Economic Model Predictive Control Scheme Research Articles

Dynamic wind farm flow control using free-vortex wake models

Abstract. A novel dynamic economic model-predictive control strategy is presented that improves wind farm power production and reduces the additional demands of wake steering on yaw actuation when compared to an industry state-of-the-art reference controller. The novel controller takes a distributed approach to yaw control optimisation using a free-vortex wake model. An actuator-disc representation of the wind turbine is employed and adapted to the wind farm scale by modelling secondary effects of wake steering and connecting individual turbines through a directed graph network. The economic model-predictive control problem is solved on a receding horizon using gradient-based optimisation, demonstrating sufficient performance for realising real-time control. The novel controller is tested in a large-eddy simulation environment and compared against a state-of-the-art look-up table approach based on steady-state model optimisation and an extension with wind direction preview. Under realistic variations in wind direction and wind speed, the preview-enabled look-up table controller yielded the largest gains in power production. The novel controller based on the free-vortex wake produced smaller gains in these conditions while yielding more power under large changes in wind direction. Additionally, the novel controller demonstrated potential for a substantial reduction in yaw actuator usage.

Efficient management of HVAC systems through coordinated operation of parallel chiller units: An economic predictive control approach

In developed countries, air conditioning systems have become major contributors to energy consumption in buildings. Cooling installations made up of independent chiller units connected in parallel pose a challenge in finding the most energy-efficient management strategy. This article proposes a novel economic model predictive control strategy to optimize the operation of multiple-chiller HVAC systems according to an economic cost comprising energy consumption as well as a thermal comfort index. Provided that the gradient of the economic cost function can be calculated or estimated, the proposed controller only entails the solution of a single quadratic programming (QP) problem at each sampling period, reducing computational requirements and thus facilitating the deployment on commercial embedded HVAC management platforms. The proposed controller improves economic performance. As our numerical analysis shows, optimal sequencing is achieved for the case of dual-chiller plants. Moreover, the controller adapts to possible variations in the economic criteria, enabling the system to respond to changes in electricity price and user preferences. A realistic case study, using a high fidelity model of a building simulated in TRNSYS, demonstrates the effectiveness of the proposed methodology. In particular, the proposed controller demonstrates to be more efficient than state-of-the-art QP-based economic predictive controllers in achieving a reduction of the energy consumption up to 5.19%, which is in line with the targeted reduction of 7.9% by 2030 in the EU Green Deal's ‘Fit for 55’ package.

MPPT Strategy of Waterborne Bifacial Photovoltaic Power Generation System Based on Economic Model Predictive Control

At present, the new energy industry represented by photovoltaics has become the main force to realize the optimization of China’s energy structure and the goal of “double carbon”; with the absence of land resources, the waterborne bifacial photovoltaic has ushered in a new opportunity. Therefore, in order to address the problem that the maximum power point tracking (MPPT) of photovoltaics (PV) could not take into account, the dynamic economic performance in the control process, an economic model predictive control (EMPC), is proposed in this work to realize the MPPT of the waterborne bifacial PV power generation system. Firstly, the model of the bifacial PV module is constructed by combining the ray-tracing irradiance model and considering the effect of water surface albedo on the irradiance absorbed by the module. Secondly, the EMPC controller is designed based on the state-space model of the system to maximize the power generation as the economic performance index, and to solve the optimal input variables time by time to achieve a rolling optimization with the operational requirements of the system itself as the constraints. Thirdly, the MATLAB/Simulink (R2022a) simulation experimental results verify that the EMPC strategy could be utilized to achieve MPPT of the waterborne bifacial PV power generation system, according to the changes of environment. Finally, it is also demonstrated that the bifacial PV power generation system that employed the EMPC strategy outperformed the traditional MPPT algorithm, with respect to both output power tracking velocity and accuracy, and the power generation could be improved by about 6% to 14.5%, which significantly enhances the system’s dynamic process economics.

