Primal-dual Subgradient Algorithm Research Articles

Distributed Real-Time Multi-Objective Control of a Virtual Power Plant in DC Distribution Systems

Due to the strong coupling between power injections and bus voltages in DC distribution networks, it is impossible to achieve economic dispatch and recover the nominal voltages of all buses, simultaneously. In light of this, a multi-objective optimization-based control model is formulated for the multiple distributed energy resources (DERs) within the DC distribution network to be dispatched as a virtual power plant. A compromise can be made between the economic efficiency and voltage distribution quality by placing different weights on the multiple objectives. To this end, a distributed real-time multi-objective control strategy is proposed. Specifically, facilitated by sequential peer-to-peer communication, the dynamics of the cyber-physical DC distribution system imitate the primal-dual subgradient algorithm to the multi-objective optimization, thus can iteratively converge to its optimal solution. A dynamic weight adjustment mechanism is further designed to bound the bus voltages within a feasible range. This strategy is plug-and-play, privacy-preserving, robust to communication failures, and responsive to change of operating conditions compared to its centralized counterpart. Simulation studies of two typical DC distribution feeders verify the effectiveness of the proposed approach in various scenarios.

Peer-to-peer decentralized energy trading framework for retailers and prosumers

The smart grid technology has increased the penetration of distributed energy resources by the development and expansion of communication infrastructures and thus arisen the need to redesign electricity markets. With the emergence of prosumers as smart agents capable of both producing and consuming energy, the modern electricity markets should be prosumer-centric. In the novel prosumer-centric approach, the prosumers can trade energy with each other in a peer-to-peer (P2P) fashion. In these new markets, the retailers can be thought of as profit-based companies that either generate electrical energy themselves or purchase it at variable prices from the wholesale market or other prosumers in the local market. The retailers then sell the generated or purchased energy to the local consumers, wholesale markets, or other prosumers. Smart grid technology allows the prosumers (the end-users) to purchase their needed power from any producer in the grid and sell their excess energy to any consumer or retailer. This article designs a competitive market consisting of prosumers and retailers such that all prosumers (as buyers and sellers) and retailers conduct peer-to-peer energy trading. A novel decentralized approach called primal–dual sub-gradient algorithm (PDSGA) is used to clear the designed market without third-party involvement or disclosure of players’ private information. The feasibility of the proposed market and the effectiveness of the novel decentralized scheme for market-clearing are demonstrated through numerical implementation.

An Improved Shrunken Primal-Dual Subgradient (i-SPDS) algorithm for Optimization in Large-Scale Networked Environment

Large-scale optimization in networked environment is challenged by scalability. To address this challenge, this paper sets its aim on developing algorithms for optimization problems with strongly coupled objective function and global constraints. As a radical improvement of the state-of-the-art algorithms, this paper develops an improved shrunken-primal-dual subgradient (i-SPDS) algorithm that establishes fast convergence and eliminates optimality gap. The developed algorithm is rather generic in that it can be readily implemented in any applications that involve large-scale networked environment. Simulations are conducted to demonstrate the efficacy and efficiency of the proposed i-SPDS.

Robust kinematic calibration for improving collaboration accuracy of dual-arm manipulators with experimental validation

Kinematic calibration has been widely employed for manipulators to promote their performance characteristics. For a dual-arm manipulator, most of the research attentions are paid to improving the absolute positioning accuracy. However, collaborative accuracy plays a critical role in mutual operations between the two arms. For example, in dangerous chemical experiments, the dual-arm manipulator is often demanded to grab a target object with the two hands, or re-grasp a test tube from one hand to the other, where the inferior collaborative accuracy may lead to the failure of experiments. Hence, in this paper, collaborative accuracy of dual-arm manipulators is well defined and fully considered as the objective for calibration. Robustness of the calibration is further guaranteed by minimizing the maximum distance error. The formulated problem is not a typical convex optimization, and gradient search algorithm does not work well for this problem. With researches on optimization moving forward, recent advances in nonlinear optimization are employed to seek for the solution effectively, and it is found that the minimax problem can be approximately linearized to a sequence quadratic programming (SQP) problem. Furthermore, a primal-dual subgradient algorithm is applied for solving the SQP problem with a fast local convergence. Finally, in order to verify the superiority of the proposed method, an experiment is performed on an IRB 14000 manipulator, and corresponding outcomes indicate that the RMS collaborative positioning and the orientation accuracies are significantly improved by 90.60% and 91.42%. To the best of our knowledge, our method has reached the best collaborative accuracy compared with existing works (Wang et al., 2014; Roncone et al., 2014; Motta et al., 2001).

