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Fuzzy cognitive maps for decision support to maintain water quality in
ageing water mains
Fuzzy cognitive maps for decision support to
maintain water quality in ageing water mains
Sadiq, R.; Kleiner, Y.; Rajani, B.B.
NRCC-47305
A version of this document is published in / Une version de ce document se trouve dans :
DMUCE 4, 4
thInternational Conference on Decision-Making in Urban
and Civil Engineering, Porto, Portugal, Oct. 28-30, 2004, pp. 1-10
1
Fuzzy cognitive maps for decision support to maintain water
quality in ageing water mains
R. Sadiq
1, Y. Kleiner
2, and B.B. Rajani
3Institute for Research in Construction, National Research Council Canada
1 rehan.sadiq@nrc-cnrc.gc.ca, 2 yehuda.kleiner@nrc-cnrc.gc.ca, 3 balvant.rajani@nrc-cnrc.gc.ca
Abstract
The prioritization of water mains for renewal requires the simultaneous consideration of their structural integrity and hydraulic performance as well as their contributions to the deterioration of water quality. Presently, several decision models exist for water main renewal. Most consider the structural integrity of the pipes as the sole decision criterion, although some consider also their hydraulic capacity and others consider only network reliability. Various attempts have been made by water utilities to consider multiple decision drivers in their prioritization process, however, these are done through simple point scoring methods, which are essentially qualitative, inherently subjective and do not consider input uncertainties. The impact of deteriorating pipes on water quality in the distribution network has not been considered in decision process.
This paper outlines a framework for assessing risk associated with water quality failures in distribution networks due to ageing mains. The available field data are both quantitative and qualitative, and when available, they are often uncertain and vague. Numerous factors affect water quality in the distribution system and the interactions amongst them are complex and often not well understood. For these and other reasons, fuzzy cognitive maps are examined as a decision support tool to develop an integrated approach for the renewal of water mains.
Keywords
Distribution networks, fuzzy cognitive maps, ageing water mains, and water quality.
1 Introduction
The level of uncertainty associated with a system is proportional to its complexity, which arises as a result of vaguely defined relationships among various entities, and randomness in the mechanisms governing the domain. The behaviour of the decision-maker and the uncertainty in the effects of a decision add to this complexity. Ross (2004) describes complex systems as those, which can not be analyzed easily and involve human judgment such as in biological processes, medicine, social issues, engineering analysis or design, economic, or political systems, where the vast arrays of inputs and outputs could not all possibly be captured analytically or controlled in any conventional sense. Moreover, the relationships between the causes and effects in these systems are generally not well understood, but often can be observed empirically. Complex systems consist of a large number of interacting entities that may be designated as subsystems, concepts, agents or components. Complex systems are highly nonlinear in nature and cannot be derived from sheer summation of the
2
behavior of individual agents. The modelling of complex dynamic systems requires methods that combine human knowledge and experience as well as expert judgment. A typical modern water supply is a complex system, that comprises water source(s), treatment plant(s), transmission mains, and the distribution network, which includes pipes, pumps and distribution tanks. While water quality can be compromised at any component, failure at the distribution level can be critical because it is closest to the point of delivery and, with the exception of rare filter devices at the consumer level, there are virtually no safety barriers before consumption. Water quality failures that compromise either the safety or the aesthetics of water in distribution networks, can generally be classified into the following major categories (Kleiner, 1998):
• Intrusion of contaminants into the distribution network through system
components whose integrity is compromised or through misuse or cross-connection or intentional introduction of harmful substances in the water distribution network.
• Regrowth of microorganisms in the distribution network.
• Microbial (and/or chemical) breakthrough and by-products, and residual
chemicals from the water treatment plant.
• Leaching of chemicals and corrosion products from system components into the
water.
• Permeation of organic compounds from the soil through system components into
the water supplies.
Water distribution networks may comprise (depending on the size of the water utility) thousands of kilometres of pipes, which can vary in age, material, installation practices and operational and environmental conditions. Since the pipes are not visible, it is relatively difficult and expensive to collect data on their performance and deterioration, and therefore few field data are typically available. Further, it is often difficult to determine or validate the exact cause of a water contamination event or an outbreak of a waterborne disease because such episodes are often investigated after the occurrence has ended. This multitude of water quality failure types, combined with the inherent complexity of the distribution networks, assures that risk analysis is a highly challenging task and subject to substantial uncertainties.