Linearly discounted economic MPC without terminal conditions for periodic optimal operation

In this work, we study economic model predictive control (MPC) in situations where the optimal operating behavior is periodic. In such a setting, the performance of a standard economic MPC scheme without terminal conditions can generally be far from optimal even with arbitrarily long prediction horizons. Whereas there are modified economic MPC schemes that guarantee optimal performance, all of them are based on prior knowledge of the optimal period length or of the optimal periodic orbit itself. In contrast to these approaches, we propose to achieve optimality by multiplying the stage cost by a linear discount factor. This modification is not only easy to implement but also independent of any system- or cost-specific properties, making the scheme robust against online changes therein. Under standard dissipativity and controllability assumptions, we can prove that the resulting linearly discounted economic MPC without terminal conditions achieves optimal asymptotic average performance up to an error that vanishes with growing prediction horizons. Moreover, we can guarantee practical asymptotic stability of the optimal periodic orbit under the additional technical assumption that dissipativity holds with a continuous storage function. We complement these qualitative guarantees with a quantitative analysis of the transient and asymptotic average performance of the linearly discounted MPC scheme in a numerical simulation study.

Offshore wind power generation system control using robust economic MPC scheme

With the increasing penetration of wind power in the modern power system, the economic model predictive control (EMPC) has been widely employed in the control of the offshore wind power generation system for promoting the dynamic economy. A robust EMPC strategy, aiming to minimizing damage to the turbine while maximizing the electric power output, is developed in this paper to enhance the dynamic economic performance under uncertain wind speed variation. A linear model is utilized to approximate the original nonlinear behavior of the wind power generation system, while the consequent linearization error is taken as the external disturbance of the control system. A linear feedback controller with a robust control invariant set is designed to restrict the deviation between the nominal linear system and the actual nonlinear wind power generation system. Plant simulations, as well as NREL FAST code testing, are given to demonstrate the proposed controller's effectiveness throughout the overall operating region.

Enhanced reinforcement learning in two-layer economic model predictive control for operation optimization in dynamic environment

Economic model predictive control (EMPC) has attracted an abundance of interest in both academic and industrial communities in recent years because it is able to increase the economic profits of dynamic systems. For greater computational efficiency, two-layer EMPC schemes are applied in some complex process control. However, the performance of two-layer EMPC is significantly influenced by the accuracy of the chosen process model. Reinforcement learning (RL) has been studied as a model-free strategy of model-based control approaches, but its safety and stability remain a concern. In order to estimate the model parameters of nonlinear dynamic systems in real time, this work introduces a unique scheme for merging two-layer EMPC and RL. In this scheme, the two-layer EMPC technique maintains closed-loop stability and recursive feasibility while operating the closed-loop dynamic system optimally. And the RL agent continually compares the observed states to the predictions made by the EMPC and modifies the time-varying parameters as necessary. The usability of the proposed scheme is shown on a chemical process of ethylene oxide production in dynamic environment. This work enables online and continuous control, optimization, and model correction, and makes process production optimization more feasible and profitable.

Stable feedback linearization-based economic MPC scheme for thermal power plant

The major concern of modern power plants is changing from tracking control to environmental and economic issues. The economic model predictive control (EMPC) scheme, which incorporates the boiler-turbine unit's dynamic tracking and economic optimization into one online framework, can well enhance the dynamic economic performance. Considering the strong nonlinearity that existed in the boiler-turbine system, this paper presents an advanced EMPC scheme based on the input/output feedback linearization (IOFL) approach. The boiler-turbine dynamics are converted via the IOFL method to a linear form, which can be readily used to constitute the standard EMPC scheme. A dual-mode method is adopted in this paper to guarantee the stability of the IOFL EMPC strategy for the boiler-turbine system, in which the first-mode optimizes the economic objective function while preserving the states of the system within a feasible region, and the second-mode moves the system state to the optimum operating point using an auxiliary controller. The simulations under the MATLAB environment demonstrate that the application of the IOFL-based EMPC scheme enhances the economic and dynamic output performance under load demand changes in comparison with fuzzy hierarchical MPC and fuzzy economic MPC schemes.

