Mixed-integer Model Predictive Control Research Articles

ReLU surrogates in mixed-integer MPC for irrigation scheduling

Efficient water management in agriculture is important for mitigating the growing freshwater scarcity crisis. Mixed-integer Model Predictive Control (MPC) has emerged as an effective approach for addressing the complex scheduling problems in agricultural irrigation. However, the computational complexity of mixed-integer MPC still poses a significant challenge, particularly in large-scale applications. This study proposes an approach to enhance the computational efficiency of mixed-integer MPC-based irrigation schedulers by employing Rectified Linear Unit (ReLU) surrogate models to describe the soil moisture dynamics of the agricultural field. By leveraging the mixed-integer linear representation of the ReLU operator, the proposed approach transforms the mixed-integer MPC-based scheduler with a quadratic cost function into a mixed-integer quadratic program, which is the simplest class of mixed-integer nonlinear programming problems that can be efficiently solved using global optimization solvers. The effectiveness of this approach is demonstrated through comparative studies conducted on a large-scale agricultural field across two growing seasons, involving other machine learning surrogate models, specifically Long Short-Term Memory (LSTM) networks, and the triggered irrigation scheduling method. The ReLU-based approach significantly reduces solution times—by up to 99.5%—while achieving comparable performance to the LSTM approach in terms of water savings and Irrigation Water Use Efficiency (IWUE). Moreover, the ReLU-based approach achieves enhanced performance in terms of irrigation water savings and IWUE compared to the triggered approach.

Robust economic–environmental dispatch problem for BESS and PV generators in MT-HVDC systems: A MI-MPC approach

The research addresses the energy management problem (EMP) in multi-terminal high-voltage direct current (MT-HVDC) systems using a mixed-integer model predictive control (MI-MPC). The MT-HVDC system consists of conventional thermal generation plants (fossil sources), renewable generation sources (solar photovoltaic plants), and battery energy storage systems (BESS). This combination creates a complex nonlinear EMP, as it introduces the power losses of the transmission networks through B-coefficients. The main contributions of this research can be summarized in three elements: (i) the formulation of a measurement- or simulation-based approach to determine B−coefficients in MT-HVDC systems for calculating power losses; (ii) the convexification of the EMP model through MI-MPC approaches, allowing to minimize economic and environmental objective functions; and (iii) the inclusion of uncertainties through the robust convex method. Numerical validations in the 11-bus grid, including the linear transportation model for the MT-HVDC system, confirm the multi-objective nature of the EMP when considering economic and environmental indices. In addition, the effect of uncertainties on generation and demand causes significant variations in the optimal Pareto front when compared to the deterministic scenario.

Taking the human out of decomposition-based optimization via artificial intelligence, Part II: Learning to initialize

The repeated solution of large-scale optimization problems arises frequently in process systems engineering tasks. Decomposition-based solution methods have been widely used to reduce the corresponding computational time, yet their implementation has multiple steps that are difficult to configure. We propose a machine learning approach to learn the optimal initialization of such algorithms which minimizes the computational time. Active and supervised learning is used to learn a surrogate model that predicts the computational performance for a given initialization. We apply this approach to the initialization of Generalized Benders Decomposition for the solution of mixed-integer model predictive control problems. The surrogate models are used to find the optimal initialization, which corresponds to the number of initial cuts that should be added in the master problem. The results show that the proposed approach can lead to a significant reduction in solution time, and active learning can reduce the data required for learning.

