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Computers & chemical engineering 157, 107610 (2022). doi:10.1016/j.compchemeng.2021.107610, Published by Elsevier Science, Amsterdam [u.a.]

21th IFAC Virtuel World Congress, IFAC, online, 11 Jul 2020 - 17 Jul 2020; IFAC-PapersOnLine 53(2), 5273-5278 (2020). doi:10.1016/j.ifacol.2020.12.1207 special issue: "21th IFAC World Congress / Edited by Rolf Findeisen, Sandra Hirche, Klaus Janschek, Martin Mönnigmann", Published by Elsevier, Frankfurt

Optimization of the energy consumption at fluctuating short-term electricity markets is a promising measure to increase the economic efficiency of energy-intense processes. This can be addressed by integrating the economic perspective directly into the process control, i.e., by using economic nonlinear model predictive control (eNMPC). We present a single-layer eNMPC framework for optimal operation of an industrial-scale nitrogen plant participating in real-time electricity markets. To achieve real-time capability, we utilize suboptimal updates as well as our reduced modeling approach for rectification columns combining compartmentalization and artificial neural networks (Schafer et al., AIChE J., doi:10.1002/aic.16568). We demonstrate the real-time capability of the approach in-silico. We explicitly account for model-plant mismatch by using a detailed full-order stage-by-stage model that is common in literature as plant replacement. Our results show that close-to-optimal savings in electricity costs are enabled via the eNMPC strategy even under consideration of inherently uncertain market forecasts whilst safely satisfying production targets. Furthermore, the disturbance rejection capability of the control structure is investigated, showing that severe unmeasured disturbances with slow dynamics can be rejected effectively without violating product requirements.

Global structural identifiability determines if it is possible to uniquely estimate unknown parameters of a model from measurements. We consider two definitions for global structural identifiability - one proposed by Walter and Lecourtier in Mathematics and Computers in Simulation, 60:472-482 (1982) and the other by Glad and Ljung in the Proceedings of the 29th Conference on Decision and Control (1990). The two definitions appear distinct because of the role of the inputs. We present here a proof of the equivalence of the two definitions. We revisit the formulation of the optimization problem to analyze global structural identifiability proposed by Asprey and Mantalaris in IFAC Computer Application in Biotechnology (2001) and propose a modification to the problem formulation. We also demonstrate with the help of an elementary model that their solution algorithm can lead to erroneous conclusions. We further analyze the global structural identifiability of a simple model using the optimization-based approach.

We present a multistage NMPC scheme with adaptive on-line scenario-tree generation. The scenario tree is assembled from predictions of worst-case uncertainty realizations that are identified based on a first-order approximation of the process model. The key property of the presented approach is that the size of the resultant optimal control problems does not scale directly with the number of uncertain model parameters. We demonstrate the applicability of the approach with an industrially relevant semi-batch polymerization process under parametric model uncertainty and noisy, incomplete state measurements. By allowing to account explicitly for estimation errors, the presented approach yields increased robustness when compared to nominal NMPC and a standard multistage NMPC scheme. Moreover, we investigate a combination of the presented approach with on-line estimation of uncertain model parameters alongside approximation of their confidence region to reduce the uncertainty range and consequently mitigate unnecessary conservatism. The results show that adaptation of model and uncertainty range yields considerable economic benefits without impairing the attained level of robustness for the considered process.

The performance of an organic Rankine cycle (ORC) relies on process design and operation. Simultaneous optimization of design and operation for a range of working fluids (WFs) is therefore a promising approach for WF selection. For this, deterministic global process optimization can guarantee to identify a global optimum, in contrast to local or stochastic global solution approaches. However, providing accurate thermodynamic models for a large number of WFs while maintaining computational tractability of the resulting optimization problems are open research questions. We integrate accurate thermodynamic and transport properties via artificial neural networks (ANNs) and solve the design problems with MAiNGO in a reduced-space formulation. We illustrate the approach for an ORC process for waste heat recovery of a diesel truck. After an automated preselection of 122 WFs, ANNs are automatically trained for the 37 selected WFs based on data retrieved from the thermodynamic library CoolProp. Then, we perform deterministic global optimization of design and operation for every WF individually. Therein, the trade-off between net power generation and investment cost is investigated by multiobjective optimization. Further, a thermoeconomic optimization finds a compromise between both objectives. The results show that, for the given conditions, monoaromatic hydrocarbons are a promising group of WFs. In future work, the proposed method and the trained ANNs can be applied to the design of a variety of energy processes.

