Order Cone Relaxation Research Articles

Compressed sensing: a discrete optimization approach

We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression, image reconstruction, and multi-label learning. We introduce an ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _2$$\\end{document} regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve small-scale instances of CS to certifiable optimality. When compared against solutions produced by three state of the art benchmark methods on synthetic data, our numerical results show that our approach produces solutions that are on average 6.22%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6.22\\%$$\\end{document} more sparse. When compared only against the experiment-wise best performing benchmark method on synthetic data, our approach produces solutions that are on average 3.10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.10\\%$$\\end{document} more sparse. On real world ECG data, for a given ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _2$$\\end{document} reconstruction error our approach produces solutions that are on average 9.95%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$9.95\\%$$\\end{document} more sparse than benchmark methods (3.88%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.88\\%$$\\end{document} more sparse if only compared against the best performing benchmark), while for a given sparsity level our approach produces solutions that have on average 10.77%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10.77\\%$$\\end{document} lower reconstruction error than benchmark methods (1.42%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.42\\%$$\\end{document} lower error if only compared against the best performing benchmark). When used as a component of a multi-label classification algorithm, our approach achieves greater classification accuracy than benchmark compressed sensing methods. This improved accuracy comes at the cost of an increase in computation time by several orders of magnitude. Thus, for applications where runtime is not of critical importance, leveraging integer optimization can yield sparser and lower error solutions to CS than existing benchmarks.

A robust optimal dispatching strategy of distribution networks considering fast charging stations integrated with photovoltaic and energy storage

With the increasing number of electric vehicles, a large number of charging loads connected to the power system will have an impact on the economic and safe operation of the power system. In this paper a day-ahead optimal dispatching method for distribution network (DN) with fast charging station (FCS) integrated with photovoltaic (PV) and energy storage (ES) is proposed to deal with the negative impact of FCS on DN. By adjusting the load distribution of DN through the optimization decision of ES and soft open points (SOP), the operation level of DN is improved. Firstly, based on the historical vehicle travel data, Monte Carlo simulation method (MCSM) is applied to realize the simulation of fast charging load. Secondly, the uncertainties of PV power is addressed via a robust optimization model of the economic operation level of DN. Based on the second order cone relaxation and duality theory, a two-stage optimal dispatching model of DN is proposed. The optimization model is divided into main problem (MP) and sub problem (SP). For MP, the access position of FCS is adjusted based on SOP. And the charging and discharging power of ES is adjusted. The load distribution is optimized. For SP, based on the uncertainty of PV, the worst scenario of DN is calculated. The robustness of the proposed strategy is guaranteed. Finally, the proposed is verified based on the IEEE 33 bus system and a road network with 34 nodes. The simulation results show that the proposed method can effectively relieve line congestion of DN. The operating range of the voltage is better optimized. And the operation cost of DN is reduced significantly.

Day-ahead optimal scheduling model of transmission–distribution integrated electricity–gas systems based on convex optimization

With the deepening of the coupling degree between power network and natural gas network, the traditional independent optimal scheduling of transmission and distribution integrated electricity–gas systems cannot make full use of the flexible control ability of various resources and the complementary support capabilities between different energy systems. Therefore, this paper puts forward a cooperative optimization scheduling model of transmission–distribution​ integrated electricity–gas systems (TD-IEGS). The coordinated operation of multi-energy coupling devices including gas-fired units, combined heat and power (CHP) units and electric boilers (EB) are considered in this paper to enhance system operation flexibility and economy. The second order cone relaxation (SOCR) method is utilized to tackle the non-linear power and gas flow constraints, and the proposed mixed-integer non-linear programming co-optimization scheduling model is transformed into a mixed integer second order cone programming model (MISOCP). Case studies demonstrate that compared with the independent operation mode, the proposed co-optimization scheduling model of TD-IEGS can reduce total operation cost by about 6.7% under 50% wind power penetration level. Moreover, the co-optimization of TD-IEGS can promote wind power utilization especially in a high share of wind energy.

