Optimal Flow Control Problems Research Articles

Optimal control of tidal flow

The tidal flow through a channel connecting two basins with different tidal regimes can be optimally controlled by means of a turbine fence or array to maximise the extracted mechanical power. The paper gives the optimal control strategy as a function of the blockage ratio $\sigma$ , i.e. the ratio of the turbine cross-section to the cross-section of the local passage of a turbine. The results presented are a physically consistent generalisation of the results of Garrett & Cummins (Proc. R. Soc. Lond. A, vol. 461, 2060, pp, 2563–2572), valid only for $\sigma =1$ and turbine efficiency of one, now for arbitrary blockage ratio $0 < \sigma \leqslant 1$ . Published research over the past decade on the same topic has taken the momentum equation and the turbine drag force as a starting point. The new approach presented here, in contrast, takes the energy equation as the starting point and uses the relative volume flow as the control variable. As the work shows, this new approach has three advantages. First, starting with the energy equation allows us to derive an optimal flow control problem resulting in an Euler–Lagrange equation using the physically consistent and experimentally validated actuator disk model for the free surface flow of Pelz et al. (J. Fluid Mech., vol. 889, 2020) in a direct and formal way. The optimal control problem is solved (a) numerically and (b) analytically. In the latter case, the turbine characteristics are approximated by a rational function in the relevant design and operating range. The analytical solution (b) validated against the numerical solution (a) is surprisingly concise and easy to apply in practice, as shown by use cases. Second, instead of the induction factor, we use the volume flow that is the same for all turbines in a cascade, i.e. a row of turbines in the direction of flow, which significantly reduces the complexity of the optimal control task of turbine arrays. Third, we obtain a well-founded energy estimate, whereas previous methods overestimate the energy yield due to inconsistent turbine disc models (for the consistency and valid parameter ranges of different models, also in comparison with experiments, see Pelz et al., J. Fluid Mech., vol. 889, 2020). The results can be used for the conceptual design of turbine arrays, but also for a sound physically realistic and consistent resource assessment of tidal power for a system consisting of two basins, a channel and a turbine fence with $0<\sigma \leqslant 1$ and operated in a complete tidal cycle.

The problem of combined optimal load flow control of main conveyor line

The problem of combined optimal load flow control of main conveyor line

Dynamic optimization analyses and algorithm design for the parallel heat exchange system in spacecraft

Dynamic optimization of the fluid loop is critical of the active thermal control system (ATCS) for future spacecraft. In this paper, the transient heat current model of the parallel heat exchanger system is constructed by the thermoelectric analogy method to analyze the optimal control problem of dynamic flow allocation. The continuous free time optimal flow control problem is transformed into a fixed-time nonlinear programming problem which can be solved by the sequential quadratic programming (SQP) algorithm through the control variable parameterization (CVP) and time-scaling methods. The simulation showed that the optimized flow allocation significantly reduced the process time required for system heat dissipation. Compared with the non-gradient particle swarm optimization (PSO) algorithm, the SQP method constructed in this paper decreased the optimization index by about 4%∼6% and the calculation time by about one order of magnitude. To diminish the sensitivity of the SQP algorithm to the initial value, this paper simplifies the model according to the system characteristics. Through the Pontryagin extremum principle, it is found and proved that, under the simplified model, the optimal flow rates are a set of time-independent variables, which significantly reduces the difficulty of the optimization problem, according to which a hybrid algorithm is designed that uses the optimal result of the simplified model as the initial value predictor. Compared with the average of using the random initial value, using the estimated initial value decreased the optimization index by more than 2% and the computation time by about 60%, under the same SQP algorithm.

Reduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient-specific data assimilation.

Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease. In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step toward matching patient-specific physiological data in patient-specific vascular geometries as best as possible. Two critical challenges that reportedly arise in such problems are: (a) lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (b) high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition-Galerkin.

