LAGRANGIAN RELAXATION-BASED PARTICLE SWARM OPTIMIZATION FOR UNIT COMMITMENT PROBLEM

This paper presents an application of a combined genetic algorithms (GAs) and Lagrangian relaxation (LR) method for the unit commitment (UC) problem. Genetic algorithms (GAs) are a general purpose optimization technique based on principle of natural selection and natural genetics. The Lagrangian relaxation (LR) method provides a fast solution but it may suffer from numerical convergence and solution quality problems. The proposed Lagrangian relaxation and genetic algorithms (LRGA) incorporates genetic algorithms into Lagrangian relaxation method to update the Lagrangian multipliers and improve the performance of Lagrangian relaxation method in solving combinatorial optimization problems such as the UC problem. Numerical results on two cases including a system of 100 units and comparisons with results obtained using Lagrangian relaxation (LR) and genetic algorithms (GAs), show that the feature of easy implementation, better convergence, and highly near-optimal solution to the UC problem can be achieved by the LRGA.

Using suitable solution techniques to the unit commitment (UC) and economic load dispatch (ELD) problems play major role on finding an optimal power scheduling in electrical power system. The UC and ELD problems are to decide when and which generating units to commit and decommit, in order to minimize the total fuel cost over specified period subject to a large number of difficult constraints. Lagrangian relaxation (LR) has been widely used to solve UC and ELD. Recently bundling method (BM) has appeared to be one of the best multipliers updated techniques for LR. This method has promising and the fastest convergence characteristics for UC and ELD problems. For that reason, this paper is to solve the UC and ELD problems via LR, BM and interior point method. This paper explores also the best possible way of updating Lagrangian multipliers with minimum computation time using fast programming techniques.

The AC-DC hybrid microgrid is a credible evolution path for the microgrid. State estimation in complex distribution network is a significant foundation for the safe operation. In AC-DC hybrid microgrid, centralised state estimation algorithms have quite a few deficiencies such as large communication capacity and privacy protection. In response to these obstacles, this paper proposed a novel distributed state estimation algorithm. First, a three-stage state estimation model was established for AC-DC hybrid microgrid, in which the second stage is a nonlinear transform for the linearisation of the nonlinear state estimation progress so that the first and third stage can solve the linear state estimation problem. Furthermore, the Lagrangian relaxation method was ultilised for the distributed problem solution. In the proposed algorithm, AC subsystems and DC subsystems respectively execute state estimation through local information, and the overall consistency of the system state estimation can be eventually achieved according to the transfer iteration of the boundary information. The computational efficiency and convergence of the proposed method was ensured owing to the combination of bilinear theory and Lagrangian relaxation. Simulation results indicate that the proposed method can achieve its goal and be superior to existing method in efficiency.

In this paper, we present a comprehensive and in-depth investigation on the optimization of Proportional-Integral (PI) controller tuning for achieving stability and desired overshoot in the step response. The main objective of this study is to compare the effectiveness of Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) techniques in finding the optimal parameters for the PI controller. The PI controller is a widely used control algorithm that plays a crucial role in many industrial processes. Its tuning greatly affects the system's performance, particularly in terms of stability and overshoot. Therefore, finding the optimal tuning parameters is of utmost importance. To address this optimization problem, we propose the utilization of two popular metaheuristic algorithms, GA and PSO. These algorithms are known for their ability to efficiently search through large solution spaces and find near-optimal solutions. By applying these algorithms to the PI controller tuning problem, we aim to determine which technique yields better results in terms of stability and overshoot tuning. In our comparative study, we provide a detailed explanation of both GA and PSO algorithms, focusing on their working principles and mathematical formulations. We also describe how these algorithms can be applied to the PI controller tuning problem. Furthermore, we highlight the key differences between GA and PSO, shedding light on their strengths and weaknesses. To assess the performance of GA and PSO, we conduct several experiments using different benchmark functions and step response models. We measure the stability and overshoot metrics for various parameter settings obtained through GA and PSO. By thoroughly analyzing the obtained results, we draw meaningful conclusions regarding the effectiveness of each technique. Our findings demonstrate that both GA and PSO exhibit promising results in optimizing PI controller tuning. These observations provide valuable insights and guidelines for choosing the appropriate algorithm based on specific control requirements. In conclusion, this comparative study is thought to contribute to the field of control systems engineering by offering a comprehensive analysis of GA and PSO techniques in the context of PI controller tuning. By highlighting their strengths and weaknesses, it is aimed to provide researchers and practitioners with valuable information for making informed decisions when optimizing control parameters for stability and overshoot reduction purposes.

