Deterministic Form Research Articles

Chance-constrained unit commitment with energy storage systems in electric power systems

The high penetration of renewables has amplified errors in forecasts of wind and solar power generation that are made to support the operation of power systems. To guarantee the security and reliability of the power system, sufficient dispatchable power generation capacity must be reserved. Various studies of the use of risk assessment to improve operational performance have been carried out. However, most methods of so doing are designed to cover a given risk level of uncertainty, which is determined from forecasts rather than actual results. As renewable generation and load capacity increase, defining the risk level without considering unit commitment (UC) may limit scheduling efficiency. In this study, chance-constrained programming is combined with goal programming to optimize risk-based UC. Solar power generation, wind power generation, and demand forecast errors are handled by the optimization of reserve schedules and adjustments of risk by imposing a penalty cost. The proposed risk-based UC model is transformed into a mixed-integer linear programming (MILP) problem using the equivalent deterministic form and piecewise linear functions to find an efficient solution. Case studies of a realistic Taiwan power system in 2025 are considered; these include peak (Summer 2025) and off-peak (Winter 2025) scenarios. Simulation results reveal that the flexible operation of a battery energy storage system (BESS) (upward/downward reserve) reduces risk costs by a significant amount (19.12%∼100%). Sensitivity analysis reveals how the risk preferences of the System Operator (SO) can be represented in terms of predefined risk levels and confidence intervals to capture the uncertainties in load, wind power generation, and solar power generation forecasts. The spinning reserve level is optimized to reduce the risk cost and the addition of BESS in flexible operations further reduces the risk cost. Lastly, pumped storage units as a downward reserve can reduce wind spillage and excess solar power generation costs.

Seismic hazard assessment of the urban area of Ambato, Ecuador, in deterministic form

Seismic microzonation of the urban area of Ambato, Ecuador, was done in 2018 in a probabilistic and a deterministic manner. This type of calculation is presented in the first part of the article. For this purpose, three geologic faults and three strong-motion equations were considered. For each geologic fault, recurrence periods are determined using two methods. It is seen that a magnitude 6.3 earthquake associated with the blind faults traversing Ambato may occur in 80 to 100 years, and one of magnitude 6.5 in the next 300 years. Geophysical and geotechnical studies of the urban area of the city of Ambato are presented. These permitted the acquisition of curves with the same period of soil vibration and equal speed of the shear wave in the first 30 m, plus the classification of soils of the city. Later, six models of strong soil movements were considered and horizontal acceleration spectra of the soil were obtained in a mesh of points separated every 500 m, for each soil profile. Average spectra were found for soil profiles C, D and E when making comparisons with the spectra found in the 2018 study. Based on the results of the present study and those from 2018, new spectral forms are proposed for the urban area of the city of Ambato (called spectral envelopes) and compared to spectra reported by seismic regulations in force in Ecuador (NEC-15).

A robust study of a piecewise fractional order COVID-19 mathematical model

In the current manuscript, we deal with the dynamics of a piecewise covid-19 mathematical model with quarantine class and vaccination using SEIQR epidemic model. For this, we discussed the deterministic, stochastic, and fractional forms of the proposed model for different steps. It has a great impact on the infectious disease models and especially for covid-19 because in start the deterministic model played its role but with time due to uncertainty the stochastic model takes place and with long term expansion the use of fractional derivatives are required. The stability of the model is discussed regarding the reproductive number. Using the non-standard finite difference scheme for the numerical solution of the deterministic model and illustrate the obtained results graphically. Further, environmental noises are added to the model for the description of the stochastic model. Then take out the existence and uniqueness of positive solution with extinction for infection. Finally, we utilize a new technique of piecewise differential and integral operators for approximating Caputo-Fabrizio fractional derivative operator for the purpose of constructing of the fractional-order model. Then study the dynamics of the models such as positivity and boundedness of the solutions and local stability analysis. Solved numerically fractional-order model used Newton Polynomial scheme and present the results graphically.

Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter

In the stage of modelling, measuring, mechanical processing and manufacturing of the nonlinear energy harvesting system, deviations and errors of system parameters are inevitable. Even slight variation of key parameters may have a significant influence on the output voltages, especially for the multi-stable nonlinear case. Therefore, the investigation of dynamic behaviors for the tristable energy harvesting system with uncertain parameters is of important value both for research and application. In this paper, the uncertainty of a tristable piezoelectric vibration energy harvester with a random coefficient ahead of the nonlinear term is studied. By using the Chebyshev polynomial approximation, this tristable energy harvesting system is first reduced into an equivalent deterministic form, the ensemble mean responses of which are derived to exhibit the stochastic behaviors. The periodic and chaotic motions, bifurcations and crises under different conditions are analyzed. The results show that the output voltage is sensitive to the uncertainty of the nonlinear coefficient, which leads to unstable behavior around the bifurcation and crisis points particularly. Exploring the influence pattern of uncertain parameters on the output voltage and avoiding the unstable parameter intervals are essential for optimizing the structure. It can further improve the efficiency of the nonlinear energy harvesting system.

Multi-Objective Capacitated Solid Transportation Problem with Uncertain Variables

This paper investigates a multi-objective capacitated solid transportation problem (MOCSTP) in an uncertain environment, where all the parameters are taken as zigzag uncertain variables. To deal with the uncertain MOCSTP model, the expected value model (EVM) and optimistic value model (OVM) are developed with the help of two different ranking criteria of uncertainty theory. Using the key fundamentals of uncertainty, these two models are transformed into their relevant deterministic forms which are further converted into a single-objective model using two solution approaches: minimizing distance method and fuzzy programming technique with linear membership function. Thereafter, the Lingo 18.0 optimization tool is used to solve the single-objective problem of both models to achieve the Pareto-optimal solution. Finally, numerical results are presented to demonstrate the application and algorithm of the models. To investigate the variation in the objective function, the sensitivity of the objective functions in the OVM model is also examined with respect to the confidence levels.

Life Interval Prediction for Aging Solid-Propellant Rocket Motors with Uncertain Chemical Parameters

Predictions of the remaining lifetime of aging solid-propellant rocket motors are affected by the uncertainties of various chemical parameters in the reaction. This paper proposes a nonprobabilistic interval method for evaluating the life bounds of aging solid-propellant rocket motors. In this method, chemical uncertainties in the aging process are represented by uncertain-but-bounded initial parameters, and the uncertainties arising in aging solid-propellant rocket motors are described by an interval model. A sparse regression Legendre inclusion function is then employed to establish an interval model for aging solid-propellant rocket motors, which can be directly solved in deterministic form using the finite element method. Finally, in conjunction with the deterministic solutions, the sparse regression Legendre interval algorithm is employed to obtain the bounds of the storage life related to the uncertain interval parameters. The effectiveness of the proposed method is verified by considering an uncertain storage life prediction problem under the accelerated thermoaging process with constant temperature conditions. The proposed method provides a reliable tool for predicting the storage life of aging solid-propellant rocket motors.

A novel cooperative multi-search benders decomposition for solving the hydrothermal unit-commitment problem

Renewable energy and modernization of power operation demand Independent System Operators (ISOs) to solve ever more complex and larger programming problems to securely and economically schedule power resources. A key step in the scheduling process is the unit commitment (UC). In a hydro-dominated system, this process also involves managing reservoirs and is called hydrothermal UC (HTUC). Due to its size, non-convex nature and strict solution-time requirements, HTUC remains a challenging problem to ISOs even in its deterministic form. Relying on a commercial solver to directly tackle this problem might require a careful configuration of the solver’s parameters, and, given the generality of commercial solvers, it might not fully exploit all characteristics of the HTUC. As an alternative, researchers have recently proposed dual-decomposition strategies for addressing the UC, which enables solving subproblems in parallel. However, dualizing a non-convex problem generally results in a duality gap and requires adhoc techniques for recovering a primal solution. In this paper, we propose a novel cooperative multi-search Benders decomposition approach to solve the deterministic HTUC. Our approach benefits from being a primal method and, at the same time, it exploits the structure of the HTUC to solve subproblems in parallel. To assess the effectiveness of our proposal, we use the state-of-the-art solver Gurobi as a benchmark, and we perform our experiments on 25 cases of a large-scale system with over 1,000 generating units and 7,000 buses. On average, our approach is more than 15 times faster than Gurobi and it is able to solve the HTUC problems in under 20 min to a 0.1% gap.

