Affine Arithmetic Method Research Articles

Dispatching optimization of city gas station district energy systems with multiple uncertainties based on an improved affine arithmetic method

According to the energy consumption characteristics of gas stations, the district energy system (DES) based on natural gas and renewable energy is a suitable form of energy supply. In the process of DES operation and dispatching, there are various uncertainties brought by the strong coupling of equipment and the fluctuation of internal and external surroundings, and these uncertain parameters are often correlated. In this paper, an improved affine arithmetic (AA) based method is proposed to solve the economic dispatching problem of DES with correlated uncertainties. Firstly, an optimal dispatching model considering economic and environmental factors is constructed. An affine algorithm is introduced and interval arithmetic (IA) is used to improve the problem of interval extension and high-order noise element. Taking a classical DES system as an example, the optimal dispatching of the system is analyzed using deterministic method, interval method and the proposed method. The application effect of the proposed method under different uncertainties scale and amplitude is also discussed. The calculation results in different scenarios show that compared with the traditional IA method, the proposed method can effectively reduce the width of the target value interval by about 13%, which can further contribute to the optimal operation decision.

A Novel Affine Arithmetic Method With Missed the Triangular Domain With Uncertainties

A novel affine arithmetic (AA) method with missed triangular domain (remaining area is convex polygons) is proposed in this paper for solving interval power flow (IPF) problem with uncertainties. An interval correlation model of random variables is established, and the original correlated interval variables are transformed into the standard space using interval arithmetic (IA) to reduce overestimation problem because of interval operation. To alleviate the errors caused by inaccurate enveloping of different intervals, an interval model with missed triangular domain is proposed and integrated into optimization model of IPF in form of additional constraints, which is developed in this paper through AA. IPF results, i.e., voltage magnitude, voltage angles and line flows are much narrower because it enhances restriction of correlated variables. Case studies on various test systems, i.e., IEEE 33-bus, 69-bus, and 118-bus system, indicate effectiveness of proposed technique compared with the other methods.

Mixed probabilistic and interval optimal power flow considering uncertain wind power and dispatchable load

Considering uncertain wind power and dispatchable load, a mixed probabilistic and interval optimal power flow model is proposed, and Monte Carlo sampling and affine arithmetic method are used to solve it. First, an uncertain optimal power flow model with mixed probabilistic variables and interval variables is established by expressing the uncertain dispatchable load as the interval model and the uncertain wind speed and node load as the probabilistic model. Then Monte Carlo sampling is used to sample the probabilistic variables in the proposed model. By this way, the mixed probabilistic and interval optimal power flow can be transformed into interval optimal power flow with sampling points, and the interval optimal power flow of each sampling point can be solved by the affine arithmetic method. Finally, a maximum probability density function and a minimum probability density function are synthesized based on the interval extremum of unknown variables in optimal power flow for each sampling point. The numerical results obtained by the IEEE‐118 and IEEE‐300 bus systems show that the mixed probabilistic and interval optimal power flow model has the merits of handling the problem including both probabilistic variables and interval variables at the same time, obtaining the probabilistic interval of optimal power flow with any value and learning the maximum probability and minimum probability of the power system's possible operating status. The proposed algorithm has the advantage of high solution efficiency. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A Modified Affine Arithmetic Method for Computational Error Analysis

Computational precision always has been a concern in digital systems design and optimization. This study presents a modified Affine Arithmetic method to calculate uncertainty range in the output of arithmetic units. We have considered five case studies, namely: quadratic equation, RGB to CrYCb system, fourth-order equation, multivariate polynomial functions and low pass filter, by which the proposed method is evaluated. Presented results indicate the modified affine arithmetic can outperform the traditional methods.

A Novel Affine Arithmetic Method to Solve Optimal Power Flow Problems With Uncertainties

An affine arithmetic (AA) method is proposed in this paper to solve the optimal power flow (OPF) problem with uncertain generation sources. In the AA-based OPF problem, all the state and control variables are treated in affine form, comprising a center value and the corresponding noise magnitudes, to represent forecast, model error, and other sources of uncertainty without the need to assume a probability density function (pdf). The proposed AA-based OPF problem is used to determine the operating margins of the thermal generators in systems with uncertain wind and solar generation dispatch. The AA-based approach is benchmarked against Monte Carlo simulation (MCS) intervals in order to determine its effectiveness. The proposed technique is tested and demonstrated on the IEEE 30-bus system and also a real 1211-bus European system.

Phase stability analysis using a modified affine arithmetic

Phase stability analysis is a crucial step in the determination of multiphase equilibrium. This analysis by the tangent plane distance (TPD) minimization is a well-known technique, as well as the difficulties in providing guarantees that the global minimum has been found. On this regard, interval methods are powerful tools since they provide such guarantees. In this work, an interval Newton method plus generalized bisection, based on a modified affine arithmetic, is used to reliably find all possible stationary points of the TPD function. Additionally, an improved convergence test is suggested as well as a special treatment for mole fraction weighted averages. Several mixtures with up to 5 components, including LLE island type ternary systems, were studied. Both activity coefficient models and cubic equations of state were considered. For all the cases tested, the proposed modified affine arithmetic method was superior to other interval-based methods.

Robust Plotting of Polar Algebraic Curves, Space Algebraic Curves, and Offsets of Planar Algebraic Curves

In this paper, based on modifications to the well-known subdivision scheme in combination with modified affine arithmetic method to guide the subdivision, we propose four robust and reliable algorithms for plotting polar algebraic curves, space algebraic curves and offsets of planar algebraic curves.

Comparison of interval methods for plotting algebraic curves

This paper compares the performance and efficiency of different function range interval methods for plotting f( x, y)=0 on a rectangular region based on a subdivision scheme, where f( x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Horner form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubin's method, Rivlin's method, Gopalsamy's method, and related methods which also take into account derivative information. Our experimental results show that the affine arithmetic method, interval arithmetic using the centred form, the Bernstein coefficient method, Taubin's method, Rivlin's method, and their related derivative methods have similar performance, and generally they are more accurate and efficient than Gopalsamy's method and interval arithmetic using the power basis, the Bernstein basis, and Horner form methods.