Optimizing the Dynamic Economic Dispatch Problem by the Distributed Consensus-Based ADMM Approach

In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements. The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition. One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter. Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.

This article introduces a relatively current method named grasshopper optimization algorithm (GOA) for explaining power system based dynamic economic load dispatch (DELD) problem. However, like other optimization approaches, GOA suffers from premature convergence and a slow convergence rate. Thus, to boost the convergence mobility of GOA, oppositional based learning (OBL) is merged with the GOA approach. Moreover, another improvization, namely, chaotic concept, is also integrated with the GOA to improve its solution quality. The suggested oppositional based chaotic grasshopper optimization algorithm (OCGOA) method is applied to handle the DELD problem in the most efficient manner. Among the renewable energies, intermittent wind energy (WE) is the most sustainable one and remains technically and economically advantageous for electrical energy generation. The thermal and wind based power systems are hybridized, so that power generation is distributed among the operating units in an equivalent manner such that overall cost and loss part are optimized until all practical constraints are fulfilled. The production cost of the unpredictable wind generation power is further incorporated in the operational cost by employing a probability density function (PDF) formula. The precision and performance of the suggested GOA and OCGOA approaches are validated on 6-unit and 10-unit DELD systems for conventional and wind based energy systems. The efficacy and performance of the suggested OCGOA is judged by comparing it with conventional GOA and other presently developed meta-heuristic optimization techniques found in the literature.

The krill herd (KH) approach is proposed in this article for tackling the dynamic economic dispatch (DED) problem with the integration of wind power generation, including constraints such as valve-point loadings and prohibited operating zones (PoZs). The purpose of resolving the DED problem is to determine the optimum feasible combination of power outputs for all allocated generating units within a given period while simultaneously meeting dynamic operational constraints and power demand. Due to the addition of valve points and PoZs to the cost characteristics of the generating unit, the DED issue becomes a highly non-linear and non-convex optimization issue. When transmission losses are taken into account, the DED problem becomes significantly more complex to solve. KH is a swarm-inspired algorithm based on krill individual herding behavior. KH employs two global and two local tactics, and these work in parallel, making KH a powerful algorithm. The effectiveness of the KH method is investigated and validated through extensive testing on various test systems, including 5-, 6-, 10-, 16-, and 30-unit test systems. Extensive studies are conducted to evaluate the efficacy of the proposed KH algorithm; the present research work compares the convergence properties of the suggested approach with those of the other recently published methods. In addition to reduced generation costs compared to other results published in recent literature, the numerical findings of the KH approach reveal that it has strong convergence qualities.

In the present scenario, optimal and reliable operation is crucial for electric power generation and supply. The optimal operation can be attained by the economic generation of electric power and is dependent on economic dispatch (ED). In this paper, the solution to the Dynamic Economic Dispatch problem is achieved by a novel technique known as Spotted Hyena Optimization (SHO) algorithm. SHO is behaviorally encouraged by the inspiration of attacking nature of Spotted Hyena. The DED optimization problem is solved by considering various non-linear constraints such as ramp rate constraints, prohibited operating zones, and valve point loading effect. The applied technique is tested for the different test systems to present the effectiveness compared to other referred methods. A comparative analysis is depicted with diverse methods to deduce the efficacy and optimality of the applied technique.

Moth flame optimization algorithm is novel nature inspired heuristic paradigm inspired by navigation method of moths in nature and based on the concept that the moth eventually converges toward the light. This paper presents the application of MFO algorithm for the solution of non-convex and dynamic economic load dispatch problem of electric power system. The performance of MFO algorithm is tested for non-convex, convex and dynamic economic load dispatch problem of seven IEEE benchmarks and the results are verified by a comparative study with lambda iteration method, particle swarm optimization algorithm, genetic algorithm (GA), artificial bee colony, evolutionary programming (EP) and grey wolf optimizer (GWO). Also, in the proposed research, the impact of renewable energy sources (i.e. wind and solar) has been taken into consideration along with conventional thermal power generating units. Also, critical analysis has been made for percentage cost saving with due consideration of solar and wind power units and it has been experimentally observed that the addition of renewable energy sources to conventional thermal power system results in significant cost saving. Comparative results show that the performance of Moth Flame Optimizer Algorithm is better than recently developed GWO algorithm and other well known heuristics and meta-heuristics search algorithms.

Implementing a model predictive control strategy on the dynamic economic emission dispatch problem with game theory based demand response programs

Distributed optimization has gained popularity in many areas because it can to cope with the increasing volume of data, and the corresponding demand for computational resources. A popular distributed solver is the alternating direction method of multipliers (ADMM). Fully decentralized (D) ADMM arises when node-to-node communications have to abide by the underlying network connectivity. DADMM’s convergence however, slows down as the network diameter grows large. To deal with this challenge, the recently proposed hybrid (H) ADMM provides considerable performance boost over DADMM by exploiting local topology information. But HADMM only applies to unweighted graphs. The present contribution develops a weighted hybrid (WH) consensus-based ADMM approach that can deal with weighted graphs. The resultant scheme further improves the performance of HADMM through graph-aware weight tuning. Theoretical analysis offers convergence guarantees and establishes linear convergence rate, while numerical tests on various graphs demonstrate the WHADMM merits.

