Probabilistic Optimization Problem Research Articles

Energy-efficient maximization algorithm for energy harvesting under imperfect channel information

In this paper, we propose a novel energy-efficient resource optimization scheme in non-orthogonal multiple access (NOMA) networks with simultaneous wireless information and power transfer (SWIPT). In this system, we consider practical imperfect channel information that accounts for random channel delays and channel estimation errors. Additionally, the small cell users (SCUs) in a heterogeneous network (HetNet) can harvest energy from the small base station (SBS) signal by energy harvesting. Based on the non-linear energy harvesting (NEH) model, the problem of maximizing energy efficiency with imperfect channel state information (CSI) is formulated. Since the formulated problem is a probabilistic mixed non-convex optimization problem, a two-stage algorithm is developed to jointly optimize sub-channel matching and power allocation. In particular, the problem is first transformed into a non-probabilistic problem through the relaxation method. For the sub-channels matching problem, a low-complexity many-to-many matching algorithm is designed to achieve dynamic matching based on the preference lists. For the power allocation problem, the closed-form transmission power of SCUs can be derived by the Dinkelbach method and Lagrangian dual approach. Simulation results show that the proposed scheme can achieve higher energy efficiency than existing linear energy harvesting (LEH)-NOMA and NEH-OFDMA schemes, with improvements of 37.99% and 84.69%, respectively.

Sequential Improvement Method in Probabilistic Criteria Optimization Problems for Linear-in-State Jump Diffusion Systems

Sequential Improvement Method in Probabilistic Criteria Optimization Problems for Linear-in-State Jump Diffusion Systems

A Data-Driven Decision-Making Model for Configuring Surgical Trays Based on the Likelihood of Instrument Usages

In order to perform a surgical procedure, substantial numbers of sterile instruments should be readily available to surgeons through the containers referred to as surgical trays and peel packs. After the procedure, all instruments in the opened containers, regardless of whether they have been used or not, must go through the labor-intensive re-sterilization process. Empirical studies have shown that the utilization rate of instruments within trays is very low due to not having optimized tray configurations. Additionally, surgical trays often include instruments that are not likely to be used but are included “just in case”, which imposes an additional cost on hospitals through unnecessary instrument re-sterilization. This study is the first analytical attempt to address the issue of configuring surgical trays based on the likelihood of instrument usage through formulating and solving a probabilistic tray optimization problem (PTOP). The PTOP model can serve as a decision support for surgeons by providing them with the tray’s probability of being used for optimally configured trays and its associated reprocessing costs. The PTOP is constructed upon an integer non-linear programming model. A decomposition-based heuristic and metaheuristic method coupled with two novel local search algorithms are developed to solve the PTOP. The application of this model can be illustrated through a case study. We discuss how hospitals could benefit from our model in reducing the costs associated with opening trays unnecessarily before a procedure. Additionally, we conducted a risk analysis to estimate the level of confidence for the recommended solution.

A smooth approximation approach for optimization with probabilistic constraints based on sigmoid function

Many practical problems, such as computer science, communications network, product design, system control, statistics and finance, etc.,can be formulated as a probabilistic constrained optimization problem (PCOP) which is challenging to solve since it is usually nonconvex and nonsmooth. Effective methods for the probabilistic constrained optimization problem mostly focus on approximation techniques, such as convex approximation, D.C. (difference of two convex functions) approximation, and so on. This paper aims at studying a smooth approximation approach. A smooth approximation to the probabilistic constraint function based on a sigmoid function is analyzed. Equivalence of PCOP and the corresponding approximation problem are shown under some appropriate assumptions. Sequential convex approximation (SCA) algorithm is implemented to solve the smooth approximation problem. Numerical results suggest that the smooth approximation approach proposed is effective for optimization problems with probabilistic constraints.

Three-stage relaxation-weightsum-correction based probabilistic reactive power optimization in the distribution network with multiple wind generators

The uncertainty of wind powers complicates the reactive power optimization in the distribution network. In this paper, the authors proposed a three-stage relaxation-weightsum-correction (TSRWC) based probabilistic reactive power optimization method to deal with the uncertainty and correlation of DFIG based wind generators in the distribution network. Firstly, by the inverse NataF transformation, an improved point estimate method is proposed to transform the uncertain wind speeds into a series of data samples with weights. The digital integral technology and root search method are employed to calculate the correlation matrix in the standard normal space. Then, with the consideration of active utilization of reactive power capability of DFIG based wind generators, the reactive power optimization problem in the distribution network corresponding to each data sample is formulated. Further, the three-stage relaxation-weightsum-correction solution method is proposed to solve the probabilistic reactive power optimization problem. Finally, the numerical simulation experiments are conducted on the real 19-node distribution network and the modified PG&E 69-node distribution network, respectively. The experimental results verified that by the proposed TSRWC based probabilistic reactive power optimization method, the expectation and deviation of power loss are more adjacent to the results from the Monte Carlo Simulation experiments than the scenario without the consideration of correlation. Meanwhile, no obvious increasement of consumed time appeared.© 2017 Elsevier Inc. All rights reserved.