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50×) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

Learning-based multi-agent MPC for irrigation scheduling

Amid concerns about freshwater scarcity, the agricultural sector faces challenges in water conservation and optimizing crop yields, highlighting the limitations of traditional irrigation scheduling methods. To overcome these challenges, this paper introduces a unified, learning-based predictive irrigation scheduler that integrates machine learning and Model Predictive Control (MPC), while also incorporating multi-agent principles. The proposed framework incorporates a three-stage management zone delineation process, utilizing k-means clustering and hydraulic parameters estimates for optimized agro-hydrological modeling. Long Short-Term Memory (LSTM) networks are employed for accurate and computationally efficient root zone soil moisture modeling. The scheduler, formulated as a mixed-integer MPC with zone control, utilizes the identified LSTM networks to maximize root water uptake while minimizing overall water consumption and fixed irrigation costs. Additionally, the learning-based scheduler adopts a multi-agent MPC paradigm, where decentralized hybrid actor–critic agents and the concept of a limiting irrigation management zone are employed to enhance computational efficiency. Evaluating the performance on a 26.4-hectare field in Lethbridge for the 2015 and 2022 growing seasons demonstrates the superiority of the proposed scheduler over the widely-used triggered scheduling approach in terms of Irrigation Water Use Efficiency (IWUE) and total prescribed irrigation. Notably, the proposed approach achieves water savings between 7 to 23%, coupled with IWUE increases ranging from 10 to 35%.

Online Ecological Gearshift Strategy via Neural Network with Soft-Argmax Operator

This paper presents a neural network optimizer with soft-argmax operator to achieve an ecological gearshift strategy in real-time. The strategy is reformulated as the mixed-integer model predictive control (MIMPC) problem to minimize energy consumption. Then the outer convexification is introduced to transform integer variables into relaxed binary controls. To approximate binary solutions properly within training, the soft-argmax operator is applied to the neural network with the fact that all the operations of this scheme are differentiable. Moreover, this operator can help push the relaxed binary variables close to 0 or 1. To evaluate the strategy effect, we deployed it to a 2-speed electric vehicle (EV). In contrast to the mature solver Bonmin, our proposed method not only achieves similar energy-saving effects but also significantly reduces the solution time to meet real-time requirements. This results in a notable energy savings of 6.02% compared to the rule-based method.

Eco-Coasting Controller Using Road Grade Preview: Evaluation and Online Implementation Based on Mixed Integer Model Predictive Control

Coasting is a common method used in eco-driving to reduce fuel consumption by utilizing kinetic energy. However, in order to avoid excessive computation induced by integer coasting maneuvers, the powertrain model used in eco-driving controllers that rely on look-ahead road information has been oversimplified. This oversimplification assumes that the engine goes to idle when coasting, which significantly limits the fuel-saving potential. To address this issue, we propose an eco-coasting strategy that calculates the optimal timing and duration of coasting maneuvers using road information preview. Different from the engine-idling method, two control-oriented coasting methods, fuel cut-off method and engine start/stop method are formulated for the model-based optimal control. To evaluate and choose the best coasting mechanism for eco-coasting strategy, dynamic programming (DP) is performed to provide the globally optimal performance (i.e., benchmark results) for evaluating the engine-idling method, fuel cut-off method, and engine start/stop method. Based on the offline simulation results, the engine start/stop method consistently outperforms the fuel cut-off method in terms of both fuel consumption and travel time. This is attributed to the engine start/stop method eliminating the engine drag torque during deceleration, despite the additional energy cost required for engine restart being taken into account in the modeling, thus providing a fair evaluation. Then, the online performance of the eco-coasting strategy with engine start/stop mechanism is evaluated using Mixed Integer Model Predictive Control (MIMPC). We propose a tailored mixed-integer programming algorithm to facilitate online implementation. Simulation results show that the proposed eco-coasting strategy achieves near-optimal performance compared to DP and outperforms the rule-based method.

