Non-convex Economic Dispatch Research Articles

Two efficient logarithmic formulations to solve nonconvex economic dispatch

To solve economic dispatch (ED) with nonconvex nonlinear generation cost functions, this paper adopts a global optimization technique where piecewise linear approximation (PLA) functions replace the nonlinear functions. However, a problem with this technique is that the PLA model requires incorporating many discrete variables into the optimization model, which may render the model computationally expensive. This paper presents two efficient representation techniques of the PLAs, which scale logarithmically, including binary zig-zag (ZZB) and general integer zig-zag (ZZI). We replace the nonconvex cost functions in the ED with their PLAs and obtain an alternative ED formulation where two logarithmic techniques model the PLAs. Then, one can efficiently solve the alternative ED model with the standard branch-and-cut/bound-based algorithms to the desired optimality gap. To verify the techniques’ efficacy, we consider test systems having 13, 38, 40, 80, and 160 units. The numerical evaluations show that the logarithmic techniques presented can yield high-quality optimal solutions in less than 2 seconds in the considered cases, illustrating the techniques’ efficiency.

Pricing Economic Dispatch With AC Power Flow Via Local Multipliers and Conic Relaxation

We analyze two locational marginal pricing schemes in electricity markets derived from an economic dispatch (ED) problem with AC power flow equations that define a non-convex feasible set. The first among these prices, termed AC-LMPs, are derived from Lagrange multipliers that satisfy Karush-Kuhn-Tucker conditions for stationarity/local optimality of the non-convex ED problem. The second, called SDP-LMPs, are derived from optimal dual multipliers of the semidefinite programming (SDP)-based convex relaxation of the ED problem. We establish that AC-LMPs and SDP-LMPs are derived from Lagrange dual-equivalent problems. Hence, they coincide under zero duality gap, but may not be equal when the gap is non-zero or the AC-LMPs are associated with a stationary/locally optimal, but not globally optimal, dispatch. SDP-LMPs share interesting parallels with convex hull prices (CHPs), being derived from the Lagrangian dual of the non-convex ED problem. However, there are important differences. For example, the epigraphs of the SDP relaxation of the ED problem and the latter, parameterized by the nodal power demands, may not enjoy the relationship that CHPs satisfy and derive their name from. Also, while CHPs minimize a form of side-payments, SDP-LMPs may not. We prove that AC-LMPs, (and SDP-LMPs under zero duality gap) guarantee revenue adequacy under a condition that is sufficient, but not necessary. Finally, the SDP-LMPs are shown to equal SOCP-DLMPs, that are distribution locational marginal prices derived with second-order cone programming-based relaxations of power flow equations over radial distribution networks. We illustrate our theoretical findings through numerical experiments.

An intelligent quasi-oppositional HBO technique to solve non-smooth non-convex economic dispatch problem

An intelligent quasi-oppositional HBO technique to solve non-smooth non-convex economic dispatch problem

A hybrid quantum-behaved particle swarm optimization solution to non-convex economic load dispatch with multiple fuel types and valve-point effects

Economic dispatch problems (EDPs) can be reduced to non-convex constrained optimization problems, and most of the population-based algorithms are prone to have problems of premature and falling into local optimum when solving EDPs. Therefore, this paper proposes a hybrid quantum-behaved particle swarm optimization (HQPSO) algorithm to alleviate the above problems. In the HQPSO, the Solis and Wets local search method is used to enhance the local search ability of the QPSO so that the algorithm can find solutions that is close to optimal when the constraints are met, and two evolution operators are proposed and incorporated for the purpose of making a better balance between local search and global search abilities at the later search stage. The performance comparison is made among the HQPSO and the other ten population-based random search methods under two different experimental configurations and four different power systems in terms of solution quality, robustness, and convergence property. The experimental results show that the HQPSO improves the convergence properties of the QPSO and finally obtains the best total generation cost without violating any constraints. In addition, the HQPSO outperforms all the other algorithms on 7 cases of all 8 experimental cases in terms of global best position and mean position, which verifies the effectiveness of the algorithm.

