Economic Cost Function Research Articles

Economic Model Predictive Control of a Point Absorber Wave Energy Converter

In this paper, we propose an economic model predictive control (EMPC) strategy for the constrained optimization problem of the wave energy converter system. Unlike standard model predictive control (MPC) that uses the tracking cost function, in EMPC, a general economic cost function that directly reflects the economic objective of the system is established. In the wave energy converter (WEC), the general economic cost function aims to maximize the energy extracted from ocean waves and minimize the operation cost. The terminal equality constraint in the optimization problem is used to steer the system state to the steady-state at the end of the optimization horizon. Then the controller solves the non-convex optimization problem in real-time. The auxiliary optimization problem is used for the stability analysis, and the convergence of the system can be proved via the Lyapunov technique. Several simulation results are presented to demonstrate the effectiveness of the proposed strategy.

Design and analysis of a renewable energy power system for shale oil exploitation using hierarchical optimization

Abstract The exploitation of shale oil resources has attracted the wide attention. Using traditional fossil fuel as power supply for shale oil in-situ electric heating mining has an impact on the environment. In this paper, we propose a renewable energy microgrid system as a power supply for in-situ heating mining system of shale oil. A hierarchical optimization model was developed to optimize the capacity allocation of the microgrid components, which employs HOMER to initial optimization, then an improved multi-objective particle swarm algorithm with the lowest the economic cost function and the annual load shortage rate index of the system is implemented. A hierarchical hybrid strategy game method is used to screen the generated Pareto set to meet the different preferences of decision-makers. The sensitivity analysis of system reserve capacity, wind resources, and wind turbine prices was carried out. The simulation results using data from a demonstration project in China verified that the proposed method optimized the use of renewable energy, reduced the operating costs and produced positive environmental benefits. The results of this study provide a useful reference for decision-makers to plan the power supply scheme of shale oil extraction by in-situ electric heating.

Fast sensitivity-based economic model predictive control for degenerate systems

We present a sensitivity-based nonlinear model predictive control (NMPC) algorithm and demonstrate it on a case study with an economic cost function. In contrast to existing sensitivity-based approaches that make strong assumptions on the underlying optimization problem (e.g. the linear independence constraint qualification implying unique multiplier), our method is designed to handle problems satisfying a weaker constraint qualification, namely the Mangasarian-Fromovitz constraint qualification (MFCQ). Our nonlinear programming (NLP) sensitivity update consists of three steps. The first step is a corrector step in which a system of linear equations is solved. Then a predictor step is computed by a quadratic program (QP). Finally, a linear program (LP) is solved to select the multipliers that give the correct sensitivity information. A path-following scheme containing these steps is embedded in the advanced-step NMPC (asNMPC) framework. We demonstrate our method on a large-scale case example consisting of a reactor and distillation process. We show that LICQ does not hold and the path-following method is able to accurately approximate the ideal solutions generated by an NLP solver.

A Modifier-Adaptation Approach to the One-Layer Economic MPC

In this paper, we address the problem of modeling error in economically optimal control. A single layer controller is proposed that integrates the economical part of the Real Time Optimization (RTO), the dynamic part of the Model Predictive Control (MPC) and the Modifier Adaptation strategy (MA), resulting in a controller with the following characteristics: a) recursive feasibility guarantee of the controller; b) asymptotic closed-loop stability for any change in the economic cost function; c) convergence guarantee to the economic optimum of the real plant (offset-free) for any change in the cost function of the controller; and d) simple implementation of the controller. We show the behaviour of the proposal by means of a motivating example that highlights the performance of the proposed algorithm.

Cooperation-Based Distributed Economic MPC for Economic Load Dispatch and Load Frequency Control of Interconnected Power Systems

In this letter, a distributed economic model predictive control (EMPC) strategy is proposed for economic load dispatch and load frequency control of interconnected power systems. A general economic cost function is used in the EMPC to calculate the control sequence directly optimizes the economic performance of the system without using a hierarchical control structure, thus achieving both the real-time optimization and transient optimization. A cooperation-based distributed EMPC algorithm is proposed and the asymptotic stability is proven. Simulation results reveal the advantages of the proposed method.

