Dynamic Optimization and Optimal Control of Hydrogen Blending Operations in Natural Gas Networks

We present a dynamic model for the optimal control problem (OCP) of hydrogen blending into natural gas pipeline networks subject to inequality constraints. The dynamic model is derived using the first principles partial differential equations (PDEs) for the transport of heterogeneous gas mixtures through long distance pipes. Hydrogen concentration is tracked together with the pressure and mass flow dynamics within the pipelines, as well as mixing and compatibility conditions at nodes, actuation by compressors, and injection of hydrogen or natural gas into the system or withdrawal of the mixture from the network. We implement a lumped parameter approximation to reduce the full PDE model to a differential algebraic equation (DAE) system that can be easily discretized and solved using nonlinear optimization or programming (NLP) solvers. We examine a temporal discretization that is advantageous for time-periodic boundary conditions, parameters, and inequality constraint bound values. The method is applied to solve case studies for a single pipe and a multi-pipe network with time-varying parameters in order to explore how mixing of heterogeneous gases affects pipeline transient optimization

Transitions from Monotonicity to Chaos in Gas Mixture Dynamics in Pipeline Networks

The blending of hydrogen generated using clean energy into natural gas pipeline networks is proposed in order to utilize existing energy systems for their planned lifetime while reducing their reliance on fossil fuels. We formulate a system of partial differential equations (PDEs) that govern the flow dynamics of mixtures of gases in pipeline networks under the influence of time-varying compressor and regulator control actions. The formulation is derived for general gas networks that can inject or withdraw arbitrary time-varying mixtures of gases into or from the network at arbitrarily specified nodes. The PDE formulation is discretized in space to form a nonlinear control system which is used to prove that homogeneous mixtures are well-behaved and heterogeneous mixtures may be ill-behaved in the sense of monotone-ordering of solutions. We use numerical simulations to compute interfaces that delimit periodic and monotone system responses and show that any solution in the monotonic operating region eventually approaches a periodic orbit. Our results are demonstrated for examples of a single pipeline and a small test network

A trust region framework for heat exchanger network synthesis with detailed individual heat exchanger designs

•Detailed heat exchanger models are embedded within a heat exchanger network synthesis optimization.•A trust region filter algorithm is used as a surrogate modelling strategy for HENS for the first time to incorporate pressure drops, numbers of shells, etc.•An integer-cut strategy algorithm is introduced to explore different topologies.•Results are compared to existing methods for solving HENS with detailed models, demonstrating excellent performance.We develop a trust region filter strategy for simultaneous optimal design of heat exchanger networks that includes detailed design of shell-and-tube heat exchangers. The strategy first solves a mixed-integer nonlinear programming (MINLP) formulation with shortcut models to generate candidate network topologies, which are then used in a non-isothermal mixing nonlinear programming (NLP) suboptimization with detailed optimal exchanger design models embedded using a modified trust region filter (TRF) algorithm. An integer cut based strategy is used to bound the solutions from MINLP and the NLP which aids in convergence to the solution of the overall simultaneous design problem. Under assumptions, the TRF based strategy can guarantee convergence to near optimal solutions of the overall design problem. The presented solution strategy is thus able to find optimal heat exchanger network designs based on the simultaneous optimization of the network topology and mass and energy balances, together with detailed shell-and-tube heat exchanger optimization, including the number of shell and tube passes, pressure drops, and tubes, tube lengths, etc. The proposed strategy is tested on three literature based case studies and their results are compared with previous studies to showcase its performance

Heat exchanger network synthesis with detailed exchanger designs: Part 1. A discretized differential algebraic equation model for shell and tube heat exchanger design

A new method for the detailed design of shell and tube heat exchangers is presented through the formulation of coupled differential heat equations, along with algebraic equations for design variables. Heat exchanger design components (tube passes, baffles, and shells) are used to discretize the differential equations and are solved simultaneously with the algebraic design equations. The coupled differential algebraic equation (DAE) system is suitable for numerical optimization as it replaces the nonsmooth log mean temperature difference (LMTD) term. Discrete decisions regarding the number of shells, fluid allocation, tube sizes, and number of baffles are determined by solving an LMTD‐based method iteratively. The resulting heat exchanger topology is then used to discretize the detailed DAE model, which is solved as a nonlinear programming model to obtain the detailed exchanger design by minimizing an economic objective function through varying the tube length. The DAE model also provides the stream temperature profiles inside the exchanger simultaneously with the detailed design. It is observed that the DAE model results are almost equal to the LMTD‐based design model for one‐shell heat exchangers with constant stream properties but shows significant differences when streams properties are allowed to vary with temperature or the number of shells are increased. The accuracy of the solutions and the required computational costs show that the model is well suited for solving heat exchanger network synthesis problems combined with detailed exchanger designs, which is demonstrated in Part 2 of the paper

