and 5 L/min, respectively. The mean value and standard deviation of flow rate are 4.8 and 3 L/min, respectively, which give an estimate of the distribution parameters q 4.95 and r Risk of Demand not Met With each specific tank capacity and a sample of size obtained by Monte-Carlo simulation, a risk curve expressing the annual probability of exceedance versus the amount of demand not met can be derived, as shown in Fig. 3. It is clear in Fig. 3 that when the rainwater tank capacity becomes larger, the quantity of demand not met becomes less; i.e. the risk curve shift to the left kl tank 6-kL tank 8-kL tank Demand not met (kl/year) Fig. 3 Annual exceedance probability versus demand not met The risk analysis method of PMRM is used in this study. Therefore, to determine the risk of a specific demand not met, in addition to the conventional expected risk, the risk function, as defined in Eq. (), is also evaluated for the exceedance probability in the range of (,.5]. An event with annual exceedance probability of.5 has a return period of years. The conventional average risks and the low-probability risks for the range of tank capacities considered are plotted in Fig. 4. Fig. 4 shows that when the tank capacity increases up to around kl, it is effective in reducing the demand not met both in the average and the low-probability risks, though with a decreased rate of effectiveness. For a tank capacity beyond kl, however, reduction of the demand not met becomes marginal, meaning that the investment of using a large tank may not be justified if the quantity of demand not met is the major concern.

23 Volumetric reliability Tank capacity (kl) 3 Average risk Risk given PE < Demand not met (kl/year) Fig. 4 Tank capacity versus the average risk and the risk given the exceedance probability <.5 7. Volumetric Reliability A frequently used performance measure in rainwater tank analysis is the volumetric reliability, defined as the total volume of water supplied divided by the total demand during a period of time. Let Y denote the water supplied by the tank and D the demand, then for a time period with a number of time steps N, the volumetric reliability R V is expressed by N N R Y D (3) V i i i i For the example investigated in this report, the volumetric reliability versus the tank capacity is plotted in Fig. 5. It shows that, similar to the average risk curve in Fig. 4, increasing tank capacity is effective for increasing R V when the tank capacity is relatively low, but when the tank capacity is greater than, say kl, the gain in R V becomes marginal. Theoretically, R V will approach when the tank capacity approaches infinity Rainwater tank capacity (kl) Fig. 5 Volumetric reliability versus tank capacity

24 Annual cost ($) Annual cost ($) 7.3 Optimal Tank Capacity This subsection investigates the choice of optimal tank capacity on the basis of minimum annual water cost. The annual water cost includes the metered water cost and the rainwater tank cost averaged over its lifetime; e.g. if acquisition of a tank costs $5 and it is used for years, then the annualized tank cost is $75. Maintenance cost is not considered in this study. Therefore, the annual tank acquisition and metered water cost, C, is a tank cost Ca mean annual demand not met in kl C w (4) tank life in which C w is the metered water cost per kl. A range of rainwater tank sizes and the associated acquisition costs listed as at July 8 by Australian Eco Shop is considered, as listed in Table 6. Suppose that the metered water cost is $./kl. Fig. 6 (a) shows the annualized cost versus tank capacity for, 3, and 5 years of tank use. It is seen that for a tank lifetime of years, the lower the tank capacity, the lower the annual cost, whereas for tank lifetimes of 3 and 5 years, tank capacities of 4 kl and 5 kl, respectively, are the most economical. If the metered water cost is doubled up to $.4/kL, as shown in Fig. 6 (b), then for tank lifetimes of, 3, and 5 years, the most economical tank sizes are 5 kl, 5 kl, and 9 kl, respectively. An alternative way of finding the minimal annual cost is via the relationship between the annual cost and the mean annual demand not met, as shown in Fig. 7. For example, when the metered water cost is $.4/kL and the tank lifetime is years, then the minimum annual cost of $7 is achieved when the tank capacity is 5 kl, with which the unmet annual demand is 3.3 kl, as shown in Fig. 7 (b). If no rainwater tank is installed, then the mean annual demand of toilet flushing and garden watering, a combination of about 84.5 kl, would cost $3 with a metered water cost of $.4/kL. Table 6 Rainwater tank capacity and acquisition cost July 8, Australian Eco Shop) Capacity (L) Cost Metered water cost: $./kl years 3 year 5 years 5 Metered water cost: $.4/kL years 3 year 5 years Tank capacity (kl) Tank capacity (kl) (a) (b) Fig. 6 Annual cost versus tank capacity for metered water cost (a) $./kl, (b) $.4/kL

