Search results for: constrained directions control

Authors: Saroj Kumar Sarangi, M. C. Ray

Abstract:

This paper deals with the analysis of active constrained layer damping (ACLD) of doubly curved laminated composite shells using active fiber composite (AFC) materials. The constraining layer of the ACLD treatment has been considered to be made of the AFC materials. A three dimensional energy based finite element model of the smart doubly curved laminated composite shell integrated with a patch of such ACLD treatment has been developed to demonstrate the performance of the patch on enhancing the damping characteristics of the doubly curved laminated composite shells. Particular emphasis has been placed on studying the effect of variation of piezoelectric fiber orientation angle in the constraining AFC layer on the control authority of the ACLD patch.

Keywords: Active constrained layer damping, Active fibercomposites, Finite element modeling, First order shear deformationtheory.

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Authors: Md. Sayeed Salam

Abstract:

This paper presents an efficient emission constrained economic dispatch algorithm that deals with nonlinear cost function and constraints. It is then incorporated into the dynamic programming based hydrothermal coordination program. The program has been tested on a practical utility system having 32 thermal and 12 hydro generating units. Test results show that a slight increase in production cost causes a substantial reduction in emission.

Keywords: Emission constraint, Hydrothermal coordination, and Economic dispatch algorithm.

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Authors: Sayeed Salam

Abstract:

This paper presents an efficient emission constrained hydrothermal scheduling algorithm that deals with nonlinear functions such as the water discharge characteristics, thermal cost, and transmission loss. It is then incorporated into the hydrothermal coordination program. The program has been tested on a practical utility system having 32 thermal and 12 hydro generating units. Test results show that a slight increase in production cost causes a substantial reduction in emission.

Keywords: Emission constraint, Hydrothermal coordination, and Hydrothermal scheduling algorithm.

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Authors: A. Shirzadeh Chaleshtari, Sh. Shadrokh

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Renewable and non-renewable resource constraints have been vast studied in theoretical fields of project scheduling problems. However, although cumulative resources are widespread in practical cases, the literature on project scheduling problems subject to these resources is scant. So in order to study this type of resources more, in this paper we use the framework of a resource constrained project scheduling problem (RCPSP) with finish-start precedence relations between activities and subject to the cumulative resources in addition to the renewable resources. We develop a branch and bound algorithm for this problem customizing precedence tree algorithm of RCPSP. We perform extensive experimental analysis on the algorithm to check its effectiveness and performance for solving different instances of the problem in question.

Keywords: Resource constrained project scheduling problem, cumulative resources, branch and bound algorithm, precedence tree.

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Authors: Myoung Ro Kim, Gilwon Yoon

Abstract:

A game using electro-oculography (EOG) as control signal was introduced in this study. Various EOG signals are generated by eye movements. Even though EOG is a quite complex type of signal, distinct and separable EOG signals could be classified from horizontal and vertical, left and right eye movements. Proper signal processing was incorporated since EOG signal has very small amplitude in the order of micro volts and contains noises influenced by external conditions. Locations of the electrodes were set to be above and below as well as left and right positions of the eyes. Four control signals of up, down, left and right were generated. A microcontroller processed signals in order to simulate a DDR game. A LCD display showed arrows falling down with four different head directions. This game may be used as eye exercise for visual concentration and acuity. Our proposed EOG control signal can be utilized in many other applications of human machine interfaces such as wheelchair, computer keyboard and home automation.

Keywords: DDR game, EOG, eye movement.

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Authors: Shewit Tsegaye, Fekadu Shewarega

Abstract:

This paper presents a survey of articles, books and reports, which articulate the recent trends and aspects of Security Constrained Economic Dispatch (SCED). The period under consideration is 2008 through 2018. This is done to provide an up-to-date review of the recent major advancements in SCED, the state-of-the-art since 2008, identify further challenging developments needed in smarter grids, and indicate ways to address these challenges. This study consists of three areas of interest, which are very important and relevant for articulating the recent trends of SCED. These areas are: (i) SCED of power system with integrated renewable energy sources (IRES), (ii) SCED with post contingency corrective actions and (iii) Artificial intelligence based SCED.

Keywords: Security constrained economic dispatch, SCED of power system with IRES, SCED with post contingency corrective actions, artificial intelligence based SCED, IRES.