2
Water Quality in Distribution Networks
Water quality is generally defined by a collection of upper and lower limits on selected indicators (Maier, 1999). A water quality failure is often defined as an exceedence of one or more water quality indicators from specific regulations, or in the absence of regulations, exceedence of guidelines or self-imposed limits driven by customer service needs (Sadiq et al., 2004). Kirmeyer et al. (2000) identified and ranked important water quality concerns with respect to public and utility perceptions and satisfaction (Table 1). In North America, microbial safety was ranked as the number one priority by both water utilities and the public.
2.1 Deterioration
mechanisms
The water quality in a distribution network changes due to a variety of physical, chemical and microbiological processes. These processes are quite complex and are often not fully understood. Indications of deteriorating water distribution networks
3
include the increased frequency of leaks and breaks, taste and odour and red water complaints, reduced hydraulic capacity and increased disinfectant demands. Fig. 1 provides a simplified graphical illustration of these interactions. Additional information about the deterioration of water distribution networks and mitigation strategies are addressed in a series of white papers available on US EPA website (http://www.epa.gov/safewater/tcr/tcr.html, 2004).
Table 1. Rank of major water quality concerns in distribution network.
Water quality indicators and associated concerns in
distribution network Public Utilities
Safety (meeting regulations especially microbial) 1 1
Free of excess chlorine residual 2 -
Taste and odour 3 3
Good appearance 4 -
Uniform water quality 5 -
Disinfectant residual - 2 Corrosion control - 4 DBP formation - 5 9 Physico-chemical 9 Biological 9 Aesthetics Effects on water quality Back flow Cross-connection In flow High breakage frequency rate Internal corrosion and leaching
Broken pipes and deteriorated gaskets Maintenance events De-pressurized and low-pressure pipes Contaminated soil Contaminated groundwater Tuberculation Formation of biofilm Water quality from treatment plant Injured bacteria AOC, TOC, Nutrients Residual disinfectant DBPs In-line disinfection pH, alkalinity, DO, etc. Detachment and sloughing Regrowth and protection of microbes Transients
Figure 1. Conceptual map of water quality failures in water mains
Deterioration of the structural integrity of pipes can have a multi-faceted impact on water quality, especially in the domain of contaminant intrusion. Frequent pipe breaks increase the possibility of intrusion through the compromised sections in several ways.
4
During repairs, intrusion can occur if flushing and local disinfection are not performed adequately. Furthermore, the possibility of contaminant intrusion through unprotected cross connections increases since pipes are pressurized during repair. This de-pressurization will also increase the likelihood of contaminant intrusion in pipes with holes, which can be especially detrimental if the surrounding soil is contaminated or if there are leaky sewers nearby.
Pipe materials can affect water quality in various ways. Materials that are susceptible to corrosion may exacerbate bacterial regrowth when corrosion tubercles shield bacterial colonies from chlorination and sloughing. This bacterial regrowth, often in the form of biofilm can have an adverse effect on the quality of water as described later in this section. Contaminants such as hydrocarbons can permeate through plastic pipe walls or elastomeric gaskets. Leaching of chemical compounds from pipe walls into the bulk water can occur: e.g., arsenic, barium, cadmium, and chromium may leach from cement-based pipes under static conditions; chemical compounds from plastisizers in plastic pipes and pipe liners may leach; internal corrosion of metallic pipes and plumbing devices increases the concentration of metal compounds in the water (Kleiner, 1998).