Since design optimization faces the challenge of solving inherently large optimization problems, the complexity of underlying dynamic systems is often reduced by applying quasi-steady state assumptions. It is thereby indispensable to identify the components whose transient behavior is essential to ensure meaningful results. In this study, we discuss a dynamic model for an illustrative hybrid energy system, which extends the quasi-steady models of Voll et al. (2013) . Based on optimal operation with fixed design, we underline the importance of the relationship between the dynamics of the model and of the input data for the adequateness of quasi-steady operation. Our results emphasize the need for suitable ramp constraints in quasi-steady models of dynamic systems within operational optimization. Moreover, the existence of a storage unit is no sufficient justification for quasi-steady state assumptions.

Global deterministic process optimization problems have recently been solved efficiently in a reduced-space by automatic propagation of McCormick relaxations (Bongartz and Mitsos, J. Global Optim, 2017). However, the previous optimizations have been limited to simplified thermodynamic property models. Herein, we propose a method that learns accurate thermodynamic properties via artificial neural networks (ANNs) and integrates those in deterministic global process optimization. The resulting hybrid process model is solved using the recently developed method for deterministic global optimization problems with ANNs embedded (Schweidtmann and Mitsos, J. Optim. Theory Appl., 2018). The optimal operation of a validated steady state model of an organic Rankine cycle is solved as a case study. It is especially challenging as the thermodynamic properties are given by the implicit Helmholtz equation of state. The results show that modeling of thermodynamic properties via ANNs performs favorable in deterministic optimization. This method can rapidly be extended to include properties from existing thermodynamic libraries, based on models or data.

We present a multistage scheme for shrinking horizon nonlinear model predictive control (NMPC). This approach generates scenario trees on-line in an adaptive manner, and assembles them from predictions of worst-case uncertainty realizations through first-order approximations of the process model. A key result of this approach is that both size and complexity of the resultant optimal control problems do not scale directly with the number of uncertain model parameters. Moreover, we apply and demonstrate this approach on a challenging industrially-relevant semi-batch polymerization process under parametric model uncertainty. The results show that adaptive scenario generation leads to improved performance, while maintaining the attained level of robustness for the considered process.

Artificial neural networks are used in various applications for data-driven black-box modeling and subsequent optimization. Herein, we present an efficient method for deterministic global optimization of optimization problems with artificial neural networks embedded. The proposed method is based on relaxations of algorithms using McCormick relaxations in a reduced space (Mitsos et al. in SIAM J Optim 20(2):573–601, 2009) employing the convex and concave envelopes of the nonlinear activation function. The optimization problem is solved using our in-house deterministic global solver. The performance of the proposed method is shown in four optimization examples: an illustrative function, a fermentation process, a compressor plant and a chemical process. The results show that computational solution time is favorable compared to a state-of-the-art global general-purpose optimization solver.

The goal of this work is an efficient control scheme for the robust satisfaction of constraints for batch processes under parametric uncertainties. The multi-stage economic nonlinear model-predictive controller (eNMPC) [S. Lucia, et al. J. Process Control, 2013] is combined with two different methods for robust dynamic optimization. The multi-stage NMPC considers a scenario tree, branching at each sampling interval. In this work, the scenarios for the branches are generated with two previously investigated methods of detecting worst-case scenarios. One method is based on discretization of the uncertain parameter set (J. Puschke, et al. Comput. Chem. Eng., Vol. 98, 2017a). With this approach, all worst-case models are considered, including points not lying at the boundary of the uncertainty set. The other method is a heuristic approach (J. Puschke, et al. Comput. Chem. Eng., 2017b), which considers only values at the boundary with a high sensitivity. These methods for robust control are evaluated on the basis of an illustrative case study. The results show that, in contrast to the nominal eNMPC, less or even no constraint violations occur.

