Distributed Optimization Approach Research Articles

Integrated energy microgrids participating in voltage regulation ancillary services: An improved ADMM based distributed optimization approach

AbstractThe integrated energy microgrid (IEM) has good potential for both active and reactive power regulation, which are gradually becoming critical resources for participating in power system ancillary services. This paper constructs a distributed optimization model for the IEMs participating in voltage regulation ancillary services by considering multiple physical network constraints. To reduce computational complexity, the DistFlow power flow model and the second‐order cone relaxation method are introduced to transform the optimization model into a second‐order cone programming problem. Moreover, an improved accelerated consensus alternating direction method of multipliers algorithm is proposed to accelerate distributed optimization solutions of the model while protecting user privacy. Finally, the proposed optimization model and solution method are tested in the modified IEEE‐33 node test case to verify their effectiveness and feasibility.

Distributed Nonconvex Optimization for Control of Water Networks with Time-coupling Constraints

AbstractIn this paper, we present a new control model for optimizing pressure and water quality operations in water distribution networks. Our formulation imposes a set of time-coupling constraints to manage temporal pressure variations, which are exacerbated by the transition between pressure and water quality controls. The resulting optimization problem is a nonconvex, nonlinear program with nonseparable structure across time steps. This problem proves challenging for state-of-the-art nonlinear solvers, often precluding their direct use for near real-time control in large-scale networks. To overcome this computational burden, we investigate a distributed optimization approach based on the alternating direction method of multipliers (ADMM). In particular, we implement and evaluate two algorithms: a standard ADMM scheme and a two-level variant that provides theoretical convergence guarantees for our nonconvex problem. We use a benchmarking water network and a large-scale operational network in the UK for our numerical experiments. The results demonstrate good convergence behavior across all problem instances for the two-level algorithm, whereas the standard ADMM approach struggles to converge in some instances. With an appropriately tuned penalty parameter, however, both distributed algorithms yield good quality solutions and computational times compatible with near real-time (e.g. hourly) control requirements for large-scale water networks.

Hierarchical optimal dispatching method of transmission system operators‐distribution system operators cooperated integrated energy system considering gas‐thermal equivalence model

AbstractThe integrated energy system (IES) is moving towards the large‐scale cooperation between transmission system operators (TSO) and distribution system operators (DSO) with the coupling of electricity, heating, and gas networks. The conventional optimization method is not time‐efficient for solving such a large‐scale problem when taking account of the cooperation of TSO‐DSO of the IES. To solve such a problem, this paper proposed a hierarchical optimization framework for TSO‐DSO cooperated IES. To further improve the computation efficiency, the equivalent models of heating and gas system that retain the heating inertia characteristic and gas pipe storage characteristic are presented. A computation method for the TSO‐DSO cooperated IES is also proposed from the hierarchical distributed optimization perspective. The case study confirms the efficacy of the proposed gas‐thermal equivalent model in significantly enhancing computational speed. Furthermore, the proven effectiveness of the hierarchical distributed optimization approach in leveraging the full potential of flexible resources in transmission and distribution systems while safeguarding internal information privacy enables the development of robust dispatching strategies for each system.

Distributed optimization for joint peer‐to‐peer electricity and carbon trading among multi‐energy microgrids considering renewable generation uncertainty

AbstractThe increasing penetration of renewable energy and the further coupling of the electricity and carbon markets have hindered the realization of efficient and low‐carbon transformation processes in new power systems. This study addresses the optimization problems of joint peer‐to‐peer (P2P) electricity and carbon trading in multi‐energy microgrids (MEMGs), taking into account the risks associated with renewable generation in a distributed manner. First, a coordinated operation model is developed to describe the joint P2P electricity and carbon trading issues among MEMGs, aiming to minimize operating costs, mitigate potential risk losses, and reduce renewable energy wastage. Second, the conditional value‐at‐risk technique, paired with stochastic programming, is employed to quantify potential risk losses arising from uncertainties. Finally, a distributed optimization approach is developed based on the alternating direction method of multipliers to maintain the privacy and independence of decision‐making in individual MEMGs. During the trading processes, the Lagrangian multipliers are used as price signals to ensure fairness in optimal trading schemes among MEMGs. Moreover, a parallel solution mechanism is implemented to improve overall operational efficiency with minimal calculation expenditure. The simulation results demonstrate that the proposed method can reduce operation costs and carbon emissions while also preventing a significant amount of renewable energy abandonment.