Approximate Scenario-Based Economic Model Predictive Control With Application to Wind Energy Conversion System

This paper considers the effective handling of uncertainty for economic model predictive control with feasibility and stability guarantees. Firstly, a stable scenario-based economic model predictive control strategy is proposed based on Lyapunov techniques. This control strategy optimizes over a sequence of control policies instead of a sequence of control inputs, so as to take feedback into account to reduce the conservativeness. More uncertainty information over the prediction horizon is incorporated by employing an augmented prediction model with a scenario tree describing the evolution of the uncertainty. Secondly, since the scenario tree structure inevitably increases the optimization problem size, a trained deep neural network, as an approximation function, is resorted to modeling the scenario-based economic model predictive control feedback control law to make online implementation tractable. The effectiveness of this approximate controller is verified through the probabilistic validation technique. Finally, the feasibility and stability of this approximate scenario-based economic model predictive control are addressed theoretically. An application of this proposed controller on wind energy conversion systems demonstrates its effectiveness.

Economic model predictive control for a rougher froth flotation cell using physics-based models

The development of an economic model predictive control (E-MPC) strategy is presented. The strategy uses a novel dynamic flotation model that incorporates the physics of the froth phase in a flotation cell. The dynamic model was previously calibrated and validated using experimental data.Sensitivity analyses were conducted to select a suitable objective function that accounted for both process economics and control variable sensitivities. While the ultimate goal of a rougher flotation cell is to maximise the metallurgical recovery at a steady state for a specified minimum grade, it was evident that the incorporation of air recovery dynamics (which can be measured in real-time) and concentrate grade dynamics (calculated through first-principle models) led to the best results. The addition of a dynamic variable that can be easily measured online, i.e. air recovery, offers great potential to improve plant performance in existing froth flotation systems. Furthermore, a minimum concentrate grade was imposed in the E-MPC strategy. This acts as an economic constraint as it allows the metallurgical recovery to be optimised while ensuring that concentrate grade requirements are met.The dynamic optimisation problem for the E-MPC strategy was discretised using orthogonal collocations, and was implemented in Matlab using automatic differentiation via CasADi. Two typical manipulated variables were considered: air flowrate and pulp height setpoints. Based on laboratory-scale data, the implementation of the E-MPC strategy resulted in improvements ranging from +8 to +22 % in metallurgical recovery, while maintaining the specified grade. This is therefore an encouraging control strategy to explore in larger flotation systems.

Economic model predictive control based on lattice trajectory piecewise linear model for wastewater treatment plants

Economic model predictive control (EMPC) is an effective control strategy for wastewater treatment plants (WWTPs), which are crucial for preventing water pollution and improving the water quality. However, industrial processes typically involve large-scale nonlinear and strongly coupled systems, and it is computationally expensive to solve nonlinear optimization problems based on nonlinear prediction models such as EMPC. In this study, to facilitate the application of EMPC to large-scale nonlinear systems such as WWTPs, the lattice trajectory piecewise linear (PWL) model is used to approximate the nonlinear system with a predefined error bound. The resulting optimization problem can be expressed as a continuous PWL programming problem if the cost function of the EMPC is linear. Therefore, an iterative descent algorithm is proposed to transform the PWL programming problem into a series of linear programming problems. The stability of the nonlinear system operating under EMPC is analyzed. The EMPC scheme based on the lattice trajectory PWL model is applied to a WWTP benchmark problem, in which the state is 78-dimensional and the EMPC cost is linear. The proposed strategy outperforms the EMPC schemes based on the nonlinear prediction model and trajectory PWL prediction model. Overall, the EMPC scheme based on the lattice trajectory PWL prediction model can improve the computational efficiency of optimization problems while ensuring control performance.

Dual-zone economic model predictive control of residential space heating for demand response using a single heat meter

Recent research has indicated a significant potential for providing demand response using Economic Model Predictive Control (E-MPC) of radiators in residential space heating systems. Previous studies indicate the need for distinct temperatures in different rooms, which would require an E-MPC with different temperature constraints across the dwelling. Studies on E-MPC concepts for this purpose often assume that it is possible to measure the heating power delivered to each room separately. However, this assumption is not valid in typical existing residential buildings heated by hydronic radiators; they only have one heat meter measuring the heating use of the entire housing unit. This study presents a simulation-based study of a novel dual-zone E-MPC scheme that relies on indoor air temperature measurements in individual rooms of the housing unit, and only one heat meter measuring the total heating use for space heating and domestic hot water. The results indicate that the proposed scheme has the same load shifting potential as an E-MPC using room-level heat metering. The proposed scheme thus reduces the additional hardware and data infrastructure needed for the practical realization of E-MPC of residential space heating systems for demand response purposes.