Approaches for smart linear regression in a difficult quasi economic dispatch problem

<span lang="EN-US">Traditional methods indispensably necessitate monotonically increasing characteristic for fuel cost of generators in a thermal power plant. However, in medium and large thermal power plants, this condition is a dream to accomplish. So, to meet out these exigencies heuristic methods like swarm optimization technique, genetic algorithm technique and bee colony based hybrid solar thermal technology (BHSTT) are used to realize the practical nonlinearities associated with valve point loading emanated out of multi-valving effect, associated with power station. However, the heuristic methods too face challenges arising out of bulky thermal power plants adopting cubic cost functions and possessing stringent non-convex economic dispatch problem following multi-valving and erratic behavior of nonlinear loads at the load center. So, at its favor function evaluation method dealing with cubic cost function is attempted in this dissertation to yield a satisfactory optimal solution for economic dispatch problem. This method deals with the real power generation of producing units as well as the complex power of units, as well as dealing with severe nonlinear stringent fuel cost characteristics that are prevalent in today’s bulky thermal power plants. In comparison to previous approaches, the findings achieved are highly encouraging.</span>

Optimization of Non-Convex Economic Dispatch Problem Using Hybrid Approach Based on Bacterial Foraging and Genetic Algorithm

Fulfillment of consumer demand is a foremost challenge for all electrical power utilities. An electrical power system consists of several generating units and each of these units owns a distinct operating set of operating parameters. A fundamental challenges lies in a fact that there may not be a correlation between operating costs of these machines and their generated output. Major reason for lack of the correlation includes the ramp-rate limits, transmission losses and manipulation due to valve-points in the generation units cost function, and thus, it becomes a non-linear optimization problem. In view of this fact, it creates a rigorous need to devise a robust solution to cater for such a non-linear optimization scenario. In this paper, bacterial foraging and genetic algorithms based hybrid technique is used to effectively tackle the economic dispatch problem. The presented technique incorporates two modifications in the original bacterial foraging algorithm including differential evolution inspired bacterial movement and genetic algorithm based bacterial reproduction. Where inspired bacterial movement involves modification in the directional movement for each bacterium in such a manner that every bacterium tries to improve its direction and position based on differential evolution. The proposed technique is applied to non-convex dynamic economic dispatch (DED) problem in order to obtain optimal solution within feasible functional limits while satisfying the load demand at the same time. The obtained results demonstrate that the proposed hybrid approach outperform the other techniques in term of optimal solution and significant reduction of computational time in the given scenarios.

A non-convex economic load dispatch problem using chameleon swarm algorithm with roulette wheel and Levy flight methods

A non-convex economic load dispatch problem using chameleon swarm algorithm with roulette wheel and Levy flight methods

Target-Value-Competition-Based Multi-Agent Deep Reinforcement Learning Algorithm for Distributed Nonconvex Economic Dispatch

With the increasing expansion of the power grid, economic dispatch problems have received considerable attention. A multi-agent coordinated deep reinforcement learning algorithm is proposed to deal with distributed nonconvex economic dispatch problems. In the algorithm, agents run independent reinforcement learning algorithms and update their local Q-functions with a newly defined joint reward. The double network structure is adopted to approximate the Q-function so that the offline trained model can be used online to provide recommended power outputs for time-varying demands in real-time. By introducing the reward network, the competition mechanism between the reward network and the target network is established to determine a progressively stable target value, which achieves coordination among agents and pledges the losses of the Q-networks to converge well. Theoretical analysis is given and case studies are conducted to prove the advantages compared with existing approaches.

