Order Optimality Conditions Research Articles

Exergoeconomic investigation and multi-objective optimization of different ORC configurations for waste heat recovery: A comparative study

This paper compares four different architectures of organic Rankine cycle (ORC) namely the Basic ORC, Recuperative ORC, Regenerative ORC, and Recuperative-Regenerative ORC (RR-ORC) from the exergoeconomy viewpoint. First, a parametric analysis is performed to understand the effect of evaporator temperature, condenser temperature, and pinch point temperature difference on the performance of each system configuration. Next, a multi-objective optimization is performed to optimize the performance of the four configurations using Pareto Envelope-based Selection Algorithm-II (PESA-II) with exergy efficiency and the system cost rate as the objective functions. Pareto fronts, representing the set of optimal solutions, are obtained for each configuration, and multi-criteria decision analysis is done using the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) to select the best optimal solution. Next, the exergetic and exergoeconomic performance of the four ORC configurations is evaluated at their corresponding optimal conditions in order to provide a comparative assessment. It was found that, under optimal conditions, the RR-ORC performs superiorly to the other three configurations. Compared to the Basic ORC, the net power and exergy efficiency of the RR-ORC are 16.19% and 15.33% higher, while the system cost rate is 1.68 % low. After, the RR-ORC, the Regenerative, and the Recuperative ORCs are the second and the third-best configurations. It was also found that in all four configurations, the cost rate of exergy loss accounts for approximately 60 % of the total system cost rate.

New optimality criteria for convex continuous-time problems of vector optimization

In this paper, vector continuous-time programming problems with inequality type of constraints are considered. All the results are obtained under convexity assumptions. The main tool employed in the derivation of the zero order optimality conditions is a new version of theorem of the alternative in functional spaces. To apply the alternative theorem, a certain regularity condition must be satisfied. No differentiability assumption is imposed.

Experimental investigation of swirl number influence on spiral vortex structure dynamics

The hydraulic turbines are recently forced to operate far away from the optimal conditions in order to balance fluctuations in electricity generation. In case of Francis, pump and propeller turbines, using only single control component of guide vanes, it means that in regions where the high residual swirl enters the draft tube, the flow is decelerated and convenient conditions for the vortex rope development are created. Such flow conditions are considered to be the triggering mechanism for occurrence of different forms of vortex structures in the Francis turbine draft tube, e.g. spiral or straight vortex rope at part load or full load respectively. Independently on the vortex rope shape the unsteady pressure fields develop producing periodic stress on turbine components and possibly resulting in noise, blade cracks, runner lift, power swing, etc. To study and mimic such flow conditions, a simplified device of vortex generator apparatus is employed. Thanks to its design, the vortex generator enables to change the ratio between fluxes of axial momentum and tangential moment of momentum of generated swirl. Then, the behavior of vortex structure changes in a similar way as the flow rate variation in the draft tube of Francis turbine. For above mentioned reasons the unsteady cavitating spiral vortex is experimentally studied using both high speed video record and particle image velocimetry (PIV). The main focus is on change of vortex dynamics regarding to the swirl number variation. The proper orthogonal decomposition (POD) together with the classical fast Fourier transformation (FFT) are employed to extract dominant modes and frequencies from experimental data.

Neumann boundary optimal control problems governed by parabolic variational equalities

Abstract We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα . We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.

Application of variational calculus to parameter estimation in a real hydrological system

This paper presents an estimation study based on calculus of variations for parameters in a real hydrological system, namely Tondi-Kiboro catchment, in Niger. The considered dynamical representation for this system is first given, from the well-known Saint-Venant equations under some simplifying assumptions. A cost function is then defined based on the gap between measured and predicted water discharges at the real sensor position, and the calculus of variations along with the first order optimality condition are used to find the gradients of Lagrangian objective functional with respect to the parameters to be estimated. Depending on the infiltration model used in the system model, two applications of this adjoint-based approach are finally presented: the first one only estimates the friction coefficient, with the assumption that infiltration is described by the physical so-called Green-Ampt model, while the second one additionally estimates the parameters of an alternative infiltration representation called Horton model. Some simulation-based validation and robustness discussion concludes the paper.

Nash equilibria in electricity pool markets with large-scale wind power integration

This work investigates the interaction between power producers with conventional and wind generation portfolios participating in a network-constrained pool-based market. A stochastic bi-level problem is introduced to model the strategic behavior of each single producer. The upper-level problem maximizes the producers’ expected profits and the lower-level problem optimizes the jointly cleared energy and balancing market under economic dispatch. Market participants’ offers are modeled using linear stepwise curves, and the stochastic wind power generation is realized through a set of plausible wind scenarios. The bi-level problem is recast into a single-level mathematical problem with equilibrium constraints with primal-dual formulation using the Karush-Kuhn-Tacker first order optimality conditions and the strong duality theorem. The joint solution of all strategic producers’ problems constitutes an equilibrium problem with equilibrium constraints. The latter is reduced into an equivalent mixed integer linear program by using disjunctive constraints. Different objective functions are applied to the final program to define the range of market equilibria, and a single-iterate diagonalization process is used to justify those equilibria that are meaningful. The model addresses several cases considering different types of market competition, transmission line congestions, and different levels of wind power penetration and volatility.

Optimal control of convection-cooling and numerical implementation

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion–convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.

Vacuum Microwave-Assisted Aqueous Extraction of Polyphenolic Compounds from Avocado (Persea Americana) Solid Waste

The extraction efficacy of avocado fruit peels (AP) and seeds (AS) with the use of vacuum microwave-assisted aqueous extraction (VMAAE) was optimized in this study, with regard to extract’s total phenolic content (TPC), maximum antioxidant activity and minimal operational cost. Temperature (79.64 and 78.11 °C for AP and 43.90 and 45.26 °C for AS), time (11.89 and 11.75 min for AP, 10.18 and 10.28 min for AS), ratio of water to raw material (16.45% and 10.02% for AP, 38.73% and 37.65% for AS) and microwave power (5708.04 and 5699.10 W for AP, 5549.08 and 4797.29 W for AS) were estimated statistically as the optimal conditions in order to achieve high rates of extracts with high TPC and antioxidant activity using the 2,2-diphenyl-1-picrylhydrazyl (DPPH●) scavenging radical methods, respectively. VMAAE performed under these conditions resulted in received extracts with TPC (0.352 gallic acid equivalent-GAE/g fresh AP/min and 0.124 GAE/g fresh AS/min). Furthermore, it was calculated the DPPH● radical scavenging activity was equal to 100 mg/L expressed in L of 0.104 L/min for AP and 0.045 L/min for AS. The results of our study may give a promising solution to avocado processing companies for further utilization of their waste.

Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.

Conditional facility location problems with continuous demand and a polygonal barrier

We consider facility location problems where n facilities are present in a convex polygon in the rectilinear plane, over which continuous and uniform demand is distributed and within which a convex polygonal barrier is located (removing all demand and preventing all travel within the barrier), and the optimal location for an additional facility is sought. We start with an in-depth analysis of the representation of the bisectors of two facilities affected by the barrier and how it is affected by the position of the additional facility. Following this, a detailed investigation into the changes in the structure of the Voronoi diagram caused by the movement of this additional facility, which governs the form of the objective function for numerous facility location problems, yields a set of linear constraints for a general convex barrier that partitions the market space into a finite number of regions within which the exact solution can be found in polynomial time. This allows us to formulate a polynomial exact algorithm that makes use of a triangular decomposition of the incremental Voronoi diagram and the first order optimality conditions.

A mixed finite element based inverse approach for residual stress reconstruction

In this paper a full-field residual stress identification procedure is proposed and explored from partially measured residual elastic strain or stress within a PDE constrained optimization framework. At first, we have constructed a penalized least-square cost functional with an appropriate regularization term. We have taken data discrepancy term in energy norm while the regularization term is a complementary strain energy functional. We then construct an appropriate Lagrangian to impose the optimization constraint. It turns out that using first order optimality condition leads to standard two field mixed, of Hellinger-Reissner type, variational formulation with residual stress and Lagrangian variable as independent field variables. This allows us to use standard two and three dimensional mixed finite elements for approximation of the stress and Lagrangian variable. Using mixed finite element set-up, we then solved these equations to obtain full-field residual stress distribution. One key issue of the proposed formulation, as in case of any inverse problem, is the choice of penalization parameter. In this context, we have used modified L-curve based approach to choose optimal penalty parameter. Finally, we assess the numerical effectiveness of the proposed procedure against a few problems of engineering interest.

A distributed resistance inverse method for flow obstacle identification from internal velocity measurements

We present a penalization parameter method for obstacle identification in an incompressible fluid flow for a modified version of the Oseen equations. The proposed method consists in adding a high resistance potential to the system such that some subset of its boundary support represents the obstacle. This allows to work in a fixed domain and highly simplify the solution of the inverse problem via some suitable cost functional. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Oseen equation with respect to the potential. Finally, several numerical experiments using Navier–Stokes flow illustrate the applicability of the method, for the localization of a bi-dimensional cardiac valve from MRI and ultrasound flow type imaging data.

Second order variational analysis of disjunctive constraint sets and its applications to optimization problems

In this paper, we examine the properly twice epi-differentiability and compute the second order epi-subderivative of the indicator function to a class of sets including the finite union of parabolically derivable and parabolically regular sets. In this way, we provide no-gap second order optimality conditions for a disjunctive constrained problem. Moreover, we derive applications of our results to some types of disjunctive programs.

Mean-field optimal control for biological pattern formation

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.

Optimization of baker`s yeast production on grape juice using response surface methodology

There is an increasing interest in improving biological processes, including fermentation processes, improving fermentation conditions is difficult, as it requires the use of an appropriate improvement method that allows operating the biological fermenter under optimal conditions in order to obtain the largest possible amount of the final product. The aim of this work was to succeed in examples of fermentation conditions to produce the largest possible quantity of dry yeast biomass Saccharomyces cerevisiae using grape juice as the sole carbon source. The optimum values of five factors that have an effect on the production of dry biomass from baker`s yeast were determined. The design of the experiments was carried out using the central composite experimental design (CCD) and the number of experiments according to the design was (54) experiments, the response surface methodology method was used to determine the best possible amount of production of yeast, and has reached (41.44 g/L) after 12 hours of fermentation, under the following optimal conditions (temperature (30.11??), pH (4.75), sugar concentration (158.36 g/L), the ratio of carbon to nitrogen (an essential nutrient for yeast growth ) is (11.9), initial concentration of yeasts (2.5 g/L), the amount of urea was 6.65 g/L and the amount of ammonium sulfate used was 6.65 g/L, so that the concentration of added urea and ammonium sulfate was (50-50)% and the required C/N ratio was achieved, and the used agitation speed was equal to 200 r.p.m during the fermentation process. The fermenter power of the obtained yeast was 470 ml. Three kinematic models (Monod, Verhulst, and Tessier) were also selected for the purpose of studying the kinetic performance of Saccharomyces cerevisiae yeast. Monod and Tessier`s models did not give satisfactory results, while the best results were according to the Verhulst model. Also, the Leudeking Piret model has also been successfully used to predict substrate consumption during fermentation time.

Optimal control of the two-dimensional Vlasov-Maxwell system

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

An efficient numerical solution for time switching optimal control problems

In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.

First and second order optimality conditions for the control of Fokker-Planck equations

In this article we study an optimal control problem subject to the Fokker-Planck equation∂tρ−ν∆ρ− div(ρB[u]) = 0The control variableuis time-dependent and possibly multidimensional, and the functionBdepends on the space variable and the control. The cost functional is of tracking type and includes a quadratic regularization term on the control. For this problem, we prove existence of optimal controls and first order necessary conditions. Main emphasis is placed on second order necessary and sufficient conditions.

Second order optimality conditions for periodic optimal control problems governed by semilinear parabolic differential equations

In this paper, we study second-order optimality conditions for some optimal control problems governed by some semi-linear parabolic equations with periodic state constraint in time. We obtain a necessary condition and a sufficient condition in terms of the second order derivative of the associated Lagrangian. These two conditions correspond to the positive definite and the nonnegativity of the second order derivative of the Lagrangian on the same cone, respectively.

Bifurcation preserving discretisations of optimal control problems

The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.