Nonlinear Programming Solver Research Articles

Model predictive automatic control of sucker rod pump system with simulation case study

This work enables accelerated fluid recovery in oil and gas reservoirs by automatically controlling fluid height and bottomhole pressure in wells. Several literature studies show significant increase in recovered oil by determining a target bottomhole pressure but rarely consider how to control to that value. This work enables those benefits by maintaining bottomhole pressure or fluid height. Moving Horizon Estimation (MHE) determines uncertain well parameters using only common surface measurements. A Model Predictive Controller (MPC) adjusts the stroking speed of a sucker rod pump to maintain fluid height. Pump boundary conditions are simulated with Mathematical Programs with Complementarity Constraints (MPCCs) and a nonlinear programming solver finds a solution in near real-time. A combined rod string, well, and reservoir model simulate dynamic well conditions, and are formulated for simultaneous optimization by large-scale solvers. MPC increases cumulative oil production vs. conventional pump off control by maintaining an optimal fluid level height.

On the Construction of Linear Approximations of Line Flow Constraints for AC Optimal Power Flow

The AC optimal power flow problem is a non-convex optimization problem that is difficult to solve quickly and reliably for large-scale grids. While various approximations have been proposed, they may lead to physically meaningless solutions. This paper presents a computationally efficient algorithm for constructing accurate linear approximations of line flow constraints. These approximations reduce the complexity of the optimization problem while ensuring that the solution is physically meaningful and has a high quality. The algorithm is based on an in-depth analysis of the feasible set of the line flow constraint. Numerical experiments are performed on ten large-scale systems using three nonlinear programming solvers. Obtained results indicate that the proposed formulation helps improve the solvers’ reliability and reduce the computation time for hard large-scale problems. At the same time, the developed algorithm provides high quality approximations of the line flow constraints.

Fast nonlinear model predictive control of a chemical reactor: a random shooting approach

Abstract This paper presents a fast way of implementing nonlinear model predictive control (NMPC) using the random shooting approach. Instead of calculating the optimal control sequence by solving the NMPC problem as a nonlinear programming (NLP) problem, which is time consuming, a sub-optimal, but feasible, sequence of control inputs is determined randomly. To minimize the induced sub-optimality, numerous random control sequences are selected and the one that yields the smallest cost is selected. By means of a motivating case study we demonstrate that the random shooting-based approach is superior, from a computational point of view, to state-of-the-art NLP solvers, and features a low level of sub-optimality. The case study involves a continuous stirred tank reactor where a fast multi-component chemical reaction takes place.

Rapid Solution of Optimal Control Problems by a Functional Spreadsheet Paradigm: A Practical Method for the Non-Programmer

We devise a practical and systematic spreadsheet solution paradigm for general optimal control problems. The paradigm is based on an adaptation of a partial-parametrization direct solution method which preserves the original mathematical optimization statement, but transforms it into a simplified nonlinear programming problem (NLP) suitable for Excel NLP solver. A rapid solution strategy is implemented by a tiered arrangement of pure elementary calculus functions in conjunction with Excel NLP solver. With the aid of the calculus functions, a cost index and constraints are represented by equivalent formulas that fully encapsulate an underlining parametrized dynamical system. Excel NLP solver is then employed to minimize (or maximize) the cost index formula, by varying decision parameters, subject to the constraints formulas. The paradigm is demonstrated for several fixed and free-time nonlinear optimal control problems involving integral and implicit dynamic constraints with direct comparison to published results obtained by fundamentally different methods. Practically, applying the paradigm involves no more than defining a few formulas using basic Excel spreadsheet skills.

Reduced Integration Optimization Model for Coupled Elevated-Pressure Air Separation Unit and Gas Turbine in Oxy-combustion and Gasification Power Plant

For the promising and green oxy-combustion and gasification power plant, the most efficient and assessable approach for further improvements on the current situation with less investment turned to be process optimization and integration. In this work, the gap of systematic analysis of air separation unit (ASU) and gas turbine (GT) integrations in integrated gasification combined cycle (IGCC) power plant has been fulfilled in the elevated-pressure ASU operation conditions beyond discrete case studies. Based on the conventional rigorous mathematical simulation, a series of model reductions have been proposed and applied to increase the computational flexibility. To validate the reduced model, a base case is built and settled as the benchmark, which is consistent with the industrial experiences. Afterwards, based on the reduced model, the effects of air integration on the thermal performance of IGCC power plant have been explored under the conditions of various nitrogen injection levels. The individual influences of nitrogen injection level on IGCC plant efficiency have been explored as well. To achieve best IGCC performance, the optimization of coupled air integration and nitrogen injection as a whole is completed. Based on the proposed reduced model, the three dimensional figure about systematics analysis of integration optimization for IGCC power plant is generated for the first time. Based on its two-dimensional top view, the feasible regions are identified and optimal solution is generated through nonlinear programming problem solver based on enhanced generalized reduced gradient method.

