Economic Nonlinear Model Predictive Control Research Articles

Distributed Economic Nonlinear Model Predictive Control for Flexible Electrified Biodiesel Production─Part II: Sequential and Iterative Architectures with Computational Delay Compensation

Distributed Economic Nonlinear Model Predictive Control for Flexible Electrified Biodiesel Production─Part II: Sequential and Iterative Architectures with Computational Delay Compensation

Self-stabilizing economic nonlinear model predictive control applied to modular systems

Recent advances have been made in self-stabilizing Economic Nonlinear Model Predictive Control (eNMPC) formulation without pre-calculated setpoints, which leverages norm-based steady-state optimality conditions to enhance system robustness. To enable practical implementation, a generalized time-domain formulation is proposed, accommodating the discrete-time nature of control instrumentation and the continuous-time nature of first-principles models. A case study involving a modular membrane reactor illustrates the applicability of self-stabilizing eNMPC in real-world industrial scenarios.

End-to-end reinforcement learning of Koopman models for economic nonlinear model predictive control

(Economic) nonlinear model predictive control ((e)NMPC) requires dynamic models that are sufficiently accurate and computationally tractable. Data-driven surrogate models for mechanistic models can reduce the computational burden of (e)NMPC; however, such models are typically trained by system identification for maximum prediction accuracy on simulation samples and perform suboptimally in (e)NMPC. We present a method for end-to-end reinforcement learning of Koopman surrogate models for optimal performance as part of (e)NMPC. We apply our method to two applications derived from an established nonlinear continuous stirred-tank reactor model. The controller performance is compared to that of (e)NMPCs utilizing models trained using system identification, and model-free neural network controllers trained using reinforcement learning. We show that the end-to-end trained models outperform those trained using system identification in (e)NMPC, and that, in contrast to the neural network controllers, the (e)NMPC controllers can react to changes in the control setting without retraining.

Real-time time-varying economic nonlinear model predictive control for wind turbines

Economic nonlinear model predictive control is a great choice to tackle the control issues appearing in the current wind energy industry. In this paper, instead of generator power, aerodynamic power is employed in the economic cost function to establish the required conditions of stability and convergence of the economic performance, such as turnpike and problem convexity. However, since aerodynamic power depends on time-varying wind speed, conventional time-invariant economic nonlinear model predictive controllers cannot guarantee the stability and convergence of economic performance. This paper proposes a new time-varying economic nonlinear model predictive controller for wind turbine control that considers an economic trajectory, instead of a steady-state, in its optimization problem. The proposed time-varying economic cost function directly considers aerodynamic power, the activity of pitch angle and generator torque, and fatigue loads on the shaft and tower. Therefore, this controller can maximize power extraction and reduce fatigue load on the tower, drivetrain, and actuators. Furthermore, a fast-parallel Newton-type method is used to implement the proposed controller in actual wind turbines. An accurate aeroelastic model is used to validate the performance of the proposed control scheme. The proposed controller is also compared with two tracking nonlinear model predictive controllers, the baseline controller, and a newly developed method, under fatigue and extreme load scenarios in the presence and absence of uncertainty. The simulation results show the superior economic performance of the proposed approach. Moreover, the real-time results verify the computational speed that the proposed controller requires to deploy actual wind turbines.

Two‐layer dynamic economic nonlinear model predictive control for a Lithium‐ion battery charge process with random disturbances

AbstractThe charging process of lithium‐ion batteries is necessary for normal operation, and improper lithium‐ion battery charging strategy can cause side reactions, significant temperature rise, performance degradation, and safety concerns. This paper proposes a two‐layer dynamic economic nonlinear model predictive control economic model predictive control (EMPC) method for lithium‐ion battery charge management in which the non‐Gaussian random noise of voltage and current signal are taken into account. In the two‐layer EMPC, the upper layer finds the optimal reference trajectory, and the power in the future prediction horizon is chosen as an economic indicator to optimize the upper layer. The lower layer tracks the optimal trajectory, and model predictive control based on generalized correntropy is used. The economic cost of the system and the dynamic changes of the process are considered to greatly reduce the computational complexity and realize rapid battery charge management. Finally, the simulation results show that the tracking error of the two‐layer EMPC method based on generalized correntropy is stable around 0, while the tracking error of the two‐layer EMPC method based on MSE is stable around 0.003. The average control action times of the proposed two‐layer EMPC and single‐layer EMPC are 0.0052 and 0.0267 s, respectively. It is verified that the proposed method performs better for the lithium‐ion battery charging process with random disturbances.