Oppositional Pigeon-Inspired Optimizer for Solving the Non-Convex Economic Load Dispatch Problem in Power Systems

Economic Load Dispatch (ELD) belongs to a non-convex optimization problem that aims to reduce total power generation cost by satisfying demand constraints. However, solving the ELD problem is a challenging task, because of its parity and disparity constraints. The Pigeon-Inspired Optimizer (PIO) is a recently proposed optimization algorithm, which belongs to the family of swarm intelligence algorithms. The PIO algorithm has the benefit of conceptual simplicity, and provides better outcomes for various real-world problems. However, this algorithm has the drawback of premature convergence and local stagnation. Therefore, we propose an Oppositional Pigeon-Inspired Optimizer (OPIO) algorithm—to overcome these deficiencies. The proposed algorithm employs Oppositional-Based Learning (OBL) to enhance the quality of the individual, by exploring the global search space. The proposed algorithm would be used to determine the load demand of a power system, by sustaining the various equality and inequality constraints, to diminish the overall generation cost. In this work, the OPIO algorithm was applied to solve the ELD problem of small- (13-unit, 40-unit), medium- (140-unit, 160-unit) and large-scale (320-unit, 640-unit) test systems. The experimental results of the proposed OPIO algorithm demonstrate its efficiency over the conventional PIO algorithm, and other state-of-the-art approaches in the literature. The comparative results demonstrate that the proposed algorithm provides better results—in terms of improved accuracy, higher convergence rate, less computation time, and reduced fuel cost—than the other approaches.

A diversity-based parallel particle swarm optimization for nonconvex economic dispatch problem

The economic dispatch (ED) problem aims to minimize the total generation cost while satisfying certain constraints, such as valve-point effects, multi-fuel options, prohibited operating zones, transmission losses, and ramp rate limits. In this paper, these constraints are considered simultaneously for the first time, resulting in a complex nonconvex ED problem. A diversity-based parallel particle swarm optimization (DPPSO) is proposed to solve the nonconvex ED problem, where the implementation details—such as evaluation function design, particle definition, and equality and inequality handling strategies—have been carefully discussed. In our approach, the population of DPPSO is divided into different groups to maintain diversity in particles so that the optimization capacity can be enhanced. An asynchronous information–sharing mechanism (AISM) helps decrease the population size. Hence, the computational burden is reduced. Moreover, information in different groups is calculated parallelly and updated asynchronously to improve computational efficiency. Benchmark functions are employed to demonstrate the effectiveness of the proposed method. Furthermore, three nonconvex ED problems are resolved by the proposed method, and state-of-the-art performance has been achieved. In addition, the proposed algorithm is highly modular, making it easy to unite other salient variants of particle swarm optimization (PSO) to improve its performance.

Forgetting velocity based improved comprehensive learning particle swarm optimization for non-convex economic dispatch problems with valve-point effects and multi-fuel options

Economic dispatch (ED) plays an essential role in the operation and planning of power systems. Mathematically, it turns to be a multi-constraint, multimodal, non-linear, and multivariate coupling optimization problem when considering the valve-point effects and multi-fuel options. In this paper, an improved comprehensive learning particle swarm optimization (CLPSO) named FV-ICLPSO is presented to solve it. FV-ICLPSO introduces three improved components to conquer the issue of slow convergence rate of CLPSO: (1) an adaptive strategy by using both the iteration and problem's dimensionality is presented to tune the learning probability; (2) an adaptive method is designed to give different particles in different levels different scopes to pick the source particles to construct their learning exemplars; and (3) an improved velocity updating formula based on forgetting the previous velocity is proposed to guide particles to fly towards more promising areas. FV-ICLPSO is first validated on thirty CEC2014 benchmark numerical functions and then applied to eight ED cases with different units from 10-Unit to 640-Unit. Simulation results demonstrate the strong competitiveness of FV-ICLPSO compared with CLPSO, other state-of-the-art algorithms, and the reported results of some recently published ED solution methods. Furthermore, the effect of the three improved components on FV-ICLPSO is also investigated.

Solution of Economic Dispatch Problem Using Hybrid Multi-Verse Optimizer

Economic Dispatch (ED) is one of the important problems among numerous vital operational problems of power economics. ED is a non-convex, non-drivable, and non-linear optimization issue along with diverse constraints. In this research work, a neoteric natural influential optimization tool Multi-verse Optimizer (MVO) hybridized with sequential quadratic programming (SQP) termed as Hybrid MVO (MVO-SQP) is proposed for the solution of multi-constrained ED problem. This contemporary effort contemplates on ED problem in the milieu of convergence features, numerical proficiency, and stoutness.The non-convex ED problem is encrypted at Hybrid Multi-verse Optimizer (HMVO) framework and employed in MATLAB for 30 trials, 300 iterations, and 30 exploration universes. The presented mechanism is verified on typical standard test systems comprising of 6, 13, 15, and 40 unit test systems, and experimental outcomes are juxtaposed with the reported approaches in the literature. The statistical analysis shows that the proposed approach saves 2.927 $/h for 6 units test system, 9.0618 $/h for 13 units test system, 1.831 $/h for 15 units test system, and 113.515 $/hr for 40 units test system with losses. It is envisioned that this work will equip the researchers with an improved tool to solve ED problem.

A Cost-Effective Solution for Non-Convex Economic Load Dispatch Problems in Power Systems Using Slime Mould Algorithm

Slime Mould Algorithm (SMA) is a newly designed meat-heuristic search that mimics the nature of slime mould during the oscillation phase. This is demonstrated in a unique mathematical formulation that utilizes adjustable weights to influence the sequence of both negative and positive propagation waves to develop a method to link food supply with intensive exploration capacity and exploitation affinity. The study shows the usage of the SM algorithm to solve a non-convex and cost-effective Load Dispatch Problem (ELD) in an electric power system. The effectiveness of SMA is investigated for single area economic load dispatch on large-, medium-, and small-scale power systems, with 3-, 5-, 6-, 10-, 13-, 15-, 20-, 38-, and 40-unit test systems, and the results are substantiated by finding the difference between other well-known meta-heuristic algorithms. The SMA is more efficient than other standard, heuristic, and meta-heuristic search strategies in granting extremely ambitious outputs according to the comparison records.

A novel modified random walk grey wolf optimisation approach for non-smooth and non-convex economic load dispatch

A novel modified random walk grey wolf optimisation approach for non-smooth and non-convex economic load dispatch

A novel modified random walk grey wolf optimisation approach for non-smooth and non-convex economic load dispatch

A novel modified random walk grey wolf optimisation approach for non-smooth and non-convex economic load dispatch

A solution to economic dispatch problem using hybrid particle chemical reaction optimization (HP-CRO) algorithm

The electrical power demand is on heave due to constant increase in demand from manufacturing firms and ménage. Therefore optimized and cost operative power generation is the requirement of the day. In power system operation and planning the Economic Dispatch (ED) problem is prevailing and noteworthy one. The ED problem of several standards is disentangled with so many classical and meta-heuristics optimization techniques. In the triumphed research work, a unique and innovative optimization procedure, Chemical Reaction Optimization (CRO) hybridized with Particle Swarm Optimization (PSO) algorithm is recommended for the solution of ED problem. Generating unit power limits, load fulfillment and prohibited operating zones are considered as constrained for different IEEE standard ED problems. The non-convex economic dispatch problem is coded at Hybrid Particle Chemical Reaction Optimization (HP-CRO) charter and employed in MATLAB for 50 trials, 50 search agents and 1000 iterations. The offered technique is applied on various IEEE standard test systems including 6, 15 and 40 generating units. The exhibited results are paralleled with stated practices mentioned in the literature for above test systems. The results obtained from proposed techniques give an edge to those obtained from other techniques from cost point of view.