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Authors: Shyam S.N. Perera

Abstract:

An optimal control problem for a mathematical model of efficiency of antiviral therapy in hepatitis B virus infections is considered. The aim of the study is to control the new viral production, block the new infection cells and maintain the number of uninfected cells in the given range. The optimal controls represent the efficiency of antiviral therapy in inhibiting viral production and preventing new infections. Defining the cost functional, the optimal control problem is converted into the constrained optimization problem and the first order optimality system is derived. For the numerical simulation, we propose the steepest descent algorithm based on the adjoint variable method. A computer program in MATLAB is developed for the numerical simulations.

Keywords: Virus infection model, Optimal control, Adjoint system, Steepest descent

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Authors: Muhammad Rana, Quazi Mamun, Rafiqul Islam

Abstract:

In the Internet of Things (IoT), many devices are connected and accumulate a sheer amount of data. These Internet-driven raw data need to be transferred securely to the end-users via dependable networks. Consequently, the challenges of IoT security in various IoT domains are paramount. Cryptography is being applied to secure the networks for authentication, confidentiality, data integrity and access control. However, due to the resource constraint properties of IoT devices, the conventional cipher may not be suitable in all IoT networks. This paper designs a robust and effective lightweight cipher to secure the IoT environment and meet the resource-constrained nature of IoT devices. We also propose a symmetric and block-cipher based lightweight cryptographic algorithm. The proposed algorithm increases the complexity of the block cipher, maintaining the lowest computational requirements possible. The proposed algorithm efficiently constructs the key register updating technique, reduces the number of encryption rounds, and adds a layer between the encryption and decryption processes.

Keywords: Internet of Things, IoT, cryptography block cipher, s-box, key management, IoT security.

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Authors: Luciano Nieddu, Giuseppe Manfredi

Abstract:

In this paper a Pattern Recognition algorithm based on a constrained version of the k-means clustering algorithm will be presented. The proposed algorithm is a non parametric supervised statistical pattern recognition algorithm, i.e. it works under very mild assumptions on the dataset. The performance of the algorithm will be tested, togheter with a feature extraction technique that captures the information on the closed two-dimensional contour of an image, on images of industrial mineral ores.

Keywords: K-means, Industrial ores classification, Invariant Features, Supervised Classification.

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Authors: Rahmad Wisnu Wardana, Eakachai Warinsiriruk, Sutep Joy-A-Ka

Abstract:

Selecting the most suitable welding process usually depends on experiences or common application in similar companies. However, this approach generally ignores many criteria that can be affecting the suitable welding process selection. Therefore, knowledge automation through knowledge-based systems will significantly improve the decision-making process. The aims of this research propose integrated data envelopment analysis (DEA) and fuzzy credibility constrained programming approach for identifying the best welding process for stainless steel storage tank in the food and beverage industry. The proposed approach uses fuzzy concept and credibility measure to deal with uncertain data from experts' judgment. Furthermore, 12 parameters are used to determine the most appropriate welding processes among six competitive welding processes.

Keywords: Welding process selection, data envelopment analysis, fuzzy credibility constrained programming, storage tank.

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Authors: M. Qorbani

Abstract:

Neuromuscular control of posture as understood through studies of responses to mechanical sudden acceleration automatically has been previously demonstrated in individuals with chronic ankle instability (CAI), but the presence of acute condition has not been previously explored specially in a sudden acceleration. The aim of this study was to determine neuromuscular control pattern in those with and without unilateral acute ankle sprains. Design: Case - control. Setting: University research laboratory. The sinker–card protocol with surface translation was be used as a sudden acceleration protocol with study of EMG upon 4 posture stabilizer muscles in two sides of the body in response to sudden acceleration in forward and backward directions. 20 young adult women in two groups (10 LAS; 23.9 ± 2.03 yrs and 10 normal; 26.4 ± 3.2 yrs). The data of EMG were assessed by using multivariate test and one-way repeated measures 2×2×4 ANOVA (P< 0.05). The results showed a significant muscle by direction interaction. Higher TA activity of left and right side in LAS group than normal group in forward direction significantly be showed. Higher MGR activity in normal group than LAS group in backward direction significantly showed. These findings suggest that compared two sides of the body in two directions for 4 muscles EMG activities between and within group for neuromuscular control of posture in avoiding fall. EMG activations of two sides of the body in lateral ankle sprain (LAS) patients were symmetric significantly. Acute ankle instability following once ankle sprains caused to coordinated temporal spatial patterns and strategy selection.