Flexible Operation of AC/DC Distribution Network Based on Improved Flexibility Demand Quantification

The AC / DC active distribution network (ADN) connected to a large number of renewable energy source (RES) and new loads such as EV will become the main form of the future ADN, which intensifies the volatility and uncertainty of the net load. To cope with this situation, considering various types of flexibility resources, this paper quantitatively evaluates the flexibility demand and supply in the system, and constructs a day-ahead flexible operation model of AC/DC ADN with a high proportion of RES. At first, the flexibility demand is quantified firstly by considering volatility and uncertainty comprehensively, and the conditional probability distribution of wind power volatility is obtained through the analysis of mutual information and other correlation coefficients. Then, a quantitative method to the time flexibility provided by orderly charging of electric vehicles (EVs) is proposed for the flexible cooperation with other resources. Besides, voltage source converter (VSC) with precise loss is considered as flexibility source to participate in dispatching. Furthermore, considering the preference of flexibility in different directions, a weighted node flexibility margin index is proposed. Considering the accuracy and computational efficiency, a second order cone relaxation method which is suitable for the elaborate VSC model is applied in this paper. In the end, Test results of an improved IEEE33-bus system with a three-terminal DC network show that the proposed model can improve the flexibility and reduce the operation cost of the ADN.

Scenario-based comprehensive expansion planning model for a coupled transportation and active distribution system

The widespread utilization of electric vehicles has inspired the emerging trend of coupled transportation and distribution system, which entails the systematic methodologies to model the new planning problems. This paper proposes a scenario-based comprehensive expansion planning model for a coupled transportation and active distribution system. With the aim of minimizing the investment and operation costs, this model determines the best alternatives, locations and sizes for candidate assets, including traffic roads, distribution lines, distribution generators, capacitor banks, static var compensators, voltage regulators, energy storage systems and charging facilities, as well as their operation strategies. First, a generated scenario method is extended to incorporate the uncertainty of the traffic flow demand. Based on multiple scenarios, the steady-state distribution of traffic flow is characterized by the Wardrop user equilibrium principle, and the corresponding equivalent constraints are derived and incorporated into the model. For active distribution system, we formulate the operation constraints for related infrastructures. Considering the interdependency between the two systems, an expansion planning model is proposed, which simultaneously optimizes the investment and operation strategies. Due to the nonlinear nature of the model, we have developed a three-dimensional piecewise linear approximation and applied second order cone relaxation to reformulate the model as a mixed-integer second order conic program; thus, the global optimal solution can be found in a reasonable time frame. Results on a test system reveal that the variations of both systems have some influences on each other, indicating the significance of considering the interdependencies between transportation and active distribution system.

Optimizing operation of integrated energy system considering distribution network and natural gas network

In this paper, an optimizing operation model of integrated energy system (IES) based on distribution network and natural gas network is proposed. While meeting the safety operation constraints of these two networks, the cost of energy for the integrated energy system is minimized at the same time. The proposed model is a comprehensive non-convex problem which is difficult to solve. In order to solve this problem, the incremental piecewise linearization as well as second order cone relaxation (SOCR) is applied to convert the original model into a convex problem. Therefore, the operational scheme of the integrated energy system can be obtained through solving the prosed model. Finally, the effectiveness of the prosed model was verified by simulation on an integrated energy system consisting of the 20-node natural gas network and IEEE 14 node system.

Two-stage Reactive Power Optimization Strategy for Active Distribution Network Based on Extreme Scenarios

Uncertainties from distributed generations (DGs) make conventional reactive power optimization (RPO) methods difficult to be applied in the distribution network (DN). To settle this issue, a novel two-stage RPO strategy for active distribution network (ADN) based on extreme scenarios is proposed in this paper. Random variables are handled through extreme scenarios, then flexible and inflexible variables are decided in two different stages to adapt to DG’s randomness. Firstly, based on second order cone relaxation (SOCR), power flow equations are convexly relaxed to transform original model into mixed integer second order cone programming (MISOCP). Then, extreme scenario method (ESM) is employed to determine DG’s deployment. Finally, numerical tests are performed on a modified IEEE 33-bus DN to demonstrate the effectiveness and feasibility of the proposed strategy.

