Probabilistic Optimization Method Research Articles

A decoupled probabilistic constrained topology optimization method based on the constraint shift

AbstractTopology optimization (TO) has recently emerged as an advanced design method. To ensure practical reliability in the design process, it is imperative to incorporate considerations of uncertainty. Consequently, performing reliability analysis (RA) during the design phase becomes necessary. However, RA itself constitutes an optimization problem. Combining these two optimization problems can result in inefficiency. To address this challenge, we propose a decoupled approach that integrates deterministic topology optimization (DTO) and RA cycles. The reliability‐based stress‐constrained TO (RBSCTO) problem is considered in this paper. The DTO constraint is derived based on shifting vectors derived from the previous cycle's RA outcomes, enabling low‐reliability constraint shift towards the feasible direction. The DTO is solved based on solid‐isotropic‐material‐with‐penalization (SIMP) and augmented Lagrangian method. Meanwhile, the optimization problem in RA is addressed using finite differences and the interior point method. To reduce the errors resulting from linear approximation and optimization in RA when the target reliability is very low, an outlier handling method is employed. Meantime, we utilize a probabilistic neural network to enhance the efficiency of reliability assessment. Comparative studies against traditional methods across four RBSCTO tasks are demonstrated to validate its effectiveness. Monte Carlo simulations are used to validate the reliability of results.

A probabilistic multi-objective design method of sail-photovoltaic-hybrid power system for an unmanned ocean surveillance trimaran

Considering the abundant wind and solar energy resources on waters, this paper integrates wind assistant propulsion and photovoltaic technology with ship electric power system as sail-photovoltaic-hybrid power system (sail-PV-HPS). When properly designed, sail-PV-HPS can present environmental and economic benefits compared with conventional sail-diesel power system. However, the optimal design of sail-PV-HPS is a complex task due to environmental uncertainties. In order to fill such gap, this paper proposes a probabilistic optimization method to determine the size parameters of sail-PV-HPS, pursing minimum GHG emission and lifecycle cost. Two pairs of joint distributions of wind speed and wind direction as well as solar irradiance and ambient temperature are established based on coupla function considering the interconnection and seasonal characteristics of the meteorology variables. The performance of the proposed approach is assessed by the retrofit of the power system of an unmanned ocean surveillance trimaran sailing in the Yellow Sea area. A deterministic optimization and a quasi-probabilistic optimization are performed to highlight the effect and importance of taking the uncertainties and their correlation into consideration. Results show that the sail-PV-HPS designed by the proposed probabilistic method outperforms the sail-PV-HPS given by the deterministic optimization and a quasi-probabilistic optimization, as well as, the original power system of the trimaran in terms of GHG emission and lifecycle cost.

An approach for joint optimization of probabilistic group test based on cost and time value: taking nucleic acid detection of COVID-19 as an example.

In recent years, the world has encountered many epidemic impacts caused by various viruses, COVID-19 has spread and mutated globally since its outbreak in 2019, causing global impact. Nucleic acid detection is an important means for the prevention and control of infectious diseases. Aiming at people who are susceptible to sudden and infectious diseases, considering the control of viral nucleic acid detection cost and completion time, a probabilistic group test optimization method based on the cost and time value is proposed. Firstly, different cost functions to express the pooling and testing costs are used, a probability group test optimization model that considers the pooling and testing costs is established, the optimal combination number of samples for nucleic acid testing is obtained, and the positive probability and the cost functions of the group testing on the optimization result are explored. Secondly, considering the impact of the detection completion time on epidemic control, the sampling ability and detection ability were incorporated into the optimization objective function, then a probability group testing optimization model based on time value is established. Finally, taking COVID-19 nucleic acid detection as an example, the applicability of the model is verified, and the Pareto optimal curve under the minimum cost and shortest detection completion time is obtained. The results show that under normal circumstances, the optimal combination number of samples for nucleic acid detection is about 10. Generally, 10 is used to calculate for the convenience of organization, arrangement and statistics, except for cases where there are special requirements for testing cost and detection completion time.

