Functions Of Uncertain Parameters Research Articles

Real time Energy Management System of a photovoltaic based e-vehicle charging station using Explicit Model Predictive Control accounting for uncertainties

This paper proposes an Explicit Model Predictive Control (eMPC) for the energy management of an e-vehicle charging station fueled by a photovoltaic plant (PV), a battery energy storage system (BESS), and the electrical grid. The method computes offline an explicit solution of the MPC, which is stored and then used for real time control. Multiparametric programming is used to calculate the explicit solution by mapping the MPC laws as a function of uncertain parameters. In this paper, the uncertainties introduced into the multiparametric programming are the photovoltaic power production, the electricity price, the battery’s state of charge, and the electric vehicle power consumption. Moreover, the environmental impact of the charging station operation is considered through the C O 2 emissions level. The explicit solution is computed for a specific range of uncertain parameters. Then, during the real-time control, their current values are measured to evaluate the control laws saved beforehand. The proposed approach, consisting of an offline MPC-based determination of the control laws followed by their online use, reduces the computational burden without affecting the MPC performance. Numerical simulations and experimental results confirm the eMPC’s performance for the proposed application. • EV charging station based on PV/BESS uses eMPC control as its EMS. • PV and EV power, CO 2 , price, and BESS’s SoC are included as uncertainties. • eMPC creates offline the control laws as function of the uncertain parameters. • The online evaluation of the control laws takes less than one second.

Parametric Transient Stability Constrained Optimal Power Flow Solved by Polynomial Approximation Based on the Stochastic Collocation Method

To better respond to the impact of power system-uncertain parameters on transient stability, a novel model named the parametric transient stability constrained optimal power flow (parametric TSCOPF) is proposed. It seeks the optimal control scheme of transient stability constrained optimal power flow (TSCOPF) expressed by the function of uncertain parameters in power systems. The key difficulty to solve this model lies in that the relationship between the parametric TSCOPF solution and uncertain parameters is implicit, which is hard to derive generally. To this end, this paper approximates the optimal solution of parametric TSCOPF by polynomial expressions of uncertain parameters based on the stochastic collocation method. First, the parametric TSCOPF model includes both uncertain parameters and transient stability constraints, in which the transient stability constraint is constructed as a set of polynomial expressions using the SCM. Then, to derive the relationship between the parametric TSCOPF solution and uncertain parameters, the SCM is applied to the parametric Karush–Kuhn–Tucker (KKT) conditions of the parametric TSCOPF model, so that the optimal solution of the parametric TSCOPF is approximated by using polynomial expressions with respect to uncertain parameters. The proposed parametric TSCOPF model has been tested on a 3-machine, 9-bus system, and the IEEE 145-bus system, which verifies the effectiveness of the proposed method.

A hybrid method for analysing stationary random vibration of structures with uncertain parameters

This paper presents a new hybrid polynomial dimensional decomposition-pseudo excitation method (PDD-PEM) for analysing stationary random vibration of structures with uncertain parameters. The power spectral density (PSD) of the structural random response is expressed as a function of uncertain parameters using the PEM and a finite hierarchical expansion is performed using the component functions of the PDD. The component functions are then represented by approximate Fourier-polynomial expansions with orthonormal polynomial bases and expressions are given for calculating the first two moments of the PSD of random structural responses, using the coefficients of the expansion. Considering that calculating the coefficients requires difficult multi-dimensional integration, a dimension-reduction integration method and a Gaussian numerical integration method are introduced to effectively reduce the computational efforts of the calculation of the coefficients. The results of five numerical examples indicate that the proposed method is nearly as accurate as but more efficient than the Monte Carlo simulation method.