Mixed Integer Second Order Cone Relaxation With Dynamic Simulation for Proper Power System Islanding Operations

Power system islanding operation acts as an important attractive corrective measure that is widely studied. However, a proper splitting strategy for the islanding operation should satisfy both steady-state and transient stability constraints. As a result, an ac power flow-based optimization model with dynamic simulation is proposed to obtain a proper splitting strategy. Specifically, the method includes two steps: 1) optimization of a splitting strategy satisfying the steady-state constraints and 2) transient stability evaluation of the strategy by dynamic simulation. Furthermore, the optimization and evaluation steps are iterated until finding a splitting strategy that satisfies both the steady-state constraints and the transient stability constraints. Finally, the effectiveness of the proposed method is verified on the WSCC 9-bus and the IEEE RTS 24-bus test systems.

Distribution Locational Marginal Pricing by Convexified ACOPF and Hierarchical Dispatch

This paper proposes a hierarchical economic dispatch (HED) mechanism for computing distribution locational marginal prices (DLMPs). The HED mechanism involves three levels: 1) the top level is the national (regional) transmission network; 2) the middle level is the distribution network; and while 3) the lowest level reflects local embedded networks or microgrids. Each network operator communicates its generalized bid functions (GBFs) to the next higher level of the hierarchy. The GBFs approximate the true cost function of a network by a series of affine functions. The concept of Benders cuts are employed in simulating the GBFs. The ac optimal power flow (ACOPF) is convexified and then used for dispatching generators and calculating GBFs and DLMPs. The proposed convexification is based on the second order cone reformulation. A sequential optimization algorithm is developed to tighten the proposed second order cone relaxation of ACOPF. The properties of the sequential tightness algorithm are discussed and proved. The HED is implemented in the GAMS grid computing platform. The GBFs and DLMPs are calculated for the modified IEEE 342 node low voltage test system. The numerical results show the utility of the proposed HED and GBF in implementing DLMP.

Decentralized Stochastic Optimal Power Flow in Radial Networks With Distributed Generation

This paper develops a power management scheme that jointly optimizes the real power consumption of programmable loads and reactive power outputs of photovoltaic (PV) inverters in distribution networks. The premise is to determine the optimal demand response schedule that accounts for the stochastic availability of solar power, as well as to control the reactive power generation or consumption of PV inverters adaptively to the real power injections of all PV units. These uncertain real power injections by PV units are modeled as random variables taking values from a finite number of possible scenarios. Through the use of second order cone relaxation of the power flow equations, a convex stochastic program is formulated. The objectives are to minimize the negative user utility, cost of power provision, and thermal losses, while constraining voltages to remain within specified levels. To find the global optimum point, a decentralized algorithm is developed via the alternating direction method of multipliers that results in closed-form updates per node and per scenario, rendering it suitable to implement in distribution networks with large number of scenarios. Numerical tests and comparisons with an alternative deterministic approach are provided for typical residential distribution networks that confirm the efficiency of the algorithm.

Optimal power flow of a distribution system based on increasingly tight cutting planes added to a second order cone relaxation

Convex relaxations of the optimal power flow (OPF) problem have received a lot of attention in the recent past. In this work, we focus on a second-order cone (SOC) relaxation applied to an OPF based on a branch flow model of a radial and balanced distribution system. We start by examining various sets of conditions ensuring the exactitude of such a relaxation, which is the main focus of the existing literature. In particular, we observe that these sets always include a requirement on the objective to be a minimization of a function increasing with the branch flow apparent powers. We consider this hypothesis to be at odds with what is to be expected of an active distribution system and demonstrate in specific case studies its counterproductive impact. We continue by introducing an objective function allowing distributed generations and storages (DGS) to take advantage of the benefits they bring to the power system as a whole. As this entails the possibility for the relaxation not to be exact, we describe and prove the theoretical convergence to optimality of an algorithm consisting in adding an increasingly tight linear cut to the SOC relaxation. In order to allow the attainment of a solution satisfying the network constraints in a finite number of steps, we continue by introducing a tailored termination criterion. Afterwards, we investigate the ability of our algorithm to obtain a satisfactory solution on several case studies, spanning various network sizes, number of nodes equipped with DGs and their level of penetration. We then conclude on the benefits brought about by this approach and reflect on its limits and the opportunities for further improvements.