A Novel Consensus-based Distributed Algorithm for Economic Dispatch Based on Local Estimation of Power Mismatch

I. INTRODUCTION

• More distributed generators, storage devices, dispatchable loads, and customer participation make optimal economic dispatch increasingly challenging.The paper notes that existing practices have difficulty accommodating these changes,,,.Source: §I · p. 1
• Centralized dispatch requires participants to disclose information to a controller, which faces heavy computational burden and is not designed for plug-and-play participation at scale.The paper also identifies privacy, connectivity, convergence speed, and complexity as concerns with existing distributed consensus algorithms.Source: §I · p. 2
• The proposed distributed algorithm shares estimated power mismatch rather than cost functions, utility functions, incremental cost, or output power.Consumers connect only to local DGs, and the stated contributions include fast convergence, scalability, and easy implementation.Source: §I · p. 3

II. SYSTEM MODELING AND PROBLEM DESCRIPTION

• Economic dispatch allocates distributed-generator resources to meet load in the most cost-effective way.The model defines consumer utility, generator cost, and surplus before formulating the overall optimization problem.Source: §II · p. 3

A. Utility Function, Marginal Benefit and Consumer’s Surplus

• Consumer utility functions represent preferences by measuring satisfaction as a function of demand, and this paper uses a customized quadratic model.The model’s parameters are selected for different consumers.Source: A. Utility Function, Marginal Benefit… · p. 4
• Marginal benefit is the additional utility from one more unit of electricity and is treated as non-increasing as consumption grows.The cited passages state that common utility functions are non-decreasing, while marginal benefit normally decreases as users consume more energy,.Source: A. Utility Function, Marginal Benefit… · p. 4

B. Cost Function and DG’s Surplus

• A quadratic cost function estimates a DG’s output-power cost, with coefficients customized for each generator.The passage identifies α, β, and γ as cost-function coefficients and P_i^Gen as generated power.Source: B. Cost Function and DG’s Surplus · p. 4
• A DG’s surplus is its profit when the energy sale price exceeds the cost of production.The paper expresses the profit for a DG selling energy at λ $/kWh in equation (6).Source: B. Cost Function and DG’s Surplus · p. 4

C. Global Optimization Problem

• The global objective maximizes consumer utility and generator welfare by maximizing summed utility and minimizing summed generation cost.The current algorithm does not consider power losses or maximum power-line capacity.Source: C. Global Optimization Problem · p. 5
• Power balance equates aggregated generation and consumption, while each DG’s output and each consumer’s demand are bounded by their maximum capacities.The passage identifies participant capacity limits as constraints in the optimization problem.Source: C. Global Optimization Problem · p. 5

III. DISTRIBUTED ALGORITHM FOR ECONOMIC DISPATCH

• The section reviews graph theory and elaborates the proposed distributed algorithm.The preceding passage frames consensus-based distributed approaches as a way to solve optimization problems such as economic dispatch.Source: §III · p. 5
• The section places the algorithm within consensus-based distributed approaches for solving optimization problems such as economic dispatch.Source: §III · p. 5
• Figure 4 depicts a DG’s surplus and cost functions.Source: Figure 4 · p. 6

• A communication network is represented as a directed or undirected graph G(V, ξ), whose nodes are agents and whose edges encode information-sharing links.Source: A. Graph Theory · p. 6
• The adjacency matrix A is symmetric for an undirected graph, with positive entries for connected nodes and zero entries otherwise.Source: A. Graph Theory · p. 6
• The Laplacian matrix L has diagonal entries equal to the sums of adjacent weights and off-diagonal entries lij = −aij.Source: A. Graph Theory · p. 6

• The method defines consensus as zero power mismatch and treats minimizing the network’s Laplacian potential as the route toward that state.Source: B. Consensus-based Distributed Protocols · p. 7
• The consensus update is a gradient-descent process on Laplacian potential, whose minimum occurs when each agent’s estimated mismatch is zero.Source: B. Consensus-based Distributed Protocols · p. 7
• Each DG shares its estimated power mismatch, not private cost, utility, incremental-cost, or output-power information; consumers need no links among themselves.Source: B. Consensus-based Distributed Protocols · p. 7
• DGs update incremental cost internally and use it to determine output power, while sharing only mismatch estimates with neighboring DGs.Source: B. Consensus-based Distributed Protocols · p. 8
• Consumers receive λ from one DG, calculate demand from their utility, and send estimated demand to a physically connected local DG without disclosing utility-function properties.Source: B. Consensus-based Distributed Protocols · p. 9