A distributed randomized method for the identification of switched ARX systems

SummaryThe identification of switched systems amounts to a mixed integer nonlinear optimization problem, where the continuous variables are associated to the model parameterizations of the different modes, and the discrete ones are related to the switching signal (each data sample is assigned to a mode, and switching occurs when the mode assignment changes over time). In the batch form of the identification problem, the combinatorial complexity increases exponentially with the size of the training set, which makes the precise identification of the switching signal the most challenging task in the identification problem. To tackle this complexity we propose a distributed optimization approach, based on the solution of multiple instances of a much simpler problem, where switching can occur only at specific time instants (different for each subproblem), and an information sharing mechanism that preserves likely switching times to improve the local solutions. We employ an adapted version of a previously developed randomized algorithm to solve the individual subproblems. Another important feature of the proposed method is an a posteriori heuristic correction method, that is applied to further refine the switching locations based on the estimated local models before the information sharing phase. The performance of the proposed algorithm is analyzed and compared with other methods on synthetic datasets.

Tailoring Gradient Methods for Differentially Private Distributed Optimization

Decentralized optimization is gaining increased traction due to its widespread applications in large-scale machine learning and multi-agent systems. The same mechanism that enables its success, i.e., information sharing among participating agents, however, also leads to the disclosure of individual agents' private information, which is unacceptable when sensitive data are involved. As differential privacy is becoming a de facto standard for privacy preservation, recently results have emerged integrating differential privacy with distributed optimization. However, directly incorporating differential privacy design in existing distributed optimization approaches significantly compromises optimization accuracy. In this paper, we propose to redesign and tailor gradient methods for differentially-private distributed optimization, and propose two differential-privacy oriented gradient methods that can ensure both rigorous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -differential privacy and optimality. The first algorithm is based on static-consensus based gradient methods, and the second algorithm is based on dynamic-consensus (gradient-tracking) based distributed optimization methods and, hence, is applicable to general directed interaction graph topologies. Both algorithms can simultaneously ensure almost sure convergence to an optimal solution and a finite privacy budget, even when the number of iterations goes to infinity. To our knowledge, this is the first time that both goals are achieved simultaneously. Numerical simulations using a distributed estimation problem and experimental results on a benchmark dataset confirm the effectiveness of the proposed approaches.

A Distributed Optimization Approach for Collaborative Object Lifting Using Multiple Aerial Robots

In the past decade, multi-robot collaborative object transport has garnered significant attention, with the majority of research targeting transport strategies. This study recasts the collaborative object lifting challenge into an optimization problem framework. Within this setup, each robot leverages a local evaluation function to determine its lifting location. Collectively, these robots strive to optimize a unified evaluation function. An intertwined equation constraint is embedded within the optimization schema, ensuring that the system’s mass center remains stable throughout the lifting process. Furthermore, we impose local feasibility constraints, thereby delimiting the optimal lifting location to a specified region. This research introduces several algorithms, differentiated based on the constraints applied to robot velocity. By harnessing these algorithms, robots can autonomously pinpoint the most apt lifting location that aligns with predetermined criteria. This methodology necessitates a robot to engage in exchanges of auxiliary variables solely with its immediate peers. Noteworthily, parameters such as location, velocity, and mass are accessed in a localized manner, reinforcing data privacy and reducing communication burdens. The paper concludes with a robust mathematical validation that underscores asymptotic convergence to the exact optimal lifting location, underpinned by numerical simulations which attest to the potency of the proposed algorithms.

Multiple Mittag-Leffler Stability of Almost Periodic Solutions for Fractional-Order Delayed Neural Networks: Distributed Optimization Approach.

This article proposes new theoretical results on the multiple Mittag-Leffler stability of almost periodic solutions (APOs) for fractional-order delayed neural networks (FDNNs) with nonlinear and nonmonotonic activation functions. Profited from the superior geometrical construction of activation function, the considered FDNNs have multiple APOs with local Mittag-Leffler stability under given algebraic inequality conditions. To solve the algebraic inequality conditions, especially in high-dimensional cases, a distributed optimization (DOP) model and a corresponding neurodynamic solving approach are employed. The conclusions in this article generalize the multiple stability of integer-or fractional-order NNs. Besides, the consideration of the DOP approach can ameliorate the excessive consumption of computational resources when utilizing the LMI toolbox to deal with high-dimensional complex NNs. Finally, a simulation example is presented to confirm the accuracy of the theoretical conclusions obtained, and an experimental example of associative memories is shown.