Sub-hourly Unit Commitment (UC) problems have been suggested as a way to improve power system efficiency. Such problems, however, are much more difficult than hourly UC problems. This is not just because of the increased number of period to consider, but also because of much reduced unit ramping capabilities leading to more complicated convex hulls. As a result, state-of-the-art and practice methods such as branch-and-cut suffer from poor performance. In this paper, our recent Surrogate Absolute-Value Lagrangian Relaxation (SAVLR) method, which overcame major difficulties of standard Lagrangian Relaxation, is enhanced by synergistically incorporating the concept of Ordinal Optimization (OO). By using OO, solving subproblems becomes much faster. Testing of Midcontinent ISO (MISO)’s problem with 15 minutes as the time interval over 36 hours involving about 1,100 units and 15000 virtuals demonstrates that the new method obtains near-optimal solutions efficiently and significantly outperforms branch-and-cut.

A central problem in multitarget, multisensor, and multiplatform tracking remains that of data association. Lagrangian relaxation methods have shown themselves to yield near optimal answers in real-time. The necessary improvement in the quality of these solutions warrants a continuing interest in these methods. These problems are NP-hard; the only known methods for solving them optimally are enumerative in nature with branch-and-bound being most efficient. Thus, the development of methods less than a full branch-and-bound are needed for improving the quality. Such methods as K-best, local search, and randomized search have been proposed to improve the quality of the relaxation solution. Here, a partial branch-and-bound technique along with adequate branching and ordering rules are developed. Lagrangian relaxation is used as a branching method and as a method to calculate the lower bound for subproblems. The result shows that the branch-and-bound framework greatly improves the resolution quality of the Lagrangian relaxation algorithm and yields better multiple solutions in less time than relaxation alone.

This study has focused on the UELB problem that aims at leveling the traffic load of a communication link in an information network by determining the path of a flow that minimizes the maximum load factor of the link. The UELB problem has been proved to be NP-hard, and an approximate solution method using the LP relaxation has been proposed to verify the results. This paper proposes another algorithm using the Lagrangian relaxation method in order to objectively evaluate the accuracy of the approximate solutions obtained so far. We have conducted comparison experiments with the exact solution. The results showed that the proposed algorithm gradually improves the accuracy of the solution as the number of calculations of the subgradient method is increased. In addition, it was found that when Lagrangian relaxation is performed, it is better to apply LP relaxation at the same time. Conducting the relaxation at the same time will lead to a reduction in computation time.

This paper presents performance comparative study of various particle swarm optimization (PSO) techniques for the placement of generator units in the distributed generation (DG) system. For the installation of generator units in the distributed generation system, it is very important to know the generator sizing and its placement in the network system for reducing the line losses and hence the cost. Various PSO techniques such as Canonical PSO, Hierarchical PSO (HPSO), Time varying acceleration coefficient (TVAC) PSO, Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients (HPSO-TVAC), Stochastic inertia weight (Sto-IW) PSO and Time varying inertia weight (TVIW) PSO have been used for comparative study. Here the main objective function (OF) is to minimize the system cost. These techniques have been tested on the standard IEEE-14 bus, IEEE-30 bus and IEEE-57 bus network system by the use of MATLAB software.

An optimal solution to the unit commitment (UC) and economic load dispatch (ELD) problems lead to remarkable saving in the power system operational cost. UC and ELD main problems are to find the maximum profit and fast computation simulation time for the scheduling program. Lagrangian relaxation (LR) has become one of the best solution methods in solving the UC and ELD problems. The approach is claimed to be more efficient and easier than other methods in solving large-scale problems. When applying LR, the dual function needed to be updated at the higher level. Lately, bundling method (BM) is one of the best solution methods to Lagrangian dual problem. This paper presents LR technique, BM and interior point method for solving the UC and ELD problems. It will also explore the best possible way of updating Lagrangian multipliers with minimum computation time using fast programming techniques.