Economic model predictive control for the operation optimization of water distribution networks with risks

AbstractFor water distribution networks (WDNs) with risks of insufficient water supply, some users' water demand is affected. Because the operation cost is positively related to the number of users whose water demand cannot be met, making the most of the limited water sources to meet the water demand of as many users as possible is the key to reduce the related cost in the operation optimization. Considering the time‐varying of users' water demand and the dynamic characteristics of WDNs, real‐time dynamic operation optimization of WDNs is required. In this paper, an economic cost function which connects the cost incurred by the risks and the controllable pressure is proposed based on the distribution of users and the corresponding water supply method. An EMPC strategy with the ability to guarantee the closed‐loop stability is adopted for the optimization problem. Performance of the proposed methods is illustrated based on the simulation of the Newtown case at different risk levels. Simulation results show that the extra cost decreases greatly in the situation with risks by applying the proposed pressure‐related economic function and the EMPC strategy.

A self-interested distributed economic model predictive control approach to battery energy storage networks

In this work, a dissipativity based distributed economic model predictive control (DEMPC) approach is developed for the operation of battery energy storage (BES) networks in residential microgrids. With the presence of a microgrid power market (MPM), control of the BES systems is formulated as a self-interested distributed control problem, as individual DEMPC controllers minimize their local economic cost functions based on the price prediction of MPM. Due to the intermittent nature of photovoltaic (PV) power generations and load demands, the DEMPC without proper coordination or constraints may lead to excessive energy trading and price oscillations in MPM. To solve this problem, dissipativity theory with dynamic supply rates is adopted in this paper to deal with the interactions between individual users and the MPM. The microgrid-wide performance requirement of attenuation of the net power fluctuations with respect to time-varying PV generation and demands, is converted into the dissipative trajectory constraints imposed on individual DEMPC controllers. The proposed approach is scalable as it does not require online iterative optimizations across the controller network. A case study is presented to illustrate the proposed method.

A Comparative Study of MPC and Economic MPC of Wind Energy Conversion Systems

In this work, we perform a comprehensive comparative study of two advanced control algorithms—the classical tracking model predictive control (MPC) and economic MPC (EMPC)—in the optimal operation of wind energy conversion systems (WECSs). A typical 5 MW wind turbine is considered in this work. The tracking MPC is designed to track steady-state optimal operating reference trajectories determined using a maximum power point tracking (MPPT) algorithm. In the design of the tracking MPC, the entire operating region of the wind turbine is divided into four subregions depending on the wind speed. The tracking MPC tracks different optimal reference trajectories determined by the MPPT algorithm in these subregions. In the designed EMPC, a uniform economic cost function is used for the entire operating region and the division of the operating region into subregions is not needed. Two common economic performance indices of WECSs are considered in the design of the economic cost function for EMPC. The relation between the two economic performance indices and the implications of the relation on EMPC performance are also investigated. Extensive simulations are performed to show the advantages and disadvantages of the two control algorithms under different conditions. It is found that when the near future wind speed can be predicted and used in control, EMPC can improve the energy utilization by about 2% and reduce the operating cost by about 30% compared to classical tracking MPC, especially when the wind speed varies such that the tracking MPC switches between operating subregions. It is also found that uncertainty in information (e.g., future wind speed, measurement noise in wind speed) may deteriorate the performance of EMPC.

Economic model predictive control based on a periodicity constraint

This paper addresses a novel economic model predictive control (MPC) formulation based on a periodicity constraint to achieve an optimal periodic operation for discrete-time linear systems. The proposed control strategy does not rely on forcing the terminal state by means of a terminal equality constraint and hence it does not require a priori knowledge of a periodic steady trajectory. Instead, at each sampling time step the economic cost function is optimized based on a periodicity constraint over all the periodic trajectories that include the current state. The recursive feasibility and the closed-loop convergence to a periodic steady trajectory are discussed. Moreover, an optimality certificate of this steady trajectory is provided based on the Karush–Kuhn–Tucker (KKT) optimality conditions. Finally, an application to a well-known water distribution network benchmark is presented to demonstrate the proposed economic MPC in which the closed-loop simulation results obtained with a linear model and a virtual–reality simulator are both provided.

A real-time optimization framework for the time-varying economic environment

In this paper, we propose a conceptual framework for nonlinear systems to integrate real-time optimization (RTO) and model predictive control (MPC) under time-varying economic environments. In the RTO layer, we introduce a lookup table including a large number of steady-state points of the nonlinear system, which are predetermined offline. Once the parameters of the economic cost function are varied, we are able to take a quick online search on the lookup table to find a point for satisfied economic performance and then send it to the MPC layer as a temporary control target. The temporary target is also employed as the initial solution for solving the optimization of the RTO layer. When the optimal target is calculated by RTO, it will replace the temporary one as the new control target of MPC. Compared to the two-layer framework which suffers from long waiting time to get the optimal operating points, the lookup-table-based RTO (LT-RTO) framework provides a quick-produced suboptimal target for MPC. It avoids unnecessary economic losses if MPC is still tracking outdated target even parameters of the cost function have already changed. We demonstrate the effectiveness through a chemical process model that the LT-RTO framework makes an improvement of the economic performance.

Planning tool for polygeneration design in microgrids

This work suggests a methodology to assist the designer during the planning phase of microgrids and eco-districts. A mixed integer linear programming model is designed to mathematically describe the different energy systems and the physical relations among them. Given the different electrical/thermal demand profiles, the micro grid’s topology and a set of boundary conditions, the model can identify the optimum mix of (poly-)generation units and energy storage systems, as well as the necessary district heating/cooling infrastructure. Both economic and energetic cost functions are defined to explore the problem from different perspectives. The described tool is applied to study an actual district of the NTU campus in Singapore, comprising 5 multi-purpose buildings and a district cooling network supplied by centralized electrical chillers. The planning tool was run to assess the optimal configuration that minimizes the overall cost (initial investment and O&M); the outcome results presented a layout and a mix of energy systems different from the present one. In particular, the optimal configuration results to be a district cooling system served by a mix of electrical chiller plant, trigeneration distributed energy system and sensible cold thermal energy storage.

Plant-Wide Control of a Base Metal Refinery: Top-Down Analysis

A top-down control structure is presented for the optimal steady-state operation of a base metal refinery. The economic cost function for the plant is derived in terms of the production rate of primary and secondary products, the losses of primary product to the secondary product streams, and the fixed and variable costs of the plant. The economic performance of the plant is largely defined by the operation of an exothermic pressure leach circuit. The leach efficiency of this circuit determines the potential production rate for the plant, its operation affects costs, and it is the source of all primary product losses. Given market prices for the primary and secondary products, and reagent and utility costs, the proposed control structure can be used to define the optimal operating region for the pressure leach circuit so as to maximize the profitability of the plant.

Economic Model Predictive Control of Bihormonal Artificial Pancreas System Based on Switching Control and Dynamic R-parameter

Blood glucose (BG) regulation is a long-term task for people with diabetes. In recent years, more and more researchers have attempted to achieve automated regulation of BG using automatic control algorithms, called the artificial pancreas (AP) system. In clinical practice, it is equally important to guarantee the treatment effect and reduce the treatment costs. The main motivation of this study is to reduce the cure burden. The dynamic R-parameter economic model predictive control (R-EMPC) is chosen to regulate the delivery rates of exogenous hormones (insulin and glucagon). It uses particle swarm optimization (PSO) to optimize the economic cost function and the switching logic between insulin delivery and glucagon delivery is designed based on switching control theory. The proposed method is first tested on the standard subject; the result is compared with the switching PID and the switching MPC. The effect of the dynamic R-parameter on improving the control performance is illustrated by comparing the results of the EMPC and the R-EMPC. Finally, the robustness tests on meal change (size and timing), hormone sensitivity (insulin and glucagon), and subject variability are performed. All results show that the proposed method can improve the control performance and reduce the economic costs. The simulation results verify the effectiveness of the proposed algorithm on improving the tracking performance, enhancing robustness, and reducing economic costs. The method proposed in this study owns great worth in practical application.

Climate change effects on denitrification and associated avoidance costs in three Baltic river basin - coastal sea systems

To fight ongoing eutrophication of the Baltic Sea, major international directives and agreements, like the EU Water Framework Directive or the Helcom Baltic Sea Action Plan, aim at a reduction of nutrient loads to the Baltic Sea. These external nutrient load reductions interact with retention and removal processes (ecosystem services) in coastal and marine waters. Beside sedimentation, denitrification is the most important regulation ecosystem service provided by the sea, because it removes nitrogen (N) to the atmosphere and controls eutrophication. Denitrification is a spatially and temporally highly variable process and largely depends on temperature and N concentrations in the water. Climate change will have strong effects on denitrification. Increasing temperatures favour denitrification and changes in precipitation and discharge will alter the availability of N, with strong differences between river basins. We combine a 3D–ecosystem model of the Baltic Sea with economic marginal cost functions for three major Baltic river basins: Odra, Vistula and Neva. Objective is to calculate the monetary consequences of climate change (scenario A1B) for the implementation of water quality objectives, taking into account both, spatial changes in N loads and denitrification. In all regions over 90% of the entering N loads are subject to denitrification mainly in coastal waters. For the climate change scenario (A1B), our approach suggests reduced riverine N-loads of 7690 t/a (Odra), 6410 t/a (Vistula) and increased loads of 3200 t/a for the Neva. At the same time our model results suggest an increased denitrification of N-loads by 9.1% (Odra) and 9.9% (Neva) and minor changes with respect to the Vistula loads. In combination, in all three regions climate change helps to reach policy targets (lower N concentrations in the sea) at lower costs. The monetary benefit of load reduction measures that do not have to be implemented because of climate change are 11,270 million €/a (Odra), 2000 million €/a (Vistula) and 92 million €/a (Neva). The valuations strongly depend on the assumed climate change scenario and the economic approach, but indicate that climate change effects differ much between river basins and should be taken into account in nutrient load reduction management plans.

Distributed Economic MPC With Separable Control Contraction Metrics

In this letter, a flexible and scalable distributed economic model predictive control scheme is proposed for large-scale networked systems consisting of nodes with local self-interest economic objectives but a shared responsibility for stability of the interconnected network. In this letter, subsystems are controlled to track reference trajectories while optimizing their own economic costs during the transitions. A reference-independent stability constraint is introduced using a sum-separable control contraction metric. The proposed control design approach is thereby independent of both reference trajectories and the economic cost functions and as such suitable for flexible manufacturing. We observe that self-interested distributed optimization with shared responsibility for stability results in the same general problem formulation as network utility maximization, and make use of the alternating direction method of multipliers to coordinate the optimization. The proposed approach is illustrated with an example of distributed control of a network with 120 nodes, each of which is a three-state nonlinear system.

Advanced-step Nonlinear Model Predictive Control Based on Contraction Analysis

Recently, an approach of advanced-step nonlinear model predictive control (NMPC) based on nonlinear programming (NLP) sensitivity was developed for controlling large-scale complex plants. It can reduce the online computation time significantly while maintaining the same nominal stability of the conventional NMPC. However, the sensitivity based fast online update method cannot be applied to those cases where the NMPC problem contains a non-convex economic cost function or constraints. In this paper, based on contraction analysis, a novel online update mechanism, differential Lyapunov based MPC, which involves solving a quadratic programming problem with stability constraints is proposed to address this issue. A case study with non-convex state constraints is presented to demonstrate the concepts.

Sensitivity-Based Economic NMPC with a Path-Following Approach

We present a sensitivity-based predictor-corrector path-following algorithm for fast nonlinear model predictive control (NMPC) and demonstrate it on a large case study with an economic cost function. The path-following method is applied within the advanced-step NMPC framework to obtain fast and accurate approximate solutions of the NMPC problem. In our approach, we solve a sequence of quadratic programs to trace the optimal NMPC solution along a parameter change. A distinguishing feature of the path-following algorithm in this paper is that the strongly-active inequality constraints are included as equality constraints in the quadratic programs, while the weakly-active constraints are left as inequalities. This leads to close tracking of the optimal solution. The approach is applied to an economic NMPC case study consisting of a process with a reactor, a distillation column and a recycler. We compare the path-following NMPC solution with an ideal NMPC solution, which is obtained by solving the full nonlinear programming problem. Our simulations show that the proposed algorithm effectively traces the exact solution.

Economic design of moving average control chart for non-normally distributed data: a comparative study

An economic cost function for moving average control chart under non-normal quality characteristic is obtained using continuous and ceased production modes. This cost function can be used to obtain optimal design parameters which minimise the loss cost per hour of the production. Burr distribution has been used to transform the non-normal data to normal. Sensitivity analysis of the economic design is also discussed. The ceased process is observed to be dominant with significant saving in the cost over the continuous process.

Hierarchical Model Predictive Control of independent systems with joint constraints

This paper describes a hierarchical scheme for the control of independent stable systems subject to joint constraints. At the higher layer of the control structure reduced order dynamic models are used to minimize an economic cost function by adopting a long sampling time, while at the lower layer independent shrinking horizon MPC controllers working at a faster rate are designed for the original models to guarantee stability and convergence. A novel model reduction procedure is developed and simulation results are reported to witness the potentialities of the approach.

Port economic cost functions: A service perspective

Ports provide services rather than producing physical products. However, port cost functions specify port cost as functions of physical products, e.g., port cargo throughput long-run cost is a function of port resource prices and cargo throughput (a physical product). In comparison, port cargo service long-run cost is a function of port resource prices, cargo service and shipper cargo received. The shipper provides cargo that is serviced by the port. A service cannot be touched and its user is involved in its provision. This paper derives port cost functions for which port outputs are “service outputs”, e.g., cargo, vessel and vehicle services.