A Differentially Private Method for Distributed Optimization in Directed Networks via State Decomposition

In this paper, we study the problem of consensus-based distributed optimization, where a network of agents, abstracted as a directed graph, aims to minimize the sum of all agents' cost functions collaboratively. In existing distributed optimization approaches (Push-Pull/AB) for directed graphs, all agents exchange their states with neighbors to achieve the optimal solution with a constant stepsize, which may lead to the disclosure of sensitive and private information. For privacy preservation, we propose a novel state-decomposition based gradient tracking approach (SD-Push-Pull) for distributed optimization over directed networks that preserves differential privacy, which is a strong notion that protects agents' privacy against an adversary with arbitrary auxiliary information. The main idea of the proposed approach is to decompose the gradient state of each agent into two sub-states. Only one substate is exchanged by the agent with its neighbours over time, and the other one is not shared. That is to say, only one substate is visible to an adversary, protecting the sensitive information from being leaked. It is proved that under certain decomposition principles, a bound for the sub-optimality of the proposed algorithm can be derived and the differential privacy is achieved simultaneously. Moreover, the trade-off between differential privacy and the optimization accuracy is also characterized. Finally, a numerical simulation is provided to illustrate the effectiveness of the proposed approach.

A distributed optimization approach via symmetric group actions under time-varying communication networks

The widespread adoption of edge computing has revolutionized data processing by decentralizing computational power, resulting in faster response times, decreased network latency, and enhanced privacy. This paradigm shift is crucial due to the increasing demand for real-time analytics and the growth of Internet of Things devices. Optimizing resource allocation and decision-making in on-board computing presents challenges, especially in dynamic and heterogeneous environments. Distributed optimization techniques address these challenges by allowing multiple agents to collaborate and make decisions based on local knowledge. This paper focuses on distributed optimization in multi-agent systems with time-varying communication networks and explores the advantages of considering symmetric group action to reach global optimization. In particular, we focus our study into a class of Integer Linear Programming (ILP) for modeling real-world optimization problems in decentralized environments. The paper introduces a novel approach that leverages group actions and probabilistic selection of initial states to enhance convergence to desirable solutions. The method guarantees feasibility while minimizing computational effort for the agents. Additionally, numerical simulations have been conducted to validate the proposed algorithm, demonstrating its efficacy in optimizing resource allocation for a class of scheduling problems.

Priority-Considered Load Shedding in Economic Dispatch: Distributed Optimization Approach

This paper studies two fundamental problems in power systems: the economic dispatch problem (EDP) and load shedding. In particular, convex optimization problems are formulated for both the EDP and the load shedding problem. For the EDP, an extension of the problem considering the transmission losses is presented. Furthermore, emphasis is placed on scheduling the load shedding when there exist some priorities on the loads. To solve the EDP and the load shedding problem in a distributed setting, we develop a method that combines the dual decomposition approach and the extragradient-based strategy. Notably, this work provides a fixed step size-based scheme for a strongly convex resource allocation problem considering general nonaffine coupled constraints. We show the proposed algorithm converges to the optimal solution under the assumption that the underlying graph is undirected. In addition, the method has an ergodic convergence rate of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1/k)</i> in terms of the optimality residuals and the constraint violations. Simulation results are presented to demonstrate the effectiveness of the proposed optimization problems and distributed algorithm.

Distributed Optimization for Graph Matching.

Graph matching, or the determination of the vertex correspondences between a pair of graphs, is a crucial task in various problems in different science and engineering disciplines. This article aims to propose a distributed optimization approach for graph matching (GM) between two isomorphic graphs over multiagent networks. For this, we first show that for a class of asymmetric graphs, GM of two isomorphic graphs is equivalent to a convex relaxation where the set of permutation matrices is replaced by the set of pseudostochastic matrices. Then, we formulate GM as a distributed convex optimization problem with equality constraints and a set constraint, over a network of multiple agents. For arbitrary labelings of the vertices, each agent only has information about just one vertex and its neighborhood, and can exchange information with its neighbors. A projected primal-dual gradient method is developed to solve the constrained optimization problem, and globally exponential convergence of the agents' states to the optimal permutation is achieved. Finally, we illustrate the effectiveness of the algorithm through simulation examples.