Hydraulic/operating conditions including flow velocities, direction and water residence time may affect water quality through consumption of residual chlorine and consequently increase microbial growth. Certain hydraulic conditions can favour deposition and accumulation of sediments and consequently help microbial growth by protecting it from disinfectants. In addition, excessive operating pressures may increase the incidences of breaks and leaks in ageing water mains and ultimately lead to repairs and thus increase the risk of contaminant intrusions (Habibian, 1992). Biofilm is a deposit consisting of microorganisms, microbial products and detritus at the surfaces of pipes or tanks. Biological regrowth may occur when injured bacteria enter from the treatment plant into the distribution network. Bacteria can attach themselves to surfaces in storage tanks and on rough inner surfaces of water mains and rejuvenate and flourish under favourable conditions, e.g., nutrient supply such as organic carbon, long residence time, etc. The regrowth of organisms in the distribution network results in increased chlorine demand, which has two adverse effects: (a) a reduction in the level of free available chlorine may hinder the network’s ability to contend with local occurrences of contamination, and (b) an increased level of disinfection to satisfy the chlorine demand of biofilms results in higher concentrations of disinfection by-products (DBPs).
2.2 Water quality monitoring
Monitoring in the water supply system helps to anticipate and mitigate any deterioration in water quality. The National Guide to Sustainable Municipal Infrastructure Canada (2003, 2004) summarizes some of the benefits of implementing a program for water quality monitoring in the distribution network:
• reduces public health risk by early detection of poor water quality;
• meets legislated requirements;
• helps to make decisions for operation and maintenance activities;
• increases consumer confidence, fosters trust between consumers and water
utilities and reduces perceived risk;
5
• provides a pro-active approach to deal with emerging water quality issues in the
distribution network.
2.3 Water quality management
The management of water quality in a distribution network encompasses a variety of factors: selection of source water and type of water treatment, monitoring protocol, corrosion control programs, condition assessment for pipe deterioration, as well as construction, maintenance, and rehabilitation practices. Preventive strategies include maintenance of water pressure, identification & replacement of leaky water mains, maintenance & monitoring of residual chlorine, cross connection control programs, disinfection of pipes after repairs and corrosion control measures (Leland, 2002). A framework for water quality management is shown in Figure 2, where factors are partitioned into categories, namely, operation/design (hydraulics), pipe characteristics, site specific factors, pipe and water quality deterioration mechanisms, mitigative actions and water quality indicators. Each category in this framework consists of numerous agents, which represent a specific variable for pipe or water property, condition, process, behaviour or mechanism.
Water quality indicator Water quality deterioration h i Mitigative actions Pipe deterioration mechanisms
Site specific factors Water quality management
Operation and design Pipe
characteristics
Figure 2. A framework for water quality management for the distribution network.
3
Modelling of Complex Systems
3.1 Alternative
techniques
Most water distribution networks have only a limited number of water quality failures each year, making statistically significant generalizations difficult. The rarity of water quality failures belie their seriousness, however, as each failure indicates the potential for harmful health effects and public complaints. In complex systems with such data-sparse circumstances, expert knowledge and belief can serve as an alternative representation of a domain.
6
Artificial neural networks (ANN) are useful for data-based hypothesis generation in domains where causal relationships among variables are unknown. However, ANN requires substantial amounts of data for their training. Decision trees can be used to make inference about a domain where data are limited and if there is knowledge of the processes that describe the domain. Generally, decision trees are described by a dichotomous classification scheme to arrive at a probable explanation for each input. Bayesian networks can represent expert knowledge in domains where knowledge is uncertain, ambiguous, and/or incomplete. Pearl (1982) claimed that classical probability theory is a reliable method of representing uncertainty, around which an expert system methodology – the Bayesian network – can be built. But, Bayesian networks have serious shortcomings in obtaining reliable conditional probabilities, and may lead to computational intractability, and inability to model vagueness and ambiguity. In such frameworks, all events in a set are considered equal and assigned the same binary value - yes or no. This approach has very little relevance to most real-world problems as pointed out by Liu (2002).
Eden et al. (1992) defined a cognitive map as a “…directed graph characterized by a hierarchical structure which is most often in the form of a means/end graph.” Cognitive maps express the judgment that certain events or actions will lead to particular outcomes. Cognitive maps have been successfully used for decision-making, prediction, explanation and strategic planning.
3.2 Fuzzy cognitive map (FCM)
Fuzzy cognitive map (FCM), an extension of cognitive map, is an illustrative causative representation of the description and model of complex systems (Kosko, 1997). FCM draws a causal representation, which models the behavior of any system. FCM is an interactive structure of concepts, each of which interacts with the rest showing the dynamics and different aspects of the behavior of the system. Human experience and knowledge of the operation of complex systems are embedded in an FCM, i.e., knowledge gained about the operation of the system by human experts and its behavior under different circumstances.
