dsoa: The implementation of a dynamic system optimization algorithm

This paper describes the ANSI C/C++ computer program dsoa , which implements an algorithm for the approximate solution of dynamics system optimization problems. The algorithm is a direct method that can be applied to the optimization of dynamic systems described by index‐1 differential‐algebraic equations (DAEs). The types of problems considered include optimal control problems and parameter identification problems. The numerical techniques are employed to transform the dynamic system optimization problem into a parameter optimization problem by: (i) parameterizing the control input as piecewise constant on a fixed mesh, and (ii) approximating the DAEs using a linearly implicit Runge‐Kutta method. The resultant nonlinear programming (NLP) problem is solved via a sequential quadratic programming technique. The program dsoa is evaluated using 83 nontrivial optimal control problems that have appeared in the literature. Here we compare the performance of the algorithm using two different NLP problem solvers, and two techniques for computing the derivatives of the functions that define the problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison of direct optimization algorithms for dynamic network flow control

Several direct optimization algorithms for dynamic non-linear optimal control problems with bounded control inputs are reviewed. The algorithms are based on well-known conjugate gradient and variable metric methods for unconstrained optimization. Scaling is shown to improve the algorithms' performance considerably. Comparison of the algorithms is provided for a real-life example of sewer network flow control.     

Optimal energy management for grid‐connected storage systems

This paper proposes three near‐optimal (to a desired degree) deterministic charge and discharge policies for the maximization of profit in a grid‐connected storage system. The changing price of electricity is assumed to be known in advance. Three near‐optimal algorithms are developed for the following three versions of this optimization problem: (1) The system has supercapacitor type storage, controlled in continuous time. (2) The system has supercapacitor or battery type storage, and it is controlled in discrete time (i.e., it must give constant power during each sampling period). A battery type storage model takes into account the diffusion of charges. (3) The system has battery type storage, controlled in continuous time. We give algorithms for the approximate solution of these problems using dynamic programming, and we compare the resulting optimal charge/discharge policies. We have proved that in case 1 a bang off bang type policy is optimal. This new result allows the use of more efficient optimal control algorithms in case 1. We discuss the advantages of using a battery model and give simulation and experimental results. Copyright © 2014 John Wiley & Sons, Ltd.     

ACADO toolkit—An open‐source framework for automatic control and dynamic optimization

In this paper the software environment and algorithm collection ACADO Toolkit is presented, which implements tools for automatic control and dynamic optimization. It provides a general framework for using a great variety of algorithms for direct optimal control, including model predictive control as well as state and parameter estimation. The ACADO Toolkit is implemented as a self‐contained C++ code, while the object‐oriented design allows for convenient coupling of existing optimization packages and for extending it with user‐written optimization routines. We discuss details of the software design of the ACADO Toolkit 1.0 and describe its main software modules. Along with that we highlight a couple of algorithmic features, in particular its functionality to handle symbolic expressions. The user‐friendly syntax of the ACADO Toolkit to set up optimization problems is illustrated with two tutorial examples: an optimal control and a parameter estimation problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Shape optimization analysis: First- and second-order necessary conditions

In this paper, first- and second-order necessary conditions for optimality are studied for a domain optimization problem. The optimization problem considered is the minimization of an objective function defined on the domain boundary through the solution of a boundary value problem. In order to derive the first and second variations of the objective function due to boundary variation, the first and second variations of the solution of the boundary value problem are calculated using a perturbation technique. An iterative shape optimization algorithm for potential flow problems in R² with Dirichlet boundary conditions is presented. In the algorithm a boundary element method (BEM) is employed to solve the Laplace equation numerically. The validity and accuracy of the algorithm have been verified on a problem where the final solution is known. Finally, the problem of designing a 90° bend for two-dimensional potential flow is solved.     