Rapid Modeling and Parameter Estimation of Partial Differential Algebraic Equations by a Functional Spreadsheet Paradigm

We present a systematic spreadsheet method for modeling and optimizing general partial differential algebraic equations (PDAE). The method exploits a pure spreadsheet PDAE solver function design that encapsulates the Method of Lines and permits seamless integration with an Excel spreadsheet nonlinear programming solver. Two alternative least-square dynamical minimization schemes are devised and demonstrated on a complex parameterized PDAE system with discontinues properties and coupled time derivatives. Applying the method involves no more than defining a few formulas that closely parallel the original mathematical equations, without any programming skills. It offers a simpler alternative to more complex environments which require nontrivial programming skill and effort.

SOLVING IRREGULAR STRIP PACKING PROBLEMS WITH FREE ROTATIONS USING SEPARATION LINES

ABSTRACT Solving nesting problems or irregular strip packing problems is to position polygons on a fixed width and unlimited length strip, obeying polygon integrity containment constraints and nonoverlapping constraints, in order to minimize the used length of the strip. To ensure non-overlapping, we use separation lines, i.e., straight lines that separate polygons. We present a nonlinear programming model that considers free rotations of the polygons and of the separation lines. This model uses a considerable smaller number of variables than the few other approaches proposed in the literature. We use the nonlinear programming solver IPOPT (an algorithm of interior points type), which is part of COIN-OR. Computational tests were run using established benchmark instances and the results were compared with the ones obtained with other methodologies in the literature that use free rotations.

Gait and Trajectory Optimization for Legged Systems Through Phase-Based End-Effector Parameterization

We present a single trajectory optimization formulation for legged locomotion that automatically determines the gait sequence, step timings, footholds, swing-leg motions, and six-dimensional body motion over nonflat terrain, without any additional modules. Our phase-based parameterization of feet motion and forces allows to optimize over the discrete gait sequence using only continuous decision variables. The system is represented using a simplified centroidal dynamics model that is influenced by the feet's location and forces. We explicitly enforce friction cone constraints, depending on the shape of the terrain. The nonlinear programming problem solver generates highly dynamic motion plans with full flight phases for a variety of legged systems with arbitrary morphologies in an efficient manner. We validate the feasibility of the generated plans in simulation and on the real quadruped robot ANYmal. Additionally, the entire solver software TOWR , which used to generate these motions is made freely available.

An Improved Method for the DCOPF With Losses

This paper discusses the marginal line loss approximation in the dc optimal power flow (DCOPF) and proposes a new methodology that improves the accuracy of current practices. In practice, the DCOPF solution may have significant differences compared to the ac base point that determines marginal losses, and this paper provides the only published example that illustrates this problem to our knowledge. The proposed methodology updates the marginal losses without solving a new ac base point so that the optimal solution accurately reflects system losses. Previously proposed methodologies require nonlinear programming solvers, additional ac power flow solutions, or are less accurate compared to actual losses, and they only give results for one or two test networks. This paper provides results on nine different test networks. The key advantage of the proposed methodology is that it can be easily integrated into current market software.

Minimisation of active power losses and number of control adjustments in the optimal reactive dispatch problem

Trade-offs between power system's optimal operational performance and the minimal number of control adjustments necessary to attain a desired operating point make optimal reactive dispatch (ORD) solutions practical to system operators. In this study, a multi-objective ORD model that provides, in terms of weighting factors, trade-offs between minimal active power losses in transmission systems and minimal number of control adjustments in generator voltages, tap ratios and shunt controls is featured. This multi-objective ORD is formulated as a mixed-integer non-linear programming (MINLP) problem, and the proposed resolution methodology is based on translating the original MINLP problem into non-linear programming (NLP) problem deploying a sigmoid function, enabling the use of NLP solvers. Both original MINLP and translated NLP models are implemented in GAMS and numerical tests with IEEE test-systems with up to 300 buses are conducted using DICOPT, KNITRO and CONOPT solvers to validate the proposed ORD model and its resolution methodology. Results demonstrate the relation between active power losses and the number of adjustments in control variables, which is valuable information for operation planning. Another fundamental result is the high computational performance of the method when compared to specialized MINLP solvers.