Solving non-convex economic dispatch with valve-point effects and losses with guaranteed accuracy

The economic dispatch problem is a fundamental problem in power system operations. An extensive body of literature has focused on providing fast and robust algorithms for solving the various instances of the economic dispatch. In order to capture physical effects such as the power losses of the network or the valve-point loading effect of combined cycle gas turbines, non-convex models of the economic dispatch have been considered. However, these features of the problem render the convergence analysis more challenging, and few methods in the literature provide insights on the global optimality of the derived solution. In this work, we propose an algorithm that efficiently provides a feasible solution, along with a lower bound, to a non-smooth and non-convex instance of the economic dispatch problem. We test our method on extensively studied test cases, and show that, in comparison with state-of-the-art methods and within a comparable computation time, it provides a solution with a much lower deviation from the power balance constraint, while furthermore producing a lower bound with an optimality gap below one percent.

A Non-Convex Economic Dispatch Problem with Point-Valve Effect Using a Wind-Driven Optimisation Approach.

This study presents the efficiency of the wind-driven optimisation (WDO) approach in solving non-convex economic dispatch problems with point-valve effect. The best economic dispatch for a power system is one wherein the system can generate energy at a low cost. The calculation of the generating cost is subject to a number of constraints, such as the power demand for the entire system and the generation limit for each generator unit in the system. In addition, the system should also produce low power loss. The WDO optimisation technique is developed based on the concept of natural wind movement, which serves as a stabiliser to equalise the inequality of air pressure in the atmosphere. One major advantage of WDO over other techniques is its search accuracy. The proposed algorithm has been implemented in two systems, namely, the 10-generator and 40-generator systems. Both systems were tested in a Matlab environment. To highlight the capabilities of WDO, the results using this proposed technique are compared with the results obtained using flower pollination algorithm, moth flame optimisation, particle swarm optimisation and evolutionary programming techniques to determine the efficiency of the proposed approach in solving economic dispatch. The simulation results show the capability of WDO in determining the optimal power generation value with minimum generation cost and low rate of power loss.

A Solution to Economic Dispatch Problem using Hybrid Particle Chemical Reaction Optimization (HP-CRO) Algorithm

The electrical control request is on stagger due to constant growth in entreaty since manufacturing firms besides ménage. Consequently augmented plus price operational power cohort is the prerequisite of the era. In power classification process plus scheduling the Economic Dispatch (ED) tricky is prevailing and noteworthy some. The ED problematic of several standards is disentangled through consequently sundry conventional and meta-heuristics aggrandizement skills. Now the triumphed investigation exertion, a unique plus innovative aggrandizement way, Chemical Reaction Optimization (CRO) mongrelised thru Particle Swarm Optimization (PSO) system is recommended aimed at the explanation of ED problematic. Generating unit power limits, load fulfillment and prohibited operating zones remain painstaking by way of guarded on behalf of diverse IEEE customary ED muddles. The non-convex economic dispatch hitch is coded at Hybrid Particle Chemical Reaction Optimization (HP-CRO) contrivance then selected the MATLAB environment intended for 50 prosecutions, 50 search agents and 1000 iterations. The offered scheme stands smeared proceeding a number of IEEE usual trial structures counting six, fifteen plus forty engendering parts. The displayed upshots stay coordinated per quantified wont declared hip the articles aimed at directly above trial structures. The costs obtained commencing wished-for modus operandi bounce a superiority toward individuals gained after new practise on one occasion price opinion.

Distributed Successive Convex Approximation for Nonconvex Economic Dispatch in Smart Grid

This article presents a distributed consensus-based successive convex approximation (DSCA) algorithm to solve nonconvex nondifferentiable economic dispatch (ED) problems. The ED model formulated incorporates generation constraints, valve-point effects, and multiple fuel types. A perturbation technique enables the proposed DSCA to tackle such a nondifferentiable and nonconvex optimization, which paves the way to solving more complicated optimization problems that occur in practical applications. The local generation constraint is taken care by a local surrogate convex optimization directly. The global equality constraint is handled based on a consensus protocol, where the local generation-demand mismatch among all dispatchable generators (DGs) is shared in a distributed manner. As a result, the power distribution of DGs is updated, and the generation cost is minimized. Several case studies show that the proposed DSCA algorithm can achieve superior ED solutions and computational efficiency over existing nonconvex optimization algorithms.