Distributed Algorithms for Robust Convex Optimization via the Scenario Approach

This paper proposes distributed algorithms to solve robust convex optimization (RCO) when the constraints are affected by nonlinear uncertainty. We adopt a scenario approach by randomly sampling the uncertainty set. To facilitate the computational task, instead of using a single centralized processor to obtain a “global solution” of the scenario problem (SP), we resort to multiple interconnected processors that are distributed among different nodes of a network to simultaneously solve the SP. Then, we propose a primal-dual subgradient algorithm and a random projection algorithm to distributedly solve the SP over undirected and directed graphs, respectively. Both algorithms are given in an explicit recursive form with simple iterations, which are especially suited for processors with limited computational capability. We show that, if the underlying graph is strongly connected, each node asymptotically computes a common optimal solution to the SP with a convergence rate $O(1/(\sum _{t=1}^k\zeta ^t))$ , where $\lbrace \zeta ^t\rbrace$ is a sequence of appropriately decreasing stepsizes. That is, the RCO is effectively solved in a distributed way. The relations with the existing literature on robust convex programs are thoroughly discussed and an example of robust system identification is included to validate the effectiveness of our distributed algorithms.

Decentralized Charging Control of Electric Vehicles in Residential Distribution Networks

Electric vehicle (EV) charging can negatively impact electric distribution networks by exceeding equipment thermal ratings and causing voltages to drop below standard ranges. In this paper, we develop a decentralized EV charging control scheme to achieve "valley-filling" (i.e., flattening demand profile during overnight charging), meanwhile meeting heterogeneous individual charging requirements and satisfying distribution network constraints. The formulated problem is an optimization problem with a non-separable objective function and strongly coupled inequality constraints. We propose a novel shrunken primal-dual subgradient (SPDS) algorithm to support the decentralized control scheme, derive conditions guaranteeing its convergence, and verify its efficacy and convergence with a representative distribution network model.

Primal-Dual Subgradient Algorithm for Distributed Constraint Optimization Over Unbalanced Digraphs

This paper investigates the distributed convex optimization problem with coupled inequality constraints over unbalanced digraphs depicted as row-stochastic matrices, where each agent in the network only has access to its local information, while the local constraint functions of all the agents are coupled in inequality constraints. To solve this kind of problems, a novel distributed iterative algorithm is proposed via consensus scheme and the projected primal-dual subgradient method. Different from the previous related results, our algorithm can not only deal with coupling inequality constraints but also conquer the unbalanced topology caused by directed graphs. Moreover, it is proved that under certain assumptions, the optimal solution of the problem can be asymptotically obtained by performing the designed algorithm. Finally, the numerical examples are presented to further illustrate the efficacy of the proposed approach.

An Adaptive Primal-Dual Subgradient Algorithm for Online Distributed Constrained Optimization.

In this paper, we consider the problem of solving distributed constrained optimization over a multiagent network that consists of multiple interacting nodes in online setting, where the objective functions of nodes are time-varying and the constraint set is characterized by an inequality. Through introducing a regularized convex-concave function, we present a consensus-based adaptive primal-dual subgradient algorithm that removes the need for knowing the total number of iterations in advance. We show that the proposed algorithm attains an [where ] regret bound and an bound on the violation of constraints; in addition, we show an improvement to an regret bound when the objective functions are strongly convex. The proposed algorithm allows a novel tradeoffs between the regret and the violation of constraints. Finally, a numerical example is provided to illustrate the effectiveness of the algorithm.

Distributed coordination regulation marginal cost control strategy for automatic generation control of interconnected power system

The area-level coordination of regulation capacity by automatic generation control (AGC) is an effective approach to mitigate wind power fluctuations in interconnected power system with high wind penetration. However, the applicability of this method is restricted by the existing AGC strategy. For this reason, power systems in China are experiencing serious wind curtailments to meet the power balance. In this study, a distributed coordination AGC strategy based on regulation marginal cost (RMC) is proposed for coordinating regulation capacity in different areas to remove frequency deviations. Detailed regulating characteristics of the AGC strategy in the existing power systems are discussed. Based on the above analysis, a distributed coordination RMC controller is designed to meet the proposed multi-area coordination rule using a partial primal-dual sub-gradient algorithm. In this proposed controller, the coordination signal containing RMC is exchanged with adjacent controllers through communication networks. The simulation results prove that the distributed coordination RMC strategy can economically coordinate the regulation capacity of an interconnected power system and satisfy the frequency control performance standard.

Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

In this paper, we present an improved subgradient algorithm for solving a general multi-agent convex optimization problem in a distributed way, where the agents are to jointly minimize a global objective function subject to a global inequality constraint, a global equality constraint and a global constraint set. The global objective function is a combination of local agent objective functions and the global constraint set is the intersection of each agent local constraint set. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. Our main focus is on constrained problems where the local constraint sets are identical. Thus, we propose a distributed primal-dual subgradient algorithm, which is based on the description of the primal-dual optimal solutions as the saddle points of the penalty functions. We show that, the algorithm can be implemented over networks with changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically converge to optimal solution with optimal value of the optimization problem under the Slater’s condition.

