Fuzzy optimization model for water quality management of a river system - Closure

A fuzzy waste-load allocation model, FWLAM, is developed for water quality management of a river system using fuzzy multiple-objective optimization. An important feature of this model is its capability to incorporate the aspirations and conflicting objectives of the pollution control agency and dischargers. The vagueness associated with specifying the water quality criteria and fraction removal levels is modeled in a fuzzy framework. The goals related to the pollution control agency and dischargers are expressed as fuzzy sets. The membership functions of these fuzzy sets are considered to represent the variation of satisfaction levels of the pollution control agency and dischargers in attaining their respective goals. Two formulations—namely, the MAX-MIN and MAX-BIAS formulations—are proposed for FWLAM. The MAX-MIN formulation maximizes the minimum satisfaction level in the system. The MAX-BIAS formulation maximizes a bias measure, giving a solution that favors the dischargers. Maximization of the bias measure attempts to keep the satisfaction levels of the dischargers away from the minimum satisfaction level and that of the pollution control agency close to the minimum satisfaction level. Most of the conventional water quality management models use waste treatment cost curves that are uncertain and nonlinear. Unlike such models, FWLAM avoids the use of cost curves. Further, the model provides the flexibility for the pollution control agency and dischargers to specify their aspirations independently.

Fuzzy Logic-Based Approaches in Water Resource System Modelling

Recent research in modelling uncertainty in water resource systems has highlighted the use of fuzzy logic-based approaches. A number of research contributions exist in the literature that deal with uncertainty in water resource systems including fuzziness, subjectivity, imprecision and lack of adequate data. This chapter presents a broad overview of the fuzzy logic-based approaches adopted in addressing uncertainty in water resource systems modelling. Applications of fuzzy rule-based systems and fuzzy optimisation are then discussed. Perspectives on the scope for further research are presented.

A fuzzy multistage evolutionary (FUME) clustering technique

n this paper, a multistage evolutionary scheme is proposed for clustering in a large data base, like speech data. This is achieved by clustering a small subset of the entire sample set in each stage and treating the cluster centroids so obtained as samples, together with another subset of samples not considered previously, as input data to the next stage. This is continued till the whole sample set is exhausted. The clustering is accomplished by constructing a fuzzy similarity matrix and using the fuzzy techniques proposed here. The technique is illustrated by an efficient scheme for voiced-unvoiced-silence classification of speech.

A Fuzzy Approximation Scheme for Sequential Learning in Pattern Recognition

An adaptive learning scheme, based on a fuzzy approximation to the gradient descent method for training a pattern classifier using unlabeled samples, is described. The objective function defined for the fuzzy ISODATA clustering procedure is used as the loss function for computing the gradient. Learning is based on simultaneous fuzzy decisionmaking and estimation. It uses conditional fuzzy measures on unlabeled samples. An exponential membership function is assumed for each class, and the parameters constituting these membership functions are estimated, using the gradient, in a recursive fashion. The induced possibility of occurrence of each class is useful for estimation and is computed using 1) the membership of the new sample in that class and 2) the previously computed average possibility of occurrence of the same class. An inductive entropy measure is defined in terms of induced possibility distribution to measure the extent of learning. The method is illustrated with relevant examples.

A fuzzy logic based dynamic wave model inversion algorithm for canal regulation

A fuzzy logic based centralized control algorithm for irrigation canals is presented. Purpose of the algorithm is to control downstream discharge and water level of pools in the canal, by adjusting discharge release from the upstream end and gates settings. The algorithm is based on the dynamic wave model (Saint-Venant equations) inversion in space, wherein the momentum equation is replaced by a fuzzy rule based model, while retaining the continuity equation in its complete form. The fuzzy rule based model is developed on fuzzification of a new mathematical model for wave velocity, the derivational details of which are given. The advantages of the fuzzy control algorithm, over other conventional control algorithms, are described. It is transparent and intuitive, and no linearizations of the governing equations are involved. Timing of the algorithm and method of computation are explained. It is shown that the tuning is easy and the computations are straightforward. The algorithm provides stable, realistic and robust outputs. The disadvantage of the algorithm is reduced precision in its outputs due to the approximation inherent in the fuzzy logic. Feed back control logic is adopted to eliminate error caused by the system disturbances as well as error caused by the reduced precision in the outputs. The algorithm is tested by applying it to water level control problem in a fictitious canal with a single pool and also in a real canal with a series of pools. It is found that results obtained from the algorithm are comparable to those obtained from conventional control algorithms.