The optimal solution in dynamic optimization of batch processes often exhibits active path constraints. The goal of this work is the robust satisfaction of path constraints in the presence of parametric uncertainties based on known worst-case formulations. These formulations are interpreted as semi-infinite programs (SIP). Two known SIP algorithms are extended to the dynamic case and assessed. One is a discretization approach and the other a local reduction approach. With these presented concepts, robust path constraint satisfaction is in principle guaranteed. In this work, however, local methods are used to approximate the global solution of the lower-level problem with local solvers thus allowing for (rather unlikely) constraint violations. Finally, the penicillin fermentation is introduced as a well-known case study with uncertainties, which is modified in this work by adding further dependencies. The adaptation of the SIP concepts to dynamic optimization problems are shown to be successful for this case study.

Existing open-source modeling frameworks dedicated to energy systems optimization typically utilize (mixed-integer) linear programming ((MI)LP) formulations, which lack modeling freedom for technical system design and operation. We present COMANDO, an open-source Python package for component-oriented modeling and optimization for nonlinear design and operation of integrated energy systems. COMANDO allows to assemble system models from component models including nonlinear, dynamic and discrete characteristics. Based on a single system model, different deterministic and stochastic problem formulations can be obtained by varying objective function and underlying data, and by applying automatic or manual reformulations. The flexible open-source implementation allows for the integration of customized routines required to solve challenging problems, e.g., initialization, problem decomposition, or sequential solution strategies. We demonstrate features of COMANDO via case studies, including automated linearization, dynamic optimization, stochastic programming, and the use of nonlinear artificial neural networks as surrogate models in a reduced-space formulation for deterministic global optimization., Comment: 24 pages, 1 graphical abstract, 13 figures, 4 tables

Mathematical programming computation : MPC 13(3), 553-581 (2021). doi:10.1007/s12532-021-00204-y, Published by Springer, Berlin

Using nonlinear process models in discrete-time scheduling typically prohibits long planning horizons with fine temporal discretizations. Therefore, we propose an adaptive grid algorithm tailored for scheduling subject to time-variable electricity prices. The scheduling problem is formulated in a reduced space. In the algorithm, the number of degrees of freedom is reduced by linearly mapping one degree of freedom to multiple intervals with similar electricity prices. The mapping is iteratively refined using a wavelet-based analysis of the previous solution. We apply the algorithm to the scheduling of a compressed air energy storage. We model the efficiency characteristics of the turbo machinery using artificial neural networks. Using our in-house global solver MAiNGO, the algorithm identifies a feasible near-optimal solution with

Optimization and engineering (2020). doi:10.1007/s11081-020-09502-1, Published by Springer Science + Business Media B.V, Dordrecht [u.a.]

Typically, the consideration of nonlinear process models in discrete-time scheduling is limited to short planning horizons and/or coarse discretizations due to a linear scaling of the problem size with the number of considered scheduling intervals. To overcome this limitation, we recently proposed a wavelet-based algorithm focusing on scheduling problems with time-variable electricity prices, which iteratively adapts the time grid (Schaferet int., Mitsos, doi:10.1016/j.compchemeng.2019.106598). In this work, we extend our approach by presenting a systematic method for the identification of promising initial aggregated time grids based on the analysis of the wavelet representation of the time series of electricity prices. We apply the procedure to a literature example addressing the scheduling of a seawater reverse osmosis (Ghobeity and Mitsos, doi: 10.1016/j.desal.2010.06.041). We demonstrate that substantial reductions in the number of optimization variables in a reduced-space formulation are possible, while furnishing feasible schedules that lead to insignificant deviations below0. 05 % in the objective value compared to the global optimum using the full time grid.

Fluctuating electricity prices create an incentive for the flexible operation of electricity intensive processes, such as air separation units (ASUs). Shutting down an ASU during times with peak electricity prices has been claimed economically attractive but requires an efficient and largely automated start-up procedure. Previous works have considered simulations of plant start-ups and dynamic optimization of load scheduling near the nominal operation mode. Discrete events like the appearance of a liquid phase have impeded any rigorous ASU start-up optimization. In this work, we formulate the optimal start-ups as dynamic optimization problems with regularized algebraic complementarity constraints (Caspari et al., 2019b) using a mechanistic dynamic process model in Modelica. Our approach captures physical effects appearing during start-up like the appearance and disappearance of phases. We solve the resulting optimization problems with direct single-shooting using the dynamic optimization framework DyOS. We perform in-silico dynamic offline optimizations of an ASU start-up and consider different process modifications. We consider cold start-up optimizations, where the process medium is initialized at cryogenic conditions just before liquefaction. The results illustrate that the proposed approach can be applied to large-scale processes. The results show further that liquid assist operation reduced the optimal start-up time by about 70 % compared to the start-up without this modification.