Optimal probabilistic reliability-oriented planning of islanded microgrids considering hydrogen-based storage systems, hydrogen vehicles, and electric vehicles under various climatic conditions

Much attention has been paid to the deployment of Hydrogen storage systems (HSSs) and Hydrogen vehicles (HVs) in the modernized energy system. However, a research gap exists in the literature about optimal probabilistic planning of microgrids (MGs) equipped with HSS, considering the uncertainties of renewable energy resources and electric vehicle (EV) and HV owners' behaviors. The main purpose of this research is to fill such a gap by developing a new probabilistic optimization problem to determine the capacity of Hydrogen-based MGs' sub-systems. Another contribution is to consider the reliability constraints and loss of energy cost (LOEC) in the MGs' total net present cost (TNPC). The Monte Carlo simulation (MCS) and Flower Pollination Algorithm (FPA) are used to model stochastic behaviors and solve the proposed probabilistic optimization problem. This paper studies different actual climates of Iran based on historical data, while various coordinated/uncoordinated charging modes of EVs and HVs are examined. Test results infer that a significant inaccuracy (more than 4.66% depends on the climate conditions and vehicle scenarios) occurs due to neglecting the uncertainties. The sensitivity analyses imply that the reliability constraints, LOEC, and their interactions might affect the MGs’ optimal design.

On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints

We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.

Power allocation for D2D aided cooperative NOMA system with imperfect CSI

Non-orthogonal multiple access (NOMA) has become one of the promising technologies for 5G, which can improve the spectrum resource utilization and system throughput along with the support of effective resource allocation algorithms. Most previous works on NOMA-enhanced cooperative relay systems assume perfect channel state information (CSI), which degrades the performance of their schemes severely when used in imperfect CSI environment. In this article, a power allocation algorithm in D2D aided cooperative NOMA communications with imperfect CSI is proposed. In the proposed algorithm, the probabilistic non-convex optimization problem is transformed into a non-probabilistic convex optimization problem by evaluating the channel gains and using the successive convex programming (SCP) which approximately gives the lower bound of the maximum transmission rate. In the SCP enhanced power allocation algorithm (SCPPAA), we iteratively obtain the sub-optimal power allocation coefficients for the optimization problem by Lagrangian dual multiplier method and Karush–Kuhn–Tucker conditions. This program is divided into two layers for updating the power allocation coefficients and the multipliers respectively. Numerical results demonstrate that our algorithm has a fast convergence performance and the algorithm in D2D-based cooperative NOMA scheme has significant sum-data-rate advantages compared with it in traditional ways.

Fatigue inspection and maintenance optimization: A comparison of information value, life cycle cost and reliability based approaches

Fatigue cracks increase structural failure risk and timely maintenance is very important. Maintenance planning is often formulated as a probabilistic optimization problem, considering uncertainties in structural and load modelling, material properties, damage measurements, etc. A decision rule or strategy, e.g. condition based maintenance (CBM), needs to be set up, and then an optimal maintenance criterion or threshold is derived via solving the optimization problem. This paper develops a probabilistic maintenance optimization approach exploiting value of information (VoI) computation and Bayesian decision optimization. The VoI based approach explicitly quantifies added values from future inspections, and gives an optimal decision (or strategy) by direct modelling decision alternatives and evaluating their expected outcomes, rather than a pre-defined strategy. A comparative study on VoI, life cycle cost (LCC) and reliability based optimization approaches is conducted. It is shown that the VoI based approach takes all available maintenance strategies into account (both with and without involving inspections), and can reliably yield optimal maintenance strategies, whether the VoI is larger than or equal to zero. When the VoI is equal to zero, LCC and reliability based CBM optimization can lead to suboptimal maintenance strategies. The differences in the approaches are illustrated on fatigue-sensitive components in a marine structure.