Economic Nonlinear Model Predictive Control of Continuous Viral Bioreactors

Viral particle systems are integral parts of modern biotechnology, finding use in vaccines, drug delivery platforms, and recombinant protein production. Continuous manufacturing of these systems can offer improved manufacturability and quality control. However, viral systems often have complex kinetics which can introduce undesirable process dynamics and lower product titers in continuous operation. This article explores the use of economic nonlinear dynamic optimization and model predictive control to achieve multiple process objectives such as maximizing productivity and/or purity. Economic nonlinear model predictive control is also demonstrated to robustly control the bioreactor under plant-model mismatch in different scenarios.

Impacts of g-g-v Constraints Formulations on Online Minimum-Time Vehicle Trajectory Planning

The g-g-v diagram is a popular representation of a vehicle’s maximum longitudinal and lateral accelerations, as a function of its speed. Recently, g-g-v diagrams have been used as performance constraints for online minimum-time vehicle trajectory planning with economic nonlinear model predictive control (E-NMPC). However, in the related literature, the modeling accuracy of the g-g-v formulations was often compromised in favor of computational efficiency for online E-NMPC. This paper compares various formulations of the g-g-v constraints, evaluating their accuracy, computational efficiency and impact on the lap times, when applied by an artificial race driver (ARD) to control a sports car model on a circuit. Also, we propose a new g-g-v diagram formulation, based on convex polytopic and non-convex polynomial constraints. Our formulation yields improved modeling accuracy while preserving a low computational burden for online E-NMPC. Finally, we analyze the vehicle trajectories, to understand how ARD achieves lower lap times through an improved knowledge of the maximum performance.

Fast two-layer nonlinear economic predictive control using machine learning for ORC systems with non-Gaussian disturbances

This study proposes a machine learning-based generalized correntropy two-layer nonlinear economic model predictive control (ML-GC-TL-NEMPC) algorithm to improve the computational speed for Organic Rankine Cycle (ORC) systems with non-Gaussian disturbances. In the upper layer, the ratio of the net output power to the total heat transfer area is used as the economic performance index to optimize the optimal reference trajectory. In the lower layer, the generalized correntropy-based MPC (GC-MPC) is used to track the optimal trajectory. To quickly obtain an optimal solution, a Deep Neural Network (DNN) neural network controller is combined with kernel support vector machine (SVM). It learns the possible optimal solution via off-line training. Then it can be quickly implemented on on-line applications, so it can save the optimal calculation time. The proposed ML-GC-TL-NEMPC is applied to ORC. Its computation and performance are faster and better than the traditional two-layer EMPC and single-layer EMPC.

Self-stabilizing economic model predictive control without pre-calculated steady-state optima: Stability and robustness

We propose a new economic nonlinear model predictive control (eNMPC) formulation that tracks the optimality conditions of the real-time optimization problem rather than any specific steady states. The proposed formulation maintains its nature of optimizing economic performance and assured stability properties with the Lyapunov inequality constraint for the closed-loop control. Under general assumptions, we prove that the proposed controller is asymptotically stable without process disturbances and is input-to-state stable when there is a process disturbance. The proposed eNMPC is demonstrated on two case studies and compared against setpoint-tracking NMPC with setpoints determined by the steady-state real-time optimizer to show improved dynamic performance. We also highlight the capability of self-stabilization of the new eNMPC with parameter updates in the process model.

Special issue on “Optimal design and operation of energy systems”

Special issue on “Optimal design and operation of energy systems”

Robustness of an Economic Nonlinear Model Predictive Control for Wind Turbines Under Changing Environmental and Wear Conditions

In this letter, the authors have assessed the robustness of an Economic Nonlinear Model-Predictive Controller (ENMPC) aimed at maximizing the power production of wind turbines. The scope of this letter is to quantify the sensitivity of this type of controller concerning wind conditions, climate, wind speed prediction unavailability, and aerodynamic performance degradation. A power production controller’s robustness is crucial for the wind turbine industry due to the extreme variability of external conditions and the wear caused by long-term continuous operativity. Model-Predictive controllers are, in principle, more prone to robustness issues concerning standard controllers, a fact that limits their adoption on actual wind turbines. The analysis is performed with the fully-aeroelastic solver OpenFAST considering a wide set of realistic load cases. It is demonstrated that the ENMPC previously developed is robust to wind prediction unavailability and change in wind turbulence intensity. Conversely, it is not robust to the modelling error due to aerodynamic degradation. Indeed, a reduction in generated power concerning the reference controller is observed, especially for operating region two and end-life blades. Finally, a significant increase in power production is achieved considering the external temperature variation thanks to the ENMPC’s direct handling of the generator temperature constraint.