Reliability-based design optimization of material orientation and structural topology of fiber-reinforced composite structures under load uncertainty

Currently, most works on synchronous optimization of composite topology and fiber orientation are all carried out under deterministic conditions, where the influence of uncertain factor on design optimization of fiber-reinforced composite structures is ignored. In this work, an effective probabilistic reliability-based topology optimization (RBTO) method is proposed for material orientation and structural topology of continuum fiber-reinforced composite structures under load uncertainty for the first time. A nested double-loop optimization algorithm of minimizing the relative volume fraction while satisfying the reliability requirement of the compliance response is established, where inner loop is for reliability analysis and outer loop is for topology optimization. The sensitivities of reliability constraint with respect to random variable and design variables are detailed using the adjoint variable method. Finally, several typical examples are performed to illustrate the effectiveness of the proposed methodology.

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.

Reliability-based topology optimization of piezoelectric smart structures with voltage uncertainty

Currently, most of the piezoelectric structures are designed under deterministic conditions, where the influence of uncertain factors on the output motion accuracy is ignored. In this work, a probabilistic reliability-based topology optimization method for piezoelectric structure is proposed to deal with the working voltage uncertainty. A nested double-loop optimization algorithm of minimizing the total volume while satisfying the reliability requirement of the displacement performance is established, where the PEMAP-P (piezoelectric material with penalization and polarization) model is used for parameterization of stiffness matrix, piezoelectric coupling matrix, and polarization direction. This strategy consists of an inner loop for reliability analysis and an outer loop for topology optimization. The reliability index approach based on most probable point (MPP) is used for realizing the evaluation of reliability constraint in reliability analysis. The sensitivities of reliability constraint with respect to the random variables and design variables are detailed using the adjoint variable method. Typical examples are performed to illustrate the effectiveness of the proposed RBTO method. A comparison of the optimization results for different reliability indexes, standard deviations of the voltage, spring stiffnesses, and displacement limits are conducted, as well as the deterministic topology optimization results.

Introducing a probabilistic definition of the target in a robust treatment planning framework

The ‘clinical target distribution’ (CTD) has recently been introduced as a promising alternative to the binary clinical target volume (CTV). However, a comprehensive study that considers the CTD, together with geometric treatment uncertainties, was lacking. Because the CTD is inherently a probabilistic concept, this study proposes a fully probabilistic approach that integrates the CTD directly in a robust treatment planning framework. First, the CTD is derived from a reported microscopic tumor infiltration model such that it explicitly features the probability of tumor cell presence in its target definition. Second, two probabilistic robust optimization methods are proposed that evaluate CTD coverage under uncertainty. The first method minimizes the expected-value (EV) over the uncertainty scenarios and the second method minimizes the sum of the expected value and standard deviation (EV-SD), thereby penalizing the spread of the objectives from the mean. Both EV and EV-SD methods introduce the CTD in the objective function by using weighting factors that represent the probability of tumor presence. The probabilistic methods are compared to a conventional worst-case approach that uses the CTV in a worst-case optimization algorithm. To evaluate the treatment plans, a scenario-based evaluation strategy is implemented that combines the effects of microscopic tumor infiltrations with the other geometric uncertainties. The methods are tested for five lung tumor patients, treated with intensity-modulated proton therapy. The results indicate that for the studied patient cases, the probabilistic methods favor the reduction of the esophagus dose but compensate by increasing the high-dose region in a low conflicting organ such as the lung. These results show that a fully probabilistic approach has the potential to obtain clinical benefits when tumor infiltration uncertainties are taken into account directly in the treatment planning process.

Multi-objective Adaptive Evolutionary Algorithm to Enhance Voltage Stability in Power Systems

Problems with two or more conflicting objectives have been handled as needing a multi-objective approach in recent years. The solution for these types of problems is normally to satisfy the conflicting objectives simultaneously in order to find tradeoffs between different criteria. Optimization in power systems is an important example of how to tackle multi-objective problems since these systems have conflicting performance indicators and normally operate close to their constraints due to the continuous increase in demand. In this paper, a multi-objective approach is applied to the voltage stability problem in power systems by using an adaptive evolutionary algorithm. The proposed method regards examining the following stability indicators of power systems: the voltage profile, the total reactive power loss, and the voltage collapse margin as requiring a multi-objective approach. Several experiments were conducted in IEEE 14, 57, and 118 busbar systems by using the proposed method and other probabilistic and heuristic optimization methods. The results showed that the proposed adaptive evolutionary algorithm enhanced the voltage stability and outperformed the other methods, especially when the size of the power system increases.

