Consensus-based Alternating Direction Method Of Multipliers Research Articles

Dynamic Consensus-Based ADMM Strategy for Economic Dispatch with Demand Response in Power Grids

This paper introduces a dynamic consensus-based economic dispatch (ED) algorithm utilizing the Alternating Direction Method of Multipliers (ADMM) to optimize real-time pricing and generation/demand decisions within a decentralized energy management framework. The increasing complexity of modern energy markets, driven by the proliferation of Distributed Energy Resources (DER) and variable demands from hybrid electric vehicles, necessitates a departure from traditional centralized dispatch methods. This research proposes a novel ADMM-based solution tailored for non-responsive and responsive demand units that integrates demand response mechanisms to adaptively manage real-time fluctuations while enhancing security and privacy through distributed data management. The testing of the algorithm on the IEEE 39 bus system under various load conditions over 24 h demonstrated the algorithm’s effectiveness in handling traditional and renewable energy sources, particularly highlighting the economic benefits of shifting controllable loads to periods of low-cost renewable availability. The findings underscore the algorithm’s potential to reduce energy costs, enhance energy efficiency, and offer a scalable solution across diverse grid systems, contributing significantly to advancing global energy policy and sustainable management practices.

Fully Distributed Economic Dispatch with Random Wind Power Using Parallel and Finite-Step Consensus-Based ADMM

In this paper, a fully distributed strategy for the economic dispatch problem (EDP) in the smart grid is proposed. The economic dispatch model considers both traditional thermal generators and wind turbines (WTs), integrating generation costs, carbon trading expenses, and the expected costs associated with the unpredictability of wind power. The EDP is transformed into an equivalent optimization problem with only an equality constraint and thus can be solved by an alternating-direction method of multipliers (ADMM). Then, to tackle this problem in a distributed manner, the outer-layer framework of the proposed strategy adopts a parallel ADMM, where different variables can be calculated simultaneously. And the inner-layer framework adopts a finite-step consensus algorithm. Convergence to the optimal solution is achieved within a finite number of communication iterations, which depends on the scale of the communication network. In addition, leveraging local and neighbor information, a distributed algorithm is designed to compute the eigenvalues of the Laplacian matrix essential for the finite-step algorithm. Finally, several numerical examples are presented to verify the correctness and effectiveness of the proposed strategy.

A consensus-based alternating direction method of multipliers approach to parallelize large-scale minimum-lap-time problems

AbstractMinimum-lap-time planning (MLTP) problems, which entail finding optimal trajectories for race cars on racetracks, have received significant attention in the recent literature. They are commonly addressed as optimal control problems (OCPs) and are numerically discretized using direct collocation methods. Subsequently, they are solved as nonlinear programs (NLPs). The conventional approach to solving MLTP problems is serial, whereby the resulting NLP is solved all at once. However, for problems characterized by a large number of variables, distributed optimization algorithms, such as the alternating direction method of multipliers (ADMM), may represent a viable option, especially when multicore CPU architectures are available.This study presents a consensus-based ADMM approach tailored to solving MLTP problems through a distributed optimization algorithm. The algorithm partitions the problem into smaller subproblems based on different sectors of a track, distributing them among multiple processors. ADMM is then used to ensure consensus among the distributed computational processes. In particular, here the term “consensus” denotes the requirement for each subproblem to achieve mutual agreement across the junction areas. The paper also outlines specific strategies leveraging domain knowledge to improve the convergence of the distributed algorithm. The ADMM approach is validated against the serial approach, and numerical results are presented for both single-lap and multilap scenarios. In both cases, the ADMM approach proves superior for problem dimensions of 70k+ variables compared to serial methods. In planning scenarios with complex vehicle models on long track horizons, i.e., for problems with 1M+ variables, the efficiency gain of the ADMM approach is substantial, and it becomes the only viable option to maintain computational times within acceptable limits.

Distributed robust operation of integrated energy system considering gas inertia and biogas–wind renewables

Multiple uncertainties caused by renewable energy and loads bring challenges to the secure operation of integrated energy system (IES). Concurrently, the centralized control methods based on global information of IES cannot preserve the data privacy of each individual subsystem. To deal with these issues, this paper proposes a distributed two-stage robust optimization (TRO) scheduling strategy for IES considering gas inertia and biogas–wind renewables. To address the uncertainties of wind turbines output and electricity loads in IES, a TRO model is first established, and the column and constraint generation (C&CG) algorithm is employed to handle the min–max–min problem of TRO model. On this basis, a distributed cooperative scheduling framework based on the consensus-based alternating direction method of multipliers (C-ADMM) for IES is developed, which requires no overall system information but only local information. Additionally, natural gas pipelines that consider gas inertia are used as gas storage facilities, which provides IES with additional scheduling resources and flexibility. Furthermore, an adaptive step-size strategy is proposed to adjust the step-size of ADMM in real time, which reduces the dependence of ADMM performance on the initial parameters selection and improves its convergence performance. The simulation results in a three-subsystem joint IES are presented to demonstrate the effectiveness of the proposed distributed TRO scheduling strategy.