Optimal Control for Wind Turbine Wake Mixing on Floating Platforms

Dynamic induction control is a wind farm flow control strategy that utilises wind turbine thrust variations to accelerate breakdown of the aerodynamic wake and improve downstream turbine performance. However, when floating wind turbines are considered, additional dynamics and challenges appear that make optimal control difficult. In this work, we propose an adjoint optimisation framework for non-linear economic model-predictive control, which utilises a novel coupling of an existing aerodynamic wake model to floating platform hydrodynamics. Analysis of the frequency response for the coupled model shows that it is possible to achieve wind turbine thrust variations without inducing large motion of the rotor. Using economic model-predictive control, we find dynamic induction results that lead to an improvement of 7% over static induction control, where the dynamic controller stimulates wake breakdown with only small variations in rotor displacement. This novel model formulation provides a starting point for the adaptation of dynamic wind farm flow control strategies for floating wind turbines.

A novel Economic Nonlinear Model Predictive Controller for power maximisation on wind turbines

Reducing the Levelized Cost of Energy (LCoE) is one of the main objectives of the wind turbine industry. There are several ways to achieve this goal: reducing construction and installation costs, reducing Operating&Maintenance costs, or increasing the power output. In this work, an Economic Nonlinear Model Predictive Control strategy is developed to maximise the power production of wind turbines. A novel three-states, non-linear Reduced Order Model is developed to predict aerodynamic power, rotor thrust and generator temperature with suitable accuracy. The control action is obtained from a constrained optimisation problem that uses the developed model, where the objective is the maximisation of the integral of the aerodynamic power. A set of constraints (including a bound on the generator temperature and the rotor thrust) are imposed. First, the turbine model is validated against high-fidelity simulations, then the controller performance and robustness are assessed in the entire wind range of operation, obtaining a significant increase of average power. Apart from the assessment of the controller performance in OpenFAST, the controller robustness is verified, introducing errors in the estimation of incoming wind, up to the case of a complete lack of information. The controller (freely downloadable from a dedicated repository) is effective in all the operating regions without the need for logical switches. Moreover, thanks to the optimised numerical solver adopted, it can be applied to actual wind turbines (which require real-time algorithmic performance).

Q-learning of the storage function in Economic Nonlinear Model Predictive Control

The closed-loop stability of an optimal policy provided by an Economic Nonlinear Model Predictive Control (ENMPC) scheme requires the existence of a storage function satisfying dissipativity conditions. Unfortunately, finding such a storage function is difficult in general. In contrast, tracking NMPC scheme uses a stage cost that is lower-bounded by a class-K∞ function and the closed-loop stability is fairly straightforward to establish. Under the dissipativity conditions, ENMPC has an equivalent tracking MPC that delivers the same optimal policy. In this paper, we use this idea and parameterize the stage cost and terminal cost of a tracking MPC with an additional parameterized storage function. We show that, if the parameterization of the tracking MPC scheme is rich enough to capture the exact optimal action-value function of the ENMPC scheme, then the parameterized storage function for the optimal parameters satisfies the dissipativity conditions for both discounted and undiscounted ENMPC schemes. In fact, we show that these conditions are met for dissipative problems. We propose to use Q-learning as a practical way of adjusting the parameters of the tracking MPC. Different numerical examples are provided to illustrate the efficiency of the proposed method, including LQR, non-dissipative, non-polynomial and a nonlinear chemical case studies. For instance, in the provided non-polynomial case study, the learning method can improve the storage function estimation by about 60% and 99.5% after 10 and 50 learning steps, respectively, compared with the Sum-of-Square method.

Economic nonlinear model predictive control of fatigue for a hybrid wind-battery generation system

An economic nonlinear model predictive controller (ENMPC) is formulated for a wind turbine-battery hybrid generation system. The controller aims to maximize the operational profit of the generator by balancing between generated wind power and turbine tower fatigue as well as battery cyclic fatigue. Other than the tower fore-aft fatigue, tower side-side fatigue is also considered to assess impact on overall economic performance. A moving horizon estimator (MHE) is formulated to provide meaningful initialization to the ENMPC in presence of plant model mismatch.The formulated controller utilizes the parametric online rainflow counting (PORFC) approach for direct cyclic fatigue cost minimization within ENMPC. The closed-loop simulation shows significantly higher profit compared to a realistic base-case scenario and relatively higher profit compared to another economic controller.

A novel Nonlinear Model Predictive Controller for Power Maximization on Floating Offshore Wind Turbines

Reducing the Levelized Cost of Energy is the main objective of wind turbine industry, in particular for the emerging sector of floating offshore turbines. In this work, a novel Economic Nonlinear Model Predictive Control (ENMPC) strategy is developed to maximize the power production of floating offshore wind turbines. The control problem is solved through an indirect method, which achieves the computational efficiency required to apply it in real world cases. A non-linear Reduced Order Model of the floating turbine predicts aerodynamic power, generator temperature and platform motions inside the controller. A set of constraints, including a bound on the generator temperature, the thrust and platform velocities are imposed. Simulations using the open-source engineering tool OpenFAST on the 5MW NREL wind turbine supported by the OC3 spar buoy platform [1] are performed to validate the turbine model and then to assess the controller performances in realistic wind and sea state conditions. With respect to the standard controller, a 4.3% increase of generated power in rated conditions is achieved with a more stable generator temperature.