Irrigation management zone delineation and optimal irrigation scheduling for center pivot irrigation systems

In this study, a three-stage irrigation management zone delineation approach and a model-based scheduler are proposed to provide variable rate irrigation schedules for a 26.4-hectare field at the Alberta Irrigation Technology Centre in Lethbridge, Alberta, Canada. After an initial demarcation of the investigated field into quadrants according to the four different crops growing in the field, the k-means clustering approach is used to delineate each quadrant further into smaller irrigation management zones on the basis of soil hydraulic parameters, elevation of the field and the variable rate irrigation resolution of the center pivot system. For each management zone, a long short-term memory (LSTM) neural network model is then developed to characterize the soil water dynamics in the root zone. Based on the LSTM models, and based on the LSTM models, a mixed-integer model predictive controller (MPC) is designed to prescribe irrigation schedules for the various management zones in each quadrant to ensure optimal crop water uptake while minimizing water consumption and irrigation costs. The applicability of the proposed approach is illustrated in simulations. The results show that the proposed approach achieves 26% water savings compared to a triggered scheduling scheme.

Learning for Online Mixed-Integer Model Predictive Control With Parametric Optimality Certificates

We propose a supervised learning framework for computing solutions of multi-parametric Mixed Integer Linear Programs (MILPs) that arise in Model Predictive Control. Our approach also quantifies sub-optimality for the computed solutions. Inspired by Branch-and-Bound techniques, the key idea is to train a Neural Network or Random Forest which, for a given parameter, predicts a strategy consisting of (1) a set of Linear Programs (LPs) such that their feasible sets form a partition of the feasible set of the MILP and (2) an integer solution. For control computation and sub-optimality quantification, we solve a set of LPs online in parallel. We demonstrate our approach for a motion planning example and compare against various commercial and open-source mixed-integer programming solvers.

Microalgae Production and Maintenance Optimization via Mixed-Integer Model Predictive Control

This paper studies the joint production and maintenance scheduling in microalgae manufacturing systems comprised of multiple machines, which are subject to coupled production demand agreements and operational maintenance constraints. Namely, there are some microalgae production demands to be met over a given horizon, and the maintenance of each microalgae manufacturing unit must be done before a given deadline. Moreover, the number of units whose maintenance can be done simultaneously over the same day is limited, and the units that undergo maintenance cannot contribute to microalgae production during their maintenance day. To solve the considered problem, we design a mixed-integer nonlinear model predictive controller, which is implemented in two optimization stages. The former regards a mixed-integer model predictive control problem, while the latter considers a nonlinear model predictive control problem. The proposed approach allows us to decouple the mixed-integer and nonlinear parts of the whole problem, and thus provides more flexibility on the optimization solvers that can be employed. In addition, the first stage also evaluates the attainability of the demand agreements, and provides a mechanism to minimally adjust such constraints so that their satisfaction can be guaranteed at the second stage. The overall model predictive control approach is based on experimental data collected at VAXA Technologies Ltd., and the effectiveness of the proposed method is validated through numerical simulations including multiple manufacturing units and uncertainties.

Model-predictive energy management system for thermal batch production processes using online load prediction

Predictive energy management systems (EMS) enable industrial plants to participate in the modern power market and reduce energy cost. In this paper, a novel modular model predictive EMS specifically designed for industrial thermal batch processes is presented. The EMS consists of a two-layer mixed-integer model predictive controller and an online load predictor, and thus solves the main challenges of EMS in industry - high implementation costs and the possible reduction of production reliability. The modular formulation of the optimization problem enables system integrators to implement the EMS without time-consuming modelling tasks and elaborate parameter tuning. The online load predictor estimates the typical pulse-like heat loads of batch processes ensuring both - reliable production and maximal flexibility of the power demand. The utilization of real-time data provides additional robustness against uncertainties caused by human operators. The performance of the EMS is evaluated in a case study of an existing food plant.

Dual dynamic programming for multi-scale mixed-integer MPC

We propose a dual dynamic integer programming (DDIP) framework for solving multi-scale mixed-integer model predictive control (MPC) problems. Such problems arise in applications that involve long horizons and/or fine temporal discretizations as well as mixed-integer states and controls (e.g., scheduling logic and discrete actuators). The approach uses a nested cutting-plane scheme that performs forward and backward sweeps along the time horizon to adaptively approximate cost-to-go functions. The DDIP scheme proposed can handle general MPC formulations with mixed-integer controls and states and can perform forward-backward sweeps over block time partitions. We demonstrate the performance of the proposed scheme by solving mixed-integer MPC problems that arise in the scheduling of central heating, ventilation, and air-conditioning (HVAC) plants. We show that the proposed scheme is scalable and dramatically outperforms state-of-the-art mixed-integer solvers.