A stability analysis on a smoking model with stochastic perturbation

PurposeThe purpose of this paper is to investigate the effects of irregular unsettling on the smoking model in form of the stochastic model as in the deterministic model these effects are neglected for simplicity.Design/methodology/approachIn this research, the authors investigate a stochastic smoking system in which the contact rate is perturbed by Lévy noise to control the trend of smoking. First, present the formulation of the stochastic model and study the dynamics of the deterministic model. Then the global positive solution of the stochastic system is discussed. Further, extinction and the persistence of the proposed system are presented on the base of the reproductive number.FindingsThe authors discuss the dynamics of the deterministic smoking model form and further present the existence and uniqueness of non-negative global solutions for the stochastic system. Some previous study’s mentioned in the Introduction can be improved with the help of obtaining results, graphically present in this manuscript. In this regard, the authors present the sufficient conditions for the extinction of smoking for reproductive number is less than 1.Research limitations/implicationsIn this work, the authors investigated the dynamic stochastic smoking model with non-Gaussian noise. The authors discussed the dynamics of the deterministic smoking model form and further showed for the stochastic system the existence and uniqueness of the non-negative global solution. Some previous study’s mentioned in the Introduction can be improved with the help of obtained results, clearly shown graphically in this manuscript. In this regard, the authors presented the sufficient conditions for the extinction of smoking, if <1, which can help in the control of smoking. Motivated from this research soon, the authors will extent the results to propose new mathematical models for the smoking epidemic in the form of fractional stochastic modeling. Especially, will investigate the effective strategies for control smoking throughout the world.Originality/valueThis study is helpful in the control of smoking throughout the world.

Multi-Objective Faculty Course Assignment Problem with Result and Feedback Based Uncertain Preferences

In this paper, a multi-objective faculty course allocation problem with result analysis and feedback analysis based on uncertain preferences mathematical model is presented. To deal with an uncertain model, three different ranking criteria are being used to develop: a) Expected value, b) Optimistic value, c) Dependent optimistic value criterion. These mathematical models are transformed into their corresponding deterministic forms using the basic concepts of uncertainty theory. The deterministic model of DOCM consists of fractional objectives which are converted into their linear form using Charnes and Cooper’s transformation. These deterministic formulations MOFCAP are converted into a single objective problem by using the fuzzy programming technique with linear and exponential membership functions. Further, the single objective problem for all the defined models is solved in the Lingo 18.0 software to derive the Pareto-optimal solution. The sensitivity of the models is also performed to examine the variation in the objective function due to the variation in parameters. Finally, a numerical example is given to exhibit the application and algorithm of the models.

A fixed charge transportation problem with damageable items under uncertain environment

An uncertain mathematical model for a fixed charge transportation problem is created in this paper. To make it closer to actuality, item damage during the transportation is considered. Besides, as some indeterministic factors may appear in the problem because of the incompletion of historical data, several parameters in this model such as purchasing price, supplies, demands and so on are regarded as uncertain variables based on uncertainty theory. After that, the original model is analyzed in two criteria of expected value criterion and chance-constrained one, and two kinds of uncertain models as well as their corresponding deterministic forms are constructed. Then, two intelligent algorithms like the genetic algorithm (GA) and the particle swarm optimization algorithm (PSO) are employed to figure out the equivalent forms of the models. Finally, the applications of the models are displayed via several numerical examples.