Scenario-Based Network Reconfiguration and Renewable Energy Resources Integration in Large-Scale Distribution Systems Considering Parameters Uncertainty

Renewable energy integration has been recently promoted by many countries as a cleaner alternative to fossil fuels. In many research works, the optimal allocation of distributed generations (DGs) has been modeled mathematically as a DG injecting power without considering its intermittent nature. In this work, a novel probabilistic bilevel multi-objective nonlinear programming optimization problem is formulated to maximize the penetration of renewable distributed generations via distribution network reconfiguration while ensuring the thermal line and voltage limits. Moreover, solar, wind, and load uncertainties are considered in this paper to provide a more realistic mathematical programming model for the optimization problem under study. Case studies are conducted on the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks, where the 59- and 83-node distribution networks are real distribution networks in Cairo and Taiwan, respectively. The obtained results validate the effectiveness of the proposed optimization approach in maximizing the hosting capacity of DGs and power loss reduction by greater than 17% and 74%, respectively, for the studied distribution networks.

A reliability-based sieve technique: A novel multistage probabilistic methodology for the damage assessment of structures

A novel multistage probabilistic methodology called the “Reliability-Based Sieve Technique (RBST)” is presented in this paper for the reliable assessment of the functionality of structures. In this method, the structural damagedetectionproblem is defined as a multistage probabilistic optimization problem. The Holistic Objective Function (HOF) based on the combination of the inherent characteristics of the structure (i.e., the vibration frequencies and mode shapes) is incorporated into the Interactive Autodidactic School (IAS) optimization algorithm, for the first time, to solve the problem. In addition, the Latin Hypercube Sampling (LHS) techniqueis used to simulate and analyzethe probabilistic damage assessment problem. In each stage of the proposed methodology,the Probability of Damage Existence (PDE) is computed in each of the structural elementsthrough a probabilistic damage detection analysis. According to the results of the PDE inthe structural elements in each stage, the elements with low PDEsare gradually sieved inthe subsequent steps. The sifted elements ineach stage are considered asintactones in the next stage. This systematic filtration of the design variables can simultaneously decrease the dimensions and increase the speed of the optimization problem. To improve the performance of the RBST, the sizesof the sieves are regularly reduced for the next stages. This multistage procedure iscontinueduntil convergence to a precise structural damage location diagnosis and intensity prognosis is achieved. Finally, to investigate the efficiency and robustness of the proposed technique, it is examined on three benchmark structures by taking the high level of uncertainties associated with both finite element modeling errors and vibration data noises into account. The obtained results confirmed that the proposed technique correctly identifies the damage indices and has consummate capability compared with the single-stage probabilistic analysis. Likewise, it is observed that the RBST accurately sieves the intact elementsfromthe vector of the design variables in highly noisy conditions. Consequently, in the presence of both mentioned uncertainty sources, the RBST not only assesses the damage site and extent properly, but also shows a reliable performance in terms of PDE.

Probability-damage approach for fail-safe design optimization (PDFSO)

Fail-safe design often leads to oversized structures as the design must be able to behave properly for the different accidental situations that could undergo throughout their lifespan. The existing research about structural optimization applied to fail-safe design does not contemplate the idea of having accidental situations with different probabilities of occurrence. This means that the structure must satisfy the design constraints in the damaged structure regardless of whether one accidental situation is much more likely than another. The objective of this research is to formulate a new optimization strategy to get fail-safe structures with minimum weight taking into account the available data about the probability of occurrence associated to each partial collapse. A multi-model probabilistic optimization problem is defined and applied to two structures: a 2D truss structure with stress constraints as well as the tail section of an aircraft fuselage with stress and buckling constraints.

Co-optimization of ampacity and lifetime with considering harmonic and stochastic parameters by Imperialist Competition Algorithm

This study presents a novel probabilistic algorithm for the optimal placement of underground power cables to maximize lifetime and ampacity and minimize total cost of the cable system at the same time. In this paper, the Imperialist Competition Algorithm (ICA) and Monte Carlo (MC) simulation are used to solve probabilistic optimization problem. In our previous study, we have just considered probabilistic parameters that are effective on ampacity. However, In fact, we need to consider all the other parameters such as cable’s lifetime and harmonic of demand side for co-optimizing the ampacity and lifetime to obtain the best arrangement of cable system. The cost function of optimization has been improved relative to the last one. Cable lifetime and harmonic of power system, as two important design factors, have been added to cost function in this paper. The novelty of this paper is consisted of economic co-optimization of ampacity and lifetime simultaneously. Furthermore, all statistical variations of cable system parameters, ambient temperature, dimensions of native/backfill soil and uncertainty in demand side such as load current fluctuation and harmonic are considered as design parameters at the same time. Since, all design parameters are stochastic; the change of seasons and climate can modify these parameters along the cable system. Hence, this paper indicates the necessity of consideration of uncertainty parameters at the design phase by taking into account cable lifetime based on a probabilistic approach and avoid using deterministic value. For a better explanation of this method, a numerical example is reported by using corrugated aluminum sheath cable model.