A novel large-scale EV charging scheduling algorithm considering V2G and reactive power management based on ADMM

Electric vehicle aggregators (EVAs) that utilize vehicle-to-grid (V2G) technologies can function as both controllable loads and virtual power plants, providing key energy management services to the distribution system operator (DSO). EVAs can also balance the grid’s reactive power as a virtual static VAR compensator (SVC) and provide voltage stability by utilizing advanced electric vehicle (EV) chargers that are capable of four-quadrant operations to provide reactive power management. Finally, managed charging can benefit EVAs themselves by minimizing power factor penalties in their electricity bills. In this paper, we propose a novel EV charging scheduling algorithm based on a hierarchical distributed optimization framework that minimizes peak load and provides reactive power compensation for the DSO by collaboration with EVAs that manage both the active and the reactive charging and discharging power of participating EVs. Utilizing the alternative direction method of multipliers (ADMM), the proposed distributed optimization approach scales well with increased EV charging infrastructure by balancing active and reactive power while decreasing computational burden. In our proposed hierarchical approach, each EVA schedules the active and reactive EV charging and discharging power for 1) reactive power compensation in order to minimize power factor penalty and electricity cost accrued by the EVA, 2) satisfaction of each EV’s energy demand at minimal charging cost, and 3) peak shaving and load management for the DSO. When compared with an uncoordinated charging model, the efficacy of this proposed model is successfully demonstrated through a 300% decreased peak EV load for the DSO, 28% lower electricity costs for EV users, and 98.55% smaller power factor penalty, along with 17.58% lower overall electricity costs, for EVAs. The performance of our approach is validated in a case study with 50 EVs at multiple EVAs in an IEEE 13-bus test case and compared the results with uncoordinated EV charging.

Event-Trigger-Based Distributed Optimization Approach for Two-Level Optimal Model of Isolated Power System With Switching Topology

Due to the uncertain output of intermittent energy resources and dynamic communication topology, it brings a great challenge for the optimal security control of isolated power system. To address this problem, this article proposes a two-level optimal control strategy with event-triggered switching mechanisms. For ensuring the security of the isolated power system, event-triggered switching mechanisms are proposed in the upper-level model to decrease potential risk of supply security and voltage stability, which can ensure system security as well as a low switching cost. In the lower-level model, a distributed optimization with switching topology is developed to minimize power generation cost under the above switching mechanisms, and the convergence ability of the proposed distributed optimization method is well proved with a uniformly globally exponentially stable condition. The obtained simulation results reveal that the proposed optimization approach can properly deal with the security issue of an isolated power system as well as dynamically minimize the economic cost.

A State-Based Potential Game-Theoretic Approach for Massive Residential Consumers Participating in Demand Response

In order to realize massive residential consumers participating in demand response (DR), load aggregator (LA) can be employed to aggregate flexible loads. In this article, a distributed optimization approach based on state-based potential game (SPG) is proposed to describe the bidding behavior among LAs in day-ahead DR market for load shifting amount. Under the consideration of DR resource's energy consumption satisfaction, profit-oriented bidding decision problem is modeled from the perspective of individual LA and DR market. Basically, the establishment and proof of an SPG model is presented in details. Then, the best-strategy-response iterative algorithm is proposed to search for the Nash equilibrium in a distributed way. Finally, simulation is carried out to verify the efficiency and effectiveness of the proposed approach. Simulation results show that the whole profit of all LAs increases 0.15%, while the running time of the algorithm reduces 59.53% compared with the conventional algorithm in the noncooperative game.

Distributed Optimization of District Heating Networks Using Optimality Condition Decomposition

The optimal operation of District Heating Networks (DHNs) is a challenging task. Current or future optimal dispatch energy management systems attempt to optimize objectives, such as monetary cost minimization, emission reduction, or social welfare maximization. Typically, this requires highly nonlinear models and has a substantial computational cost, especially for large DHNs. Consequently, it is difficult to solve the resulting nonlinear programming problem in real time. In particular, as typical applications allow for no more than several minutes of computation time. However, a distributed optimization approach may provide real time performance. Thereby, the solution of the central optimization problem is obtained by solving a set of small-scale, coupled optimization problems in parallel. At runtime, information is exchanged between the small-scale problems during the iterative solution procedure. A well-known approach of this class of distributed optimization algorithms is Optimality Condition Decomposition (OCD). Important advantages of this approach are the low amount of information exchange needed between the small-scale problems and that it does not require the tuning of parameters, which can be challenging. However, the DHNs model equation structure brings along many difficulties that hamper the application of the OCD approach. Simulation results demonstrate the applicability range of the presented method.