Synthesis of Combined Heat and Mass Exchange Networks Via a Trust Region Filter Optimisation Algorithm Including Detailed Unit Designs

Mass and heat integration are important to achieving economically and environmentally sustainable processes through increased efficiency. Typically, heat and mass exchange networks are solved separately using process integration techniques such as pinch technology or formulating nonconvex mixed-integer nonlinear programming (MINLP) problems, which are challenging to solve. To simplify the MINLP, shortcut models are employed, which can result in under/overestimation of the real network, leading to suboptimal or infeasible designs. We introduce a new optimisation algorithm for combined heat and mass exchanger network synthesis (CHAMENS), including detailed design models. The method uses shortcut models in an MINLP to find network topology, followed by a nonlinear programming (NLP) suboptimisation. The NLP allows non-isothermal and non-isocompositional mixing, uses detailed unit models of packed columns based on orthogonal collocation on finite elements (OCFE), and detailed shell and tube heat exchanger designs. We incorporate a differential-algebraic equation (DAE) based shell and tube heat exchanger design model via surrogates in a trust region filter (TRF) framework, guaranteeing optimal solutions for the detailed exchanger models are found by the surrogate models. We demonstrate the proposed approach on a case study, showcasing its performance and the need to incorporate detailed unit models in topology optimisation to find practical optimal designs

Heat exchanger network synthesis with detailed exchanger designs—2. Hybrid optimization strategy for synthesis of heat exchanger networks

We propose a new strategy to synthesize heat exchanger networks with detailed designs of individual heat exchangers. The proposed strategy uses a multistep approach by first obtaining a heat exchanger network topology through solving a modified version of the mixed integer nonlinear programming (MINLP) stage‐wise superstructure of Yee and Grossmann, which includes a smoothed LMTD approximation and pressure drops. In a second nonlinear programming (NLP) suboptimization step, we allow for nonisothermal mixing to solve problems with or without exchanger bypasses. The selected heat exchangers along with the mass and energy balances obtained are then used to design the network with detailed exchanger designs through solving a sequence of NLPs for individual heat exchanger designs. The NLPs are based on the detailed discretized optimization models of Kazi et al., which solve quickly and reliably to obtain heat exchangers based on rigorous, first‐principles derived coupled differential equations. These models solve a differential algebraic equation system and do not rely on usual assumptions associated with other heuristic‐based exchanger design methods, such as log mean temperature difference and FT correction factors. These detailed exchanger designs are then used to update the network optimization model through sets of correction factors on heat exchanger area, number of shells, heat transfer coefficients, and pressure drops of each exchanger design, in a method based on that of Short et al. The method solves reliably, guaranteeing feasible exchangers for every potential network generated by the shortcut models, through validation with rigorous heat exchanger models at every iteration. In addition, the method does not increase the nonlinearity of the MINLP model, nor does it require any manual intervention or initialization from the user. Three examples are solved and the results are compared to those obtained in the literature

An Optimisation Algorithm for Detailed Shell-and-Tube Heat Exchanger Designs for Multi-Period Operation

Heat exchangers (HEs) are crucial processing units in industrial plants. Heat exchanger networks (HENs) are often designed for nominal operation. However, processes are becoming increasingly dynamic and should be able to operate over a range of operational periods to address changes in market, seasonality, and start-up and shutdown. HEN synthesis has been widely studied, however most approaches use simplified HE models for optimisation and analysis of the structures, assuming the largest area across all periods of operation results in a feasible HE. The detailed HE design, which includes many more practical constraints, such as variable heat transfer coefficients that are a function of velocity, may result in certain exchangers being infeasible in some operational periods. In this study, an HE design algorithm is proposed, which finds optimal shell-and-tube HEs that are feasible across any number of operational periods, which may involve different duties and fluid properties. This is the first such design algorithm presented in literature. The algorithm works via a smart enumeration algorithm, which solves a nonlinear programming (NLP) optimisation subproblem for each combination of discrete decisions (number of baffles, stream allocation, tube diameters, etc.). Each NLP solves Bell Delaware design equations across all considered periods and allows stream splitting and HE bypassing to find an optimal multi-period HE. If no feasible HE is found, a permutation algorithm is used to find the optimal combination of HEs that can fulfil the required heat duties. The algorithm is demonstrated on two examples, showcasing its performance. Future work is suggested to increase the algorithm’s computational efficiency and to include it in multi-period HEN synthesis