Keywords: Neuromuscular response, sEMG, Lateral Ankle Sprain, posture.

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Authors: Viraj Tyagi, Ajai Jain, P. K. Jain, Aarushi Jain

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This paper presents a genetic algorithm based loading methodology for a capacity constrained job-shop with the consideration of alternative process plans for each part to be produced. Performance analysis of the proposed methodology is carried out for two case studies by considering two different manufacturing scenarios. Results obtained indicate that the methodology is quite effective in improving the shop load balance, and hence, it can be included in the frameworks of manufacturing planning systems of job-shop oriented industries.

Keywords: Manufacturing planning, loading, genetic algorithm, Job-Shop

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Authors: Jo-Han Chang, Chung-Jung Wu

Abstract:

This study involves a design proposal. The objective of is to create a seat arrangement model for public reading spaces that enable free arrangement without disturbing the users. Through a subjective perception scale, this study explored whether distance between seats and direction of seats influence individual subjective perceptions in a public reading space. This study also involves analysis of user subjective perceptions when reading in the settings on 3 seats at different directions and with 5 distances between seats. The results may be applied to public chair design. This study investigated that (a) whether different directions of seats and distances between seats influence individual subjective perceptions and (b) the acceptable personal space between 2 strangers in a public reading space. The results are shown as follows: (a) the directions of seats and distances between seats influenced individual subjective perceptions. (b) subjective evaluation scores were higher for back-to-back seat directions with Distances A (10cm) and B (62cm) compared with face-to-face and side-by-side seat directions; however, when the seat distance exceeded 114cm (Distance C), no difference existed among the directions of seats. (c) regarding reading in public spaces, when the distance between seats is 10cm only, we recommend arranging the seats in a back-to-back fashion to increase user comfort and arrangement of face-to-face and side- by-side seat directions should be avoided. When the seatarrangement is limited to face-to-face design, the distance between seats should be increased to at least 62cm. Moreover, the distance between seats should be increased to at least 114cm for side- by-side seats to elevate user comfort.

Keywords: Individual Subjective Perceptions, Personal Space, Seat Arrangement.

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Authors: R. Karbalaei, A. Ghaffari, R. Kazemi, S. H. Tabatabaei

Abstract:

An integrated vehicle dynamics control system is developed in this paper by a combination of active front steering (AFS) and direct yaw-moment control (DYC) based on fuzzy logic control. The control system has a hierarchical structure consisting of two layers. A fuzzy logic controller is used in the upper layer (yaw rate controller) to keep the yaw rate in its desired value. The yaw rate error and its rate of change are applied to the upper controlling layer as inputs, where the direct yaw moment control signal and the steering angle correction of the front wheels are the outputs. In the lower layer (fuzzy integrator), a fuzzy logic controller is designed based on the working region of the lateral tire forces. Depending on the directions of the lateral forces at the front wheels, a switching function is activated to adjust the scaling factor of the fuzzy logic controller. Using a nonlinear seven degrees of freedom vehicle model, the simulation results illustrate considerable improvements which are achieved in vehicle handling through the integrated AFS/DYC control system in comparison with the individual AFS or DYC controllers.

Keywords: Intelligent strategy, integrated control, fuzzy logic, AFS/DYC.

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Authors: Seong-Mo An, Rohit Ramesh, Young-Sook Lee, Wan-Young Chung

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The proper assessment of interaxial distance and convergence control are important factors in stereoscopic imaging technology to make an efficient 3D image. To control interaxial distance and convergence for efficient 3D shooting, horizontal 3D camera rig is designed using some hardware components like 'LM Guide', 'Goniometer' and 'Rotation Stage'. The horizontal 3D camera rig system can be properly aligned by moving the two cameras horizontally in same or opposite directions, by adjusting the camera angle and finally considering horizontal swing as well as vertical swing. In this paper, the relationship between interaxial distance and convergence angle control are discussed and intensive experiments are performed in order to demonstrate an easy and effective 3D shooting.

