Approximation of System Components for Pump Scheduling Optimisation

© 2015 The Authors. Published by Elsevier Ltd.The operation of pump systems in water distribution systems (WDS) is commonly the most expensive task for utilities with up to 70% of the operating cost of a pump system attributed to electricity consumption. Optimisation of pump scheduling could save 10-20% by improving efficiency or shifting consumption to periods with low tariffs. Due to the complexity of the optimal control problem, heuristic methods which cannot guarantee optimality are often applied. To facilitate the use of mathematical optimisation this paper investigates formulations of WDS components. We show that linear approximations outperform non-linear approximations, while maintaining comparable levels of accuracy

The Maritime Economy of Ancient Cyprus in Terms of the New Institutional Economics

"The Maritime Economy of Ancient Cyprus in Terms of the New Institutional Economics deals with the maritime economy of ancient Cyprus from 1450 BC to 295 BC, and comprises three parts which correspond to three distinct economic cycles: first economic cycle during the age of internationalism 1450–1200 BC second economic cycle during political volatility, economic growth and transformation 1200–525 BC third economic cycle in the Persian Empire until annexation into the Ptolemaic kingdom. The principles of New Institutional Economics are used to trace the island’s institutions and their continuity and to reconstruct its maritime history. A unique feature is that for the first time a traditional descriptive and cultural approach is complemented by systematic and mathematical analysis and marketing documentation which results in meaningful examination of economic performance. This new approach highlights and explains the maritime economic activity of Ancient Cyprus and the Eastern Mediterranean in general. It brings together, for the first time, three distinct disciplines, that is History, Archaeology and Economic theory, in order to create a balanced explanation and reconstruction of the maritime economy of ancient Cyprus and of the challenges which confronted the ancient seafarers and traders of the Eastern Mediterranean. The approach and methodology is influenced by the author’s engineering, business background and training.

Demonstrating demand response from water distribution system through pump scheduling

Significant changes in the power generation mix are posing new challenges for the balancing systems of the grid. Many of these challenges are in the secondary electricity grid regulation services and could be met through demand response (DR) services. We explore the opportunities for a water distribution system (WDS) to provide balancing services with demand response through pump scheduling and evaluate the associated benefits. Using a benchmark network and demand response mechanisms available in the UK, these benefits are assessed in terms of reduced green house gas (GHG) emissions from the grid due to the displacement of more polluting power sources and additional revenues for water utilities. The optimal pump scheduling problem is formulated as a mixed-integer optimisation problem and solved using a branch and bound algorithm. This new formulation finds the optimal level of power capacity to commit to the provision of demand response for a range of reserve energy provision and frequency response schemes offered in the UK. For the first time we show that DR from WDS can offer financial benefits to WDS operators while providing response energy to the grid with less greenhouse gas emissions than competing reserve energy technologies. Using a Monte Carlo simulation based on data from 2014, we demonstrate that the cost of providing the storage energy is less than the financial compensation available for the equivalent energy supply. The GHG emissions from the demand response provision from a WDS are also shown to be smaller than those of contemporary competing technologies such as open cycle gas turbines. The demand response services considered vary in their response time and duration as well as commitment requirements. The financial viability of a demand response service committed continuously is shown to be strongly dependent on the utilisation of the pumps and the electricity tariffs used by water utilities. Through the analysis of range of water demand scenarios and financial incentives using real market data, we demonstrate how a WDS can participate in a demand response scheme and generate financial gains and environmental benefits

Decomposition-Based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems

We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in [1, 2] that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decompositionbased method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs [3] .Polynomial optimization, Semidefinite programming, Sparse SDP relaxations, Benders decomposition

Partitioning Procedure for Polynomial Optimization: Application to Portfolio Decisions with Higher Order Moments

We consider the problem of finding the minimum of a real-valued multivariate polynomial function constrained in a compact set defined by polynomial inequalities and equalities. This problem, called polynomial optimization problem (POP), is generally nonconvex and has been of growing interest to many researchers in recent years. Our goal is to tackle POPs using decomposition. Towards this goal we introduce a partitioning procedure. The problem manipulations are in line with the pattern used in the Benders decomposition [1], namely relaxation preceded by projection. Stengle’s and Putinar’s Positivstellensatz are employed to derive the so-called feasibility and optimality constraints, respectively. We test the performance of the proposed method on a collection of benchmark problems and we present the numerical results. As an application, we consider the problem of selecting an investment portfolio optimizing the mean, variance, skewness and kurtosis of the portfolio.Polynomial optimization, Semidefinite relaxations, Positivstellensatz, Sum of squares, Benders decomposition, Portfolio optimization