25 Annual cost ($) Annual cost ($) 5 Metered water cost: $./kl yrs 3 yrs 5 yrs 5 Metered water cost: $.4/kL yrs 3 yrs 5 yrs Demand not met (kl/year) Demand not met (kl/year) (a) (b) Fig. 7 Annual cost versus demand not met for metered water cost (a) $./kl, (b) $.4/kL 7.4 Resilience of Rainwater Tank Use Assessment of resilience for water resource systems has been an active research area (e.g. Carpenter et al. 5; Walker and Salt 6; Wang and Blackmore 9). The essential aspect of a resilient system is that it has an adequate capability to avert adverse consequences under disturbances, and a capacity of self-organization and adaptation, and thus displays a greater capacity to provide wanted services. Thus it is regarded as a key property of water resource systems if sustainable development is to be achieved. A quantitative resilience assessment framework for water resource systems has been developed (Wang and Blackmore 9), which considers (a) resilience against crossing a system performance threshold, and (b) resilience for response and recovery after a disturbance. Both types of resilience can be modified by self-organization, redundancy, human actions and governance, etc; therefore, they are dependent to some extent upon the system s adaptive capacity supported by either endogenous or exogenous forces, or both. This subsection illustrates a simple example of quantitative resilience assessment of the use of a rainwater tank in a household based on the framework by Wang and Blackmore (9). The household, with an allotment of 45 m, has four occupants (two adults and two children) and is located in an area in which water is supplied by Yarra Valley Water, Melbourne (annual rainfall 734 mm). It has a roof area of m for collecting rainwater to the rainwater tank. Water pumped from the tank is used for garden irrigation and toilet flushing. Two 6/3-litre dual flush toilets are installed in the house. A garden/lawn area of about 4 m is irrigated using hand held hoses. The characteristics of rainfall events, toilet flushing, and garden/lawn watering are assumed to conform to those discussed in the previous sections. Monte-Carlo simulation for supply-demand analysis of the installed rainwater tank for a period of one year and a sample of size 3 is conducted. The volumetric reliability, defined in Eq. (3), is used as a measure for the resilience against crossing a performance threshold. In addition, the resilience for response/recovery, R T, is defined as the inverse of mean sojourn time (days) in the empty state (Hashimoto et al. 98), as follows, N e R N sojourn time in the empty state (5) T e i i or defined as the inverse of the maximum sojourn time (days) in the empty state (Moy et al. 986), as follows R max sojourn time in the empty state (6) T i,, Ne i

26 where N e in Eqs. (5) and (6) is the number of times the tank enters the empty state in one year. A sojourn time is a duration in which the tank stays empty. In this example, R V.5 is prescribed as the threshold for regime change; i.e., if RV is not greater than.5, the installed tank is deemed unacceptable and a tank of larger capacity is required. Consider that a, L rainwater tank was installed. The average R V from the simulation results is.5, barely above the prescribed threshold of.5; R T.74, R T.59, and a mean number of empty states of 8. The, L rainwater tank is thus regarded as having no resilience against performance threshold and a higher-capacity system is required. Accordingly, a, L tank was used and its estimated R V, R T, and R T are.665,.68, and.6, respectively, with a mean number of empty states of 7.5. The, L tank has an R V value significantly above.5, R T.68.59, and a much lower number of empty states, though the difference of the R T values of.74 and.6 is not statistically significant. It is concluded, therefore, that the, L system has higher resilience than the L system with respect to the resilience against performance threshold and with respect to that for recovery from the maximum sojourn time in the empty state. Instead of replacing the, L tank with one of larger capacity, some adaptive measure may be adopted to improve the resilience measures. For example, the garden vegetables and lawn grass may be replaced with species that require less water and are more drought resistant such that the weekly watering frequency is halved, i.e.,.6 times/week in summer. It is found that R V.64.5, R T.5, R T.67, and mean number of empty states equals 9. A similar conclusion is drawn for this case as for the L tank when compared to the L tank with typical garden plants and lawn. One operational strategy to improve R T is to top the tank with tap water, an exhibition of adaptive operational capacity, when it is short in rainwater supply. The simulated results for R V and R T for the three cases are plotted in Fig. 8. Note that the behavior of this simple rainwater tank system can be easily modeled and its performance is highly correlated with the climate parameter, therefore in a sense this system exhibits essentially a linear behavior. If a system involves more complicated nonlinear behavior, then such behavior needs to be taken into account in the system modeling. Furthermore, since the properties and operations of this rainwater tank can be expressed in probabilistic terms, similar methods may be applied to other systems at cluster or regional scale provided their system properties and behavior can be captured satisfactorily using probabilistic methods. Otherwise, if the system involves uncertainties beyond the realm of probability theory, particularly those related to adaptive capacity/management, other uncertainty modeling tools, e.g., fuzzy sets, evidence theory, and monotone measures (Ayyub and Klir 6), may be needed. Applicability of such tools will be a subject of future studies.

