FUZZY RELIABILITY ANALYSIS FOR THE EVALUATION OF WATER RESOURCE SYSTEMS PERFORMANCE

Transcription

1 FUZZY RELIABILITY ANALYSIS FOR THE EVALUATION OF WATER RESOURCE SYSTEMS PERFORMANCE Ibrahim El-Baroudy, Shohan Ahmad and Slobodan P. Simonovic Department o Civil and Environmental Engineering, Institute or Catastrophic Loss Reduction, University o Western Ontario, London, ON, Canada N6A 5B9 Abstract One o the main goals o water resources systems design is to ensure that a system perorms satisactorily under a wide range o possible uture conditions. Water resource systems exhibit a high level o spatial complexity. Many elements o complex water resource systems are vulnerable to temporary disruption in service due to natural hazards or human error whether unintentional as in the case o operational errors and mistakes or due to intentional causes such as a terrorist act. Engineering risk and reliability analysis is a general methodology or the quantiication o uncertainty and the evaluation o its consequences or the saety o engineering systems. The irst step in any risk analysis is to identiy the risk, clearly detailing all sources o uncertainty that may contribute to the risk o ailure. The quantiication o risk is the second step where the eects o the uncertainties are measured using dierent system perormance indices and igures o merit. This presentation explores the utility o uzzy set theory in the ield o water resource reliability analysis and proposes three uzzy reliability indices: () a combined reliability-vulnerability index, () a robustness index, and (3) a resiliency index. These indices were successully tested using a case study o London regional water supply system. Extension o these three indices to spatial evaluation o uncertainty involved in lood plain management is presented using a case study o the Medway creek (London, Ontario). Introduction Large and complex engineering systems are subject to a wide range o possible uture conditions. Most o these conditions cannot be controlled or predicted with an acceptable degree o accuracy. Uncertainty associated with the quantiication o these potential conditions is imposing a great challenge to systems design, planning and management. Water resource systems in general include dierent types o interconnected components serving broad geographic regions. Thereore, water resource systems are at risk o ailure due to natural hazards or anthropogenic causes, whether unintentional (operational errors and mistakes) or intentional (terrorist act). Engineering risk and reliability analysis is a general methodology or the quantiication o uncertainty and the evaluation o its consequences or the saety o engineering systems (Ganoulis 994). The probabilistic reliability analysis has been extensively used in the past. It depends on nondeterministic models that incorporate a measure o randomness as a way to express uncertainty (Klir and

2 Yuan 995). Randomness is not the only source o uncertainty. Simonovic (997) states that the two major sources o uncertainty are randomness and lack o knowledge. As a result, uncertainty is either an objective act o the phenomenon under consideration or a subjective impression o human perception (Zimmermann 00). The probable perormance o the water resource system is oten described by dierent perormance indices and igures o merit such as reliability, vulnerability, robustness and resiliency. The early work o Hashimoto et al. (98a, 98b) is the basis or use o perormance indices or evaluating the risk to and reliability o water resource systems. The characteristics o many inputs to the system cannot always be measured precisely or ormulated in a proper probabilistic conceptualization, especially in the absence o the necessary data. The subjective judgment o the decision maker to estimate the probability distribution o a random event (subjective probability approach o Vick, 00) is another approach to deal with data insuiciency. The third approach is Bayes s theory where engineering judgment is integrated with observed inormation. The choice o a Bayesian approach or any subjective probability distribution presents real challenges. For instance, it is diicult to translate prior knowledge into a meaningul probability distribution, especially in the case o multi-parameter problems (Press, 003). Thereore, the probabilistic approach usually ails to address the problems o human error, subjectivity, and the lack o system perormance history and records. The concept o uzzy sets is a conceptual and mathematical ramework in which imprecise and vague phenomena can be studied (Zimmermann, 00). Thereore, uzzy set theory and uzzy logic are used to overcome ambiguity or lack o knowledge in human conception o real lie phenomena as a source o uncertainty. El-Baroudy and Simonovic (004) proposed three uzzy indices or system perormance evaluation. The proposed uzzy indices quantiy the reliability, vulnerability, robustness and resiliency o multi-component systems relecting dierent systems conigurations in the uzzy environment. This study explores the utility o uzzy perormance indices or evaluating the perormance o a complex water resource system. Regional water supply system or the City o London, Ontario, Canada is used as the irst case study. The system lacks previous experience o signiicant ailure events. Thereore the probabilistic reliability analysis can not be conducted. The second case study is used to present spatial extension o uzzy perormance indices to spatial evaluation o uncertainty involved in lood plain management. Data rom the Medway creek (London, Ontario) are used in this study. The paper briely introduces three proposed uzzy reliability indices. This introduction is ollowed by the presentation o two case studies. The results are ollowed by the conclusions o the perormed analysis.