Tuning BARON using derivative-free optimization algorithms

Optimization solvers include many options that allow users to control algorithmic aspects that may have a considerable impact on solver performance. Tuning solver options is often necessary to reduce execution time and improve solution quality. Previous studies of solver tuning techniques have focused on mixed-integer linear programming and local nonlinear programming solvers. In this paper, we investigate the potential of tuning a global optimization solver for nonlinear and mixed-integer nonlinear programming problems. In particular, derivative-free optimization (DFO) algorithms are used to find optimal values for options of the global optimization solver BARON. A set of 126 problems from the GLOBALLib and MINLPLib collections are utilized in a computational study from which we conclude that tuning options can improve the default performance of BARON for individual problems and an entire library. Additionally, we present a systematic comparison of 27 DFO solvers in terms of their ability to improve the performance of the global solver. We find that several DFO implementations are much better than others in terms of finding optimal tuning parameters.

Trajectory optimization for lunar rover performing vertical takeoff vertical landing maneuvers in the presence of terrain

This study presents a trajectory optimization framework for lunar rover performing vertical takeoff vertical landing (VTVL) maneuvers in the presence of terrain using variable-thrust propulsion. First, a VTVL trajectory optimization problem with three-dimensional kinematics and dynamics model, boundary conditions, and path constraints is formulated. Then, a finite-element approach transcribes the formulated trajectory optimization problem into a nonlinear programming (NLP) problem solved by a highly efficient NLP solver. A homotopy-based backtracking strategy is applied to enhance the convergence in solving the formulated VTVL trajectory optimization problem. The optimal thrust solution typically has a “bang-bang” profile considering that bounds are imposed on the magnitude of engine thrust. An adaptive mesh refinement strategy based on a constant Hamiltonian profile is designed to address the difficulty in locating the breakpoints in the thrust profile. Four scenarios are simulated. Simulation results indicate that the proposed trajectory optimization framework has sufficient adaptability to handle VTVL missions efficiently.

Novel Spreadsheet Direct Method for Optimal Control Problems

We devise a simple yet highly effective technique for solving general optimal control problems in Excel spreadsheets. The technique exploits Excel’s native nonlinear programming (NLP) Solver Command, in conjunction with two calculus worksheet functions, namely, an initial value problem solver and a discrete data integrator, in a direct solution paradigm adapted to the spreadsheet. The technique is tested on several highly nonlinear constrained multivariable control problems with remarkable results in terms of reliability, consistency with pseudo-spectral reported answers, and computing times in the order of seconds. The technique requires no more than defining a few analogous formulas to the problem mathematical equations using basic spreadsheet operations, and no programming skills are needed. It introduces an alternative, simpler tool for solving optimal control problems in social and natural science disciplines.

Adaptive Horizon Model Predictive Control

Adaptive Horizon Model Predictive Control (AHMPC) is a scheme for varying the horizon length of Model Predictive Control (MPC) as needed. Its goal is to achieve stabilization with horizons as small as possible so that MPC methods can be used on faster and/or more complicated dynamic process. Beside the standard requirements of MPC including a terminal cost that is a control Lyapunov function, AHMPC requires a terminal feedback that turns the control Lyapunov function into a standard Lyapunov function for the closed loop dynamics in some domain around the operating point. But this domain need not be known explicitly. MPC does not compute off-line the optimal cost and the optimal feedback over a large domain instead it computes these quantities on-line when and where they are needed. In a similar fashion, AHMPC does not compute off-line the domain on which the terminal cost is a control Lyapunov function instead it computes on-line when a state is in this domain. AHMPC verifies in real time that stabilization is being achieved. This is particularly important when dealing with nonlinear systems because the nonlinear programs that MPC passes to the nonlinear program solver are typically nonconvex. Therefore the solver may return local rather than global solutions and these may fail to be stabilizing. AHMPC detects when the solutions are not stabilizing. If this happens then there is a need to pass different initial guesses to the solver to find global solutions or at least local solutions that are stabilizing.