This paper proposes a deterministic two-stage mixed integer linear programming (TSMILP) approach to solve the reserve constrained dynamic economic dispatch (DED) problem considering valve-point effect (VPE). In stage one, the nonsmooth cost function and the transmission loss are piecewise linearized and consequently the DED problem is formulated as a mixed integer linear programming (MILP) problem, which can be solved by commercial solvers. In stage two, based on the solution obtained in stage one, a range compression technique is proposed to make a further exploitation in the subspace of the whole solution domain. Due to the linear approximation of the transmission loss, the solution obtained in stage two dose not strictly satisfies the power balance constraint. Hence, a forward procedure is employed to eliminate the error. The simulation results on four test systems show that TSMILP makes satisfactory performances, in comparison with the existing methods.

In this work, we consider the asynchronous distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Asynchronous Approximate Distributed Alternating Direction Method of Multipliers (AsyAD-ADMM), using finite-time asynchronous approximate ratio consensus, to solve the multi-node convex optimization problem, in which every node performs iterative computations and exchanges information with its neighbors asynchronously. More specifically, at every iteration of AsyAD-ADMM, each node solves a local convex optimization problem for the one of the primal variables and utilizes a finite-time asynchronous approximate consensus protocol to obtain the value of the other variable which is close to the optimal value, since the cost function for the second primal variable is not decomposable. If the individual cost functions are convex, but not-necessarily differentiable, the proposed algorithm converges at a rate of O(1/k), where k is the iteration counter. The efficacy of AsyAD-ADMM is exemplified via a proof-of-concept distributed least square optimization problem with different performance-influencing factors investigated.

Optimal dynamic economic dispatch of generation: A review

Energy management system (EMS) in microgrids (MGs) encompasses both dynamic economic dispatch (DED) and dynamic emission dispatch problems, known as dynamic economic emission dispatch (DEED). In this paper, this system is implemented for an MG with different distributed generations. The aim of 24-hour-cycle scheduling is to determine the generation level of diesel generators (DGs), participation rate of responsive loads and their incentives, charging/discharging status of the battery, and the amount of exchanged energy with the main grid. The scheduling task is performed with the objectives of minimizing operation costs and emissions and maximizing the benefit of MG operator (MGO) resulting from the demand response (DR). Due to their interactive operation, the customers’ behavioral model and the relevant cost functions have a significant impact on the optimal operation. In this paper, the cost function of customers is developed in the incentive-based DR program (DRP) with the aim of receiving a more realistic incentive and then it is combined with DEED. Finally, the zero energy balance (ZEB) interaction between the MG and the main grid is proposed as a solution. Regarding the multi-objective and nonlinear optimization problem, the proposed method, with the help of whale optimization algorithm (WOA), is implemented on a small-scale MG in MATLAB software, where electrical loads are also taken into account. Simulation results highlight the fact that the integration of DR into DEED along with import/export ZEB between the MG and the main grid leads to optimal operation and reduced power imported from the main grid, and helps maintain the load/generation balance.

Hybrid weighted probabilistic neural network and biogeography based optimization for dynamic economic dispatch of integrated multiple-fuel and wind power plants

The cross-entropy based hybrid membrane computing method is proposed in this paper to solve the power system unit commitment problem. The traditional unit commitment problem can be usually decomposed into a bi-level optimization problem including unit start-stop scheduling problem and dynamic economic dispatch problem. In this paper, the genetic algorithm-based P system is proposed to schedule the unit start-stop plan, and the biomimetic membrane computing method combined with the cross-entropy is proposed to solve the dynamic economic dispatch problem with a unit start-stop plan given. The simulation results of 10–100 unit systems for 24 h day-ahead dispatching show that the unit commitment problem can be solved effectively by the proposed cross-entropy based hybrid membrane computing method and obtain a good and stable solution.

We formulate the control of reactive power generation by photovoltaic inverters in a power distribution circuit as a constrained optimization that aims to minimize reactive power losses subject to finite inverter capacity and upper and lower voltage limits at all nodes in the circuit. When voltage variations along the circuit are small and losses of both real and reactive powers are small compared to the respective flows, the resulting optimization problem is convex. Moreover, the cost function is separable enabling a distributed, on-line implementation with node-local computations using only local measurements augmented with limited information from the neighboring nodes communicated over cyber channels. Such an approach lies between the fully centralized and local policy approaches previously considered. We explore protocols based on the dual ascent method and on the Alternating Direction Method of Multipliers (ADMM) and find that the ADMM protocol performs significantly better.

This paper proposes an efficient chaotic differential harmony search (CDHS) algorithm to solve the dynamic economic and emission dispatch (DEED) problem with security constraints. The main objective of DEED problem is to determine the optimal power generation schedule for the online generating units over a time horizon at the same time satisfying the environmental and security constraints along with other generator constraints. The DEED problem is formulated as a nonlinear constrained multi-objective optimization problem. The fuel cost and emission are considered as two competing objectives. It is converted into a single objective function by introducing a price penalty factor. The price penalty factor blends the emission with the fuel cost. In the proposed algorithm, chaotic variables are introduced instead of random number generator in the differential mutation operator of differential harmony search (DHS) algorithm to accelerate the DHS algorithm to reach the global optimal solution quickly. The effectiveness of the proposed CDHS algorithm is demonstrated on IEEE-30 bus system with valve point effects, ramp rate limits, prohibited operating zones, and security constraints for a period of 24 hours. The simulation results obtained by the proposed CDHS algorithm are compared with DHS algorithm. The optimal solution obtained by the proposed algorithm is better than DHS algorithm in terms of solution quality, speed of convergence and computational time.