3 Methodology or the application o uzzy perormance indices Acceptable level o perormance The uzzy reliability analysis quantiies the uncertainty in system inputs and the accepted unsatisactory perormance threshold through the use o an appropriate uzzy membership unction. The uzzy ailure event is represented by a uzzy membership unction that includes dierent ailure levels. This membership unction is a ormal mathematical representation o the acceptable level o system perormance. The acceptable level o perormance, also, relects the decision-makers ambiguous and imprecise perception o risk. The acceptable level o perormance is quantiied by: LR = x x x - x () where: LR is the reliability index o the acceptable level o perormance; and x and x are the bounds o the acceptable ailure region, as shown in Figure. System-state and system-ailure membership unctions System reliability analysis uses load and resistance as the undamental concepts to deine the risk o system ailure, (Simonovic, 997). In water resource systems, load and resistance are replaced by the corresponding system inputs. For example, demand and supply are used in the context o water supply system. The system demand is deined as the variable that relects dierent water requirements that may be imposed over the useul lie o the system (Ang and Tang, 984). System supply is deined as the system characteristic variable which describes the capacity o the system to meet the demand. The uzzy reliability analysis uses membership unctions to express uncertainty in both, load and resistance, o each system component. Construction o the membership unction is based on the system design data and choice o a suitable shape determined by the application context. Triangular and trapezoidal shapes are the simplest membership unction shapes. Membership unctions o system components are aggregated to calculate uzzy system perormance indices using uzzy operators. Aggregation o these membership unctions results in the system-state membership unction o the wholesystem (El-Baroudy and Simonovic, 004). The system-ailure membership unction is used to represent the system s time o recovery rom the ailure state. Multi-component systems have several system-ailure membership unctions representing 3

4 the system-ailure or each component. Aggregation o these membership unctions, also, results in a system-ailure membership unction o the whole system. Fuzzy perormance indices Fuzzy combined reliability-vulnerability index - The compatibility between the system-state and the acceptable level o perormance is the basis or the calculation o the uzzy combined reliabilityvulnerability perormance index. It is illustrated in Figure and as ollows: Weighted overlap area Compatibility Measure (CM) = Weighted area o system -state unction () Thereore, the uzzy combined reliability-vulnerability index can be expressed as ollows: RE max { CM, CM,...CM i } max{ LR, LR,...LR } max i K = (3) i K i LR where: RE is the combined uzzy reliability-vulnerability index; LR max is the reliability index o the acceptable level o perormance with which the system-state has the maximum compatibility value(cm); LR i is the reliability index o the i-th acceptable level o perormance; CM i is the compatibility index or system-state with the i-th acceptable level o perormance; and K is the total number o deined acceptable levels o perormance. Fuzzy robustness index - The uzzy robustness measures the system s ability to adapt to a wide range o possible uture load conditions. The uzzy orm o change in uture conditions is obtained through the redeinition o acceptable level o perormance and the change in system-state membership unction. As a result, the system s uzzy robustness index is deined as the change in the compatibility measure: RO = (4) CM CM 4