The practical economic dispatch (ED) problems have many non-convex characteristics, which makes the searching of the global optimum difficult when using traditional mathematical methods. This paper presents a novel hybrid algorithm (HA) to the ED problems based on the particle swarm optimization (PSO) technique and differential evolution (DE) algorithm. Since the standard PSO has the adversity of premature convergence, the mutation and crossover operators of the DE, as well as the chaotic sequences, are considered to be integrated into the PSO to improve the global searching ability. Moreover, the dynamic penalty function is applied to deal with the constraints. The proposed method is tested on a 15-unit system considering the ramp rate limits, operating zones, and network losses. Also, the results are compared with those of other optimization methods.

The problem of data association remains central in multitarget, multisensor, and multiplatform tracking. Lagrangian relaxation methods have been shown to yield near optimal answers in real-time. The necessity of improvement in the quality of these solutions warrants a continuing interest in these methods. A partial branch-and-bound technique along with adequate branching and ordering rules are developed. Lagrangian relaxation is used as a branching method and as a method to calculate the lower bound for subproblems. The result shows that the branch-and-bound framework greatly improves the solutions in less time than relaxation alone.

Application of Lagrangian relaxation to computer network control

Lagrangian Relaxation Technique for Solving Scheduling Problems by Decomposition of Timed Petri Nets

This paper deals with the configuration of a wireless network with the aim of minimizing the overall cost of both operation and network installation. The trade-off between the operation cost and the installation cost is the key consideration when designing cellular telecommunication networks, and can save costs and improve the performance of the network. In this paper, we propose an integrated framework for selecting Mobile Switching Center (MSC) among the candidate MSCs and assigning Base Stations (BSs) to the selected MSCs with the objective function of minimizing the cost of MSC setup, BS to MSC cabling, as well as the cost of handover. Capacity constraint for the selected MSC is also considered in the problem. The problem is expressed in an integer programming model and the Lagrangian relaxation method is proposed to solve the problem by dualizing some constraints. The Lagrangian relaxed problem is decomposed into subproblems that can be resolved optimally. The Lagrangian heuristic algorithm is suggested to find feasible solutions to the original problem. Computational experiments are performed to test the effectiveness and efficiency of the proposed heuristic algorithm. In the experiments, Lagrangian bounds on the optimal solution are used to show the effectiveness of the algorithm. The results of the proposed algorithm are also compared with those of some conventional meta-heuristics, Tabu search (TS) and Genetic algorithm (GA). The computational experiments show that the performance of the proposed heuristics is satisfactory in both the speed and the quality of the solution generated.

Dynamic economic dispatch (DED) over multiple time periods is a large-scale coupled spatial-temporal optimization problem. Therefore, the Lagrangian relaxation method has been widely used to split the large-scale optimization problem with coupled structure into several small sub-problems. In order to bring robustness for updating the dual multipliers and yielding convergence without strong assumptions, the augmented Lagrangian relaxation method is introduced in this paper. However, the added penalty term in an augmented Lagrangian function is non-separable, which leads to the difficulty in achieving full decomposition for parallel computation. To address this problem, a diagonal quadratic approximation method is employed to yield an approximated block separation of the non-separable penalty term. Furthermore, the ramp rate constraints are relaxed in this paper, so that the DED model is decomposed into several single-period economic dispatch models that can be efficiently handled in parallel, called the parallel augmented Lagrangian relaxation method. Particularly, the proposed relaxation strategy has a high separability feature which theoretically leads to sound convergence property. Numerical results on the IEEE 118-bus and a practical Polish 2383-bus test system over a different number of time periods show the effectiveness of the proposed method. In addition, the proposed method can be extended to other coupled spatial-temporal scheduling problems in power systems, such as energy storage dispatch.