Evaluation of Changes in Feed Particle Size within an Economic Model Predictive Control Strategy for Froth Flotation

This study presents the evaluation of the impact of feed particle size on an Economic Model Predictive Control (E-MPC) strategy for a flotation bank. The effect of particle size was assessed under two scenarios: (1) assuming constant floatability with no dependency on particle size, and (2) assuming variable floatability as a function of particle size. The E-MPC strategy uses a dynamic model that includes froth physics, which was previously calibrated and validated using experimental data. Two typical control variables were considered: air flowrate and pulp height setpoints. The proposed objective function depends on three flotation variables: (1) dynamic air recovery, a measurable variable used to quantify froth stability and is directly related to flotation performance, (2) metallurgical recovery at steady-state, and (3) dynamic concentrate grade. A moving horizon estimator (MHE) was implemented to estimate the model states in both scenarios. Simulation results showed that the estimation of metallurgical indicators (concentrate grade and recovery) was significantly affected by changes in the floatability parameter. A poor estimate of floatability is likely to lead to very different results for the control strategy. Future research should focus on estimating and updating the most significant parameters of the dynamic model, such as floatability, with an appropriate sampling time.

Safe and Robust Cloud-based Predictive Control for Water Distribution Networks

Water distribution networks with storage equalize the operating pressure in the system. Apart from providing pressure, elevated reservoirs also allow for exploiting fluctuation in electricity prices. In this work, we use an Economic Model Predictive Control scheme for planning the filling times of storage in water distribution networks. This controller needs access to electricity prices via an Information Technology (IT) connection to a power-trading system. Therefore, we propose a control architecture, where a Global Controller is deployed on a cloud server, whereas a locally placed data-acquisition and supervisory controller provide the safety guarantees in terms of an Operational Technology (OT) layer. The approach is demonstrated numerically on a real-world network architecture where real data is used to verify the robustness of the method.

Economic Linear Parameter Varying Model Predictive Control of the Aeration System of a Wastewater Treatment Plant.

This work proposes an economic model predictive control (EMPC) strategy in the linear parameter varying (LPV) framework for the control of dissolved oxygen concentrations in the aerated reactors of a wastewater treatment plant (WWTP). A reduced model of the complex nonlinear plant is represented in a quasi-linear parameter varying (qLPV) form to reduce computational burden, enabling the real-time operation. To facilitate the formulation of the time-varying parameters which are functions of system states, as well as for feedback control purposes, a moving horizon estimator (MHE) that uses the qLPV WWTP model is proposed. The control strategy is investigated and evaluated based on the ASM1 simulation benchmark for performance assessment. The obtained results applying the EMPC strategy for the control of the aeration system in the WWTP of Girona (Spain) show its effectiveness.

A stabilizing cooperative-distributed gradient-based economic model predictive control strategy for constrained linear systems

This paper proposes a stabilizing distributed gradient-based economic MPC strategy for cooperative control of constrained linear systems. This strategy is based on quadratic programming to limit the computational cost, and one key feature is the evaluation of an economically optimal steady-state directly in the control law, leading the closed-loop system towards it. The control law includes three ingredients to enlarge its domain of attraction and potentially avoid feasibility problems, namely: artificial steady-state, slack variables, and terminal equality constraints solely on non-stable states. A four-tank system is used to address the properties of the formulation, especially its stability in the nominal case, the capacity to track the desired target for a nonlinear performance economic function, and tackle unmeasured and persistent unmeasured disturbances.

Large-scale wind farm control using distributed economic model predictive scheme

The reliable control of the large-scale wind farm is crucial for the stability and security of the renewable power system with high wind power penetration. Due to the uncertain and variable nature of wind power, the traditional control strategy is difficult to work. Regarding the large-scale, geographically dispersed wind farm, an efficient distributed economic model predictive control strategy is proposed, which integrates the power tracking and economic optimization of the wind farm into one optimal control framework. By adopting the global economic cost function, the Nash optimal solutions under distributed framework approach the Pareto optimum. Thus, the reference power from the transmission system operator is accurately tracked, while the global dynamic economic optimality is guaranteed. The simulation results under step wind speed and practical wind speed variations verify the efficiency and reliability of the proposed control strategy.