Extension of the Hybrid Ant Colony Optimization Algorithm for Layout and Size Optimization of Sewer Networks

In this paper, the incremental solution building capability of Ant Colony Optimization Algorithm (ACOA) is exploited using a Tree Growing Algorithm (TGA) augmented with the efficiency of Nonlinear Programming (NLP) methods leading to a hybrid ACOA-TGA-NLP algorithm for the effective layout and pipe size optimization of pumped/gravitational sewer networks. Solution of layout and pipe size optimization of sewer network requires the determination of pipe locations, pipe diameters, average pipe cover depths, drop and pump heights minimizing the total cost of the sewer network subject to operational constraints. The resulting problem is a highly constrained Mixed-Integer Nonlinear Programming (MINLP) problem presenting a challenge even to the modern heuristic search methods. In the proposed method, the TGA is used to construct feasible tree-like layouts out of the base layout defined for the sewer network, the ACOA is used to optimally determine the pipe diameters of the constructed layout, and finally NLP is used to determine the pipe slopes from which the remaining characteristics of the network such as pump/drop locations and heights are determined. In the NLP stage of the model, the velocity and flow depth constraints are expressed in terms of the slope constraints which are easily enforced as box constraint of the NLP solver leading to a considerable reduction of the search space size. The proposed hybrid ACOA-TGA-NLP has two significant advantages over other available methods. First, this method can be used for both pumped and gravitational sewer networks. Second, the computational effort is significantly reduced compared to alternative methods. Another method is also proposed here in which the layout of the network is determined by an ad-hoc method based on engineering judgment while the component design of the network is carried out by ACOA-NLP method as defined above. Proposed hybrid methods are used to solve a benchmark example from the literature and a hypothetical test example and the results are presented and compared with those produced by the existing methods such as SPST-DDDP, SDM-DDDP and GA-DDDP. The results indicate the efficiency and effectiveness of the proposed methods and in particular the ACOA-TGA-NLP method. In fact, the optimal solution of ACOA-TGA-NLP is 149, 64.1, 22.2 and 13.6% cheaper than those of ACOA-NLP, SPST-DDDP, SDM-DDDP and GA-DDDP methods, respectively, for the benchmark text example. Furthermore, ACOA-TGA-NLP yields a solution 80% cheaper than that of ACOA-NLP method for hypothetical test example.

A study of the difference-of-convex approach for solving linear programs with complementarity constraints

This paper studies the difference-of-convex (DC) penalty formulations and the associated difference-of-convex algorithm (DCA) for computing stationary solutions of linear programs with complementarity constraints (LPCCs). We focus on three such formulations and establish connections between their stationary solutions and those of the LPCC. Improvements of the DCA are proposed to remedy some drawbacks in a straightforward adaptation of the DCA to these formulations. Extensive numerical results, including comparisons with an existing nonlinear programming solver and the mixed-integer formulation, are presented to elucidate the effectiveness of the overall DC approach.

An algorithm based on semidefinite programming for finding minimax optimal designs

An algorithm based on a delayed constraint generation method for solving semi-infinite programs for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax optimization problem is solved using a semidefinite programming based approach that requires the design space be discretized. A nonlinear programming solver is then used to solve the inner program to determine the combination of the parameters that yields the worst-case value of the design criterion. The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it (i) produces designs that compare well with minimax D−optimal designs obtained from semi-infinite programming method in the literature; (ii) can be applied to semidefinite representable optimality criteria, that include the common A−,E−,G−,I− and D-optimality criteria; (iii) can tackle design problems with arbitrary linear constraints on the weights; and (iv) is fast and relatively easy to use.

Numerical Aspects of Data Reconciliation in Industrial Applications

Data reconciliation is a model-based technique that reduces measurement errors by making use of redundancies in process data. It is largely applied in modern process industries, being commercially available in software tools. Based on industrial applications reported in the literature, we have identified and tested different configuration settings providing a numerical assessment on the performance of several important aspects involved in the solution of nonlinear steady-state data reconciliation that are generally overlooked. The discussed items are comprised of problem formulation, regarding the presence of estimated parameters in the objective function; solution approach when applying nonlinear programming solvers; methods for estimating objective function gradients; initial guess; and optimization algorithm. The study is based on simulations of a rigorous and validated model of a real offshore oil production system. The assessment includes evaluations of solution robustness, constraint violation at convergence, and computational cost. In addition, we propose the use of a global test to detect inconsistencies in the formulation and in the solution of the problem. Results show that different settings have a great impact on the performance of reconciliation procedures, often leading to local solutions. The question of how to satisfactorily solve the data reconciliation problem is discussed so as to obtain improved estimates.

