Nested Decomposition Method Research Articles

Characteristics of the Evolution of China’s Regional Economic Differences Based on the Perspective of Urban Agglomerations

Urban agglomerations are becoming an important spatial carrier for China’s industrialisation and new-type urbanisation, and an important engine to drive the national economic development, and regional economic differences based on urban agglomerations have also become a new feature of China’s regional differences in the new period. Based on the urban agglomeration perspective, we quantitatively measure China’s regional economic differences from 2001 to 2020 through the two-stage nested decomposition method of the Theil Index, and analyse the evolutionary characteristics of the economic differences within the 19 urban agglomerations in the country by decomposing them into four major regions: East, Central, West and Northeast, and the differences between the urban agglomerations within the regions and the economic differences within the urban agglomerations. The study found that: (1) Overall the overall differences of the region with urban agglomerations as the core were gradually shrinking. In terms of the trend of changes in the differences at all levels, the economic differences between regions and within urban agglomerations were significantly reduced, and the changes in the economic differences between urban agglomerations were not obvious. In terms of contribution rate, the contribution rate of intra-urban agglomeration differences had always remained at a high level, reaching about half of the overall economic differences in the country, and was the primary contributing factor to the overall economic differences. (2) In terms of absolute differences, the economic differences among urban agglomerations in the four regions showed a clear pattern of “East > West > Central ≈ Northeast”. (3) The intra-cluster economic differences of each urban agglomeration in 20 years were at different levels and showed a differentiated trend of evolution, and the areas with relatively more prominent intra-cluster economic differences were the Pearl River Delta urban agglomeration in the east, and the Lan-xi urban agglomeration and the North Slope of Tianshan Mountain urban agglomeration in the northwestern region. The economic differences of urban agglomerations are the result of a variety of factors, and the reasons for the formation of regional economic differences are analysed from the factors of economic location, national regional development policies, human resource factors and high-speed traffic differences within the urban agglomerations.

A Nested Decomposition Method for the AC Optimal Power Flow of Hierarchical Electrical Power Grids

The interconnected electrical power grids are operated by multilevel control centers with hierarchical architecture. Since large-scale renewables are integrated in different voltage levels, the traditional isolated operation of interconnected power grids is uneconomical and meets operational risk due to lack of coordination. Therefore, the coordination between control centers in different levels and areas are indispensable. However, existing distributed approaches may either encounter numerical problems or converge slowly when the nonlinear AC optimal power flow (ACOPF) models are applied. Moreover, the total iteration number presents an exponential increase with the levels of hierarchical grids. In order to improve computational efficiency, this paper proposes a nested decomposition method for the coordinated operation of multilevel ACOPF problem, which has superlinear convergence and can achieve the optimal solution (KKT point). During each iteration, a projection function, which embodies the optimal objective value of a lower level power grid projected onto its boundary variable space, is computed with second-order exactness. Thus, the proposed method can be applied to nonlinear continuous optimizations with high efficiency. The paper also provides a rigorous proof for its convergence under the condition that the lower level optimization problems are convex with continuously differentiable objectives and constraints. Numerical tests are conducted with three trilevel power grid of different scales, which verify that the computational efficiency and scalability of the proposed algorithm are superior to those of existing methods.

Multi-Period Active Distribution Network Planning Using Multi-Stage Stochastic Programming and Nested Decomposition by SDDIP

This paper presents a multi-period active distribution network planning (ADNP) with distributed generation (DG). The objective of the proposed ADNP is to minimize the total planning cost, subject to both investment and operation constraints. The paper proposes a multi-stage stochastic optimization model to address DG uncertainties over several periods, in which the decisions are made sequentially by only using the present-stage information. A nested decomposition method is proposed which applies the stochastic dual dynamic integer programming (SDDIP) method to address computational intractabilities of the proposed ADNP approach. The presented numerical results and discussions on a 33-bus distribution system and a large-scale 906-bus system verify the effectiveness of the proposed ADNP method and its solution method.