Production Control with Price, Cost, and Demand Uncertainty

An optimal production flow control problem of a make-to-stock manufacturing firm with price, cost, and demand uncertainty is studied. The objective of the flow rate control problem is maximizing the average profit that is the difference between the expected revenue and the expected production, inventory holding, and backlog costs. The uncertainties in the system are captured jointly in discrete environment states. In each environment state, the price, cost, and demand take different levels. The transitions between different environment states evolve according to a time-homogenous Markov chain. By using a continuous flow model, the optimal production policy is stated as a state-dependent hedging policy. The performance of the system where the production cost alternates between a high and a low cost level and the demand is either constant or also alternates between a high and a low level is analyzed under the double-hedging policy. According to this policy, the producer produces only when the cost is low and the surplus is between the two hedging levels. However when the backlog is below the lower hedging level, the producer produces with the maximum capacity regardless of the cost. The effects of production cost, production capacity, demand variability, and the dependence of the demand and the cost on the performance of the system are analyzed analytically and numerically. It is shown that controlling the production rate optimally allows producers respond to the fluctuations in price, cost, and demand in an effective way and maximize their profits.

Revisiting the single-phase flow model for liquid steel ladle stirred by gas

Ladle stirring is an important step of the steelmaking process to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions before casting. Gas is injected from the bottom of the bath to induce a turbulent flow of the liquid steel. Multiphase modeling of ladle stirring can become computationally expensive, especially when used within optimal flow control problems. This paper focuses therefore on single-phase flow models. It aims at improving the existing models from the literature. Simulations in a 2d axial-symmetrical configuration, as well as, in a real 3d laboratory-scale ladle, are performed. The results obtained with the present model are in a relative good agreement with experimental data and suggest that it can be used as an efficient model in optimal flow control problems.

Intermodal freight transport planning – A receding horizon control approach

This paper investigates intermodal freight transport planning problems among deep-sea terminals and inland terminals in hinterland haulage for a horizontally fully integrated intermodal freight transport operator at the tactical container flow level. An intermodal freight transport network (IFTN) model is first developed to capture the key characteristics of intermodal freight transport such as the modality change phenomena at intermodal terminals, physical capacity constraints of the network, time-dependent transport times on freeways, and time schedules for trains and barges. After that, the intermodal freight transport planning problem is formulated as an optimal intermodal container flow control problem from a system and control perspective with the use of the proposed IFTN model. To deal with the dynamic transport demands and dynamic traffic conditions in the IFTN, a receding horizon intermodal container flow control (RIFC) approach is proposed to control and to reassign intermodal container flows in a receding horizon way. This container flow control approach involves solving linear programming problems and is suited for transport planning on large-sized networks. Both an all-or-nothing approach and the proposed RIFC approach are evaluated through simulation studies. Simulation results show the potential of the proposed RIFC approach.

Dynamic Pricing Control for Constrained Distribution Networks With Storage

This paper studies the problem of optimal flow control in dynamic inventory systems. A dynamic optimal distribution problem, including time-varying supply and demand, capacity constraints on the transportation lines, and convex flow cost functions of Legendre-type, is formalized and solved. The time-varying optimal flow is characterized in terms of the time-varying dual variables of a corresponding network optimization problem. A dynamic feedback controller is proposed that regulates the flows asymptotically to the optimal flows and achieves in addition a balancing of all storage levels.

Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations

This paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties–with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates–and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

Optimal flow control in acyclic networks with uncontrollable routings and precedence constraints

This paper introduces a novel optimal flow control problem that seeks to convey a specified amount of fluid to each of the nodes of an acyclic digraph with a single source node, while minimizing the total amount of fluid inducted into the network. Two factors complicating the aforementioned task are (i) the presence of nodes with uncontrollable routing of the traversing flow and (ii) a set of precedence constraints regarding the satisfaction of the nodal fluid requirements. It is shown that the considered problem can be naturally formulated as a continuous-time optimal control problem that can be reduced to a hybrid optimal control problem with controlled switching. This property subsequently enables the solution of the considered problem through a Mixed Integer Programming formulation. Additional results in the paper establish the NP-hardness of the considered problem, highlight its affinity to some well known scheduling problems, and offer guidelines that can alleviate the increased problem complexity.