Impacts of carbon trading and wind power integration on carbon emission in the power dispatching process

Currently, many low-carbon approaches are being applied in the power dispatching process, such as dynamic economic emission dispatch (DEED), wind power integration and carbon trading. Furthermore, these approaches may influence the carbon emission from electricity. To investigate such joint influence, this paper firstly builds a stochastic DEED model in the forms of chance-constrained programming related to wind power uncertainties. After that, the established model is transformed to a deterministic form by discussing relevant statistical characteristics. Next, an auxiliary decision-making method is designed to draw the optimal dispatching solution upon real-time price of carbon trading. Finally, empirical analysis shows that: (1) the minimization of carbon emissions and fuel costs comes at the expense of the reliability of wind power integrated system, for reasons that the decrement rate of thermal power output is about 60.8% from higher reliability level to lower level, (2) the decrement rate of carbon emissions (0.01%–0.012%) is superior to the growth rate of fuel costs (0.0011%–0.0022%) as the carbon trading price increases, which proves that carbon emission reduction can be effectively realized by carbon trading rather than substantial increase in fuel costs, and (3) carbon trading is conducive to the improvement of carbon emission efficiency.

Uncertain random mean–variance–skewness models for the portfolio optimization problem

In the face of complex financial phenomena, describing the high uncertainties in financial markets remains a challenging issue in modelling and decision making. In this paper, chance theory is used to analyse the hybrid uncertainty that combines random returns and uncertain returns. We regard the total return as an uncertain random variable and study the uncertain random portfolio optimization problem. We first define the skewness of an uncertain random variable and derive some important properties and explicit expressions with deterministic distributions. These theoretical results help transform the model into a deterministic form. Then, uncertain random mean–variance–skewness portfolio optimization models and their corresponding equivalents are established to meet the diverse needs of investors. Finally, we provide two numerical experiments to illustrate the applicability of the proposed model, one of which demonstrates that the model can be applied in the area of energy finance. It is shown that when uncertain asset returns are asymmetric, this model can be used to make investment decisions for investors. We also find that a higher return is accompanied by higher risk and higher skewness.

OPTIMIZING RELIABILITY OF LINEAR FRACTIONAL DIFFERENCE SYSTEMS UNDER UNCERTAINTY AND RANDOMNESS

Some complex systems may suffer from failure processes arising from soft failures and hard failures. The existing researches have shown that the reliability of a dynamic system is not constant under uncertain random environments. First, two types of uncertain random optimization models are proposed where reliability index is quantified by chance measure based on whether soft and hard failures are independent or not. It is considered that internal degradation is driven by left Caputo fractional linear difference equation, while shocks are defined as discrete i.i.d. random variables. The shocks may generate additional uncertain degradation shifts when considering the competing dependent failure processes. Then, two proposed optimization reliability problems may be transformed into their equivalent deterministic forms on the basis of [Formula: see text]-path, and improved gradient descent method is applied to obtain optimal solutions. Finally, the numerical example of a micro-engine indicates that the optimization models are beneficial to the reliability of engineering systems.

Outage Constrained Robust Secure Beamforming in Cognitive Satellite-Aerial Networks

This letter proposes a robust beamforming scheme to enhance the physical layer security (PLS) of multicast transmission in a cognitive satellite and aerial network (CSAN) operating in the millimeter wave frequency band. Based on imperfect channel state information (CSI) of both eavesdroppers (Eves) and primary users (PUs), we maximize the minimum achievable secrecy rate (ASR) of the secondary users (SUs) in the aerial network under the constraints of the interference to the PUs in the satellite network, the quality of service (QoS) requirements of the SUs and per-antenna power budget of the aerial platform. To tackle this mathematically intractable problem, we first introduce an auxiliary variable and outage constraints to simplify the complex objective function. We then convert the non-convex outage constraints into deterministic forms and adopt penalty function approach to obtain a semi-definite problem such that it can be solved in an iterative fashion. Finally, simulation results show that with the transmit power increase, the minimal ASR of SUs obtained from the proposed BF scheme well approximate the optimal value.