Keywords: Interaxial, Convergence, Stereoscopic, Horizontal 3D Camera Rig

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Authors: Songtao Chang, Yongji Wang, Lei Liu, Dangjun Zhao

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Reentry trajectory optimization is a multi-constraints optimal control problem which is hard to solve. To tackle it, we proposed a new algorithm named CDEN(Constrained Differential Evolution Newton-Raphson Algorithm) based on Differential Evolution( DE) and Newton-Raphson.We transform the infinite dimensional optimal control problem to parameter optimization which is finite dimensional by discretize control parameter. In order to simplify the problem, we figure out the control parameter-s scope by process constraints. To handle constraints, we proposed a parameterless constraints handle process. Through comprehensive analyze the problem, we use a new algorithm integrated by DE and Newton-Raphson to solve it. It is validated by a reentry vehicle X-33, simulation results indicated that the algorithm is effective and robust.

Keywords: reentry vehicle, trajectory optimization, constraint optimal, differential evolution.

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Authors: Miloš Šeda, Tomáš Březina

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In this paper, we focus on the use of knowledge bases in two different application areas – control of systems with unknown or strongly nonlinear models (i.e. hardly controllable by the classical methods), and robot motion planning in eight directions. The first one deals with fuzzy logic and the paper presents approaches for setting and aggregating the rules of a knowledge base. Te second one is concentrated on a case-based reasoning strategy for finding the path in a planar scene with obstacles.

Keywords: fuzzy controller, fuzzification, rule base, inference, defuzzification, genetic algorithm, neural network, case-based reasoning

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Authors: Charles D. Otine, Samuel B. Kucel, Lena Trojer

Abstract:

Selecting the data modeling technique for an information system is determined by the objective of the resultant data model. Dimensional modeling is the preferred modeling technique for data destined for data warehouses and data mining, presenting data models that ease analysis and queries which are in contrast with entity relationship modeling. The establishment of data warehouses as components of information system landscapes in many organizations has subsequently led to the development of dimensional modeling. This has been significantly more developed and reported for the commercial database management systems as compared to the open sources thereby making it less affordable for those in resource constrained settings. This paper presents dimensional modeling of HIV patient information using open source modeling tools. It aims to take advantage of the fact that the most affected regions by the HIV virus are also heavily resource constrained (sub-Saharan Africa) whereas having large quantities of HIV data. Two HIV data source systems were studied to identify appropriate dimensions and facts these were then modeled using two open source dimensional modeling tools. Use of open source would reduce the software costs for dimensional modeling and in turn make data warehousing and data mining more feasible even for those in resource constrained settings but with data available.

Keywords: About Database, Data Mining, Data warehouse, Dimensional Modeling, Open Source.

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Authors: Minghui Wang, Luping Xu, Juntao Zhang

Abstract:

In this paper, according to the classical algorithm LSQR for solving the least-squares problem, an iterative method is proposed for least-squares solution of constrained matrix equation. By using the Kronecker product, the matrix-form LSQR is presented to obtain the like-minimum norm and minimum norm solutions in a constrained matrix set for the symmetric arrowhead matrices. Finally, numerical examples are also given to investigate the performance.

Keywords: Symmetric arrowhead matrix, iterative method, like-minimum norm, minimum norm, Algorithm LSQR.

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Authors: Qintao Shen, Lei Luo, Jun Ma, Jie Yu, Qingbo Wu, Yongqi Ma, Zhengji Liu

Abstract:

Control Flow Integrity (CFI) is one of the most promising technique to defend Code-Reuse Attacks (CRAs). Traditional CFI Systems and recent Context-Sensitive CFI use coarse control flow graphs (CFGs) to analyze whether the control flow hijack occurs, left vast space for attackers at indirect call-sites. Coarse CFGs make it difficult to decide which target to execute at indirect control-flow transfers, and weaken the existing CFI systems actually. It is an unsolved problem to extract CFGs precisely and perfectly from binaries now. In this paper, we present an algorithm to get a more precise CFG from binaries. Parameters are analyzed at indirect call-sites and functions firstly. By comparing counts of parameters prepared before call-sites and consumed by functions, targets of indirect calls are reduced. Then the control flow would be more constrained at indirect call-sites in runtime. Combined with CCFI, we implement our policy. Experimental results on some popular programs show that our approach is efficient. Further analysis show that it can mitigate COOP and other advanced attacks.

Keywords: Contex-sensitive, CFI, binary analysis, code reuse attack.