Optimal stochastic scenario-based allocation of smart grids’ renewable and non-renewable distributed generation units and protective devices

The smart grid reliability is dramatically affected due to system uncertainties. Although much efforts have been devoted to developing the Monte Carlo simulation (MCS)-based or analytical methods for reliability-based optimal allocation distributed generation (DGs) and protective devices (PDs), there is a research gap about developing the probabilistic scenario-based optimization methods. This paper proposes a novel stochastic scenario-based reliability evaluation method for optimal allocation of smart grids’ PDs and DGs. The scenario reduction is applied using the k-means algorithm and modified system state, including the clusters of renewable-based DGs. The malfunction of PDs is concerned, which is one of the most important contributions of the introduced method. The introduced clustering-based reliability evaluation method is applied to IEEE 33-bus test system. Test results infer that around 10% inaccuracy occurs in deterministic approaches without considering uncertainties of DGs and PDs. Obtained test results also imply that the impacts of renewable DGs’ uncertainties are more considerable than eventual malfunctions of PDs. The MCS-based methods are used to verify the precision of the introduced method. Moreover, by comparing the introduced method with other available analytical methods, it is shown that the obtained results are 1.2% more precise than current analytical ones.

Cross-Entropy Method in Application to the SIRC Model

The study considers the usage of a probabilistic optimization method called Cross-Entropy (CE). This is the version of the Monte Carlo method created by Reuven Rubinstein (1997). It was developed in the context of determining rare events. Here we will present the way in which the CE method can be used for problems of optimization of epidemiological models, and more specifically the optimization of the Susceptible–Infectious–Recovered–Cross-immune (SIRC) model based on the functions supervising the care of specific groups in the model. With the help of weighted sampling, an attempt was made to find the fastest and most accurate version of the algorithm.

Optimal probabilistic scenario‐based operation and scheduling of prosumer microgrids considering uncertainties of renewable energy sources

AbstractUncertainties of renewable energy sources (RESs) such as wind turbine (WT) and photovoltaic (PV) units are one of the considerable challenges of prosumer microgrids (PMGs) for the optimal day‐ahead operation. In this study, a new probabilistic scenario‐based method of optimal scheduling and operation of PMGs is developed. In this regard, different scenarios are generated using Monte Carlo Simulations (MCS). Furthermore, k‐means, k‐medoids, and differential evolution algorithms (DEA) are deployed to cluster the scenarios in the proposed method. A realistic commercial PMG in Iran is selected to apply the introduced method. The validity of the developed probabilistic optimization method for PMG operation is examined by comparing the results under various scenario reduction algorithms and MCS ones. The comparison of the obtained results and those of other existing deterministic methods highlights the advantages of the presented method. Furthermore, the sensitivity analyses are carried out to investigate the robustness of the developed method against the increase in the system uncertainty level. According to the test results, it is concluded that the k‐medoids algorithm has the best performance in comparison with the k‐means and the DEA‐based clustering under various conditions.

Probabilistic Energy Efficiency Analysis in Buildings Using Statistical Methods

Energy efficiency analysis has been presented recently to minimize the energy consumption of buildings by designing the optimal structural parameters. Because of uncertain parameters incorporated in the energy analysis of a building, computationally effective methods are needed to model the uncertainties in the analysis. In this paper, a probabilistic multi-objective optimization method is proposed based on the statistical methods, i.e., the empirical rule method and two-point estimate method, for analyzing the energy efficiency in buildings. The energy consumption density and thermal discomfort function of residents are considered as the objective functions of the optimization problem. The model of a 12-story commercial building is used to examine the proposed method. Probabilistic methods are compared with the deterministic one to show the effect of considering uncertainties on the results. According to the results, there is a substantial difference between the optimal values of the building’s parameters by applying probabilistic and deterministic methods.