Comparisons of PNEC derivation logic flows under example regulatory schemes and implications for ecoTTC

Derivation of Predicted No Effect Concentrations (PNECs) for aquatic systems is the primary deterministic form of hazard extrapolation used in environmental risk assessment. Depending on the data availability, different regulatory jurisdictions apply application factors (AFs) to the most sensitive measured endpoint to derive the PNEC for a chemical. To assess differences in estimated PNEC values, two PNEC determination methodologies were applied to a curated public database using the EnviroTox Platform (www.EnviroToxdatabase.org). PNECs were derived for 3647 compounds using derivation procedures based on example US EPA and a modified European Union chemical registration procedure to allow for comparisons. Ranked probability distributions of PNEC values were developed and 5th percentile values were calculated for the entire dataset and scenarios where full acute or full chronic data sets were available. The lowest PNEC values indicated categorization based on chemical attributes and modes of action would lead to improved extrapolations. Full acute or chronic datasets gave measurably higher 5th percentile PNEC values. Algae were under-represented in available ecotoxicity data but drove PNECs disproportionately. Including algal inhibition studies will be important in understanding chemical hazards. The PNEC derivation logic flows are embedded in the EnviroTox Platform providing transparent and consistent PNEC derivations and PNEC distribution calculations.

Исследование взаимосвязи продовольственных, энергетических и водных ресурсов с помощью трехсекторальной модели Лоренца

Работа посвящена проблеме создания новых методов для комплексного моделирования и управления риском, которые позволят исследовать синергетические взаимодействия между источниками рисков различного происхождения в условиях неопределенности. Предложен подход к исследованию взаимосвязи продовольственных, водных и энергетических ресурсов с помощью трехсекторальной модели Лоренца, которая объединяет в единой структуре однотипным образом описанные сектора экономики, каждый из которых рассматривается в сроках уровня производительности, количества рабочих мест. и уровня структурных нарушений В результате математического моделирования определены условия возникновения детерминированного хаоса в минимальной модели экономического развития и выявлены возможные причины возрастающей уязвимости глобальной экономики к малым изменениям параметров управления. Рассмотрена задача определения эффективных управлений с целью минимизации суммарных структурных нарушений за выбранный интервал времени. В результате модельных экспериментов обнаружены траектории изменения параметров управления, позволяющие уменьшить число структурных нарушений. Это достигается за счет изменений соотношения уровней пропозиции и спроса продукции, спроса и предложения на создание рабочих мест. Рассмотрено влияние случайных возмущений на стохастическую деформацию детерминированных аттракторов модели Лоренца. Показано, что при случайных возбуждениях траектории стохастической системы покидают детерминированный аттрактор и образуют вокруг него некоторый пучок с соответствующим вероятностным распределением. Рассмотрена возможность дальнейшего усложнения модели за счет учета других секторов экономики с помощью модели Лоренца в комплексной форме. Задача исследования n секторов экономик в этом случае сводится к рассмотрению поведения ансамбля n связанных осцилляторов, генерирующих колебания с частотами ωn соответственно. Коллективная синхронизация данных осцилляторов может быть исследована с помощью модели Курамото. Задача управления социально-экономическим развитием в условиях возникновения хаотических режимов сводится для комплексной модели к управлению частотой поля с ненулевым средним, генерируемым связанными осцилляторами.

Economic Emission Dispatch for Wind Power Integrated System with Carbon Trading Mechanism

Nowadays, the power system is faced with some new changes from low-carbon approaches, though these approaches have proved to be effective in developing low-carbon electricity. Specifically, wind power integration and carbon trading influence the traditional economic emission dispatch (EED) mode, allowing for the disturbance of wind power uncertainties and the fluctuation of carbon trading price. Aiming at the above problems, this study firstly builds a stochastic EED model in the form of chance-constrained programming associated with wind power reliability. Next, wind power features are deduced from the statistic characteristics of wind speed, and thus the established model is converted to a deterministic form. After that, an auxiliary decision-making method based on the technique for order preference by similarity to an ideal solution (TOPSIS) is designed to draw the optimal solution based upon the specific requirements of carbon emission control. The simulation results eventually indicate that the minimization of fuel costs and carbon emissions comes at the expense of wind power reliability. Meanwhile, carbon emission reduction can be effectively realized by carbon trading rather than a substantial increase in fuel costs, and carbon trading may help to improve power generation efficiency. Furthermore, carbon trading prices could be determined by the demands of carbon emission reduction and power generation efficiency improvement.