The Maritime Economy of Ancient Cyprus in Terms of the New Institutional Economics

"The Maritime Economy of Ancient Cyprus in Terms of the New Institutional Economics deals with the maritime economy of ancient Cyprus from 1450 BC to 295 BC, and comprises three parts which correspond to three distinct economic cycles: first economic cycle during the age of internationalism 1450–1200 BC second economic cycle during political volatility, economic growth and transformation 1200–525 BC third economic cycle in the Persian Empire until annexation into the Ptolemaic kingdom. The principles of New Institutional Economics are used to trace the island’s institutions and their continuity and to reconstruct its maritime history. A unique feature is that for the first time a traditional descriptive and cultural approach is complemented by systematic and mathematical analysis and marketing documentation which results in meaningful examination of economic performance. This new approach highlights and explains the maritime economic activity of Ancient Cyprus and the Eastern Mediterranean in general. It brings together, for the first time, three distinct disciplines, that is History, Archaeology and Economic theory, in order to create a balanced explanation and reconstruction of the maritime economy of ancient Cyprus and of the challenges which confronted the ancient seafarers and traders of the Eastern Mediterranean. The approach and methodology is influenced by the author’s engineering, business background and training.

Multilevel approximate robust principal component analysis

Robust principal component analysis (RPCA) is currently the method of choice for recovering a low-rank matrix from sparse corruptions that are of unknown value and support by decomposing the observation matrix into low-rank and sparse matrices. RPCA has many applications including background subtraction, learning of robust subspaces from visual data, etc. Nevertheless, the application of SVD in each iteration of optimisation methods renders the application of RPCA challenging in cases when data is large. In this paper, we propose the first, to the best of our knowledge, multilevel approach for solving convex and non-convex RPCA models. The basic idea is to construct lower dimensional models and perform SVD on them instead of the original high dimensional problem. We show that the proposed approach gives a good approximate solution to the original problem for both convex and non-convex formulations, while being many times faster than original RPCA methods in several real world datasets

Fast multilevel algorithms for compressive principal component pursuit

Recovering a low-rank matrix from highly corrupted measurements arises in compressed sensing of structured high-dimensional signals (e.g., videos and hyperspectral images among others). Robust principal component analysis (RPCA), solved via principal component pursuit (PCP), recovers a low-rank matrix from sparse corruptions that are of unknown value and support by decomposing the observation matrix into two terms: a low-rank matrix and a sparse one, accounting for sparse noise and outliers. In the more general setting, where only a fraction of the data matrix has been observed, low-rank matrix recovery is achieved by solving the compressive principal component pursuit (CPCP). Both PCP and CPCP are well-studied convex programs, and numerous iterative algorithms have been proposed for their optimisation. Nevertheless, these algorithms involve singular value decomposition (SVD) at each iteration, which renders their applicability challenging in the case of massive data. In this paper, we propose a multilevel approach for the solution of PCP and CPCP problems. The core principle behind our algorithm is to apply SVD in models of lower-dimensionality than the original one and then lift its solution to the original problem dimension. Hence, our methods rely on the assumption that the low rank component can be represented in a lower dimensional space. We show that the proposed algorithms are easy to implement, converge at the same rate but with much lower iteration cost. Numerical experiments on numerous synthetic and real problems indicate that the proposed multilevel algorithms are several times faster than their original counterparts, namely PCP and CPCP

Paramater estimation for the McKean-Vlasov stochastic differential equation

We consider the problem of parameter estimation for a stochastic McKean-Vlasov equation, and the associated system of weakly interacting particles. We first establish consistency and asymptotic normality of the offline maximum likelihood estimator for the interacting particle system in the limit as the number of particles $N\rightarrow\infty$. We then propose an online estimator for the parameters of the McKean-Vlasov SDE, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. We prove that this estimator converges in $\mathbb{L}^1$ to the stationary points of the asymptotic log-likelihood of the McKean-Vlasov SDE in the joint limit as $N\rightarrow\infty$ and $t\rightarrow\infty$, under suitable assumptions which guarantee ergodicity and uniform-in-time propagation of chaos. We then demonstrate, under the additional assumption of global strong concavity, that our estimator converges in $\mathbb{L}^2$ to the unique maximiser of this asymptotic log-likelihood function, and establish an $\mathbb{L}^2$ convergence rate. We also obtain analogous results under the assumption that, rather than observing multiple trajectories of the interacting particle system, we instead observe multiple independent replicates of the McKean-Vlasov SDE itself or, less realistically, a single sample path of the McKean-Vlasov SDE and its law. Our theoretical results are demonstrated via two numerical examples, a linear mean field model and a stochastic opinion dynamics model