Economic nonlinear model predictive control of fatigue—Formulation and application to wind turbine control

AbstractIn this article, the estimation of fatigue is implemented in the cost function of a gradient‐based model predictive controller (MPC). This is a challenging problem, because calculating fatigue leads to a non‐standard and discontinuous cost function. Based on a brief previous publication, in the present work the method is derived, explained, and assessed in detail. The key enablers of the proposed method are a sequential implementation of MPC, the periodic substitution of discontinuous aspects of the cost function by linear functions, and the assumption of a sufficiently infrequent switching of this substitution. Fatigue cost gradients are obtained efficiently by automatic differentiation. The method is implemented in an economic nonlinear model predictive controller (ENMPC), which optimally balances revenue and fatigue cost. This novel formulation is applicable to a very wide range of domains, and it is demonstrated here on the control of a wind turbine. The proposed ENMPC is fully parameterized by physical and monetary variables, and outperforms a conventional ENMPC based on the literature. The method is assessed by considering various metrics, including frequency spectra, damage estimation, switching, and gradient dynamics, which together provide useful insight into its main characteristics and an initial assessment of its performance.

Hierarchical nonlinear model predictive control of offshore hybrid power systems

This paper presents an approach for controlling ofshore hybrid power systems consisting of gas turbines, offshore wind, and batteries for satisfying an exogenous power demand. A hierarchical controller is developed comprising a high-level economic nonlinear model predictive controller that distributes the power demand according to some economic objective, a low-level nonlinear tracking model predictive controller that actuates on the hybrid power system, and a nonlinear moving horizon estimator to estimate the system state. Simulation results and concluding remarks reveal the advantage of such a hierarchical approach for a simple simulation study.

Energy Efficient Thermal Comfort Control for Residential Building Based on Nonlinear EMPC

For purpose of achieving the desired thermal comfort level and reducing the economic cost of maintaining the thermal comfort of green residential building, an energy efficient thermal comfort control strategy based on economic model predictive control (EMPC) for green residential buildings which adopts household heat metering is presented. Firstly, the nonlinear thermal comfort model of heating room is analyzed and obtained. A practical nonlinear thermal comfort prediction model is obtained by using an approximation method. Then, the economic cost function and optimization problem of energy-saving under the necessary thermal comfort requirements are constructed to realize the optimal economic performance of the dynamic process. The energy efficient thermal comfort MPC (EETCMPC) is designed. Finally, the comparison and analysis between EETCMPC and Double-layer Model Predictive Control (DMPC) is simulated. The simulation results reveal that when the clothing insulation is typical, the energy efficiency of EETCMPC is 8.9% and 11.6%, respectively, in the two simulation scenarios. When the clothing insulation varies with temperature, the energy efficiency of EETCMPC is 7.29% and 9.15%, respectively, and the total energy consumption is reduced by about 1.65% and 14.6%, respectively, compared with the typical clothing insulation. The economic performance is improved in the thermal comfort dynamic process of heating room.

Energy‐efficient cabin climate control of electric vehicles using linear time‐varying model predictive control

AbstractA cabin climate control system, often referred to as a heating, ventilation, and air conditioning (HVAC) system, is one of the largest auxiliary loads of an electric vehicle (EV), and the real‐time optimal control of HVAC brings a significant energy‐saving potential. In this article, a linear‐time‐varying (LTV) model predictive control (MPC)‐based approach is presented for energy‐efficient cabin climate control of EVs. A modification is made to the cost function in the considered MPC problem to simplify the Hessian matrix in utilizing quadratic programming for real‐time computation. A rigorous parametric study is conducted to determine optimal weighting factors that work robustly under various operating conditions. Then, the performance of the proposed LTV‐MPC controller is compared against a rule‐based (RB) controller and a nonlinear economic MPC (NEMPC) benchmark. Compared with the RB controller benchmark, the LTV‐MPC reaches the target cabin temperature at least 69 s faster with 3.2% to 15% less HVAC system energy consumption, and the averaged cabin temperature difference is 0.7°C at most. Compared with the NEMPC, the LTV‐MPC controller can achieve comparable performance in temperature regulation and energy consumption with fast computation time: the maximum differences in temperature and energy consumption are 0.4°C and 2.6%, respectively, and the computational time is reduced 72.4% on average with the LTV‐MPC.