Optimal design of an off-grid hybrid renewable energy system considering generation and load uncertainty: the case of Zanjan city, Iran

In this paper, designing a hybrid stand-alone photovoltaic/wind energy system with battery storage (PV/WT/Batt) is presented to minimize the total cost of the hybrid system and considering reliability constraints for Zanjan city in Iran country considering generation and load uncertainties. The total cost includes the cost of the system components and load losses cost and also deficit power probability-hourly interruption probability index, which is defined as the reliability constraint. The optimal size of the system components is found using an improved crow search algorithm based on mutation and crossover operators of the genetic algorithm to prevent premature convergence. The optimal designing of the hybrid system is implemented considering the uncertainty of photovoltaic and wind resources as well as the load demand of the system based on the probability distribution functions of uncertain parameters using Monte Carlo simulation. The simulations are implemented in two deterministic and probabilistic scenarios. In the deterministic scenario, the optimal design of various combinations of the hybrid system is investigated in view of the system cost and deficit power probability probability-hourly interruption probability index based on the improved crow search algorithm with and without the cost of load loss. The results of the deterministic scenario showed that the system cost is increased considering the cost of load loss, but the system load is supplied with higher reliability. Also, the deterministic results cleared the superiority of the improved crow search algorithm compared with conventional crow search algorithm and particle swarm optimization methods. Moreover, the results of the system designing showed that considering the uncertainty, the system cost increases, and the reliability is significantly improved. The results cleared that the system cost in deterministic and probabilistic designing is obtained 1.0108 M$ and 1.210 M$, respectively for load supply equal to 153.87 MWh. So, due to the inherent nature of the uncertainty in the renewable resources power generation as well as continuous load changes, a probabilistic approach of the hybrid system designing is necessary to know the accurate cost and reliability.Article highlightsEffective performance of the proposed method in system designing with lowest cost and higher reliability.Achieving to a more reliable hybrid energy system with considering cost of load losses in designing.Increasing the system the cost and improving the reliability considering uncertainties of generation and load.

TOWARDS A SUSTAINABLE MICROGRID ON ALDERNEY ISLAND USING A PYTHON-BASED ENERGY PLANNING TOOL

In remote or islanded communities, the use of microgrids (MGs) is necessary to ensure electrification and resilience of supply. However, even in small-scale systems, it is computationally and mathematically challenging to design low-cost, optimal, sustainable solutions taking into consideration all the uncertainties of load demands and power generations from renewable energy sources (RESs). This paper uses the open-source Python-based Energy Planning (PyEPLAN) tool, developed for the design of sustainable MGs in remote areas, on the Alderney island, the 3 rd largest of the Channel Islands with a population of about 2000 people. A two-stage stochastic model is used to optimally invest in battery storage, solar power, and wind power units. Moreover, the AC power flow equations are modelled by a linearised version of the DistFlow model in PyEPLAN, where the investment variables are here-and-now decisions and not a function of uncertain parameters while the operation variables are wait-and-see decisions and a function of uncertain parameters. The k-means clustering technique is used to generate a set of best (risk-seeker), nominal (risk-neutral), and worst (risk-averse) scenarios capturing the uncertainty spectrum using the yearly historical patterns of load demands and solar/wind power generations. The proposed investment planning tool is a mixed-integer linear programming (MILP) model and is coded with Pyomo in PyEPLAN.

Fast enclosing the solution set of the parametric Sylvester matrix equations

ABSTRACT In this work, we study the parametric Sylvester matrix equation whose elements are linear functions of uncertain parameters varying within some intervals. We first explore some properties of its solution set and then present some sufficient conditions for boundedness of the solution set. The main work is developing a modified variant of the parametric Krawczyk operator which reduces the computational complexity of enclosing the solution set significantly, compared to the parametric Krawczyk operator on the Kronecker product form. Some numerical experiments are given to illustrate the effectiveness of the proposed method.

Enclosing the solution set of the parametric generalised Sylvester matrix equation A(p)XB(p) + C(p)XD(p) = F(p)

In this paper, we investigate the parametric generalised Sylvester matrix equation , whose elements are linear functions of uncertain parameters varying within intervals. This model generalises both interval matrix equations and parametric interval linear systems, so it is a quite general model. First, we give some sufficient conditions under which the solution set of this parametric equation is bounded. We then propose several approaches for enclosing the solution set that acquire tighter enclosures than those obtained by relaxing the parametric system to an interval system. Some special cases of interval systems that inherently are parametric (have dependent structure) are considered, too. Finally, numerical experiments are given to illustrate the effectiveness of the proposed methods.

Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansions

In this paper, we develop a numerical approach based on chaos expansions to analyze the sensitivity and the propagation of epistemic uncertainty through a queueing systems with breakdowns. Here, the quantity of interest is the stationary distribution of the model, which is a function of uncertain parameters. Polynomial chaos provide an efficient alternative to more traditional Monte Carlo simulations for modeling the propagation of uncertainty arising from those parameters. Furthermore, polynomial chaos expansion affords a natural framework for computing Sobol’ indices. Such indices give reliable information on the relative importance of each uncertain entry parameters.

A Multistage Robust Transmission Expansion Planning Model Based on Mixed Binary Linear Decision Rules—Part I

This two-paper series introduces a new multistage robust approach for solving the transmission expansion planning (TEP) problem under the uncertainty of loads and wind power productions. The proposed multistage robust TEP model is formulated using mixed binary and linear decision rules, which aims at finding a solution withstanding any realization of the uncertain parameters belonging to a set-based uncertainty model. The robustness of the optimal expansion plan can be adjusted by the budget of uncertainty. The first part of this two-paper series presents the extended and compact formulations for a deterministic TEP problem. Then, the compact formulation of the TEP problem under the uncertainty of loads and wind power productions is introduced as a multistage robust optimization problem, which is reformulated using mixed binary and linear decision rules. The extended formulation of these decision rules, as functions of uncertain parameters, is presented in this part as well.

The Unbounded Parametric Tolerable Solution Set

We consider a linear algebraic system A(p)x=b(q), where the elements of the matrix and the right-hand side vector are linear functions of uncertain parameters p, q varying within given intervals [p], [q]. The so-called parametric tolerable solution set is studied for unboundedness. Basing on a characterization of the parametric tolerable solution set as a convex polyhedron, we present necessary and sufficient conditions (in both general and computable forms) for a nonempty parametric tolerable solution set to be unbounded. Every parametric tolerable solution set is represented as a sum of a linear subspace and a bounded convex polyhedron. The latter implies better estimations (outer and inner) for the unbounded parametric tolerable solution set. Numerical examples illustrate the discussed methodology and the solution sets.

Probabilistic Assessment of Voltage Stability of Power System with Wind Farm Based on Credibility Theory

The static voltage stability of power system with wind farm is simulated by fuzzy simulation. This method is able to calculate not only the expectations and standard deviations of diagonal elements of Jacobian reduction matrix with the uncertain parameters but also the static voltage stability indicator based on the credibility theory. The method is characterized in that the impact of the assumptions of distribution function type of uncertain parameters and parametric statistics is avoided by this method. The calculated results are impacted only according to the actual changed range and the types of membership functions of uncertain parameters. Finally the calculation results based on that method are compared with the calculation results based on two-point estimation method and Monte Carlo simulation method. The results show that the superiority and accuracy based on the method is proved.

Linear systems with a polynomial uncertainty structure: Stability margins and control

In this paper, the stability and control of linear systems with parameter uncertainty are considered. The characteristic equations of such systems are derived as functions of uncertain parameters. The cases in which the coefficients are polynomial functions of the uncertain parameters are studied and an algorithm is presented for the calculation of stability margins in the parameter space in general lp-norms. The stability is defined with respect to arbitrary regions in the complex plane (so-called D-stability). The algorithm requires a search on the contour of the region (frequency search) and to solve a system of equations at each step. Using the above algorithm, a method is also presented for the design of robust controllers for systems with uncertain parameters and robust pole placement inside the regions of interest is addressed.

An Unsupervised Learning Problem Using Limited Storage Capacity

In unsupervised learning pattern recognition problems, the need arises for updating conditional density functions of uncertain parameters using probability density function mixtures. In general, the form of the density mixtures is not reproducing, invoking the need for unlimited system storage requirements. One suboptimal method for achieving limited storage is to restrict the uncertain parameters in question to come from finite sets of values. An alternate method is proposed for a class of problems and its performance is shown to converge to that of the optimum unlimited storage system. A generalization of the procedure is also discussed.