A fuzzy dynamic flood routing model for natural channels

A fuzzy dynamic flood routing model (FDFRM) for natural channels is presented, wherein the flood wave can be approximated to a monoclinal wave. This study is based on modification of an earlier published work by the same authors, where the nature of the wave was of gravity type. Momentum equation of the dynamic wave model is replaced by a fuzzy rule based model, while retaining the continuity equation in its complete form. Hence, the FDFRM gets rid of the assumptions associated with the momentum equation. Also, it overcomes the necessity of calculating friction slope (S-f) in flood routing and hence the associated uncertainties are eliminated. The fuzzy rule based model is developed on an equation for wave velocity, which is obtained in terms of discontinuities in the gradient of flow parameters. The channel reach is divided into a number of approximately uniform sub-reaches. Training set required for development of the fuzzy rule based model for each sub-reach is obtained from discharge-area relationship at its mean section. For highly heterogeneous sub-reaches, optimized fuzzy rule based models are obtained by means of a neuro-fuzzy algorithm. For demonstration, the FDFRM is applied to flood routing problems in a fictitious channel with single uniform reach, in a fictitious channel with two uniform sub-reaches and also in a natural channel with a number of approximately uniform sub-reaches. It is observed that in cases of the fictitious channels, the FDFRM outputs match well with those of an implicit numerical model (INM), which solves the dynamic wave equations using an implicit numerical scheme. For the natural channel, the FDFRM Outputs are comparable to those of the HEC-RAS model.

Structural Damage Detection Using Modal Curvature and Fuzzy Logic

A fuzzy logic system (FLS) with a new sliding window defuzzifier is proposed for structural damage detection using modal curvatures. Changes in the modal curvatures due to damage are fuzzified using Gaussian fuzzy sets and mapped to damage location and size using the FLS. The first four modal vectors obtained from finite element simulations of a cantilever beam are used for identifying the location and size of damage. Parametric studies show that modal curvatures can be used to accurately locate the damage; however, quantifying the size of damage is difficult. Tests with noisy simulated data show that the method detects damage very accurately at different noise levels and when some modal data are missing.

Estimation of fuzzy memberships from histograms

Based on the conclusions drawn in the bijective transformation between possibility and probability, a method is proposed to estimate the fuzzy membership function for pattern recognition purposes. A rational function approximation to the probability density function is obtained from the histogram of a finite (and sometimes very small) number of samples. This function is normalized such that the highest ordinate is one. The parameters representing the rational function are used for classifying the pattern samples based on a max-min decision rule. The method is illustrated with examples.

Damage assessment of structures with uncertainty by using mode-shape curvatures and fuzzy logic

Uncertainties associated with the structural model and measured vibration data may lead to unreliable damage detection. In this paper, we show that geometric and measurement uncertainty cause considerable problem in damage assessment which can be alleviated by using a fuzzy logic-based approach for damage detection. Curvature damage factor (CDF) of a tapered cantilever beam are used as damage indicators. Monte Carlo simulation (MCS) is used to study the changes in the damage indicator due to uncertainty in the geometric properties of the beam. Variation in these CDF measures due to randomness in structural parameter, further contaminated with measurement noise, are used for developing and testing a fuzzy logic system (FLS). Results show that the method correctly identifies both single and multiple damages in the structure. For example, the FLS detects damage with an average accuracy of about 95 percent in a beam having geometric uncertainty of 1 percent COV and measurement noise of 10 percent in single damage scenario. For multiple damage case, the FLS identifies damages in the beam with an average accuracy of about 94 percent in the presence of above mentioned uncertainties. The paper brings together the disparate areas of probabilistic analysis and fuzzy logic to address uncertainty in structural damage detection.

An imprecise fuzzy risk approach for water quality management of a river system

Uncertainty plays an important role in water quality management problems. The major sources of uncertainty in a water quality management problem are the random nature of hydrologic variables and imprecision (fuzziness) associated with goals of the dischargers and pollution control agencies (PCA). Many Waste Load Allocation (WLA)problems are solved by considering these two sources of uncertainty. Apart from randomness and fuzziness, missing data in the time series of a hydrologic variable may result in additional uncertainty due to partial ignorance. These uncertainties render the input parameters as imprecise parameters in water quality decision making. In this paper an Imprecise Fuzzy Waste Load Allocation Model (IFWLAM) is developed for water quality management of a river system subject to uncertainty arising from partial ignorance. In a WLA problem, both randomness and imprecision can be addressed simultaneously by fuzzy risk of low water quality. A methodology is developed for the computation of imprecise fuzzy risk of low water quality, when the parameters are characterized by uncertainty due to partial ignorance. A Monte-Carlo simulation is performed to evaluate the imprecise fuzzy risk of low water quality by considering the input variables as imprecise. Fuzzy multiobjective optimization is used to formulate the multiobjective model. The model developed is based on a fuzzy multiobjective optimization problem with max-min as the operator. This usually does not result in a unique solution but gives multiple solutions. Two optimization models are developed to capture all the decision alternatives or multiple solutions. The objective of the two optimization models is to obtain a range of fractional removal levels for the dischargers, such that the resultant fuzzy risk will be within acceptable limits. Specification of a range for fractional removal levels enhances flexibility in decision making. The methodology is demonstrated with a case study of the Tunga-Bhadra river system in India.