GPU Parallelization Nested Decomposition Method for Solving Large Linear Systems in Reservoir Numerical Simulation

This paper designs a highly parallel Nested Factorization (NF) to solve large linear equations generated in reservoir numerical simulation problems. The NF method is a traditional linear solution preprocessing method for reservoir numerical simulation problems, and has regained attention in recent years due to its potential to extend to parallel architectures such as GPUs (Graphics Processor Units). The parallel algorithm of this paper is based on the MPNF (Massively Parallel Nested Factorization) framework proposed by Appleya (Appleyard, Appleyard, Wakefield, & Desitter, 2011). The MPNF algorithm designed in this paper focuses on its efficient implementation on the GPU parallel architecture. Its features include: using a custom matrix structure to achieve merge access, improving access bottlenecks and improving the efficiency of the SpMV algorithm. It is also applicable to the two-stage preprocessing method CPR. (Constrain Pressure Residual) pressure solution and global preprocessing stage; the MPNF method is extended to the solution of 2.5-dimensional unstructured grid problem. The parallel algorithm in this paper has been integrated into the reservoir numerical simulator. For the SPE10 (million grid, highly heterogeneous) standard example, the GPU-based parallel NF algorithm is in the structured grid model and the equivalent 2.5-dimensional non- On the structured grid model, compared with the serial version of the NF method, the acceleration ratios of 19.8 and 17.0 times were obtained respectively; compared with the mainstream serial solution method, the efficiency was also improved by 2 to 3 times.

A multistage stochastic programming model for a multi-period strategic expansion of biofuel supply chain under evolving uncertainties

We develop a multistage, stochastic mixed-integer model to support biofuel supply chain expansion under evolving uncertainties. By utilizing the block-separable recourse property, we reformulate the multistage program in an equivalent two-stage program and solve it using an enhanced nested decomposition method with maximal non-dominated cuts. We conduct extensive numerical experiments and demonstrate the application of the model and algorithm in a case study based on the South Carolina settings. The value of multistage stochastic programming method is also explored by comparing the model solution with the counterparts of an expected value based deterministic model and a two-stage stochastic model.

Accelerating techniques on nested decomposition

In this paper, we consider the nested decomposition method in the context of multistage stochastic problems with scenario trees where we contribute with improvements in accelerating convergence to optimum. We first propose a hybrid protocol for transmitting cuts across nodes aiming at a trade-off between the inherent benefits and drawbacks of the single (aggregated) versus multi-cut (desegregated) versions. The idea we aim to explore is to reduce the size of the problem solved at each iteration by applying the aggregated version of the cuts on the furthest nodes of the tree. We then turn our attention towards the selection of first-stage solutions by employing sampling and simulation aiming to carefully choose those first-stage solution that are potentially capable to drastically decrease the gap between the upper and lower bound at each iteration. The problems on which the latter technique is applied present a special structure, according to which the first-stage variables are linked to all further stages. We test both accelerating techniques on a generic problem which displays a blend of two-stage and multistage structure and we report significant savings in terms of CPU time and number of iterations to convergence.

Temporal and spatial differences and imbalance of China’s urbanization development during 1950–2006

A better understanding of the regional disparity and imbalance characteristics of China’s urbanization development is the important premise for constituting correct policy and strategy and promoting the healthy and sustainable development of urbanization in the 21st century. The regional differences of China’s urbanization level have close relations with natural conditions of landform and climate etc., the urbanization level reduces with the elevation of topography and decrease of precipitation. According to the statistical data set of urbanization in 1950–2006, the temporal change course of inter-provincial disparity of Chinese urbanization level since the founding of New China in 1949 was studied, and then the inter-regional and intra-regional disparities of urbanization development were analyzed by the Theil index and its nested decomposition method, to grasp the dynamic change of spatial disparities of China’s urbanization level on the whole. Using the imbalance index model, the imbalance status of urban population distribution relative to total population, grain output, total agricultural output value, gross output value of industry, tertiary industrial output value as well as gross regional product was discussed, to hold the balance characteristics of urbanization development relative to the regional development conditions from the macroscopic scales.

Multistage stochastic programming with fuzzy probability distribution

In this paper, we introduce the multistage stochastic program with fuzzy probability distribution. We focus on the case where fuzzy probability distribution is defined by (triangular) fuzzy numbers. We extend Ben Abdelaziz and Masri [Stochastic programming with fuzzy linear partial information on probability distribution, European Journal Operational Research 162 (2005) 619–629] solution strategy, for the two-stage stochastic program with fuzzy probability distribution, to solve the multistage model. The proposed solution strategy is based on two transformation steps. In the first step, the fuzzy transformation step, we propose to use the α -cut defuzzification technique. The level α relates to the DM credibility degree on information sources. This step ends with a certainty equivalent program. In the second step, the stochastic transformation step, we decompose the certainty equivalent program based on a minimax approach. The obtained problem is then solved using a modified version of the nested decomposition method. The modification on the nested decomposition method concerns the way in which we generate optimal constraints. The modified nested decomposition algorithm may be used to solve the multistage problem with interval probability distribution.