Optimal design of trajectory parameters and position tracking with balance for riderless bicycle

Riderless bicycles are typically nonholonomic, underactuated, and nonminimum‐phase systems. The instability and complex dynamic coupling make the trajectory generation and tracking of the bicycles more challenging. In this paper, we consider both the trajectory generation and position tracking of a riderless bicycle. To achieve smooth motion performances, the desired planar trajectory of the contact point of the bicycle's rear wheel is constructed using a parameterized polynomial curve that can connect two given endpoints with associated tangent angles. The optimal parameters of the polynomial curve are obtained by minimizing the maximum of the roll angle's quasistatic trajectory of the bicycle, and this problem is solved by the particle swarm optimization algorithm. Then, position tracking of the desired planar trajectory with balance is converted into an optimization problem subject to the dynamic constraints. The cost function is designed as the combination of the position errors and the roll acceleration of the bicycle, in order to achieve an accurate tracking performance and to prevent the bicycle from falling down. This optimization problem is solved by the Gauss pseudospectral method. Simulation results are presented to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Decentralized stochastic adaptive nash games

The optimization of stochastic systems with unknown parameters and multiple decision-makers or controllers each having his own objective is considered. Two explicit self-tuning type algorithms are proposed for decentralized stochastic adaptive Nash games under the ‘one-step-delay sharing pattern’. The first algorithm is an ad hoc constraint on the policy form, whereas the second one is based on an extension from static Nash game theory. Simulation results on a simplified economic system indicate that these algorithms are capable of stabilizing a system along targeted paths.     

Design and optimization of a novel fast distributed Kalman consensus filtering algorithm

A new type of fast distributed Kalman consensus filtering algorithm based on local information feedback is presented to tackle filtering problems in wireless sensor networks. First, this fast filtering issues are transformed into a stochastic stability problem of the dynamic estimation errors, which can be solved by Lyapunov's second method and matrix theory. Then, two sufficient conditions about the proportional-like feedback (double gains regulation) method and incremental Proportional-Integral-Derivative (PID) feedback method for the asymptotical stability of the systems are presented, respectively. Moreover, to achieve a faster convergence rate, a novel optimal method is given by combing a genetic algorithm and incremental PID. Finally, an illustrative example is presented to give a comparison of the convergence speed between the three filtering algorithms in the same condition, and verify the effectiveness and advantage of the proposed theoretical results in this article.     

Distributed estimation for spatial rigid motion based on dual quaternions

This paper proposes 2 distributed optimization algorithms for the estimation of spatial rigid motion using multiple image sensors in a connected network. The objective is to increase the estimation precision of translational and rotational motion based on dual quaternion models and the cooperation between connected sensors. The dual decomposition subgradient method and distributed Newton optimization method are applied to decompose the filtering task into a series of suboptimal problems and then solve them individually to achieve the global optimality. Our approach assumes that each sensor can communicate with its neighboring sensors to update the individual estimates. Discussion on converging speed of both methods are provided. Simulation examples are demonstrated to compare the 2 distributed algorithms with the traditional extended Kalman filter in terms of estimation accuracy and converging rate.     

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

The evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.     

An adaptive chaos particle swarm optimization for tuning parameters of PID controller

An adaptive chaos particle swarm optimization (ACPSO) is presented in this paper to tune the parameters of proportional‐integral‐derivative (PID) controller. To avoid the local minima, we introduced a constriction factor. Meanwhile, the chaotic searching is combined with the particle swarm optimization to improve the ability of the proposed algorithm. A series of experiment is performed on 6 benchmark functions to confirm its performance. It is found that the ACPSO can get better solution quality in solving the global optimization problems and avoiding the premature convergence. Based on it, the proposed algorithm is applied to tune the PID controller's parameters. The performances of the ACPSO are compared with different inspired algorithms, and these results show that the ACPSO is more robust and efficient when it is used to find the optimal parameters of PID controller.     

Hybrid BOA-GWO-PSO algorithm for mitigation of congestion by optimal reactive power management