Early Termination of Convex QP Solvers in Mixed-Integer Programming for Real-Time Decision Making

The branch-and-bound optimization algorithm for mixed-integer model predictive control (MI-MPC) solves several convex quadratic program relaxations, but often the solutions are discarded based on already known integer feasible solutions. This letter presents a projection and early termination strategy for infeasible interior point methods to reduce the computational effort of finding a globally optimal solution for MI-MPC. The method is shown to be also effective for infeasibility detection of the convex relaxations. We present numerical simulation results with a reduction of the total number of solver iterations by 42% for an MI-MPC example of decision making for automated driving with obstacle avoidance constraints.

Quasi-translations for fast hybrid nonlinear model predictive control

We present the quasi-translation (QT) strategy for mixed-integer model predictive control (MPC) problems. This strategy handles the complexity introduced by discrete controls in a simple manner that takes advantage of the sequential nature of MPC optimization problems. Some general criteria is provided guiding their application. The QT strategy can be used to balance controller performance, chatter and turnaround time. We also introduce an approach to handle hybrid models with state-dependent switches for use with a symbolic formulation of the MPC problem that integrates seamlessly with the QT strategy.We investigate the QT strategy for a variety of hybrid nonlinear MPC problems including the classic benchmark Lotka–Volterra fishing problem, a diesel airpath control problem and longitudinal vehicle control. Results show the QT strategy consistently yields a reduction in turnaround times with excellent or minimally degraded performance as compared to competing strategies.

Mixed-integer model predictive control for large-area MR-HIFU hyperthermia in cancer therapy

In hyperthermia treatments, cancer tissue is heated to enhance the desired effects of radio- and chemotherapies. A powerful technology for noninvasive feedback-controlled hyperthermia is magnetic-resonance-guided high-intensity focused ultrasound (MR-HIFU), which enables fast and millimeter-accurate heating inside the body. Electronic beam steering allows for volumetric heating, but due to its limited steering range can only be used to treat small tumors. For the treatment of larger tumors, the transducer itself must be mechanically relocated as well. Due to system limitations, however, the admissible transducer positions must be restricted to a finite set that is chosen a priori. Moreover, non-negligible time is needed for transducer relocation, during which no heating is possible. In this paper, we present a mixed-integer model predictive controller that simultaneously optimizes over the power deposition by electronic beam steering - a continuous subproblem - as well as the mechanical transducer motions - a discrete subproblem. By incorporating model knowledge of the tissue’s thermal response and of the transducer carrier motion system into the predictive algorithm, the controller optimizes treatment temperature while respecting temperature and actuation constraints. The performance of the proposed feedback control setup is demonstrated by means of simulation.

A whole-year simulation study on nonlinear mixed-integer model predictive control for a thermal energy supply system with multi-use components

This work presents a whole-year simulation study on nonlinear mixed-integer Model Predictive Control (MPC) for a complex thermal energy supply system which consists of a heat pump, stratified water storages, free cooling facilities, and a large underground thermal storage. For solution of the arising Mixed-Integer Non-Linear Programs (MINLPs) we apply an existing general and optimal-control-suitable decomposition approach. To compensate deviation of forecast inputs from measured disturbances, we introduce a moving horizon estimation step within the MPC strategy. The MPC performance for this study, which consists of more than 50,000 real-time suitable MINLP solutions, is compared to an elaborate conventional control strategy for the system. It is shown that MPC can significantly reduce the yearly energy consumption while providing a similar degree of constraint satisfaction, and autonomously identify previously unknown, beneficial operation modes.