Decentralized optimal scheduling for integrated community energy system via consensus-based alternating direction method of multipliers

To transform the original centralized scheduling framework of integrated community energy systems (ICES), a decentralized scheduling method is proposed for the optimal operation of ICES. Firstly, linearized mathematical models of the electric distribution system and natural gas distribution system are established to eliminate the nonconvexity and nonlinearity of the multi-energy systems. Meanwhile, an improved energy hub model is proposed by introducing the state variable, which is able to obtain the optimal solution with better accuracy and higher efficiency. Secondly, a decentralized scheduling framework is developed via an improved consensus-based alternating direction method of multipliers (ADMM). By introducing several coupling links as consensus variables, the optimal operation of the ICES is decoupled according to the physical interactions among each sub-system, in which the original model can be solved parallelly in a decentralized manner. Only the consensus variables are exchanged among sub-systems so that the privacy and confidential items of each sub-system operated by different entities can be ensured. Finally, two different ICES test cases with different system scales are utilized to demonstrate the effectiveness of the proposed method, which is able to provide scheduling scheme same as the centralized method. Dynamic step size modification is further considered in the consensus-based ADMM. By updating the step size during each iteration, the computation expense of optimization process is reduced with less consuming time, iteration and apparent oscillation, by which the convergence performance of ADMM is improved with higher computation efficiency.

Effects of network topology on the performance of consensus and distributed learning of SVMs using ADMM.

The Alternating Direction Method of Multipliers (ADMM) is a popular and promising distributed framework for solving large-scale machine learning problems. We consider decentralized consensus-based ADMM in which nodes may only communicate with one-hop neighbors. This may cause slow convergence. We investigate the impact of network topology on the performance of an ADMM-based learning of Support Vector Machine using expander, and mean-degree graphs, and additionally some of the common modern network topologies. In particular, we investigate to which degree the expansion property of the network influences the convergence in terms of iterations, training and communication time. We furthermore suggest which topology is preferable. Additionally, we provide an implementation that makes these theoretical advances easily available. The results show that the performance of decentralized ADMM-based learning of SVMs in terms of convergence is improved using graphs with large spectral gaps, higher and homogeneous degrees.

Distributed robust synergistic scheduling of electricity, natural gas, heating and cooling systems via alternating direction method of multipliers

Due to the development of multi-energy conversion technologies and the higher public awareness of the sustainability, the energy system has begun a transition toward the energy hub (EH) based integrated energy system (IES). This paper proposes a distributed robust synergistic scheduling method for multi-EH-based IES coupled with electricity, natural gas, heating and cooling, to minimize the operating cost of the IES while considering the uncertainties of renewable generations (RGs) and multi-energy load demands. The nonconvexity of energy network constraints, uncertainties in multiple EHs as well as limitation of information exchange lead to significant challenges for IES scheduling. To deal these issues, the original robust scheduling model of IES is first reformulated as a mixed integer second-order cone programming (MISOCP) by convex relaxation method. Furthermore, to improve the economy and flexibility of robust scheduling solution, a light robust model is developed to address the multiple uncertainties from EHs. Finally, a consensus-based alternating direction method of multipliers (ADMM) approach is developed to tackle the robust scheduling model in MISOCP form, where EH operators parallelly solve their respective EH subproblems with limited information exchange, and subsequently the EH subproblem in MISOCP form is solved by a nonconvex ADMM (NC-ADMM) approach to guarantee the convergence of the consensus-based ADMM. Simulation results in a three-EH IES are presented to illustrate the effectiveness of proposed method for optimal synergy of multiple EHs with uncertainties.