Controlled networks and their applications

A new kind of networks is considered in which the flow is controlled by reconfiguring the network. Flow control models in the network are considered, and optimal flow control problems in the network are stated.

Optimal Control of Open-Channel Flow Using Adjoint Sensitivity Analysis

An optimal flow control methodology based on adjoint sensitivity analysis for controlling nonlinear open channel flows with complex geometries is presented. The adjoint equations, derived from the nonlinear Saint-Venant equations, are generally capable of evaluating the time-dependent sensitivities with respect to a variety of control variables under complex flow conditions and cross-section shapes. The internal boundary conditions of the adjoint equations at a confluence (junction) derived by the variational approach make the flow control model applicable to solve optimal flow control problems in a channel network over a watershed. As a result, an optimal flow control software package has been developed, in which two basic modules, i.e., a hydrodynamic module and a bound constrained optimization module using the limited-memory quasi-Newton algorithm, are integrated. The effectiveness and applicability of this integrated optimal control tool are demonstrated thoroughly by implementing flood diversion controls in rivers, from one reach with a single or multiple floodgates (with or without constraints), to a channel network with multiple floodgates. This new optimal flow control model can be generally applied to make optimal decisions in real-time flood control and water resource management in a watershed.

A paradox in optimal flow control of M/M/n queues

Optimal flow control problems of multiple-server (M/M/n) queueing systems are studied. Due to enhanced flexibility of the decision making, intuitively, we expect that grouping together separated sy...

A paradox in optimal flow control of M/M/ [formula omitted] queues

Optimal flow control problems of multiple-server (M/M/ n ) queueing systems are studied. Due to enhanced flexibility of the decision making, intuitively, we expect that grouping together separated systems into one system provides improved performance over the previously separated systems. This paper presents a result counter-intuitive against such an expectation. We consider a non-cooperative optimal flow control scheme M/M/ n queueing systems where each of multiple players strives to optimize unilaterally its own power where the power of a player is the quotient of the throughput divided by the mean response time for the player. We report a counter-intuitive case where the power of every user degrades after grouping together K ( > 1 ) separated M/M/ N systems into a single M/M/ ( K × N ) system. Some numerical results are presented.

Optimal sliding-window strategies in networks with long round-trip delays

A method commonly used for packet flow control over connections with long round-trip delays is “sliding windows”. In general, for a given loss rate, a larger window size achieves a higher average throughput, but also a higher rate of spurious packet transmissions, rejected by the receiver merely for arriving out-of-order. This paper analyzes the problem of optimal flow control quantitatively, for a connection that has a cost per unit time and a cost for every transmitted packet. The optimal strategy is defined as one that minimizes the expected cost/throughput ratio, and is allowed to transmit several copies of a packet within a window. We present an algorithm for computing the optimal strategy and study its properties; in particular, we derive bounds on the optimal strategy cost/throughput performance, and show that it increases merely logarithmically with the time price, whereas the cost/throughput of the `traditional' classic window scheme is linear in the time price.