5 where: RO is the uzzy robustness index; CM is the compatibility measure beore the change in conditions; and CM is the compatibility measure ater the change in conditions. Complete Failure Region Acceptable Failure Region Complete Saety Region.0 Acceptable Level o perormance Membership value System-state Universe o discourse Figure. Fuzzy representation o the acceptable level o perormance and system-state Fuzzy resiliency index - The time required to recover rom the ailure state is represented as a uzzy set. System recovery time varies depending on the type o ailure. Thereore, a series o uzzy membership unctions are developed to represent recovery rom dierent types o ailure. The center o gravity o the maximum uzzy recovery time is used as a real number representation o the system recovery time. The system uzzy resiliency index is determined to be the inverse value o the center o gravity (Klir et al. 997): RS = t T ~ t (t) dt T ~ (t) dt t t t (5) 5

6 where: RS is the uzzy resiliency index; T ~ (t) is the system uzzy maximum recovery time; t is the lower bound o the support o the system recovery time; and t is the upper bound o the support o the system recovery time. Results and discussion Regional water supply system or the city o London case study The City o London regional water supply system consists o two main components; (i) the Lake Huron Primary Water Supply System (LHPWSS), and (ii) the Elgin Area Primary Water Supply system (EAPWSS). Three uzzy perormance indices are used to asses the perormance o the water supply system or two dierent shapes o membership unction. Table shows that the uzzy combined reliability-vulnerability index or the LHPWSS system is higher than that o the EAPWSS system or both triangular and trapezoidal shapes o membership unctions (i.e. at least 0 times higher). This is supported by the act that LHPWSS system has more redundant components, than that o EAPWSS system, which increases its reliability. Additionally, the components o the LHPWSS system are designed with an overload supply capacity o 35% that aects positively the reliability o the system. In conclusion, the LHPWSS system is more reliable and less vulnerable to disruption in service than the EAPWSS system. Robustness index shows a similar behavior or both membership unction shapes. Thereore, EAPWSS system is more sensitive to possible change in the demand conditions than LHPWSS. The uzzy resiliency index or the EAPWSS system is 4 times higher than or the LHPWSS system. However, the resiliency index is not sensitive to the shape o the membership unction. This is due to the act that the resiliency index uses the center o gravity (COG) o the system-ailure membership unction, and the change in shape does not aect its COG. Table. The LHPWSS and EAPWSS systems uzzy perormance indices Fuzzy Perormance Index LHPWSS EAPWSS Triangular Trapezoidal Triangular Trapezoidal Combined Reliability-Vulnerability Robustness (level level ) NA NA Robustness (level 3 level ) NA NA Robustness (level 3 level ) Resiliency

7 Medway creek looding case study The southern part o the Medway creek (London, Ontario) loodplain is selected or the spatial extension o the uzzy reliability analysis and its application to lood management. Fuzzy risk analysis o Medway creek is carried out rom Arva to the conluence o Medway creek and the North Thames river. London is experiencing tremendous urban development. Due to rapid urbanization, land is becoming more impervious. This change results in an increase in excess runo that makes the area more vulnerable to loods. I a severe lood occurs in the city o London, it can cause large damage to residential and agricultural lands within the loodplain. Proper assessment o lood risk and damage is essential or sustainable lood risk management. Spatial uzzy perormance indices - Integration o uzzy perormance indices computation with GIS allows or spatial uzzy reliability analysis. Each cell in GIS map is considered a decision making location or which the computation o uzzy indices is done using equations () (5). The uzzy lood damage membership unctions or agricultural land, residential land, one and two story buildings are developed based on the lood damage data. The compliance o the lood damage membership unction with dierent acceptable levels lood damage is assessed or every grid cell. The maximum value o the compatibility measures or every grid cell in space is combined into a single raster image. The designated maps are used to determine the uzzy combined reliability-vulnerability, uzzy robustness and uzzy resiliency indices. Computational procedure or the spatial uzzy combined reliability-vulnerability index is illustrated in Figure and the inal map o uzzy reliability-vulnerability index is in Figure 3(a). The GIS maps containing the inverse o the dierence in compatibility measures between two acceptable levels o lood damage represent the spatial uzzy robustness index. The uzzy robustness index measures the adaptability to change in the acceptable level o lood damage. Figure 3(b) shows the values o uzzy robustness index or Medway creek. Time to recover rom lood damage is determined using recovery time-damage relationship. Uncertainty in the value o recovery time is accounted or using a triangular uzzy membership unction. A single lood damage recovery time is used to determine the uzzy resiliency index or each grid cell and the inal map is shown in Figure 3(c). Generally, lands close to Medway Creek s (low level lands) are ound to be more vulnerable to looding. Areas close to the conluence o Medway creek and North Thames river are the most vulnerable to lood damage. 7