Consensus and Sectioning-Based ADMM With Norm-1 Regularization for Imaging With a Compressive Reflector Antenna

This paper presents three distributed techniques to find a sparse solution of the underdetermined linear problem $\textbf{g}=\textbf{Hu}$ with a norm-1 regularization, based on the Alternating Direction Method of Multipliers (ADMM). These techniques divide the matrix $\textbf{H}$ in submatrices by rows, columns, or both rows and columns, leading to the so-called consensus-based ADMM, sectioning-based ADMM, and consensus and sectioning-based ADMM, respectively. These techniques are applied particularly for millimeter-wave imaging through the use of a Compressive Reflector Antenna (CRA). The CRA is a hardware designed to increase the sensing capacity of an imaging system and reduce the mutual information among measurements, allowing an effective imaging of sparse targets with the use of Compressive Sensing (CS) techniques. Consensus-based ADMM has been proved to accelerate the imaging process and sectioning-based ADMM has shown to highly reduce the amount of information to be exchange among the computational nodes. In this paper, the mathematical formulation and graphical interpretation of these two techniques, together with the consensus and sectioning-based ADMM approach, are presented. The imaging quality, the imaging time, the convergence, and the communication efficiency among the nodes are analyzed and compared. The distributed capabitities of the ADMM-based approaches, together with the high sensing capacity of the CRA, allow the imaging of metallic targets in a 3D domain in quasi-real time with a reduced amount of information exchanged among the nodes.

Distributed tracking-ADMM approach for chance-constrained energy management with stochastic wind power in smart grid

This paper proposes a distributed strategy to the chance-constrained energy management for smart grid with penetration of stochastic wind power. The energy management model is constructed including chance constraints of spinning reserves for the sake of guaranteeing the maximum utilization of wind power on the basis of reliability. With the available wind power characterizing by Weibull distribution, the chance constraints can be converted into deterministic ones by the derived analytical form of inverse cumulative distribution function. Although the original problem is transformed into a typical convex optimization problem, the tight coupling of constraints presents challenges to the design of distributed strategy. As such, we formulate the problem into a compact form with each generator unit depending on individual decision variables, instead of the common form with decision vector being the collection of all local decision variables. Then, by developing a new initialization method and an adaptive weight matrix selection method, a distributed strategy based on tracking Alternating Direction Method of Multipliers (ADMM) is proposed to solve the model. The simulation results indicate that the proposed distributed strategy achieves comparable performance to the corresponding centralized scenario, and better performance than distributed consensus-based ADMM in the related literature. Moreover, the validity of the proposed distributed strategy is confirmed in one day chance-constrained energy management with stochastic wind power.

A New Distributed Energy Management Strategy for Smart Grid With Stochastic Wind Power

In this article, a distributed energy management strategy is proposed for smart grid with stochastic wind power. The different Weibull probability distribution function (PDF) of wind power during different operation intervals, coupled with ramp rate limits of thermal generators across operation intervals, is considered in this operation cost optimization problem. Under multiagent system communication framework, we first decompose the problem into subproblems associated to operation interval. For each operation interval, parameters of Weibull PDF are estimated by maximum likelihood method according to wind speed obtained by wind speed forecast model of committed wind turbine. Then, a consensus-based alternating direction method of multipliers (ADMM) is proposed to get the optimal operation plan for all committed generators, where Armijo line search algorithm is employed to adaptively obtain the step size in the iterations. The simulation results show that the proposed consensus-based ADMM outperforms distributed projected gradient method in previous study on energy management with wind power; the proposed distributed strategy is valid for short-term energy management for smart grid with stochastic wind power.

Optimal Energy Management in Combined Heat and Power System via A Decentralized Consensus-Based ADMM

Abstract This paper proposes a comprehensive optimization model that considers not only the economic dispatch (ED) of the combined heat and power (CHP) units, but also the demand response (DR) of consuming units in the energy management system, where each individual unit can exchange the information with its neighbours. In this optimization problem, there are energy balance constraints and individual local constraints. Particularly, the progresses of the power dispatch and the heat dispatch of each CHP unit are coupled through a feasible polygon region constraint, and the power demands of each consumer among different periods are also coupled due to the requirement of the total power consumption. To achieve the optimal energy coordination of the underlying system, we propose a decentralized alternating direction method of multipliers (ADMM), under connected communication network of individuals, such that each CHP unit and consumer can simultaneously implement their own optimal strategies based on an agreed energy price derived by a consensus protocol. The convergence and optimality of the proposed method are guaranteed under certain conditions. Simulation results are shown to demonstrate the developed results.