Automata and logics over finitely varying functions

We extend some of the classical connections between automata and logic due to Büchi (1960) [5] and McNaughton and Papert (1971) [12] to languages of finitely varying functions or “signals”. In particular, we introduce a natural class of automata for generating finitely varying functions called View the MathML source’s, and show that it coincides in terms of language definability with a natural monadic second-order logic interpreted over finitely varying functions Rabinovich (2002) [15]. We also identify a “counter-free” subclass of View the MathML source’s which characterise the first-order definable languages of finitely varying functions. Our proofs mainly factor through the classical results for word languages. These results have applications in automata characterisations for continuously interpreted real-time logics like Metric Temporal Logic (MTL) Chevalier et al. (2006, 2007) [6] and [7].

Fuzzy waste load allocation model: a multiobjective approach

Fuzzy Waste Load Allocation Model (FWLAM), developed in an earlier study, derives the optimal fractional levels, for the base flow conditions, considering the goals of the Pollution Control Agency (PCA) and dischargers. The Modified Fuzzy Waste Load Allocation Model (MFWLAM) developed subsequently is a stochastic model and considers the moments (mean, variance and skewness) of water quality indicators, incorporating uncertainty due to randomness of input variables along with uncertainty due to imprecision. The risk of low water quality is reduced significantly by using this modified model, but inclusion of new constraints leads to a low value of acceptability level, A, interpreted as the maximized minimum satisfaction in the system. To improve this value, a new model, which is a combination Of FWLAM and MFWLAM, is presented, allowing for some violations in the constraints of MFWLAM. This combined model is a multiobjective optimization model having the objectives, maximization of acceptability level and minimization of violation of constraints. Fuzzy multiobjective programming, goal programming and fuzzy goal programming are used to find the solutions. For the optimization model, Probabilistic Global Search Lausanne (PGSL) is used as a nonlinear optimization tool. The methodology is applied to a case study of the Tunga-Bhadra river system in south India. The model results in a compromised solution of a higher value of acceptability level as compared to MFWLAM, with a satisfactory value of risk. Thus the goal of risk minimization is achieved with a comparatively better value of acceptability level.

Classification of Indian meteorological stations using cluster and fuzzy cluster analysis, and Kohonen artificial neural networks

The present study deals with the application of cluster analysis, Fuzzy Cluster Analysis (FCA) and Kohonen Artificial Neural Networks (KANN) methods for classification of 159 meteorological stations in India into meteorologically homogeneous groups. Eight parameters, namely latitude, longitude, elevation, average temperature, humidity, wind speed, sunshine hours and solar radiation, are considered as the classification criteria for grouping. The optimal number of groups is determined as 14 based on the Davies-Bouldin index approach. It is observed that the FCA approach performed better than the other two methodologies for the present study.

Fuzzy-logic-based health monitoring and residual-life prediction for composite helicopter rotor

A health-monitoring and life-estimation strategy for composite rotor blades is developed in this work. The cross-sectional stiffness reduction obtained by physics-based models is expressed as a function of the life of the structure using a recent phenomenological damage model. This stiffness reduction is further used to study the behavior of measurable system parameters such as blade deflections, loads, and strains of a composite rotor blade in static analysis and forward flight. The simulated measurements are obtained using an aeroelastic analysis of the composite rotor blade based on the finite element in space and time with physics-based damage modes that are then linked to the life consumption of the blade. The model-based measurements are contaminated with noise to simulate real data. Genetic fuzzy systems are developed for global online prediction of physical damage and life consumption using displacement- and force-based measurement deviations between damaged and undamaged conditions. Furthermore, local online prediction of physical damage and life consumption is done using strains measured along the blade length. It is observed that the life consumption in the matrix-cracking zone is about 12-15% and life consumption in debonding/delamination zone is about 45-55% of the total life of the blade. It is also observed that the success rate of the genetic fuzzy systems depends upon the number of measurements, type of measurements and training, and the testing noise level. The genetic fuzzy systems work quite well with noisy data and are recommended for online structural health monitoring of composite helicopter rotor blades.

Fuzzy-expert approach for voltage-reactive power dispatch

The voltage stability control problem has become an important concern for utilities transmitting power over long distances. This paper presents an approach using fuzzy set theory for reactive power control with the purpose of improving the voltage stability of a power system. To minimize the voltage deviations from pre-desired values of all the load buses, using the sensitivities with respect to reactive power control variables form the basis of the proposed fuzzy logic control (FLC). Control variables considered are switchable VAR compensators, On Load Tap Changing (OLTC) transformers and generator excitations. Voltage deviations and controlling variables are translated into fuzzy set notations to formulate the relation between voltage deviations and controlling ability of controlling devices. The developed fuzzy system is tested on a few simulated practical Indian power systems and some IEEE standard test systems. The performance of the fuzzy system is compared with conventional optimization technique and results obtained are encouraging. Results obtained for a 24 - node equivalent EHV system of part of Indian southern grid and IEEE New England 39-bus system are presented for illustration purposes. The proposed Fuzzy-Expert technique is found suitable for on-line applications in energy control centre as the solution is obtained fast with significant speedups.