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Authors: K. Krishna Veni, S. Raja Balachandar

Abstract:

This paper presents a heuristic to solve large size 0-1 Multi constrained Knapsack problem (01MKP) which is NP-hard. Many researchers are used heuristic operator to identify the redundant constraints of Linear Programming Problem before applying the regular procedure to solve it. We use the intercept matrix to identify the zero valued variables of 01MKP which is known as redundant variables. In this heuristic, first the dominance property of the intercept matrix of constraints is exploited to reduce the search space to find the optimal or near optimal solutions of 01MKP, second, we improve the solution by using the pseudo-utility ratio based on surrogate constraint of 01MKP. This heuristic is tested for benchmark problems of sizes upto 2500, taken from literature and the results are compared with optimum solutions. Space and computational complexity of solving 01MKP using this approach are also presented. The encouraging results especially for relatively large size test problems indicate that this heuristic can successfully be used for finding good solutions for highly constrained NP-hard problems.

Keywords: 0-1 Multi constrained Knapsack problem, heuristic, computational complexity, NP-Hard problems.

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Authors: Lotfi Nabli, Achraf Jabeur Telmoudi, Radhi M'hiri

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This paper proposes a modeling method of the laws controlling manufacturing systems with temporal and non temporal constraints. A methodology of robust control construction generating the margins of passive and active robustness is being elaborated. Indeed, two paramount models are presented in this paper. The first utilizes the P-time Petri Nets which is used to manage the flow type disturbances. The second, the quality model, exploits the Intervals Constrained Petri Nets (ICPN) tool which allows the system to preserve its quality specificities. The redundancy of the robustness of the elementary parameters between passive and active is also used. The final model built allows the correlation of temporal and non temporal criteria by putting two paramount models in interaction. To do so, a set of definitions and theorems are employed and affirmed by applicator examples.

Keywords: Manufacturing systems control, flow, quality, robustness, redundancy, Petri Nets.

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Authors: Alpana Tiwari, N. K. Gaur

Abstract:

We have incorporated the translational rotational (TR) coupling effects in the framework of three body force shell model (TSM) to develop an extended TSM (ETSM). The dynamical matrix of ETSM has been applied to compute the phonon frequencies of orientationally disordered mixed crystal (ND4Br)x(KBr)1-x in (q00), (qq0) and (qqq) symmetry directions for compositions 0.10≤x≤0.50 at T=300K.These frequencies are plotted as a function of wave vector k. An unusual acoustic mode softening is found along symmetry directions (q00) and (qq0) as a result of translation-rotation coupling.

Keywords: Orientational glass, phonons, TR-coupling.

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Authors: O. S. Ebrahim, M. A. Badr, Kh. H. Gharib, H. K. Temraz

Abstract:

This paper presents a methodology based on genetic algorithm (GA) to tune the parameters of proportional-integral-differential (PID) controllers utilized in the automatic voltage regulator (AVR) and speed governor of a brushless synchronous generator driven by three-stage steam turbine. The parameter tuning is represented as a nonlinear optimization problem solved by GA to minimize the integral of absolute error (IAE). The problem of integral windup due to physical system limitations is solved using simple anti-windup scheme. The obtained controllers are compared to those designed using classical Ziegler-Nichols technique and constrained optimization. Results show distinct superiority of the proposed method.

Keywords: Brushless synchronous generator, Genetic Algorithm, GA, Proportional-Integral-Differential control, PID control, automatic voltage regulator, AVR.

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Authors: S. Jarernprasert, E. Bazan-Zurita, P. C. Rizzo

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Currently, seismic probabilistic risk assessments (SPRA) for nuclear facilities use In-Structure Response Spectra (ISRS) in the calculation of fragilities for systems and components. ISRS are calculated via dynamic analyses of the host building subjected to two orthogonal components of horizontal ground motion. Each component is defined as the median motion in any horizontal direction. Structural engineers applied the components along selected X and Y Cartesian axes. The ISRS at different locations in the building are also calculated in the X and Y directions. The choice of the directions of X and Y are not specified by the ground motion model with respect to geographic coordinates, and are rather arbitrarily selected by the structural engineer. Normally, X and Y coincide with the “principal” axes of the building, in the understanding that this practice is generally conservative. For SPRA purposes, however, it is desirable to remove any conservatism in the estimates of median ISRS. This paper examines the effects of the direction of horizontal seismic motion on the ISRS on typical nuclear structure. We also evaluate the variability of ISRS calculated along different horizontal directions. Our results indicate that some central measures of the ISRS provide robust estimates that are practically independent of the selection of the directions of the horizontal Cartesian axes.