Probabilistic Techno-Economic Optimization in Medium Voltage Distribution Networks with Fault Passage Indicators and Fault Locators

Fault passage indicators (FPIs) and fault locators (FLs) are employed in modern distribution networks in order to enhance the process of fault localization, thus resulting in reduction of interruption time and improving the reliability of power supply. In this paper, a novel probabilistic techno-economic optimization method is proposed for determining the number and positions of FPIs that lead to maximum reduction of interruptiontime and investment costs in medium voltage (MV) distribution networks with and without FLs. The proposed method is basedon a probabilistic non-sequential Monte Carlo simulation model of the real network, which is a proper compromise between complicated sequential simulation models and too simplified analytical models. The main goal of the method is to obtain maximum improvement of the network reliability indices while using the minimum number of FPIs. The method is tested on a combinedurban/rural MV distribution network in Bosnia and Herzegovina and results are thoroughly discussed.

Probabilistic investigations into the value of information: A comparison of condition-based and time-based maintenance strategies

Optimal maintenance of marine structures is challenging due to numerous fatigue-prone components, serious failure consequences, high maintenance costs, harsh sea environments, difficult access, and uncertainties in fatigue loading, resistance, and inspection and maintenance activities. Time-based maintenance (TBM) is convenient to implement. However, condition-based maintenance (CBM) is proved to be more cost-effective. This paper assesses the value of information (VoI) by inspections in CBM, compared to TBM, and investigates the conditions for CBM to outperform TBM in terms of fatigue reliability. A probabilistic maintenance optimization method and a life cycle cost analysis framework are established to derive optimal CBM and TBM strategies with the objective of maximizing lifetime reliability and evaluating their life cycle costs. The advantages of CBM, in comparison to TBM, and the VoI depends strongly on the inspection time. The CBM can achieve higher reliability with fewer maintenance costs than the TBM. An illustrative example is provided using the established probabilistic method and framework to support optimal maintenance planning. This example serves as a basis to explain the benefits of CBM to lifetime fatigue reliability and cost reduction, the conditions when CBM is more beneficial than TBM, the conditions for beneficial, ineffective and unbeneficial repair, and the VoI by inspections.

A Probabilistic Incremental Proximal Gradient Method

In this letter, we propose a probabilistic optimization method, named probabilistic incremental proximal gradient (PIPG) method, by developing a probabilistic interpretation of the incremental proximal gradient algorithm. We explicitly model the update rules of the incremental proximal gradient method and develop a systematic approach to propagate the uncertainty of the solution estimate over iterations. The PIPG algorithm takes the form of Bayesian filtering updates for a state-space model constructed by using the cost function. Our framework makes it possible to utilize well-known exact or approximate Bayesian filters, such as Kalman or extended Kalman filters, to solve large-scale regularized optimization problems.

Probabilistic optimization of dose coverage in radiotherapy.

Probabilistic optimization is an alternative to margins for handling geometrical uncertainties in treatment planning of radiotherapy where uncertainties are explicitly incorporated in the optimization. We present a novel probabilistic method based on the same statistical measures as those behind conventional margin based planning. Percentile Dosage (PD) was defined as the dose coverage that a treatment plan meet or exceed to a given probability. For optimization, we used the convex measure Expected Percentile Dosage (EPD) defined as the average dose coverage below a given PD. An iterative method gradually adjusted the constraint tolerance associated with the EPD until the desired target PD was met. It was applied to planning of cervical cancer patients focusing on systematic uncertainty caused by organ deformation. The resulting plans were compared to margin based plans using target and organ at risk PDs. The EPD tolerance converged in less than ten iterations to produce a PD within 0.1 Gy of the requested. The PD was on average within 0.5% of the requested PD when validated versus independent scenarios. The rectum volume, extracted from the PDs, receiving 90% of the intended target dose was decreased with 16% for the same target PD in comparison to margin based plans. The proposed probabilistic optimization method enabled prescription of a dose volume histogram metric to a chosen confidence. The probabilistic plans showed improved target dose homogeneity and decreased rectum dose for the same target dose coverage compared to margin based plans.