27 Resilience measure Resilience measure Resilience measure RV RT 3 Sample number (a) RV RT 3 Sample number (b) RV RT 3 Fig. 8 R V and T Sample number (c) R values for (a) L, (b) L, and (c) L (with water-efficient garden/lawn), rainwater tanks 8 Concluding Comments This report demonstrates the use of a risk assessment technique for rainwater tank system installed in a single dwelling located in a suburb in Yarra Valley, Melbourne. Because of limited data obtained, both on demand and supply, the time frame under consideration in the analysis is assumed to be in 3, the time during which the usage survey and measurements in Yarra Valley were conducted (Roberts 4 and 5). Projection of risk into the future is highly

28 uncertain largely because of the effect of envisaged climate change on temperature and rainfall patterns as implied in an Intergovernmental Panel on Climate Change report (IPCC 7) that no global climate models have been validated to be able to predict accurately the future climates. In addition, the water usage patterns in households are likely to alter as well after mandatory water restriction policies were implemented in response to the varying severity of water shortages. Despite the limitations and inherent uncertainties described above, dependent on the availability of quantity observational data, the technique used in this report can be applied to any locations in Australia, and to a community at cluster or regional scales. Similarly, it can be expanded to consider integrated urban water systems composed of potable water supply, storm and waste water discharges. An important implication for risk assessment and management is in support of the objective of sustainable development (Brundtland 987) that is stated as development that meets the needs of the present without compromising the ability of future generations to meet their own needs. To achieve sustainable development of urban water systems, a key property to consider is resilience (Blackmore and Plant 8; Wang and Blackmore 8). The essential aspect of a resilient system is a system that has an adequate capacity to avert adverse consequences under disturbances and therefore has a greater capacity to provide wanted services. This report gives a simple example of quantitative resilience assessment of the use of a rainwater tank in a household. Though resilience per se remains an active research area (e.g. Bruneau and Reinhorn 6; Hollnagel et al. 6; Walker and Salt 6; Wang and Blackmore 9), the undertakings of both risk management and resilience analysis are to enhance the sustainability of water resource systems. Studies on finding methodologies that integrate consideration of both risk and resilience as tools for enhancing sustainable water resources are warranted. Appendix A. Probability Distribution Functions As a reference, this section describes the probability distribution functions used in this report. Appendix A.. The Beta Distribution In some engineering applications, there are random variables that possess finite lower and upper bound values; in these cases, probability distributions with finite lower and upper bounds would be appropriate. The beta distribution is one of the few distributions appropriate for a random variable whose possible values are bounded. If the lower and upper limits are a and b, respectively, then the probability density function (PDF) of X with a beta distribution is X q r q r x a b x f x a x b q r q r b a (7) otherwise in which q and r are the parameters of the distribution, and by By integration-by-parts, we obtain k is the gamma function defined k x k x e dx when k. (8) k k k (9)

29 The cumulative distribution function (CDF) of the beta distribution x q r x q r x a b x F x f x dx dx X a X q r q r a b a can only be evaluated numerically. The mean value and variance of X with a beta distribution between a and b are X () q x a b a q r () X b a qr q r q r () Also the mode of X, x, is x a b a q q r (3) Appendix A.. The Exponential Distribution The PDF of the exponential distribution is x f X x e x otherwise where λ is the distribution parameter. The corresponding CDF is x F x e x X otherwise The mean and variance of X with an exponential distribution are X and respectively. (4) (5) X, Appendix A.3. The Shifted Exponential Distribution In Eqs. (4) and (5), the exponential distribution starts at x. In general, the distribution can start at any positive value of x. The corresponding PDF and CDF starting at a are x a f X x e x a otherwise x a F x e x X otherwise The mean value and variance of X with a shifted exponential distribution are X a and X, respectively. Appendix A.4. The Pareto Distribution With a location parameter a, a, and a shape parameter λ,, the PDF and CDF of a random variable X with Pareto distribution are a f x x a X (6) (7) x (8) otherwise

30 The mean value of X is the median of X is and the variance of X is x FX x x a a (9) otherwise X a x (3) xm a (3) X a The main difference between a shifted exponential distribution and a Pareto distribution is that the Pareto distribution has a heavier tail. (3) Appendix B. Non-stationary Poisson Process A stochastic process can be modeled as a Poisson process if each occurrence of the process is independent of previous occurrences. It is a stationary process if its rate of occurrence is constant in time; otherwise, it is non-stationary (or non-homogeneous). Ignoring non-stationarity in input stochastic processes to simulation models can lead to model invalidity and thus erroneous results. For instance, when garden irrigation is necessary as discussed in this report, if the rate of watering is assumed to be uniform in the day and night, rather than as that shown in Fig. 6 which clearly displays a diurnal variation, then the chance of garden irrigation happening in the early morning, say around am, would be the same as that in the evening around 6 pm, which is obviously untrue to a typical household. Mathematically, a non-stationary Poisson process is defined as follows (Ross ). If events occur randomly in time, and N t denotes the number of events that occur by time t, then N t, t constitutes a non-stationary Poisson process with intensity function t, t, if the following conditions are satisfied: N ; The numbers of events that occur in disjoint time intervals are independent; P exactly event between t and t x lim x x P or more events between t and t x lim x x A direct simulation algorithm proposed by Harrod and Kelton (6) is used in this study. As an example to see the effect of non-stationarity, Fig. 9 shows from a simulation result the instants of gardening watering in a six-day period, in which the instants of watering occur either in the afternoon or in the evening. This example shows as expected that we observe the occurrences of events more around the time intervals when t is high, and of less events in the intervals when t is low. t