Optimal scheduling of integrated demand response-enabled integrated energy systems with uncertain renewable generations: A Stackelberg game approach

In order to balance the interests of integrated energy operator (IEO) and users, a novel Stackelberg game-based optimization framework is proposed for the optimal scheduling of integrated demand response (IDR)-enabled integrated energy systems with uncertain renewable generations, where the IEO acts as the leader who pursues the maximization of his profits by setting energy prices, while the users are the follower who adjusts energy consumption plans to minimize their energy costs. Taking into account the inherent uncertainty of renewable generations, the probabilistic spinning reserve is written in the form of a chance constraint; in addition, a district heating network model is built considering the characteristics of time delay and thermal attenuation by fully exploiting its potential, and the flexible thermal comfort requirements of users in IDR are considered by introducing a predicted mean vote (PMV) index. To solve the raised model, sequence operation theory is introduced to convert the chance constraint into its deterministic equivalent form, and thereby, the leader–follower Stackelberg game is tackled into a mixed-integer quadratic programming formulation through Karush-Kuhn-Tucker optimality conditions and is finally solved by the CPLEX optimizer. The results of two case studies demonstrate that the proposed Stackelberg game-based approach manages to achieve the Stackelberg equilibrium between IEO and users by the coordination of renewable generations and IDR. Furthermore, the study on a real integrated energy system in China verifies the applicability of the proposed approach for real-world applications.

WITHDRAWN: Multi-level multi-objective fully quadratic fractional optimization model with trapezoidal fuzzy numbers using Rouben Ranking function and fuzzy goal programming

Abstract This paper offers an opportunity to solve multi-level multi-objective fully quadratic fractional optimization model(MLMOFQFOM) with trapezoidal fuzzy numbers(TrFN) and soft fuzzy quadratic constraints. Rouben-Ranking function and interactive Fuzzy Goal Programming(FGP) approach have been considered to attain a compromised solution. Firstly, fuzzy constraints are replaced by fuzzy goals using tolerance limits. After which objective functions are converted into deterministic form. Non-fractional models are constructed to get choices for decision variables. The weighted sum of under deviation variables has been implemented as an objective in the equivalent models. An algorithm is presented with numerical to support the authenticity of the explained approach.

A New Optimization Framework for Harmonic Compensation Considering Plug‐in Electric Vehicle Penetration Using Adaptive Particularly Tunable Fuzzy Chaotic Particle Swarm Optimization

Plug‐in electric vehicles (PEVs) can contribute to eliminating undesirable harmonics generated by nonlinear loads. In this study, a novel stochastic optimization approach for harmonic compensation is proposed which is capable of optimizing contrary objectives, including total harmonic distortion and harmonic inject current, simultaneously, while meeting the relevant constraints. This problem can be influenced by the uncertainty of PEVs which is reflected in the force outage rate concept. The Monte–Carlo simulation technique is implemented to consider the uncertainty associated with PEVs by generating plausible scenarios with the aim of converting the mentioned framework to the respective deterministic equivalents. Afterward, adaptive particularly tunable fuzzy chaotic particle swarm optimization (APTFCPSO) is utilized, based on the weighted sum method, and the acquired results are compared with those obtained by other implemented swarm intelligence‐based algorithms. Accordingly, at first, several benchmark optimization functions are considered to verify the performance of the APTFCPSO. Afterward, active power line conditioners (APLCs) and PEVs are separately employed for harmonics cancellation in the deterministic form. After adopting the scenario reduction technique, the optimization framework is solved for each remaining scenario by the mentioned procedure. The statistical analysis reveals that PEVs outperform APLCs to cancel harmonic orders defined in a 14‐node micro‐grid.