The generation of power with load optimization, particularly in the current deregulated electricity market conditions, is a very important process for improved planning and operation of the grid. In addition, it is very important for the system not to experience problems due to congestion, have tensile stability, and protection to increase the share of electricity from renewable sources with the current supply system. This article presents load balancing with the butterfly optimization algorithm (BOA) in a hybridized form to minimize and maximize loads when used in pool and hybrid markets. The methods have been designed to prevent the drawbacks of BOA and generate a better trade-off between exploration and exploitation abilities by hybridizing it with particle swarm optimization (PSO) and gray wolf optimizer (GWO). Empirical research on other algorithms shows that proposed hybrid BOA-GWO-PSO algorithm performs better and shows potential in diverse problems. These studies give it a significant advantage over BOA in general, and when it is employed to solve complex optimization problems validated on benchmark IEEE 30 bus system. A comparative analysis has been conducted to validate the potency of the hybrid BOA-GWO-PSO approach with some conventional meta-heuristic algorithms. Analysis of results by mathematical validation on 23 benchmark functions and application in congestion management by optimal reactive power management (RPM) reveal that the proposed technique has the potent to solve real world optimization problems and is competitive with recent methods reported in state-of- art literature.     

Neighbouring extremals of dynamic optimization problems with parameter variations

Neighbouring extremals of dynamic optimization problems with a known parameter vector θ and an unknown parameter vector π are considered in this paper. The parameter vector π and the control are to be optimally determined to minimize a cost functional with a given θ. With some simplifications, the neighbouring extremal problem is reduced to one of solving a linear, time-varying, two-point boundary value problem with integral path equality constraints. A modified backward sweep method is used to solve this problem. Example problems are solved to illustrate the validity and usefulness of the solution technique.     

Numerical solution of time-delayed optimal control problems with terminal inequality constraints

In this short communication we consider an approximation scheme for solving time-delayed optimal control problems with terminal inequality constraints. Time-delayed problems are characterized by variables x (t - τ) with a time-delayed argument. In our scheme we use a Páde approximation to determine a differential relation for y (t), an augmented state that represents x (t - τ). Terminal inequality constraints, if they exist, are converted to equality constraints via Valentine-type unknown parameters. The merit of this approach is that existing, well-developed optimization algorithms may be used to solve the transformed problems. Two linear/non-linear time-delayed optimal control problems are solved to establish its usefulness.     

Optimal control of a revenue management system with dynamic pricing facing linear demand

This paper considers a dynamic pricing problem over a finite horizon where demand for a product is a time‐varying linear function of price. It is assumed that at the start of the horizon there is a fixed amount of the product available. The decision problem is to determine the optimal price at each time period in order to maximize the total revenue generated from the sale of the product. In order to obtain structural results we formulate the decision problem as an optimal control problem and solve it using Pontryagin's principle. For those problems which are not easily solvable when formulated as an optimal control problem, we present a simple convergent algorithm based on Pontryagin's principle that involves solving a sequence of very small quadratic programming (QP) problems. We also consider the case where the initial inventory of the product is a decision variable. We then analyse the two‐product version of the problem where the linear demand functions are defined in the sense of Bertrand and we again solve the problem using Pontryagin's principle. A special case of the optimal control problem is solved by transforming it into a linear complementarity problem. For the two‐product problem we again present a simple algorithm that involves solving a sequence of small QP problems and also consider the case where the initial inventory levels are decision variables. Copyright © 2006 John Wiley & Sons, Ltd.     

Minimum fuel powered dynamic soaring of unmanned aerial vehicles utilizing wind gradients

This paper studies optimal powered dynamic soaring flights of unmanned aerial vehicles (UAVs) that utilize low‐altitude wind gradients for reducing fuel consumptions. Three‐dimensional point‐mass UAV equations of motion are used, and linear wind gradients are assumed. Fundamental UAV performance parameters are identified through the normalization of the equations of motion. In particular, a single wind condition parameter is defined that represents the combined effect of air density, UAV wing loading, and wind gradient slope on UAV flight. An optimal control problem is first used to determine bounds on wind conditions over which optimal powered dynamic soaring is meaningful. Then, powered UAV dynamic soaring flights through wind gradients are formulated as non‐linear optimal control problems. For a jet‐engined UAV, performance indices are selected to minimize the average thrust required per cycle of powered dynamic soaring that employs either variable or constant thrust. For a propeller‐driven UAV, in comparison, performance indices are selected to minimize the average power required per cycle of powered dynamic soaring with either variable or constant power. All problem formulations are subject to UAV equations of motion, UAV operational constraints, proper initial conditions, and terminal conditions that enforce a periodic flight. These optimal control problems are converted into parameter optimization with a collocation method and solved numerically using the parameter optimization software NPSOL. Analytical gradient expressions are derived for the numerical solution process. Extensive numerical solutions are obtained for a wide range of wind conditions and UAV performance parameters. Results reveal basic features of powered dynamic soaring flights through linear wind gradients. Copyright © 2004 John Wiley & Sons, Ltd.     