Optimization of axial catalyst loading in transient-operated zone-structured monoliths: Reduction of cumulative emissions in automotive oxidation catalysts

A model based mathematical optimization methodology to optimize the precious metal loading profile (PGM loading) in zone-structured catalytic converters is developed. To carry out this task, a multi zone-structured optimization formulation, where the catalyst is divided into N zones axially to obtain a non uniform optimal PGM loading profile, which can be tested experimentally, is used. The effects of the PGM loading on washcoat diffusion limitations is also considered. The objective is to optimize the spatial distribution of loading for a fixed amount of precious metal to maximize the chemical conversion efficiency under transient operation. To achieve this, the transient 1D+1D model is solved with the help of implicit solver DASPKADJOINT and translated into a non-linear optimization problem that can be solved with any derivative based nonlinear programming (NLP) solvers. The model is applied to two example cases: CO oxidation on a Pt/Al2O3 based Diesel Oxidation Catalyst catalyst (minimizing cold-start emissions) and CH4 oxidation on Pd/Al2O3 (minimizing deactivation effects). In both the cases it was observed that the optimal solution with maximum PGM loading in the channel entrance region improved the performance of the catalysts. The methodology presented is generic and can be transferred to different systems with different chemistries, which may result in significantly different optimization results and loading patterns.

Simultaneous Optimization Approach for Optimal Electrode Design of Lithium-Ion Batteries

The first attempt to improve battery performance through model-based design optimization was made by W. Tiedemann and J. Newman in 19751. They varied the electrode thickness and porosity and then observed the effects on the effective capacity. Newman later applied the reaction-zone model to maximize the specific energy of the system using the same varying and observing method2. V. Ramadesigan et al.3developed optimal spatially varying porosity for porous electrodes by considering ohmic resistance model in the electrode. They minimized the ohmic resistance in the positive electrode by varying porosity using control vector parameterization (CVP) method. They also introduced the idea of using graded electrode to further improve the performace. Since the introduction of the pseudo-2D (P2D) model by the Newman group, many efforts have been made in utilizing the P2D model in battery design. W. Du et al. applied an optimization framework based on surrogate method4. S. Golmon et al. performed the multi-objective and multi-design-parameter optimization problem with adjoint sensitivity analysis5 incorporating mechanical stress-strain relationship. S. De and his coworkers6 used a reformulated model, to perform simultaneous optimization of multiple design parameters. N. Xue et al. 7 applied the gradient-based algorithm framework to optimize the cell design. More recently, Y. Dai and V. Srinivasan8used a gradient-free direct search method to maximize the specific energy. Z. Du et al. studies the effects of various design parameters on thick electrode performance9. Most of the aforementioned optimization research, used the sequential optimization approach, also known as CVP. CVP takes the differential algebraic equation (DAE) system describing the battery physics, applies a certain nonlinear programming (NLP) solver to them, discretizes only the control variables, and solves the model. The only exception being the effort by Golmon et al which used an indirect approach. Note that as of today, global optimization cannot be guaranteed when CVP type approaches are used with P2D type models. In general optimization approaches can be classified as indirect and direct approach. In the direct approach there are two main categories – sequential and simultaneous approach. The simultaneous approach10, on the other hand, discretizes both the control variables and the design variables before solving the problem. When used for optimization, the DAE system will only be solved once at the optimal point, compared to the repeated numerical integration needed for sequential optimization. By using high order discretization scheme on both the state and the design variables, simultaneous optimization approach results in faster determination of the optimum with fewer time steps. Furthermore, this approach offers more flexibility over constraints on the state variables. Since the state variables are also discretized, it is possible to apply equality/inequality constraints on their internal values. In this work, we will first apply the simultaneous optimization approach to a resistance model similar to the one mentioned earlier3, and compare its performance with the sequential approach. We will also explore the application of the same optimization approach to the P2D model, and identify the challenges. It is viewed as a controversial topic in the literature wherein different authors disagree on the relevance of graded porosity for improved performance8. By using a simultaneous optimization approach and guaranteeing global optimization for convex problem statements, we hope to provide additional insights. Acknowledgements The authors are thankful for the financial support by the Clean Energy Institute at the University of Washington and the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium).