Economic Cost Function Research Articles

Optimal switching of MPC cost function for changing active constraints

Model predictive control (MPC) allows for dealing with multivariable interactions, known future changes and dynamic satisfaction of constraints. Standard MPC has a cost function that aims at keeping selected controlled variables at constant setpoints. This work considers systems where the steady-state optimal active constraints change during operation. This situation is not handled optimally by standard MPC which uses fixed controlled variables for the unconstrained degrees of freedom. We propose a simple framework that detects the constraint changes and updates the controlled variables accordingly. The unconstrained controlled variables are chosen to be the reduced cost gradients, which when controlled to zero minimizes the steady-state economic cost. In this paper, the nullspace method for self-optimizing control is used to estimate the cost gradient using a static combination of the measurements. This estimated gradient is also used for detecting the current set of active constraints, which in particular allows for giving up constraints that were previously active. The proposed framework, here referred to as “region-based MPC”, is shown to be optimal for linear constrained systems with a quadratic economic cost function, and it allows for good economic performance in nonlinear systems in a neighborhood of the considered design points.

Output feedback distributed economic model predictive control for parallel system in process networks

Abstract This paper proposes a distributed economic model predictive control algorithm for parallel systems which uses only output feedback. Such parallel systems are a fundamental system architecture frequently encountered in process networks where, in many cases, the state of the plant is not measurable. Economic performance is a key consideration in the operation of such industrial plants and it is of interest to develop theoretically rigorous approaches to tackle what is a practically very relevant scenario. The competitive couplings and competitive constraints inherent in parallel systems are explicitly addressed in the proposed controller design framework. Three measures are considered for optimization of such parallel systems including an economic stage cost function, energy efficiency and tracking accuracy. An economic cost function is optimized by the resulting distributed controller which can realize global control performance while also reducing the computational time for large-scale parallel systems. Stability of the system is formally proved using the principles of dissipativity. Finally, the effectiveness of the proposed theoretical approach is verified by both numerical simulation and experimentation to demonstrate practical relevance.

Innovative air supply management in fuel cells: An economic predictive control approach without pre-measured OER

Existing fuel cell air supply control methods improve the economic performance of Proton Exchange Membrane Fuel Cells (PEMFC) under steady-state conditions through oxygen excess ratio curve, while ignoring the transient process of the air supply system under varying operating conditions. Furthermore, the oxygen excess ratio curve may exhibit deviations due to fuel cell aging and measurement inaccuracies, which would further diminish the economic performance derived from the control process. Not relying on pre-measuring the oxygen excess ratio curve, an EMPC (Economic Model Predictive Control) method is proposed for fuel cell air supply systems. It incorporates an economic cost function reflecting the economic objectives of the air supply system. By dynamically optimizing the parasitic power of the air compressor, this method enhances the overall economic performance by achieving performance optimization for both transient and steady-state processes of the fuel cell system. The asymptotic stability of PEMFC is achieved through the utilization of terminal penalty functions and terminal domain constraints. Stability and convergence analysis is carried out by employing auxiliary optimization problems and the Lyapunov method. In transient processes, EMPC method can achieve a maximum 4.97 % improvement in the economic performance of PEMFC. A hardware in loop (HIL) experiment was finally conducted to show the real-time capacity of the proposed EMPC method.

Data-driven economic MPC with asymptotic stability and strong duality verification using Hankel matrix

We consider the problem of dynamic regulation with an economic cost function to control unknown linear systems, in which improving the economic performance and guaranteeing the stability of economical optimal equilibrium point are control objectives. A data-driven economic MPC scheme is presented using measured input-output trajectories without a prior system identification step. Our method uses Hankel matrices which include one input-output data trajectory for prediction in economic MPC, while persistently exciting of the input generating the data is needed. One of the novelties of the presented framework is directly verifying the strong duality property from input-output trajectory with the general cost function, considered as the supply rate. This is used to find a Lyapunov function for data-driven economic MPC. Under the strong duality assumption, asymptotic stability of the economical optimal equilibrium point for the closed-loop system with terminal equality constraint is guaranteed. The proposed data-driven economic MPC approach needs only persistently exciting data trajectory along with an upper bound on the system order and need no model description and no online parameter estimation. The proposed scheme applicability compared to the existing model-based economic MPC and data-driven MPC is illustrated for continuous stirred tank reactor (CSTR) and a numerical example and the robustness of the proposed scheme is evaluated in the case of measurement noise, as well as nonlinear model for CSTR system.

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.

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.

Agent-based sensor location strategy for smart irrigation of large crop fields

Efficient monitoring of large crop fields is important to ensure the optimal use of resources such as water in irrigation policies, but at the same time represents a challenge to determine the structure of the sensor network. A balance must be accomplished between the acquisition, operation, and maintenance costs of this sensor network with the amount of information that can be collected in real-time to support the optimal use of the resource, e.g., an optimal irrigation policy. In this study, a sensor location strategy is proposed based on an agent-based model (ABM) of the crop–soil system, a state estimation algorithm reconstructing non-measured variables, and an objective function balancing the convergence of the estimation technique and the costs of the sensor network. The ABM model describes the crop–soil dynamics and allows conveniently representing uneven landscapes where water exchanges take place between different portions of the land. Various state estimation techniques can be considered and an extended Kalman filter is implemented in the present study, whose error covariance matrix can be exploited to assess practical observability and observer convergence. Finally, an economic cost function combines the observability measure with the sensor costs in order to select an optimal or suboptimal sensor array. For validation purposes, a numerical simulation case study, corresponding to a rugged land located in Colombia, is used to test various scenarios including the variability of climatic inputs.

Optimal design and operation of reactive extrusion processes: Application to the production and scale‐up of polyurethane rheology modifiers for paints

AbstractIn this work, the methodology for the optimal design, operation and scale‐up of reactive extrusion processes in twin‐screw extruders previously presented in Reference (Cegla and Engell, 2023). is applied to the production of hydrophobically ethoxylated urethanes (HEURs). The new process is a promising alternative to the current batch technology in large reactors with long residence times. We demonstrate the use of model‐based design and scale‐up for this case. A novel mechanistic finite volume twin‐screw extruder model is used as the process model, which is adapted to the process at hand by embedding a detailed description of the HEUR chemistry and rheology. An economic cost function is used to examine the scale‐up process from an 18 mm extruder to a 27 and 75 mm extruders, considering a selected range of products. The comparison between the optimization results obtained for the individual products with the optimization results for the production of multiple material grades using the same screw setup shows the high flexibility of the extruder‐based process. Significant energy savings compared with the conventional batch process can be achieved using reactive extrusion. To quantify the effort for the transition to a purely continuous production in terms of flexibility and process logistics, product changeovers are investigated.Highlights Intensification of the HEUR production by reactive extrusion. Detailed model for the twin‐screw extruder and the chemistry. Optimization of extruder, screw design, and operating conditions. Model‐based scale‐up from laboratory to industrial scale. Investigation of flexible industrial production of different HEURs.

The configurational optimization of the artificial reefs based on the fish-egg dynamics

Artificial reefs are frequently employed in fisheries management to improve the local ecological environment and to contribute to the enhancement of fishery resources. The placement of artificial reefs can attract adult fish, but they also serve as a good environment for sticky fish eggs to hatch and grow. Fish-egg dynamics are highly dependent on the surrounding environment, which is greatly influenced by the configuration of reefs. The scientific design of the reef block number is significant from both an ecological view and economic view. This study sets up a numerical model to simulate the flow field and viscous fish eggs in a reef area, based on the mesoscopic lattice Boltzmann framework coupled with Eulerian and Lagrangian perspectives. This model is verified by particle image velocimetry (PIV), which is a measurement technique for visualizing fluids by displaying velocity vectors in a two-dimensional manner, and obtains the spatial distribution characteristics of fish eggs with various configurations of artificial reefs. To achieve the best configuration of fish reefs in the applied waters, the economic cost and ecological function of fish reefs are balanced using objective analysis. This is a reliable and practical way to take into account capital efficiency and the impact of restoration when designing reef zones.

A Fractional-Order On-Line Self Optimizing Control Framework and a Benchmark Control System Accelerated Using Fractional-Order Stochasticity

This paper presents a design and evaluation of a fractional-order self optimizing control (FOSOC) architecture for process control. It is based on a real-time derivative-free optimization layer that adjusts the parameters of a discrete-time fractional-order proportional integral (FOPI) controller according to an economic cost function. A simulation benchmark is designed to assess the performance of the FOSOC controller based on a first order plus dead time system. Similarly, an acceleration mechanism is proposed for the fractional-order self optimizing control framework employing fractional-order Gaussian noise with long-range dependence given by the Hurst exponent. The obtained results show that the FOSOC controller can improve the system closed-loop response under different operating conditions and reduce the convergence time of the real-time derivative-free optimization algorithm by using fractional-order stochasticity.

Discrete‐time contraction constrained nonlinear model predictive control using graph‐based geodesic computation

AbstractModern chemical processes need to operate around time‐varying operating conditions to optimize plant economy, in response to dynamic supply chains (e.g., time‐varying specifications of product and energy costs). As such, the process control system needs to handle a wide range of operating conditions whilst optimizing system performance and ensuring stability during transitions. This article presents a reference‐flexible nonlinear model predictive control approach using contraction based constraints. Firstly, a contraction condition that ensures convergence to any feasible state trajectories or setpoints is constructed. This condition is then imposed as a constraint on the optimization problem for model predictive control with a general (typically economic) cost function, utilizing Riemannian weighted graphs and shortest path techniques. The result is a reference flexible and fast optimal controller that can trade‐off between the rate of target trajectory convergence and economic benefit (away from the desired process objective). The proposed approach is illustrated by a simulation study on a CSTR control problem.

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.

Online Convex Optimization for Data-Driven Control of Dynamical Systems

We propose an algorithm based on online convex optimization for controlling discrete-time linear dynamical systems. The algorithm is data-driven, i.e., does not require a model of the system, and is able to handle a priori unknown and time-varying cost functions. To this end, we make use of a single persistently exciting input-output sequence of the system and results from behavioral systems theory which enable it to handle unknown linear time-invariant systems. Moreover, we consider noisy output feedback instead of full state measurements and allow general economic cost functions. Our analysis of the closed loop reveals that the algorithm is able to achieve sublinear regret, where the measurement noise only adds an additional constant term to the regret upper bound. In order to do so, we derive a data-driven characterization of the steady-state manifold of an unknown system. Moreover, our algorithm is able to asymptotically exactly estimate the measurement noise. The effectiveness and applicational aspects of the proposed method are illustrated by means of a detailed simulation example in thermal control.

Economic Model Predictive Control for Post-Combustion CO2 Capture System Based on MEA

For the post-combustion CO2 capture (PCC) system, the time variability of the economic performance is key to the production process of such an actual industrial process. However, the performance index used by the conventional model predictive control (MPC) does not reflect the economy of the production process, so the economic cost function is used instead of the traditional performance index to measure the economy of the production process. In this paper, a complete dynamic model of the PCC system is constructed in Aspen Plus Dynamics. The effectiveness of the model is verified by dynamic testing; subspace identification is carried out using experimental data, a state-space equation between flue gas flow and lean solvent flow; the CO2 capture rate is obtained; and dynamic models and control algorithm models of accused objects are established in Matlab/Simulink. Under the background of the environmental protection policy, an economic model predictive control (EMPC) strategy is proposed to manipulate the PCC system through seeking the optimal function of the economic performance, and the system is guaranteed to operate under the economic optimal and excellent quality of the MPC control strategy. The simulation results verify the effectiveness of the proposed method.

Periodically time-varying economic model predictive control with applications to nonlinear continuous stirred tank reactors

The economic cost functions can be periodically time-varying due to various economic factors such as the variations of product prices, energy consumptions, operating costs and customer demand changes. These time-varying parameters can lead to instability because the optimal steady state may not be asymptotically stable. In this paper, a periodically time-varying economic model predictive control algorithm is proposed. As the time-varying parameters in the economic cost functions vary, the average constraints are used to enforce the convergence without modifying the original economic cost functions so the transient economic performance can be improved. The time-varying terminal regions are included in the economic optimization problem in order to ensure recursive feasibility. The asymptotic convergence is proved by using the Lyapunov-like analysis. The developed predictive control algorithm is applied to the nonlinear continuous stirred tank reactors in order to gain the highest economic profits.

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.

Multi-objective optimization model for fuel cell-based poly-generation energy systems

Although the layout of combined cooling, heat and power (CCHP) systems is well known from a technical perspective, the optimal operating strategy and sizing of such systems are relatively complex, due to the high number of variables and constraints involved. To address this research gap, the present paper describes the formalization, implementation, and validation of an innovative multi-objective optimization model for CCHP poly-generative energy systems. As a novelty, the model uses two levels of optimization: a first level to optimize the best operating strategy of the system for different unit sizes, minimizing an economic cost function, maximizing the performance of the plant, and minimizing polluting emissions; the second level to optimize the size of the plant, through a technical-economic optimization function based on the Net Present Value. The model is highly flexible, allowing the execution of sensitivity analyses. The analyzed energy system is composed of a cogeneration unit, namely solid oxide and polymer electrolyte membrane fuel cells, and of heat pumps, electrical storage, and traditional auxiliary boilers. In order to prove the reliability of the CCHP units, the modeled performance curves for the fuel cell systems have been validated with existing commercial units, resulting in a degree of correlation mostly over 90%.

Oracle-based economic predictive control

This paper presents an economic model predictive controller, under the assumption that the only measurable signal of the plant is the economic cost to be minimized. In order to forecast the evolution of this economic cost for a given input trajectory, a prediction model with a NARX structure, the so-called oracle, is proposed. Sufficient conditions to ensure the existence of such oracle are studied, proving that it can be derived for a general nonlinear system if the economic cost function is a Morse function. Based on this oracle, economic model predictive controllers are proposed, and their stability is demonstrated in nominal conditions under a standard dissipativity assumption. The viability of these controllers in practical settings (where the oracle may provide imperfect predictions for generic inputs) is proven by means of input-to-state stability. These properties have been illustrated in a case study based on a continuously stirred tank reactor.

Optimal Load Frequency Control for Networked Power Systems Based on Distributed Economic MPC

This article proposes an economic model predictive algorithm for optimal load frequency control, in which both the frequency regulation and economic load dispatch (ELD) are considered, in interconnected power systems. Two-layer hierarchical control can be achieved through one level by EMPC. An economic stage cost function, including ELD and frequency regulation, which can be written in general convex form, is optimized by the controller. The distributed way is utilized to realize the control of large-scale power systems. Each subsystem-based controller works cooperatively with neighboring subsystems to achieve system-wide control performance. Asymptotic stability of the system is guaranteed by the proper terminal cost function. The efficiency and advantages of the proposed method are manifested by the simulation.

Steady-state target calculation integrating economic optimization for constrained model predictive control

Real-time optimization (RTO) in cascade model predictive control (MPC) is a classic two-layer architecture for economic optimization and dynamic control. The two-layer architecture is always integrated with steady-state target calculation (SSTC) to guarantee the feasibility of the control target. In this paper, we present a modified SSTC algorithm to recalculate the steady-state point for MPC when the optimal point calculated by RTO is no longer attainable. The proposed algorithm takes economic performance into account and retains the feasibility of solutions. Based on a classic SSTC formulation, ours has a new objective function with two terms, one for optimizing economic performance and the other for ensuring the feasibility. Moreover, the objective function of the proposed SSTC is rearranged, and the SSTC problem is formulated by quadratic programming, which can be solved in an instant for real applications. Furthermore, we provide a detailed 4-in-4-out model of the Fischer esterification reaction, which is similar to the benchmark continuous stirred tank reactor (CSTR) model, to demonstrate the effectiveness of our SSTC algorithm. The proposed algorithm is able to improve economic performances, whether a convex or non-convex economic cost function is concerned.