• The convergence argument requires ||1 − P_i|| ≤ ε < 1; a smaller ||1 − P_i|| yields faster power-mismatch convergence.Source: C. The Optimality Analysis · p. 10
• When the consensus condition holds, all DGs’ incremental costs are identical and their estimated power mismatches converge to zero.Source: C. The Optimality Analysis · p. 10
• With strictly convex objective and affine constraints satisfying Slater’s condition, KKT conditions are necessary and sufficient for optimality.Source: C. The Optimality Analysis · pp. 10–11
• The algorithm’s fixed point satisfies the KKT conditions and is presented as a global optimum; the paper also states that this optimum is unique.Source: C. The Optimality Analysis · p. 12

• The assessment uses three case studies: software simulations and an experimental test on the VOLTTRON platform with low-cost single-board PCs.Source: §IV · p. 12
• The small-scale numerical case evaluates accuracy by comparing the distributed algorithm with centralized economic-dispatch benchmark results.Source: §IV · p. 12
• A 1,400-agent network case study evaluates the algorithm’s scalability and convergence rate.Source: §IV · p. 12
• The experimental case tests practical performance using VOLTTRON and a group of low-cost credit-card-size single-board PCs.Source: §IV · p. 12

A. Case Study I (Evaluation of Accuracy)

• The evaluation uses an IEEE 39-bus system with 10 DGs and 19 consumers, with controller parameters selected by trial and error.The case-study setup uses randomly selected initial λ values; KI and KP can be set between zero and one.Source: A. Case Study I (Evaluation of Accuracy) · p. 12
• Less than 0.00201% of the average: the distributed and centralized methods’ solution mismatch in case study I.The comparison covers DG outputs and consumer demands.Source: A. Case Study I (Evaluation of Accuracy) · p. 13, Table II · p. 14, Table III · p. 14
• 36 iterations and about 1.69 seconds: the distributed solution converges in case study I.The incremental cost converges to 8.175 $/kWh, and social welfare is 5,211.5 $/hr.Source: A. Case Study I (Evaluation of Accuracy) · p. 13
• The proposed algorithm reaches the same economic dispatch results in 36 iterations, compared with 500 iterations in published work.The paper describes its convergence to the near-optima as faster than other distributed methods,,.Source: A. Case Study I (Evaluation of Accuracy) · p. 14

B. Case Study II (Evaluation of Scalability and Fast Convergence)

• Approximately 40 iterations: incremental costs reach consensus in the 1,400-agent system, with execution time of 192.579 seconds.The case includes 1,000 consumers and 400 DGs.Source: B. Case Study II (Evaluation of… · p. 14
• From 29 to 1,400 agents, the number of iterations is almost constant under a 0.001 kW power-mismatch tolerance.Simulations also include systems of 350, 700, and 1,050 agents.Source: B. Case Study II (Evaluation of… · p. 14
• Agents share only estimated power mismatch; consumers need no peer-to-peer links, and updating that mismatch reduces computational cost.Consumers need not communicate with one another or with more than one DG.Source: B. Case Study II (Evaluation of… · p. 14

C. Case Study III (Evaluation of Practical Performance)

• Local agents can update estimated power mismatch through a PI or I controller; consumers adjust demand based on the change in incremental cost.A consumer’s surplus is maximized as Δλ becomes zero.Source: C. Case Study III (Evaluation of… · pp. 14–15
• The proof-of-concept test uses six DGs and ten consumers, with each Cubieboard emulating a local agent controller.A PC running VOLTTRON serves as the information-exchange bus; the agents are implemented in Python.Source: C. Case Study III (Evaluation of… · pp. 15, 17
• 421 kW: experimental total generation satisfies the total load demand, with the centralized approach used as a benchmark.The incremental cost converges to 7.371 $/kWh at the 42nd iteration.Source: C. Case Study III (Evaluation of… · p. 17

V. CONCLUSIONS

• The paper formulates a social-welfare problem using DG cost functions and consumer utility functions.It develops a distributed algorithm that seeks the global optimum through iterative coordination between consumers and DGs.Source: §V · p. 18
• Agents share only estimated power mismatch, which the paper says contains no private information and contributes to a fair electricity market.Source: §V · p. 18
• Software simulations and experimental tests evaluate accuracy, privacy, effectiveness, convergence, scalability, and ease of implementation under varied conditions.Source: §V · p. 18