The integration of renewables into the portfolio of energy sources implies that dynamic operation of energy intensive processes, such as air separation, may give an economic advantage. Dynamic process operation can be achieved by applying economic nonlinear model predictive control (eNMPC). In this work, we present an in-silico case study of dynamic operation of an air separation process under fluctuating electricity prices. Using a day-ahead electricity price profile, an offline dynamic optimization (DO) problem is solved, which is used as an initial guess to start a fast update method deployed by the eNMPC. The eNMPC uses the same objective and constraints as the offline DO. However, the prediction horizon is shorter and current states and disturbances are taken into account. A first-principle air separation process model implemented in Modelica is used in all optimization problems. All optimization problems are solved using the DO framework DyOS. The process is flexibly operated by the application of the eNMPC, which leads to near-optimal economic process behavior during operation. This work demonstrates the contribution of model based process control to the integration of renewable energy sources in the supply chain of the process industry.

Optimal processes often exhibit active path constraints. Parametric uncertainties in the process model might thus lead to constraint violations. A heuristic approach is presented to overcome this challenge. The nominal model is optimized with additional path constraints due to worst-case models. A heuristic method of choosing these models is proposed based on sensitivities of the constraints with respect to the uncertain parameters. The presented approximation does not guarantee robust feasibility, but path constraint violations are less likely to occur compared to the optimization using the nominal model solely. Two case studies are presented: a complex emulsion copolymerization process (DAE with 139 equations) and the penicillin formation (four differential equations and two algebraic equations). The results of both case studies show that, in contrast to the optimization in the nominal case, the multi-model approach does not violate the path constraints for different scenarios of the parametric uncertainty set.

A discretization-based algorithm for the global solution of semi-infinite programs (SIPs) is proposed, which is guaranteed to converge to a feasible, $$\varepsilon $$ź-optimal solution finitely under mild assumptions. The algorithm is based on the hybridization of two existing algorithms. The first algorithm (Mitsos in Optimization 60(10---11):1291---1308, 2011) is based on a restriction of the right-hand side of the constraints of a discretized SIP. The second algorithm (Tsoukalas and Rustem in Optim Lett 5(4):705---716, 2011) employs a discretized oracle problem and a binary search in the objective space. Hybridization of the approaches yields an algorithm, which leverages the strong convergence guarantees and the relatively tight upper bounding problem of the first approach while employing an oracle problem adapted from the second approach to generate cheap lower bounds and adaptive updates to the restriction of the first approach. These adaptive updates help in avoiding a dense population of the discretization. The hybrid algorithm is shown to be superior to its predecessors both theoretically and computationally. A proof of finite convergence is provided under weaker assumptions than the assumptions in the references. Numerical results from established SIP test cases are presented.

AIChE journal (2019). doi:10.1002/aic.16788, Published by Wiley, Hoboken, NJ

Monoclonal antibodies (mAbs) represent a significant class of biopharmaceutics with a wide range of diagnostic and therapeutic applications. Typically, mAbs are produced by cultivated mammalian cells, to meet the high quality product specifications. For new products and cell lines, an adaptation of cultivation conditions is required. This is usually performed experimentally. Model-based approaches can be a powerful tool to reduce experimental efforts and accelerate process development. We present optimizations using the process model in Kappatou et al. (2018), which is based on Quiroga et al. (2016). In particular, we perform fed-batch optimizations following a sequential and a simultaneous approach. In the sequential approach, we first find optimal initial conditions for the batch process using different objectives, and then we optimize the feeding for constant initial conditions. In the simultaneous approach, we directly optimize for initial conditions and appropriate feed rates. The optimizations lead to significant improvements compared to the base case presented in Quiroga et al. (2016). The results indicate that the sequential approach is sometimes able to outperform the simultaneous one by overcoming limitations of the local optimization used. This may be due to the flexibility of the sequential approach to use different objectives for the two steps (batch and fed-batch). Therefore, the results further highlight the importance of utilizing good initialization procedures in local optimization.