Scenario-based probabilistic multi-stage optimization for transmission expansion planning incorporating wind generation integration

Integrated transmission expansion planning (TEP) and generation expansion planning (GEP) with Wind Farms (WFs) is addressed in this paper. The optimal number of expanded lines, the optimal capacity of WFs installed capacity, and the optimal capacity of wind farms lines (WFLs) are determined through a new TEP optimization model. Furthermore, the optimum capacity additions including conventional generating units is obtained in the proposed model. The Benders decomposition approach is used for solving the optimization problem, including a master problem and two sub-problems with internal scenario analysis. In order to reduce the computational burden of the multi-year and multi-objective expansion planning problem, a multi-stage framework is presented in this paper. The uncertainties of wind speed and system demand along with contingency scenarios lead to a probabilistic optimization problem. Moreover, in the proposed model, the planning time horizon is divided into three predefined stages. This multi-stage approach is used to increase the proposed model accuracy in a power system with a high level of wind power penetration. Hence, in this paper a scenario-based probabilistic multi-stage model for transmission expansion planning is proposed, incorporating optimal WFs integration. It is recognized that high wind penetration increases the transmission expansion investment cost, but based on the reduction of the investment cost of conventional units, the total system cost will be smaller. This result emphasizes the main advantage of wind generating system over the conventional generating system. This planning methodology is applied to the modified IEEE 24-bus test system and simplified Iran 400-kV real system to show the feasibility of the proposed algorithm.

Optimization of fastener pattern in airframe assembly

PurposeThe authors consider the problem of optimizing temporary fastener patterns in aircraft assembly. Minimizing the number of fasteners while maintaining final product quality is one of the key enablers for intensifying production in the aerospace industry. The purpose of this study is to formulate the fastener pattern optimization problem and compare different solving approaches on both test benchmarks and rear wing-to-fuselage assembly of an Airbus A350-900.Design/methodology/approachThe first considered algorithm is based on a local exhaustive search. It is proved to be efficient and reliable but requires much computational effort. Secondly, the Mesh Adaptive Direct Search (MADS) implemented in NOMAD software (Nonlinear Optimization by Mesh Adaptive Direct Search) is used to apply the powerful mathematical machinery of surrogate modeling and associated optimization strategy. In addition, another popular optimization algorithm called simulated annealing (SA) was implemented. Since a single fastener pattern must be used for the entire aircraft series, cross-validation of obtained results was applied. The available measured initial gaps from 340 different aircraft of the A350-900 series were used.FindingsThe results indicated that SA cannot be applicable as its random character does not provide repeatable results and requires tens of runs for any optimization analysis. Both local variations (LV) method and MADS have proved to be appropriate as they improved the existing fastener pattern for all available gaps. The modification of the MADS' search step was performed to exploit all the information the authors have about the problem.Originality/valueThe paper presents deterministic and probabilistic optimization problem formulations and considers three different approaches for their solution. The existing fastener pattern was improved.

Probabilistic approach for optimal portfolio selection using a hybrid Monte Carlo simulation and Markowitz model

In this paper, a probabilistic form of the portfolio selection problem is established in which the uncertainty of risky assets is considered through a probabilistic optimization problem. To this end, by taking seven portfolios of Shanghai stock as a case study, the mean and standard deviation of daily return values are calculated based on five years of real data. The optimal values corresponding to each random case were then stored as a comprehensive database of system responses. Then, by sorting the resulting optimal values from best to worst, the exceedance probabilities of return and risk rankings were calculated for each portfolio and presented in the form of probabilistic pie charts. The results showed that the portfolio with the highest deterministic rate of return has the highest probability of getting the best return ranking as well. However, since the probability of risk in all cases was calculated, the probability of each portfolio to place in the lower rankings (i.e., ranking 2–7) could also be discussed. Additionally, to check the convergence of the model, the probability values calculated by the Monte Carlo method against the sample size were plotted to ensure the accuracy of the final answer. Eventually, by generating Gaussian random noise and importing it into the model input, probability changes were calculated to assess the robustness of the proposed algorithm.