Optimizing the Dynamic Economic Dispatch Problem by the Distributed Consensus-Based ADMM Approach

This paper proposes a novel distributed approach to solve a new dynamic economic dispatch problem (DEDP) in which environmental cost function and ramp rate constraints are taken into consideration in islanded microgrid. In our proposed optimization model, the environmental cost function with E-exponential term and ramp rate constraints are considered to make the optimization problem more practical. Then a novel fully distributed algorithm is proposed to address the DEDP based on the alternating direction method of multipliers (ADMM) and distributed consensus theory of the multiagents system. A Lambert $W$ function is employed to tackle the E-exponential term in the environmental cost function, which is different from most existing papers which discuss the DEDP only with quadratic cost functions. A parallel projection method on account of ADMM is used to deal with the ramp rate constraints in this paper. In addition, the power balance can be guaranteed every time when the sum of initial powder output is equal to the total demand. Therefore, the proposed algorithm can deal with the supply-demand constraints, capacity limit constraints, and ramp rate constraints. Finally, simulations on IEEE14-bus are introduced to further illustrate the effectiveness of the proposed algorithm.

Distributed Inexact Consensus-Based ADMM Method for Multi-Agent Unconstrained Optimization Problem

Recently, the alternating direction method of multipliers (ADMM) has been used effectively to solve the multi-agent unconstrained optimization problems, where the objective function is the sum of privately known local objective functions of agents. In this paper, first, with the help of the edge-node incidence matrix, an unconstrained optimization problem is transformed into an equivalent optimization problem with only equality constraint and, thus, can be dealt with the ADMM conveniently. Second, a novel distributed inexact consensus ADMM is proposed to enable the agents to reach consensus on the optimal solution of the optimization problem. At the same time, the analysis of the linear convergence of the proposed algorithm is also provided under some mild conditions. Finally, some simulation results are presented to demonstrate the better effectiveness of the proposed algorithm than the standard consensus-based ADMM algorithm.

Synergistic Operation of Electricity and Natural Gas Networks via ADMM

The increasing growth in the installation of natural-gas fired units has raised necessitates on synergy between the electricity and natural gas networks. This paper discusses how synergistic operation of electricity and natural gas networks can be achieved in a distributed fashion using alternating direction method of multipliers (ADMM). A standard ADMM approach and a consensus-based ADMM approach are developed, respectively, to solve the gas-electric integrated optimal power flow problem with and without a coordination operator. In both cases, the optimization formulation for each operator is modeled, and the solution procedure as well as data exchange among multiple decision-makers are explained. Case studies on both small and large gas-electric integrated systems show that the two ADMM-based distributed approaches have significant efficiencies over the traditional Lagrange relaxation (LR) and augmented LR-based methods.

A Fully-Decentralized Consensus-Based ADMM Approach for DC-OPF With Demand Response

This paper discusses a consensus-based alternating direction method of multipliers (ADMMs) approach for solving the dynamic dc optimal power flow (DC-OPF) problem with demand response in a distributed manner. In smart grid, emerging techniques together with distributed nature of data and information, significantly increase the complexity of power systems operation and stimulate the needs for distributed optimization. In this paper, the distributed DC-OPF approach solves local OPF problems of individual subsystems in parallel, which are coordinated via global consensus variables (i.e., phase angles on boundary buses of adjacent subsystems). Three distributed DC-OPF algorithms are discussed with different convergence performance and/or communication requirement. All three distributed algorithms can effectively handle prevailing constraints for the transmission network, generating units, and demand response in individual subsystems, while the global convergence can be guaranteed. In particular, based on the traditional distributed ADMM approach, a fully decentralized algorithm without the central controller is proposed in Algorithm 2 with a new communication strategy, in which only limited information on boundary buses are exchanged among adjacent subsystems. In addition, the accelerated ADMM is discussed in Algorithm 3 for improving the convergence performance. In recognizing distributed OPF approaches in literature, one major research focus on this paper is to quantify the impact of key parameters and subsystem partitioning strategies on the convergence performance and the data traffic via numerical case studies. A general guidance for subsystem partitioning is proposed and verified via large-scale power systems.

Norm-1 Regularized Consensus-Based ADMM for Imaging With a Compressive Antenna

This paper presents a novel norm-one-regularized, consensus-based imaging algorithm, based on the Alternating Direction Method of Multipliers (ADMM). This algorithm is capable of imaging composite dielectric and metallic targets by using limited amount of data. The distributed capabilities of the ADMM accelerates the convergence of the imaging. Recently, a Compressive Reflector Antenna (CRA) has been proposed as a way to provide high-sensing-capacity with a minimum cost and complexity in the hardware architecture. The ADMM algorithm applied to the imaging capabilities of the Compressive Antenna (CA) outperforms current state of the art iterative reconstruction algorithms, such as Nesterov-based methods, in terms of computational cost; and it ultimately enables the use of a CA in quasi-real-time, compressive sensing imaging applications.