Cell Zooming With Masked Data for Off-Grid Small Cell Networks: Distributed Optimization Approach

Cell zooming has been becoming an essential enabler for off-grid small cell networks. Traditional models often utilize the numbers of active users in order to determine cell zooming strategies. However, such confidential measurement data must be concealed from others. We therefore propose a novel cell zooming method with masking noise. The proposed algorithm is designed based on distributed optimization, in which each SBS locally solves a divided optimization problem and learns how much a global constraint is satisfied or violated for temporal solutions. The important feature of this distributed control method is robustness against masking noise. We analyze the tradeoff between confidentiality and optimization accuracy, using the notion of differential privacy. Numerical simulations show that the proposed distributed control method outperforms a standard centralized control method in the presence of masking noise.

Distributed Optimization Algorithm for Multi-Robot Formation with Virtual Reference Center

In this letter, we use a distributed optimization approach to solve a class of multi-robot formation problem with virtual reference center. We investigate the design and analysis of the constrained consensus algorithm to solve the optimization problem with a sum of objective functions with some local constraints. In the multi-robot system with virtual reference center, each robot has messages on its own constraints and objective function, as well as the message about the formation that interacts with the virtual reference center. At the same time, all the robots collaborate to find the minimum value of the function defined by the formation. To find the optimal formation, we propose an algorithm with fixed step size with better performance. In addition, we use a combination of the Hungarian assignment algorithm and the proposed formation algorithm to get the optimal matching formation of the multi-robot system.

Nonlinear Distributed Model Predictive Control for multi-zone building energy systems

This paper presents a distributed Model Predictive Control (MPC) approach for multi-zone building energy systems based on nonlinear Modelica controller models. The method considers both thermal and hydraulic coupling among different building zones. The iterative and parallel distributed optimization approach builds upon an uncooperative approach for thermal coupling using the Nash equilibrium approach and a cooperative approach for the hydraulic coupling using the Alternating Direction Method of Multipliers (ADMM). Apart from thermal coupling through walls, the modeling takes thermal coupling through doors into account using a data-driven approach, which calculates the inter-zone air exchanges based on temperature differences between door-coupled zones. The hydraulic coupling enables consideration of interactions between the zones introduced by a shared, central Heating, Ventilation and Air Conditioning (HVAC) system. The distributed MPC framework is structured in an easy-scalable, plug-and-play composition, where local systems are automatically assigned to the global coordination scheme. The distributed MPC method is applied to a simulative nonlinear case study, consisting of a six-room-building Modelica model considering both thermal and hydraulic interactions. The benefits of the proposed approach are demonstrated and compared against centralized and decentralized control concepts in terms of energy consumption, discomfort and computation time.

Frequency regulation and congestion management by Virtual Storage Plants

The potential role of energy storage as flexible resource to help meet the challenging energy and environmental targets is well recognized. It can provide both steady-state and dynamic support to the power networks. A Virtual Storage Plant (VSP) is an aggregation of heterogeneous energy storage units that act as a single storage device to provide system-level support. This paper introduces an innovative and scalable hierarchical control framework for the optimal coordination of a large number of distributed storage devices through VSPs to provide frequency regulation and congestion management in multi-area power networks. The proposed approach combines distributed optimization and consensus-based control. At a lower level, a consensus-based control is designed to track power setpoints and to maintain a balanced state-of-charge among storage units. This facilitates the modeling of VSPs at higher level, where a distributed optimization approach is applied to achieve cross-regional coordination of multiple VSPs to deliver frequency support to interconnected power systems, while avoiding congested power flows over the tie lines. The proposed control framework is assessed on a realistic case study implemented by using MATLAB and the high-fidelity power system simulator DigSILENT. The IEEE 39 Bus system and a real regional power network in South-East Europe are considered in the case study. The simulation results demonstrate the promising performance of the proposed approach and the effectiveness of VSP in promoting large-scale storage integration.