8 Fuzzy calculation GIS maps Membership value STEP membership unction 0 Lower bound o lood damage Modal value o lood damage Upper bound o lood damage STEP Weighted area o lood damage membership unction Membership value Area under the triangular uzzy lood damage membership unction 0 Calculation o the weighted area o the lood damage membership unction STEP 3 Acceptable level o lood damage Membership value 0 First acceptable level o lood damage Second acceptable level o lood damage Third acceptable level o lood damage Weighted area approach STEP 4 Calculation o the overlap area & weighted overlap areas Membership value Common overlap area Area o high signiicance Area o low signiicance Weighted overlap area or st acceptable level o lood damage Weighted overlap area or nd acceptable level o lood damage Weighted overlap area or 3 rd acceptable level o lood damage 0 Weighted Overlap area Compatibil ity Measure ( CM ) = Weighted area o lood damage membership unction STEP 5 Calculation o compatibility Membership value Commom overlap area Area o high signiicance Area o low signiicance Compatibility measure (CM) or st acceptable level o lood damage Compatibility measure (CM) or nd acceptable level o lood damage Compatibility measure (CM3) or 3 rd acceptable level o lood damage 0 Reliability index STEP 5 Fuzzy reliability index RE = i K { } m ax { L R, L R,...L R i} m ax C M, C M,...C M L R i K i m ax Figure. Spatial uzzy combined reliability-vulnerability index calculation 8

9 (a) (b) (c) Figure 3. Spatial uzzy indices or Medway creek looding 9

10 Conclusions Water resource systems are vulnerable to a wide variety o hazards that could potentially limit their ability to perorm satisactorily. The diversity o uncertainty sources presents a great challenge to water resource systems design, planning and management. The probabilistic approach usually ails to address the problems o human error, subjectivity, and the lack o system perormance history and records. This paper illustrates the use o new uzzy perormance indices as criteria or system perormance evaluation using the city o London regional water supply system. The authors extended the application o uzzy indices to account or spatial uncertainty in lood plain management o the Medway creek. Reerence () Ang, H-S and Tang, H Probability concepts in engineering planning and design. John Wiley & Sons, Inc, USA. () El-Baroudy, I. and S. Simonovic 004. Fuzzy criteria or the evaluation o water resource systems perormance. Water Resource Research, 40: W0503, doi:09/003wr0088. (3) Ganoulis, J. G Engineering risk analysis o water pollution: probabilities. VCH, Weinheim, The Netherlands. (4) Hashimoto, T., J. R. Stedinger and D. P. Loucks 98a. Reliability, resiliency, and vulnerability criteria or water resources system perormance evaluation. Water Resources Research, 8, No. : 4-0. (5) Hashimoto, T., D. P. Loucks, and J. R. Stedinger 98b. Robustness o water resources systems. Water Resources Research, 8, No. : -6. (6) Klir, G. and B. Yuan 995. Fuzzy sets and uzzy logic: theory and applications. Prentice-Hall, Inc, NJ, USA. (7) Klir, G. J., St. Clair,U. H. and Yuan, B Fuzzy set theory: oundations and applications. Prentice-Hall, Inc., NJ, USA. (8) Press, S. J. (003), Subjective and Objective Bayesian Statistics: Principles, Models, and Applications, John Wiley & Sons, Inc, USA. (9) Simonovic, S. P Risk in sustainable water resources management, Sustainability o Water Resources Under Increasing Uncertainty, Fith Scientiic Assembly o IAHS, Proceedings o the Rabat Symposium S, IAHS Publication, No. 40, pp.3-7. (0) Vick, S. (00), Degrees o Belie: Subjective Probability and Engineering Judgment, ASCE Press, Reston, VA, USA. () Zimmermann, H.-J. 00. Fuzzy sets theory and applications, Kluwer Academic Publisher, USA. 0