A reinforcement learning-based economic model predictive control framework for autonomous operation of chemical reactors

Economic model predictive control (EMPC) is a promising methodology for optimal operation of dynamical processes that has been shown to improve process economics considerably. However, EMPC performance relies heavily on the accuracy of the process model used. As an alternative to model-based control strategies, reinforcement learning (RL) has been investigated as a model-free control methodology, but issues regarding its safety and stability remain an open research challenge. This work presents a novel framework for integrating EMPC and RL for online model parameter estimation of a class of nonlinear systems. In this framework, EMPC optimally operates the closed loop system while maintaining closed loop stability and recursive feasibility. At the same time, to optimize the process, the RL agent continuously compares the measured state of the process with the model’s predictions (nominal states), and modifies model parameters accordingly. The major advantage of this framework is its simplicity; state-of-the-art RL algorithms and EMPC schemes can be employed with minimal modifications. The performance of the proposed framework is illustrated on a network of reactions with challenging dynamics and practical significance. This framework allows control, optimization, and model correction to be performed online and continuously, making autonomous reactor operation more attainable.

Offset-Free Economic MPC Based on Modifier Adaptation: Investigation of Several Gradient-Estimation Techniques

Various offset-free economic model predictive control schemes that include a disturbance model and the modifier-adaptation principle have been proposed in recent years. These schemes are able to reach plant optimality asymptotically even in the presence of plant–model mismatch. All schemes are affected by a major issue that is common to all modifier-adaptation formulations, namely, plant optimality (note that convergence per se does not require perfect plant gradients) requires perfect knowledge of static plant gradients, which is a piece of information not known in most practical applications. To address this issue, we present two gradient-estimation techniques, one based on Broyden’s update and the other one on linear regression. We apply these techniques for the estimation of either the plant gradients or the modifiers directly. The resulting economic MPC schemes are tested in a simulation and compared on two benchmark examples of different complexity with respect to both convergence speed and robustness to measurement noise.

Economic model predictive control of nonlinear systems using a linear parameter varying approach

AbstractThis article proposes an economic model predictive control (EMPC) approach for linear parameter varying (LPV) systems. An efficient implementation of the associated MPC optimization problem is introduced based on transforming the LPV model into a linear time‐varying one by using an estimation of the scheduling variables along the prediction horizon. This estimation is based on the optimal states/inputs determined from the solution of the previous optimization problem while running the receding horizon strategy. Using this approach, the solution of the proposed LPV‐based EMPC scheme would be possible by solving a series of quadratic programming problems at each time instant. This approach allows reducing the computational burden compared with the solution of a nonlinear optimization problem that would result naturally from the LPV‐based formulation. The stability of the proposed approach is guaranteed by forcing the terminal state to converge towards the optimal equilibrium/working point of the system. Moreover, the terminal constraint is relaxed by using a constraint set around the terminal state instead of a constraint value and adding a penalty on the terminal state into the cost function. Besides, strict dissipativity is established as a sufficient condition to prove stability. Finally, the effectiveness of the LPV‐based EMPC strategy is shown by controlling a small‐scale pasteurization system in simulation. The comparison between the proposed control approach and standard MPC approaches is performed. Results show the advantages of the proposed LPV‐based EMPC controller in terms of economic cost minimization.

A distributed network‐based economic model predictive control for networked smart energy systems considering communication network constraints

Generally, the modern networked energy systems are constructed from some coupled sub-systems interconnected via their states and share their information through a communication network featuring time delays. This article suggests an efficient multi-agent based cooperative-distributed networked economic model predictive control scheme for energy consumption optimisation and reducing the cost of consumed electricity in network-based smart energy systems considering communication network delays. In this regard, a buffer-based moving horizon strategy with estimator is suggested to estimate the own states of every sub-system and the coupled states, that is, the states in which their information exchange among sub-systems and their values are not accessible due to communication time delays. Moreover, the boundedness of the estimation error as well as the stability of the closed-loop system are established. The usability and proficiency of the suggested scheme are proved by applying the developed approach on two case studies.