Mixed-integer model predictive control of variable-speed heat pumps

Heating and cooling buildings cause a large and growing portion of the world’s carbon emissions. These emissions could be reduced by replacing inefficient or fossil-fueled equipment with efficient electric heat pumps. Recent research has shown that variable-speed heat pumps (VSHPs) could also provide a variety of power system services, such as price- or carbon-based load shifting, peak demand reduction, emergency demand response, and frequency regulation. By providing some or all of these services, VSHPs could facilitate the integration of wind and solar power into the grid, accelerating decarbonization of other economic sectors. However, the provision of power system services poses new and challenging VSHP control problems. Model predictive control (MPC) is a promising approach to these problems. MPC combines weather and load predictions, online optimization, and a mathematical model of a VSHP’s performance under varying ambient conditions. With few exceptions, existing VSHP models neglect two important features that arise in practice: (1) dependence of efficiency and capacity on compressor speed and indoor and outdoor temperatures, and (2) inability of VSHPs to operate at low compressor speeds. In this paper, we develop a new VSHP control model that considers both features. The model expresses the VSHP’s power consumption and heat production as affine functions of the driving temperatures and compressor speed. The model also includes a binary variable that constrains the VSHP to be either turned off, or operating within a range of acceptable compressor speeds using mixed-integer programming (MIP). This VSHP model and optimization strategy is combined with neural network based heat load prediction in MPC simulations. These simulations suggest that this approach could reduce energy costs by 9–22% and carbon emissions by up to 22%, relative to MPC with existing VSHP models.

LMI-based robust mixed-integer model predictive control for hybrid systems

ABSTRACT This paper proposes a linear matrix inequality (LMI) approach to mixed-integer model predictive control (MPC) of uncertain hybrid systems with binary and real valued control inputs. The stability condition of the hybrid system is obtained by using the Lyapunov function and then a sufficient state feedback control law is achieved so that guarantees the closed-loop stability and also minimises an infinite horizon performance index. The primal optimisation problem is convex, therefore, a convex relaxation is investigated by introducing the Lagrange dual function. The real and binary control inputs are obtained by solving the dual of the dual problem in the framework of LMIs. The performance and effectiveness of the proposed MPC approach and the duality gap of the convex relaxation are studied through simulation results of a hybrid system with mixed real and binary inputs.

A hierarchical MPC for multi-objective mixed-integer optimisation applied to redundant refrigeration circuits

Temperature control of an insulated cool box (ICB) can be efficiently provided by model predictive control. However, if the cooling capacity is supplied by redundant refrigeration circuits (RCs) with both continuous and switched control variables, the overall control design becomes complex: 1) Time constants of the ICB and RCs differ by a factor of approximately 100. 2) Switched and continuous control variables require mixed-integer optimisation. 3) Power consumption, wear, control performance and output tracking call for multi-objective optimisation. In this work a global linear Model Predictive Controller (MPC) is designed to compute a desired cooling capacity at a low sampling frequency for a long prediction horizon, while a mixed-integer MPC (MI-MPC) provides the actual cooling capacity utilising a fast sampling frequency and a short prediction horizon. The control concept is presented, a stability prove is given and simulation results demonstrate the performance of the concept.

A branch and bound approach for building cooling supply control with hybrid model predictive control

In this article a branch and bound approach for hybrid model predictive control of a building cooling supply is presented. This specific algorithm is the core part of a mixed-integer model predictive controller (MI-MPC) for energy efficient management of a cooling supply incorporating active storage connected to a switching chiller. The MI-MPC is applied to the model of the cooling system of a large office building in Salzburg, Austria. An efficient modelling strategy combines data-driven black-box identification and analytically derived white-box modelling for the purpose of predictive control. The validation of the resulting linear and hybrid energy supply models is done by an open loop simulation. Simulation results of the proposed control structure show excellent performance compared to the implemented conventional control strategy resulting in an increased usage of at least 50% of renewable energy sources, a cut in energy costs of about 50%, and an optimised operation of the active storage while accurately delivering the prescribed cooling power trajectory. The approach is thus promising for industrial application.