Dynamic optimization problems arise in many fields of engineering. Typically, they are subject to models of differential-algebraic equations and further process constraints. To promote and investigate the application of methods based on dynamic optimization, an efficient and modular implementation of numerical algorithms for their solution is essential. We present the current status of the open-source software DyOS for the solution of large-scale dynamic optimization problems. DyOS has been applied to optimal operation, model-predictive control and process design problems in various case studies. DyOS is based on direct adaptive shooting algorithms and it allows for multi-stage problem formulations including binary decision making. Models can either be imported as standardized functional mock-up units, flat Modelica models, or C++ models. The modular implementation of DyOS enables the use of various open-source and commercial integrators and nonlinear programming solvers based on various numerical methods. DyOS can be accessed via Matlab or Python interfaces. As an illustrative large-scale application, we present the results of optimal operation of an air separation process under fluctuating electricity prices. An open-source version of DyOS including several parts of the framework presented is available at http://permalink.avt.rwth-aachen.de/?id=295232 .

We recently demonstrated the potential of deterministic global optimization in a reduced-space formulation for flowsheet optimization. However, the consideration of implicit unit operations such as flash calculations is still challenging and the solution of complex flowsheets incorporating such operations can be intractable. We show that the solution of flash equations can be integrated in global optimization via artificial neural networks (ANNs). Thus, flash calculations are no longer performed within the flowsheet optimization. Instead, flash equations are solved offline and then learned using ANNs. ANNs have been used successfully in the literature to learn flash equilibria but have not yet been included in deterministic global optimization for this task. We embed the ANNs in a hybrid model and use deterministic global optimization to solve it. In addition, we utilize deterministic global optimization to calculate a guaranteed worst-case accuracy of ANNs compared to a rigorous model. We demonstrate the proposed approach on an illustrative five-component vapor-liquid equilibrium flash using our in-house solver MAiNGO.

Deterministic global optimization of process flowsheets has so far mostly been limited to simplified thermodynamic models. Herein, we demonstrate a way to integrate accurate thermodynamic models for the optimal process design of an organic Rankine cycle (ORC) via the use of artificial neural networks (ANNs). We generate training data using the thermodynamic library Coolprop and learn thermodynamic properties. The ANNs are subsequently embedded in the process design optimization problem in a reduced-space, which is solved to global optimality using our in-house optimization software MAiNGO. The importance of accurate thermodynamics is illustrated for the design of an ORC for geothermal power generation. We show that the use of an accurate thermodynamic model leads to different design decisions in comparison to a simplified model. Furthermore, we investigate the influence of the network architecture and complexity on accuracy, optimization results and computational performance.

Numerical optimization is very useful for design and operation of energy processes. As the design has a major impact on the economics of the system, it is desirable to find a global optimum in the presence of local optima. So far, deterministic global optimization with detailed thermodynamic models incorporated has been limited to single fluid energy systems. We extend our previously presented approach [Schweidtmann et int Mitsos, COMPUT CHEM ENG (2019)] from single-species working fluids to binary mixtures with variable composition. First, we create accurate thermodynamic data for two selected binary mixtures using a thermodynamic library. Using this data, we train artificial neural networks, select them based on desired accuracy, and include them in the process model. The resulting hybrid model is then optimized with the open-source solver MAiNGO. We perform thermodynamic optimizations of geothermal power plants, considering both organic Rankine and Kalina cycle. For each cycle, we identify the globally optimal design, operation, and working fluid composition for the selected binary fluid mixtures within tractable CPU times. We show how a second mixture component enables improved ORC performance.

Organic Rankine cycles (ORCs) offer a high structural design flexibility. The best process structure can be identified via the optimization of a superstructure, which considers design alternatives simultaneously. In this contribution, we apply deterministic global optimization to a geothermal ORC superstructure, thus guaranteeing to find the best solution. We implement a hybrid mechanistic data-driven model, employing artificial neural networks as thermodynamic surrogate models. This approach is beneficial as we optimize the problem in a reduced space using the optimization solver MAiNGO. We further introduce redundant constraints that are only considered for the lower-bounding problem of the branch-and-bound algorithm. We perform two separate optimizations, one maximizing power output and one minimizing levelized cost of electricity. The optimal solutions of both objectives differ from each other, but both have three pressure levels. Global optimization is necessary as there exist suboptimal local solutions for both flowsheet configuration and design with fixed configurations.

Optimized exothermic semi-batch emulsion polymerization typically exhibits active arcs of the constraints for the reactor temperature. Given parametric uncertainties in the process model, this might lead to constraint violations which are a safety concern. Therefore, an approach is investigated, such that constraint violations are less likely and is hence robust feasible. The two-model approach is presented as an approximate solution. Therein, two models, the nominal and a worst-case model are optimized simultaneously. Because of the challenges in defining the worst-case, a heuristic method is presented to define the worst-case parameter and its parameter value. The results of the process optimization with the two-model approach are compared with the results of the optimization with the nominal model solely. In addition the feasibility of both optimization strategies is compared by simulating hundred different scenarios with random parameter values from the uncertainty set. The presented approximation does not guarantee robust feasibility, but path constraint violations are less likely due to the introduced conservatism compared to the original optimization.

The aim of the present paper is to provide (n-l)-reliability to a power grid, guaranteeing nominal operation after the failure of any one out of n present grid components. Building on previous work (Fliscounakis et al., IEEE Transactions on Power Systems, 2013), a hierarchical programming problem is proposed to characterize the worst-case behavior of a power grid under a given contingency. The formulation is a mixed-integer linear generalized semi-infinite program with a max-min program embedded. The different levels correspond to the choice of preventive actions, realization of uncertainties in the power supply and demand, and the choice of corrective actions. In order to model active components of the grid, models are proposed for load balancing and the behavior of phase-shifting transformers. Since no rigorous solution approaches are published for the problem at hand, the possibility of extending generalized semi-infinite programming approaches to the present problem is discussed.

Journal of global optimization xx, xx (2018). doi:10.1007/s10898-018-0611-8, Published by Springer Science + Business Media B.V, Dordrecht [u.a.]

4th International Seminar on ORC Power Systems September 13-15th 2017, Politecnico Di Milano Bovisa Campus Milano, Italy / Edited by Vincenzo Dossena, Alberto Guardone and Marco Astolfi 4th International Seminar on ORC Power Systems, ORC 2017, Milano, Italy, 13 Sep 2017 - 15 Sep 2017 ; Amsterdam [u.a.] : Elsevier, Energy procedia, 129, 50-57(2017). doi:10.1016/j.egypro.2017.09.181, Published by Elsevier, Amsterdam [u.a.]

Journal of global optimization 69(4), 761-796 (2017). doi:10.1007/s10898-017-0547-4, Published by Springer Science + Business Media B.V, Dordrecht [u.a.]

Deterministic global methods for flowsheet optimization have almost exclusively relied on equation-oriented formulations. The automatic propagation of McCormick relaxations and subgradients enables an alternative formulation similar to sequential modular infeasible path methods in local optimization that operate in a reduced space while moving most model variables and equations to external functions. The application of this reduced-space formulation is demonstrated for the Williams-Otto process. For suitable choices of tear streams and additional variables and equations left to the optimizer, it enables significant reductions in computational time compared to equation-oriented formulations.

The algorithm proposed in Mitsos (Optimization 60(10---11):1291---1308, 2011) for the global optimization of semi-infinite programs is extended to the global optimization of generalized semi-infinite programs. No convexity or concavity assumptions are made. The algorithm employs convergent lower and upper bounds which are based on regular (in general nonconvex) nonlinear programs (NLP) solved by a (black-box) deterministic global NLP solver. The lower bounding procedure is based on a discretization of the lower-level host set; the set is populated with Slater points of the lower-level program that result in constraint violations of prior upper-level points visited by the lower bounding procedure. The purpose of the lower bounding procedure is only to generate a certificate of optimality; in trivial cases it can also generate a global solution point. The upper bounding procedure generates candidate optimal points; it is based on an approximation of the feasible set using a discrete restriction of the lower-level feasible set and a restriction of the right-hand side constraints (both lower and upper level). Under relatively mild assumptions, the algorithm is shown to converge finitely to a truly feasible point which is approximately optimal as established from the lower bound. Test cases from the literature are solved and the algorithm is shown to be computationally efficient.

Unfortunately, the copyright information of the article in Springerlink appears as “© Springer Science+Business Media, LLC, part of Springer Nature 2018” this has been corrected to “© The Author(s) 2018.”

Models used in process systems engineering often contain nonconvex and nonlinear functions. In this work we consider the logarithmic mean temperature difference used in heat exchanger networks. Building on the work of Mistry and Misener (2015), we present an approach to construct tight convex and concave relaxations of this function when the temperatures are not degrees of freedom but rather calculated as a function of the optimization variables. We make use of the multivariate McCormick theorem presented by Tsoukalas and Mitsos (2014) and support this approach by a simple case study where we additionally conclude results on the convergence orders of the obtained relaxations. Finally we also briefly compare with other relaxation techniques including interval extensions and α BB relaxations.

Rigorous satisfaction of path constraints in dynamic optimization is important since they often reflect the safety and quality limits of a process. A difficulty that arises is that they have to be fulfilled during the whole time horizon. Most existing methods discretize these infinite constraints and cannot guarantee the satisfaction at all times. Fu et al. (2015) proposed an algorithm that finitely returns an approximate KKT-optimal point that satisfies the path constraints rigorously. Only a finite number of interior-point constraints is needed due to an adaptive restriction of the right-hand side of the path constraints. However, this algorithm may require many iterations, as only one interior-point constraint is added per iteration. In this work, it is shown that adding more constraints at each iteration leads to improvements both in CPU time and number of iterations. One considered method is the inclusion of all local violation maxima time points. Another idea is to detect the segments, where the constraints are violated and include the middle and end-points of these segments. The algorithm is extended to treat differential-algebraic equation systems of index 1. The Williams-Otto semi-batch reactor is used as a numerical case study to demonstrate the effectiveness of the faster population methods.

In this article, a new model and a biomimetic pattern for heliostat field layout optimization are introduced. The model, described and validated herein, includes a detailed calculation of the annual average optical efficiency accounting for cosine losses, shading and blocking, aberration and atmospheric attenuation. The model is based on a discretization of the heliostats and can be viewed as ray tracing with a carefully selected distribution of rays. The prototype implementation is sufficiently fast to allow for field optimization. Parameters are introduced for the radially staggered layout and are optimized with the objective of maximizing the annual insolation weighted heliostat field efficiency. In addition, inspired by the spirals of the phyllotaxis disc pattern, a new biomimetic placement heuristic is described and evaluated, which generates layouts of both higher insolation-weighted efficiency and higher ground coverage than radially staggered designs. Specifically, this new heuristic is shown to improve the existing PS10 field by 0.36% points in efficiency while simultaneously reducing the land area by 15.8%. Moreover, the new pattern achieves a better trade-off between land area usage and efficiency, i.e., it can reduce the area requirement significantly for any desired efficiency. Finally, the improvement in area becomes more pronounced with an increased number of heliostats, when maximal efficiency is the objective. While minimizing the levelized cost of energy (LCOE) is typically a more practical objective, results of the case study presented show that it is possible to both reduce the land area (i.e. footprint) of the plant and number of heliostats for fixed energy collected. By reducing the capital cost of the plant at no additional costs, the effect is a reduction in LCOE.

An algorithm is proposed for locating a feasible point satisfying the KKT conditions to a specified tolerance of feasible inequality-path-constrained dynamic programs (PCDP) within a finite number of iterations. The algorithm is based on iteratively approximating the PCDP by restricting the right-hand side of the path constraints and enforcing the path constraints at finitely many time points. The main contribution of this article is an adaptation of the semi-infinite program (SIP) algorithm proposed in Mitsos (2011) to PCDP. It is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first-order KKT conditions of the PCDP to a specified tolerance. The main assumptions are: (i) availability of a nonlinear program (NLP) local solver that generates a KKT point of the constructed approximation to PCDP at each iteration if this problem is indeed feasible; (ii) existence of a Slater point of the PCDP that also satisfies the first-order KKT conditions of the PCDP to a specified tolerance; (iii) all KKT multipliers are nonnegative and uniformly bounded with respect to all iterations. The performance of the algorithm is analyzed through two numerical case studies.

An algorithm for the global optimization of nonlinear bilevel mixed-integer programs is presented, based on a recent proposal for continuous bilevel programs by Mitsos et al. (J Glob Optim 42(4):475---513, 2008). The algorithm relies on a convergent lower bound and an optional upper bound. No branching is required or performed. The lower bound is obtained by solving a mixed-integer nonlinear program, containing the constraints of the lower-level and upper-level programs; its convergence is achieved by also including a parametric upper bound to the optimal solution function of the lower-level program. This lower-level parametric upper bound is based on Slater-points of the lower-level program and subsets of the upper-level host sets for which this point remains lower-level feasible. Under suitable assumptions the KKT necessary conditions of the lower-level program can be used to tighten the lower bounding problem. The optional upper bound to the optimal solution of the bilevel program is obtained by solving an augmented upper-level problem for fixed upper-level variables. A convergence proof is given along with illustrative examples. An implementation is described and applied to a test set comprising original and literature problems. The main complication relative to the continuous case is the construction of the parametric upper bound to the lower-level optimal objective value, in particular due to the presence of upper-level integer variables. This challenge is resolved by performing interval analysis over the convex hull of the upper-level integer variables.

Two algorithms for the general case of parametric mixed-integer linear programs (MILPs) are proposed. Parametric MILPs are considered in which a single parameter can simultaneously influence the objective function, the right-hand side and the matrix. The first algorithm is based on branch-and-bound on the integer variables, solving a parametric linear program (LP) at each node. The second algorithm is based on the optimality range of a qualitatively invariant solution, decomposing the parametric optimization problem into a series of regular MILPs, parametric LPs and regular mixed-integer nonlinear programs (MINLPs). The number of subproblems required for a particular instance is equal to the number of critical regions. For the parametric LPs an improvement of the well-known rational simplex algorithm is presented, that requires less consecutive operations on rational functions. Also, an alternative based on predictor–corrector continuation is proposed. Numerical results for a test set are discussed.

The global solution of bilevel dynamic optimization problems is discussed. An overview of a deterministic algorithm for bilevel programs with nonconvex functions participating is given, followed by a summary of deterministic algorithms for the global solution of optimization problems with nonlinear ordinary differential equations embedded. Improved formulations for scenario-integrated optimization are proposed as bilevel dynamic optimization problems. Solution procedures for some of the problems are given, while for others open challenges are discussed. Illustrative examples are given.

Finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi-infinite programs (SIPs) involving smooth, potentially nonconvex objective function and constraints. The lower bounding problem is obtained by a formulation that combines the first- and second-order KKT necessary conditions of the lower-level problem with a discretization of the parameter set. The resulting mathematical program with equilibrium constraints (MPEC) is a relaxation of the original SIP and furnishes valid lower bounds. If the parameter set is subdivided, the optimal solution value of the lower bounding problem converges to the optimal solution value of the SIP. The upper bounding problem is based on convex and linear relaxations of the lower-level problem resulting in a restriction of the SIP. If the parameter set is subdivided, the constructed relaxations converge to the original lower-level program. The existence of SIP Slater points ensures convergence of the upper bounding problems to the optimal solution value of the SIP. Several alternatives for the upper bounding problem are presented and discussed. Numerical results are presented for a number of test problems from the literature.

A bounding algorithm for the global solution of nonlinear bilevel programs involving nonconvex functions in both the inner and outer programs is presented. The algorithm is rigorous and terminates finitely to a point that satisfies ?-optimality in the inner and outer programs. For the lower bounding problem, a relaxed program, containing the constraints of the inner and outer programs augmented by a parametric upper bound to the parametric optimal solution function of the inner program, is solved to global optimality. The optional upper bounding problem is based on probing the solution obtained by the lower bounding procedure. For the case that the inner program satisfies a constraint qualification, an algorithmic heuristic for tighter lower bounds is presented based on the KKT necessary conditions of the inner program. The algorithm is extended to include branching, which is not required for convergence but has potential advantages. Two branching heuristics are described and analyzed. Convergence proofs are provided and numerical results for original test problems and for literature examples are presented.