Probabilistic life-cycle optimization of durability-enhancing maintenance actions: Application to FRP strengthening planning

The large number of deteriorating bridges and the goal of sustainable development mean that maintenance actions should not only effectively improve the current condition of bridges but also enhance post-maintenance durability. In this paper, a probabilistic life-cycle optimization method is proposed for the planning of this type of maintenance actions. A physics-based deterioration model is directly employed in the optimization process. The computational efficiency of the proposed method is ensured by efficient sampling algorithms, multi-objective particle swarm optimization, and a bookkeeping technique. The proposed method can provide bridge maintenance schedules that account for two conflicting objectives, i.e. the maximization of life-cycle performance and the minimization of maintenance cost. A deteriorating RC bridge superstructure under chloride-induced corrosion is used to illustrate the proposed method. Point-in-time and cumulative-time failure probabilities are compared as life-cycle performance indicators. As an example of maintenance actions that enhance durability, FRP strengthening implementations for different structural components of the superstructure are scheduled during the service life of the deteriorating superstructure.

Probabilistic design optimization of wind turbine gear transmission system based on dynamic reliability

In this paper, a probabilistic design optimization method for wind power gear drive system has been put forward based on the dynamic reliability. A pure torsional dynamic model of 1.5 MW wind turbine gear transmission system is developed which is based on lumped parameter method with incorporation of gear meshing stiffness and transmission error. By taking the wind load and the gear parameters as random variables, the variation in dynamic reliability of the system with time is obtained. The probabilistic optimization design model of the wind turbine gear transmission system with minimum displacement and volume is built. The parameters affecting the dynamic performance and reliability are taken as optimization design variables. The system reliability, the effects of random parameters on the reliability, the normal meshing condition and gear strength are taken as constraint conditions. We choose the gear transmission system of a 1.5 MW wind turbine for the system optimization and experimental verification. The results show that the system optimization based on dynamic reliability can reduce the volume of the system by 17.94 % and the system reliability is also guaranteed. Moreover, the vibration of each component of the system is reduced which is beneficial for smooth operation of the gear transmission system of wind turbine.

Optimizing a Distributed Wind-Storage System Under Critical Uncertainties Using Benders Decomposition

A method for calculating the optimal size of an energy storage system (ESS) under wind uncertainties is presented based on Benders decomposition for stochastic programming in this paper. The ESSs are becoming essential components in microgrids due to significantly higher penetration of renewable energy sources. Integrating renewable energy sources coupled with ESSs in a power system enhances the power system reliability by increasing its availability and reducing its total cost of operation and maintenance. In addition, the ESS connected to a microgrid should be optimally sized to be able to provide the necessary power and minimize the total cost of investment and operation. In order to optimally size a storage system, a constrained optimization problem is solved using a probabilistic optimization method because the forecast of their output power cannot be determined accurately. In this paper, a probabilistic optimization problem is solved using the Benders decomposition for stochastic programming method to optimally size an ESS. This ESS will be integrated and connected to a grid-connected microgrid that has wind power generation. The simulation results prove the effectiveness of the proposed optimal sizing methodology.

Adaptive Evolutionary Algorithm Applied to the Voltage Stability Problem in Power Systems

In recent years, research studies have sought methods and devices which enhance the performance of power systems, since the more these systems are expanded, the greater the economic and environmental constraints. With the development of the power electronics applied to power systems, flexible alternating current transmission systems (FACTSs) have sprung up. A problem related to the use of a FACTS device is where best to install and adjust its parameters in order to improve the performance of the system. This paper sets out a method for automatically allocating FACTS devices in power systems for which an adaptive evolutionary algorithm (EA) is used. The proposed method seeks to determine not only where best to install FACTS devices and to supply them with reactive power but also to adjust other decision variables of a power system in order to enhance the voltage stability while taking some indicators into account. The method was tested in experimental studies by using standard IEEE systems, and comparing the results from these with those of the probabilistic and heuristic optimization methods, including the standard EA. The results showed that the proposed method enhanced the voltage stability of the systems and outperformed the other methods.