Exact penalty function approach to constrained optimal control problems

Given the well known concept that optimal control problems may be solved either by the maximum principle or by the dynamic programming technique which employs many numerical algorithms, this paper attempts to show that the exact penalty function method may be used to transform a constrained optimal control problem into an unconstrained optimal control problem. Under certain conditions the constrained optimal control problem is shown to be equivalent to an unconstrained optimal control problem, which can be easily solved by a numerical technique.     

Optimal patterns of glider dynamic soaring

This paper presents optimal patterns of glider dynamic soaring utilizing wind gradients. A set of three‐dimensional point‐mass equations of motion is used and basic glider performance parameters are identified through normalizations of these equations. In particular, a single parameter is defined that represents the combined effects of air density, glider wing loading, and wind gradient slope. Glider dynamic soaring flights are formulated as non‐linear optimal control problems and three performance indices are considered. In the first formulation, the completion time of one cycle of dynamic soaring is minimized subject to glider equations of motion, limitations on glider flights, and appropriate terminal constraints that enforce a periodic dynamic soaring flight. In the second formulation, the final altitude after one cycle of dynamic soaring is maximized subject to similar constraints. In the third formulation, the least required wind gradient slope that can sustain an energy‐neutral dynamic soaring flight is determined. Different terminal constraints are used to produce basic, travelling, and loiter dynamic soaring patterns. These optimal control problems are converted into parameter optimization via a collocation approach and solved numerically with the software NPSOL. Different patterns of glider dynamic soaring are compared in terms of cycle completion time and altitude‐increasing capability. Effects of wind gradient slope and wind profile non‐linearity on dynamic soaring patterns are examined. Copyright © 2004 John Wiley & Sons, Ltd.     

A trajectory‐based sampling strategy for sequentially refined metamodel management of metamodel‐based dynamic optimization in mechatronics

Dynamic optimization problems based on computationally expensive models that embody the dynamics of a mechatronic system can result in prohibitively long optimization runs. When facing optimization problems with static models, reduction in the computational time and thus attaining convergence can be established by means of a metamodel placed within a metamodel management scheme. This paper proposes a metamodel management scheme with a dedicated sampling strategy when using computationally demanding dynamic models in a dynamic optimization problem context. The dedicated sampling strategy enables to attain dynamically feasible solutions where the metamodel is locally refined during the optimization process upon satisfying a feasibility‐based stopping condition. The samples are distributed along the iterate trajectories of the sequential direct dynamic optimization procedure. Algorithmic implementation of the trajectory‐based metamodel management is detailed and applied on two case studies involving dynamic optimization problems. These numerical experiments illustrate the benefits of the presented scheme and its sampling strategy on the convergence properties. It is shown that the acceleration of the solution time of the dynamic optimization problem can be achieved when evaluating the metamodel that is lower than 90% compared to the computationally expensive model.     

Modified whale optimization algorithm for fractional‐order multi‐input SSSC‐based controller design

In this paper, the design of a fractional‐order (FO) multi‐input–single‐output (MISO)–type static synchronous series compensator (SSSC) is proposed with a goal to improve the power system stability using modified whale optimization algorithm (MWOA). The proposed MWOA achieves an appropriate balance between exploitation and exploration stages of the original whale optimization algorithm. The performance of MWOA is validated by employing the benchmark test functions and further contrasted with whale optimization algorithm and other heuristic algorithms like gravitational search algorithm, particle swarm optimization, differential evolution, and fast evolutionary programming algorithms to demonstrate its strength. The proposed FO MISO SSSC controller is optimized by the MWOA technique and tested under single‐machine infinite bus system and further extended to a multi‐machine framework. To demonstrate the superiority of MISO‐type SSSC controller, the results obtained from it are compared with particle swarm optimization and differential evolution–based conventional single‐input–single‐output structured SSSC controllers. The comparison of results of MWOA with that of other methods validates its superiority in the present context.