FCM consists of nodes (concepts, agents) and weighted arcs (connection, edge), which are graphically illustrated as signed weighted graph with optional feedback loops. Nodes on the graph represent concepts describing behavioral characteristics of the system. Concepts can be inputs, outputs, variables, states, events, actions, goals, and trends of the system. Signed weighted arcs represent causal relationships (cause and effect) that exist among concepts, whose graphic display shows clearly which relationship and to what degree they are related.
Figure 3 illustrates a simple FCM consisting of six concepts Ci (i = 1, …, 6). The
value of Ci is denoted by A'i. Weight wij ∈ [-1, 1] represents the causal relationship
between concept i and concept j, where a negative sign represents inverse causation. This scheme may give rise to the following three types of interactions:
wij > 0 a positive causality, where an increase in the value of the i
th
concept causes an
increase in the value of the jth concept;
wij < 0 a negative causality, where an increase in the value of the i
th
concept causes a
decrease in the value of the jth concept;
wij = 0 no causal relationship between the i
th
7 C1 C2 C3 C5 C6 C4 w14 w54 w12 w56 w36 w23 w32 w42 w24 w52 w43 w16 w61 w46 w35
An edge matrix representing weights of connections in FCM
C1 C2 C3 C4 C5 C6 C1 0 +0.25 0 -0.25 0 +.25 C2 0 0 +0.5 +0.75 0 0 C3 0 +.25 0 0 +0.5 +0.5 C4 0 +0.5 -1 0 0 +0.75 C5 0 +1 0 +1 0 -0.25 C6 +0.75 0 0 0 0 0 W =
Figure 3. A simple cognitive map and its edge matrix
It should be noted that while A’i is the value of Ci over its ‘universe of discourse’,
actual calculations in FCM are done using Ai, where A’i is normalized (mapped) in the
interval [0, 1]. The transformation between Ai and A’i can be a linear or a non-linear
function, depending on the nature of the concept. Ai can be expressed either by a crisp
value or a fuzzy (linguistic) constant such as ‘high’, ‘medium’, ‘very low’, etc.
Kosko (1997) proposed a rule to calculate the value of each concept based on the influence of the interconnected concepts, where the content of the following function is normalized in the interval [-1, 1]:
1 0 1 0 1 2 1 1 2 1 1 ≤ ≤ ≤ ≤ ∑ + = ≠ = − − i j n j i i t j j ij t i i t j f k A w k A k k A (1)
where Atj is the normalized (Atj∈ [0, 1]) value (a.k.a. activation level) of concept Cj at
time step t, and f(x) is a threshold function. Generally, a sigmoid function
( )
x e x f −λ + = 1 1is used to constrain the value of f (x) in the interval [0, 1], where λ > 0
determines the steepness of f (x). The coefficient ki1 expresses the influence of
interconnected concepts in the configuration of the new value of concept Ai. For
example, in Figure 3 the concept C6 receives inputs from concepts C1, C3, C4 and C5.
If experts perceive that that C4 and C5 interact in such a way that both are fully
participating in impacting C6 then the ki1 associated with them will be closer to 1.
Similarly, kj2 accounts for the importance of C6 being at its activation level in the
previous time step. The selection of coefficients ki1 and kj2 depends on the nature and
8
suggested that the previous value of each concept did not participate in the calculation
of the new value of a concept, and thus assumed kj2 = 0 and proposed the use of only
coefficient ki1.
3.3 Simulating with FCMs
Let, A0 = [0, 0, 1, 0, 0, 0, 0] be an initial vector state in the FCM depicted in Figure 3,
i.e., concept C3 is ‘activated’ or ‘fired’, and let ki1 = 1 and kj2 = 0. Let the sigmoid
function with λ = 1 be used as a threshed function. Figure 4 represents the results of
Equation 1 simulated iteratively seven times. It can be seen that the FCM reaches an equilibrium state, (i.e. no change in activation levels is observed in two consecutive iterations) approximately after 5 iterations.