Keywords: Seismic, Directionality, In-Structure Response Spectra, Probabilistic Risk Assessment.

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Authors: Fatemeh Azimi, Razeeh Sadat Aboutalebi, Amir Abbas Najafi

Abstract:

In this paper the multi-mode resource-constrained project scheduling problem with discounted cash flows is considered. Minimizing the makespan and maximization the net present value (NPV) are the two common objectives that have been investigated in the literature. We apply one evolutionary algorithm named multiobjective particle swarm optimization (MOPSO) to find Pareto front solutions. We used standard sets of instances from the project scheduling problem library (PSPLIB). The results are computationally compared respect to different metrics taken from the literature on evolutionary multi-objective optimization.

Keywords: Evolutionary multi-objective optimization makespan, multi-mode, resource constraint, net present value.

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Authors: O. Castellanos Díaz, F. Schoeggl, H. W. Yarranton, M. A. Satyro, T. M. Lovestead, T. J. Bruno

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The composition, vapour pressure, and heat capacity of nine biodiesel fuels from different sources were measured. The vapour pressure of the biodiesel fuels is modeled assuming an ideal liquid phase of the fatty acid methyl esters constituting the fuel. New methodologies to calculate the vapour pressure and ideal gas and liquid heat capacities of the biodiesel fuel constituents are proposed. Two alternative optimization scenarios are evaluated: 1) vapour pressure only; 2) vapour pressure constrained with liquid heat capacity. Without physical constraints, significant errors in liquid heat capacity predictions were found whereas the constrained correlation accurately fit both vapour pressure and liquid heat capacity.

Keywords: Biodiesel fuels, Fatty acid methyl ester, Heat capacity, Modeling, Vapour pressure

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Authors: Oğuzhan Urhan

Abstract:

In this paper, an improved motion estimation (ME) approach based on weighted constrained one-bit transform is proposed for block-based ME employed in video encoders. Binary ME approaches utilize low bit-depth representation of the original image frames with a Boolean exclusive-OR based hardware efficient matching criterion to decrease computational burden of the ME stage. Weighted constrained one-bit transform (WC‑1BT) based approach improves the performance of conventional C-1BT based ME employing 2-bit depth constraint mask instead of a 1-bit depth mask. In this work, the range of constraint mask is further extended to increase ME performance of WC-1BT approach. Experiments reveal that the proposed method provides better ME accuracy compared existing similar ME methods in the literature.

Keywords: Fast motion estimation, low-complexity motion estimation, video coding.

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Authors: A. K. Sharma, N. P. Patidar, G. Agnihotri, D. K. Palwalia

Abstract:

This paper presents the performance state analysis of Self-Excited Induction Generator (SEIG) using Artificial Bee Colony (ABC) optimization technique. The total admittance of the induction machine is minimized to calculate the frequency and magnetizing reactance corresponding to any rotor speed, load impedance and excitation capacitance. The performance of SEIG is calculated using the optimized parameter found. The results obtained by ABC algorithm are compared with results from numerical method. The results obtained coincide with the numerical method results. This technique proves to be efficient in solving nonlinear constrained optimization problems and analyzing the performance of SEIG.

Keywords: Artificial bee colony, Steady state analysis, Selfexcited induction generator, Nonlinear constrained optimization.

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Authors: A.K.M. Khaled Ahsan Talukder, Taibun Nessa, Kaushik Roy

Abstract:

The weight constrained shortest path problem (WCSPP) is one of most several known basic problems in combinatorial optimization. Because of its importance in many areas of applications such as computer science, engineering and operations research, many researchers have extensively studied the WCSPP. This paper mainly concentrates on the reduction of total search space for finding WCSP using some existing Genetic Algorithm (GA). For this purpose, some controlled schemes of genetic operators are adopted on list chromosome representation. This approach gives a near optimum solution with smaller elapsed generation than classical GA technique. From further analysis on the matter, a new generalized schema theorem is also developed from the philosophy of Holland-s theorem.

Keywords: Genetic Algorithm, Evolutionary Optimization, Multi Objective Optimization, Non-linear Schema Theorem, WCSPP.

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