A Novel Multiagent Control Scheme for Voltage Regulation in DC Distribution Systems

This paper proposes a novel multiagent control scheme to mitigate the voltage regulation challenges of dc distribution systems (DCDSs) with high penetration of distributed and renewable generation (DG). The proposed control scheme consists of two sequential stages. In the first stage, a distributed state estimation algorithm is implemented to estimate the voltage profile in a DCDS, thus enhancing the ac/dc converter operation to keep the system voltages within specified limits. The second stage is activated only when the ac/dc fails to regulate the system voltages. Two distributed power management control strategies are proposed in the second stage. The first is based on a distributed equal curtailment, at which all DG units responsible for the voltage violation are equally curtailed. The second strategy aims to optimize the output power in order to maximize the revenue of DG units. The formulated problem in the second strategy is classified as a convex optimization problem under global constraints. A distributed Lagrangian primal-dual subgradient (DLPDS) algorithm is proposed in order to obtain the global optimal solution of the formulated problem. Various case studies are performed to prove the effectiveness, robustness, and convergence characteristics of the proposed control schemes.

Minimum-Energy Wireless Real-Time Multicast by Joint Network Coding and Scheduling Optimization

For real-time multicast services over wireless multihop networks, to minimize the energy of transmissions with satisfying the requirements of a fixed data rate and high reliabilities, we construct a conflict graph based framework by joint optimizing network coding and scheduling. Then, we propose a primal-dual subgradient optimization algorithm by random samplingKmaximal stable sets in a given conflict graph. This method transforms the NP-hard scheduling subproblem into a normal linear programming problem to obtain an approximate solution. The proposed algorithm only needs to adopt centralized technique for solving the linear programming problem while all of the other computations can be distributed. The simulation results show that, comparing with the existing algorithm, this algorithm can not only achieve about 20% performance gain, but also have better performance in terms of convergence and robustness.

Distributed optimal dispatch of virtual power plant based on ELM transformation

To implement the optimal dispatch of distributed energy resources (DER) in the virtual power plant (VPP), a distributed optimal dispatch method based on ELM (Extreme Learning Machine) transformation is proposed. The joint distribution of maximum available outputs of multiple wind turbines in the VPP is firstly modeled with the Gumbel-Copula function. A VPP optimal dispatch model is then formulated to achieve maximum utilization of renewable energy generation, which can take into account the constraints of electric power network and DERs. Based on the Gumbel-Copula joint distribution, the nonlinear functional relationship between the wind power cost and wind turbine output is approximated using ELM. The approximated functional relationship is then transformed into a set of equality constraints, which can be easily integrated with the optimal dispatch model. To solve the optimal dispatch problem, a distributed primal-dual sub-gradient algorithm is proposed to determine the operational strategies of DERs via local decision making and limited communication between neighbors. Finally, case studies based on the 15-node and the 118-node virtual power plant prove that the proposed method is effective and can achieve identical performance as the centralized dispatch approach.

Distributed Optimal Dispatch of Virtual Power Plant via Limited Communication

In this letter, a distributed optimal dispatch method based on the distributed primal-dual sub-gradient algorithm is proposed. By coordinating individual decision-making of distributed energy resources (DERs) in the virtual power plant (VPP) via limited communication, the profit of VPP can be maximized. It can be proven that the proposed distributed algorithm has similar performance to the centralized dispatch.

Discrete tomography by convex–concave regularization and D.C. programming

We present a novel approach to the tomographic reconstruction of binary objects from few projection directions within a limited range of angles. A quadratic objective functional over binary variables comprising the squared projection error and a prior penalizing non-homogeneous regions, is supplemented with a concave functional enforcing binary solutions. Application of a primal-dual subgradient algorithm to a suitable decomposition of the objective functional into the difference of two convex functions leads to an algorithm which provably converges with parallel updates to binary solutions. Numerical results demonstrate robustness against local minima and excellent reconstruction performance using five projections within a range of 90 ∘ . Our approach is applicable to quite general objective functions over binary variables with constraints and thus applicable to a wide range of problems within and beyond the field of discrete tomography.

A class of convergent primal-dual subgradient algorithms for decomposable convex programs

In this paper we develop a primal-dual subgradient algorithm for preferably decomposable, generally nondifferentiable, convex programming problems, under usual regularity conditions. The algorithm employs a Lagrangian dual function along with a suitable penalty function which satisfies a specified set of properties, in order to generate a sequence of primal and dual iterates for which some subsequence converges to a pair of primal-dual optimal solutions. Several classical types of penalty functions are shown to satisfy these specified properties. A geometric convergence rate is established for the algorithm under some additional assumptions. This approach has three principal advantages. Firstly, both primal and dual solutions are available which prove to be useful in several contexts. Secondly, the choice of step sizes, which plays an important role in subgradient optimization, is guided more determinably in this method via primal and dual information. Thirdly, typical subgradient algorithms suffer from the lack of an appropriate stopping criterion, and so the quality of the solution obtained after a finite number of steps is usually unknown. In contrast, by using the primal-dual gap, the proposed algorithm possesses a natural stopping criterion.