0.42 0.63 0.76 0 1 0.72 0.55 0.68 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 7 t A C1 C2 C3 C4 C5 C6
Figure 4. Results of FCM simulations
Once the FCM reaches equilibrium, the activation values provide the “triggering or firing” strength of those concepts for a given scenario. Generally, when the FCM reaches equilibrium, the activation levels are transformed back to the corresponding real values. These activation levels may be interpreted ‘quantitatively’ or ‘qualitatively’. For example an FCM shown in Figure 4 reaches an equilibrium state
vector A7 = [0.63, 0.76, 0.42, 0.72, 0.55, 0.68], which implies that, concept C1, for
example, is 63% (fired) of its maximum normalized value. The activation levels can
also be interpreted ‘qualitatively’, e.g., C1 can be described as ‘
moderately-to-significantly’ active and C2 can be described as ‘significantly’ active, etc.
In the water quality context, more than one agent (concept) in an FCM may provide a body-of-evidence for a proof of a particular water quality failure. For example, HPC and/or total coliforms both imply a risk of microbiological water quality failure. Similarly, various agents in FCM may contribute to aesthetic water quality (i.e., taste, colour, and odour). The activation levels at equilibrium provide ‘soft evidence’, which is further processed (fused) to make an inference of a specific outcome - a risk of water quality failure.
9
4 Proposed
Framework
The following five steps will be used to establish a model to assess risk of water quality failure in water distribution networks due to ageing water mains.
4.1 Knowledge acquisition
Knowledge acquisition consists of four distinct activities: preliminary analysis; literature review; survey/interview and solicitation of opinions of an expert panel. A preliminary analysis will help obtain an overview of the problem and determine potential modular categories that would be useful to classify various concepts. A preliminary analysis breaks down the items along categorical lines, which will help identify contributory factors. An in-depth literature review will follow a preliminary analysis. The result of this analysis will provide a more comprehensive understanding of items associated with water quality. With a more comprehensive understanding, questionnaires and interview sessions can be designed to query the knowledge of utility personnel and other professionals working in the water industry. Finally, experts will be asked to discuss and organize the available information to the extent possible to help fill identified knowledge gaps.
4.2 Knowledge
aggregation
This step allows the aggregation of knowledge acquired from various sources to develop a comprehensive FCM, which will represent the understanding of the experts about a water distribution network. Concepts are established and causal relationships determined and defined in an adjacency (edge) matrix. The mathematical formulations for ‘transformation functions’ are established to normalize the scale for each concept. The type of threshold functions, causal relationships and other coefficients are also defined.
4.3 Learning/training of FCM
Learning in FCM involves updating strengths of causal links. A learning strategy is to improve FCM by fine-tuning its initial causal link or edge strengths by applying training algorithms similar to those of ANN. An FCM is simulated for known scenarios and in case of discrepancies in the outcomes, weights are readjusted to avoid counter intuitive results.
4.4 Scenario analysis
Once an FCM is finalized and ‘reasonably’ trained, multiple scenarios are generated to study hidden patterns stored in the proposed model. For this purpose, various concepts are activated through random number generators and their effects on other concepts are studied. Various statistical techniques (e.g., using principal component analysis, cluster analysis) may be used to analyze these scenarios and sensitivity analysis (e.g., using rank correlation coefficients) is performed to determine the percentage contributions of concepts that could have caused given water quality failures.
4.5 Interpreting FCM
The outcome of an FCM is in the form of concepts being ‘activated’ at different levels after reaching equilibrium. The interpretation of these concepts will determine the risk of water quality failures. Two inferencing methods, namely, the fuzzy rule-based
10
model (Mamdani, 1977) and the fuzzy measures theory (Sugeno, 1977) are proposed as potential candidates to translate FCM outputs into a risk of water quality failure. Both techniques require expert judgement to define rules and weights, respectively. These rules (or weights) are policy-driven to reflect the preferences of regulators and decision-makers.
The proposed methodology will lead to a decision support tool, which will help to study the effects of ageing water mains on water quality, and quantify their impact to enable the consideration of water quality in the planning of water main renewal.
5 Acknowledgements
This paper presents results of a preliminary investigation of an on-going research project, which is co-sponsored by the American Water Works Association Research Foundation (AwwaRF) and National Research Council of Canada (NRC).
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