Comparative analysis of probabilistic and deterministic approach to tune the power system stabilizers using the directional bat algorithm to improve system small-signal stability

The modern power system consists of a large mixture of renewable energy sources (RES), varying and flexible loads, and is also experiencing a situation where a significant number of conventional generators are being replaced by power electronics based sources. All of these factors may lead to a decrease in the small-signal stability (SSS) margin of the system which in turn can cause power system instability. Power system stabilizers (PSSs) are widely used to improve the low-frequency oscillatory stability of power systems. Currently, there are two popular methods to coordinate multiple PSSs to improve SSS: deterministic and probabilistic method. This paper first formulates the problem of coordinating multiple PSSs to improve SSS as a deterministic and a probabilistic optimization problem which is then solved using the new directional bat algorithm. A detailed comparison between the two design methods under different scenarios based on obtained results is carried out afterward. All of the analysis was carried out on a modified IEEE 68 bus system. The obtained results provide key insights as well as the advantages and limitations of the probabilistic and deterministic approach to optimize the PSS parameters.

Thieves and Police, a New Optimization Algorithm: Theory and Application in Probabilistic Power Flow

ABSTRACT This paper introduces a new meta-heuristic algorithm called Thieves and Police Algorithm (TPA), for solving probabilistic optimization problems. The proposed algorithm is designed in five different phases each following a certain objective. In this algorithm, conventional matters between the thieves’ team and the police are inspired as a social phenomenon. Motivation of thieves for stealing more properties is utilized in this algorithm to seek the optimum solution, and police track that result in fear and no risk taking of the thieves to enter areas under police surveillance is inspired to omit areas with weaker solutions. Algorithm optimization procedure ends with apprehending the leader thief by police as the optimum solution. The performance of the TPA is evaluated in terms of local optima avoidance, exploration, exploitation, effectiveness, and convergence properties using a set of classical and modern test functions. Comparing the outcomes of other meta-heuristic optimization algorithms, the successful performance of the TPA in solving different optimization problems is confirmed. Moreover, the efficiency of the TPA in terms of solving problems with probabilistic nature is assessed by solving an optimization problem in power engineering area, known as Probabilistic Power Flow (PPF), in a Micro-Grid with renewable energy resources. Based on the outcomes, the proficiency of the proposed algorithm is confirmed by solving the PPF problem and producing timely acceptable results. Hence, the TPA algorithm is suggested to solve complicated problems, with probabilistic nature, in the optimization field and other optimization problems.

Maximum Loadability of Islanded Microgrids With Renewable Energy Generation

A microgrid (MG) is a small-scale power system which is fed by constrained distributed generation (DG) units and its continuous operation is affected by the variability of available generation resources. In this paper, a global sensitivity analysis (GSA) method is proposed to evaluate the impact of variable energy resources on the maximum loadability of islanded MGs (IMGs). First, a probabilistic optimization problem is formulated to calculate the IMG load margin considering the droop characteristics of DG units and uncertainties of renewable generation, loads and distribution feeder parameters. Next, the global sensitivity (GS) is introduced that can identify the impact of independent and correlated variables on the IMG loadability. Last, the sparse polynomial chaos expansion method is used to obtain the probabilistic models for IMG load margins, and an efficient GSA method is proposed to calculate the GS of IMG loadability to prevailing variables. The probabilistic models are considered for IMG input variables and the impact of variable correlations on IMG loadability is analyzed. The proposed method for calculating the maximum loadability is tested using a 33-bus IMG and the results are compared with those of other GSA and local sensitivity analysis methods.

On-board knock probability map learning–based spark advance control for combustion engines

This article presents an on-board map learning–based spark advance control framework for combustion engines. The proposed control framework addresses the knock probabilistic constrained thermal efficiency optimization problem with three layers. First, in the upper layer, maps of knock event distribution and thermal efficiency are learned with manifold pressure and combustion phase as inputs. Second, the middle layer generates the knock probability constrained optimal combustion phase reference that is subsequently tracked by a hypothesis test-based feedback controller. Third, the lower layer employs a partial likelihood-based knock controller that retards the spark advance in case of the frequent knock events. The key contributions of this work are the three-layer control framework and the knock event distribution map learning in the upper layer. The knock event is supposed to obey binomial distribution, and the distribution is modeled by beta distribution and learned in the perspective of Bayesian learning. Moreover, the normalization algorithm is proposed for online feedfoward map update. The proposed map learning–based spark advance control framework is experimentally validated in a test bench equipped with a spark-ignition engine.