Perspectives in Flow Control and Optimization

11R11. Perspectives in Flow Control and Optimization. - MD Gunzburger (Iowa State Univ, Ames IA). SIAM, Philadelphia. 2003. 261 pp. ISBN 0-89871-527-X. $70.00.Reviewed by HG Wood, III (Dept of Mech and Aerospace Eng, Univ of Virginia, Thornton Hall, McCormick Rd, Charlottesville VA 22903-2442).For anyone interested in flow control or optimization, this book is a must read. Max Gunzburger has done a truly commendable job of providing his perspectives on the subject, and his perspectives are very well informed. He states in the preface that his first goal is “to present an introduction to the development and analysis of several methods,” and this is done very well. He also raises issues that arise in the field, and the book has an outstanding bibliography that can lead the reader to other sources. He also points out that this book is not a complete treatment of the subject, as that would take several times the size of this book. However, the book is large enough to provide a really good introduction to the subject but small enough not to be intimidating. The book consists of seven chapters with the first being a brief history and introduction to the subject. Chapter 2 discusses the different approaches to optimal control and optimization that include the one-shot method and the sensitivity and adjoint based methods. The discussion is clear and provides information in a way that the reader could develop an algorithm from the presentation. Chapter 3 provides illustrations of the approaches discussed in Chapter 2. Chapter 4 deals with accuracy and consistency, and Chapter 5 discusses how to reduce the costs of optimization. Chapter 6 presents analysis and numerical analysis of optimal flow control problems. One of the examples is the analysis of a shape control problem for the stationary Navier-Stokes equations. The examples are very well presented and allow the reader to get a very good foundation that provides the necessary information to attack new more complicated problems. Chapter 7 is a brief discussion of feedback control for fluid flows. In conclusion, this is a very well written book that should be part of the library of anyone interested in fluid dynamics or optimization. This book could also be used as a textbook for a graduate course on the subject.

Necessary and sufficient conditions for optimal flow control in multirate multicast networks

The authors consider the optimal flow control problem in multirate multicast networks where all receivers of the same multicast group can receive service at different rates with different QoS. The objective is to achieve the fairness transmission rates that maximise the total receiver utility under the capacity constraint of links. They first propose necessary and sufficient conditions for the optimal solution to the problem, and then derive a new optimal flow control strategy using the Lagrangian multiplier method. Like the unicast case, the basic algorithm consists of a link algorithm to update the link price, and a receiver algorithm to adapt the transmission rate according to the link prices along its path. In particular if some groups contain only one receiver and become unicast, the algorithm will degrade to their previously proposed unicast algorithm.

Numerical Approximation of Optimal Flow Control Problems by a Penalty Method: Error Estimates and Numerical Results

The purpose of this paper is to present numerically convenient approaches to solve optimal Dirichlet control problems governed by the steady Navier--Stokes equations. We will examine a penalized Neumann control approach for solving Dirichlet control problems from numerical and computational points of view. The control is affected by the suction or injection of fluid through the boundary or by boundary surface movements in the tangential direction. The control objective is to minimize the vorticity in the flow or to drive the velocity field to a desired one. We develop sequential quadratic programming methods to solve these optimal control problems. The effectiveness of the optimal control techniques in flow controls and the feasibility of the proposed penalized Neumann control approaches for flow control problems are demonstrated by numerical experiments for a viscous, incompressible fluid flow in a two-dimensional channel and in a cavity geometry.

Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems

This article is concerned with penalty methods for solving optimal Dirichlet control problems governed by the steady-state and time-dependent Navier-Stokes equations. We present, in two different versions, the penalized methods for solving the steady-slate Dirichlet control problems. These approaches are implemented and compared numerically. We also generalize the penalty methods to the time-dependent case. Scmidiscrete and fully discrete approximations of time-dependent Dirichlet control problems are discussed and implemented. Numerical results for solving both the steady-state and the time dependent Dirichlet control problems are reported.

An optimal production flow control problem with impulsive demand

We consider a system consisting of a single machine cell and producing one part type. The machine cell has a finite capacity and is reliable. We consider a finite planning horizon containing N impulsive demands. The demand occurs instantaneously. Each such demand is known by its size and when it occurs in the planning horizon. In order to keep the production as close to the demand as possible there is a trade off between building up inventory and letting customers wait. An optimal control problem is formulated to determine the optimal production strategy. Solution technique is developed via Pontryagin's Minimum Principle.