Economic Optimization Problem Research Articles

An Introduction to Lagrange Multipliers: Theory and Applications in Economics

The purpose of this paper is to explore the basic applications of the Lagrange multiplier method in economics and to help beginners build their understanding of this mathematical tool. Tesla Inc. is used as a case study to examine how it maximizes profits by optimizing pricing and marketing expenses under given market conditions and constraints. The context of the study includes the importance of maintaining Tesla’s leadership position as the world’s leading electric car manufacturer in a competitive market. In this paper, the Lagrange multiplier method is used to incorporate specific market share and marketing expense constraints into the profit function and to find the extreme values by constructing the Lagrange function and solving for the derivatives. However, the results of the study show that the solutions obtained are extreme minima rather than the expected maxima. This result highlights the possible limitations of the Lagrange multiplier method in economic optimization problems. Nonetheless, this paper provides valuable lessons for understanding and applying the Lagrange multiplier method and points the way for further research on the application of optimization methods in economic decision making.

MILP and PSO approaches for solving a hydropower reservoirs intraday economic optimization problem

AbstractShort-term hydropower generation with several water reservoirs requires deciding, for each moment in time, the volume of water (outflow) that is released from every reservoir to be turbined and generate energy. Knowing the price of energy at every time period, the objective is to maximize the income earned from the generated energy. In this paper, we present (1) a Hydropower Reservoirs Operation Optimization problem with a higher level of detail than those found in the literature, encompassing temporal delays, water hammer effects, and increased temporal discretization, among others features, and (2) two distinct approaches for addressing this problem: MILP and PSO. These methods are compared across instances of varying nature to evaluate their performance. We make our code available on GitHub: https://github.com/baobabsoluciones/flowing-basin.

Decision-focused learning under decision dependent uncertainty for power systems with price-responsive demand

Contemporary power systems are experiencing a growing integration of flexible resources on the consumer side. Different from flexible demand that submits specific bids to energy markets, price-responsive demand (PRD) adjusts its power consumption without notice, simply based on the resulting electricity prices as well as internal priorities and limitations. In this paper, we demonstrate how a high penetration of PRD (the behavior of which is invisible to the operator during the day-ahead unit commitment stage) results in systematic inefficiency costs and formulate the so-termed decision-focused learning problem of learning to provide a demand forecast which, once fed as an input to the operator’s economic dispatch optimization problem, results in an efficient dispatch. Interestingly, the prescribed demand forecast affects the resulting prices, which in turn affect the actual demand realization, giving rise to a decision-dependent uncertainty. Motivated by the problem’s hard-to-evaluate objective function, we solve it using Bayesian optimization. The empirical evaluations demonstrate significant savings in the effective real-time system cost, compared to the current practice of using the default demand forecast. Moreover, the method is shown to achieve a system cost that is fairly close to the one achieved by a system that fully integrates PRD into the day-ahead process; but without requiring any change in the operator’s existing dispatch algorithm while avoiding all efforts necessary for the integration of flexible demand, which is a widely pursued field of ongoing research.

Economic Optimal Scheduling of Integrated Energy System Considering Wind–Solar Uncertainty and Power to Gas and Carbon Capture and Storage

With the shortage of fossil energy and the increasingly serious environmental problems, renewable energy based on wind and solar power generation has been gradually developed. For the problem of wind power uncertainty and the low-carbon economic optimization problem of an integrated energy system with power to gas (P2G) and carbon capture and storage (CCS), this paper proposes an economic optimization scheduling strategy of an integrated energy system considering wind power uncertainty and P2G-CCS technology. Firstly, the mathematical model of the park integrated energy system with P2G-CCS technology is established. Secondly, to address the wind power uncertainty problem, Latin hypercube sampling (LHS) is used to generate a large number of wind power scenarios, and the fast antecedent elimination technique is used to reduce the scenarios. Then, to establish a mixed integer linear programming model, the branch and bound algorithm is employed to develop an economic optimal scheduling model with the lowest operating cost of the system as the optimization objective, taking into account the ladder-type carbon trading mechanism, and the sensitivity of the scale parameters of P2G-CCS construction is analyzed. Finally, the scheduling scheme is introduced into a typical industrial park model for simulation. The simulation result shows that the consideration of the wind uncertainty problem can further reduce the system’s operating cost, and the introduction of P2G-CCS can effectively help the park’s integrated energy system to reduce carbon emissions and solve the problem of wind and solar power consumption. Moreover, it can more effectively reduce the system’s operating costs and improve the economic benefits of the park.

Research on Dynamic Economic Dispatch Optimization Problem Based on Improved Grey Wolf Algorithm

The dynamic economic dispatch (DED) problem is a typical complex constrained optimization problem with non-smooth, nonlinear, and nonconvex characteristics, especially considering practical situations such as valve point effects and transmission losses, and its objective is to minimize the total fuel costs and total carbon emissions of generating units during the dispatch cycle while satisfying a series of equality and inequality constraints. For the challenging DED problem, a model of a dynamic economic dispatch problem considering fuel costs is first established, and then an improved grey wolf optimization algorithm (IGWO) is proposed, in which the exploitation and exploration capability of the original grey wolf optimization algorithm (GWO) is enhanced by initializing the population with a chaotic algorithm and introducing a nonlinear convergence factor to improve weights. Furthermore, a simple and effective constraint-handling method is proposed for the infeasible solutions. The performance of the IGWO is tested with eight benchmark functions selected and compared with other commonly used algorithms. Finally, the IGWO is utilized for three different scales of DED cases, and compared with existing methods in the literature. The results show that the proposed IGWO has a faster convergence rate and better global optimization capabilities, and effectively reduces the fuel costs of the units, thus proving the effectiveness of IGWO.

A New Hybrid Three-Term HS-DY Conjugate Gradient In Solving Unconstrained Optimization Problems

Conjugate Gradient (CG) method is an interesting tool to solve optimization problems in many fields, such design, economics, physics and engineering. Until now, many CG methods have been developed to improve computational performance and have applied in the real-world problems. Combining two CG parameters with distinct denominators may result in non-optimal outcomes and congestion.In this paper, a new hybrid three-term CG method is proposed for solving unconstrained optimization problems. The hybrid threeterm search direction combines Hestenes-Stiefel (HS) and Dai-Yuan (DY) CG parameters which standardized by using a spectral to determine the suitable conjugate parameter choice and it satisfies the sufficient descent condition. Additionally, the global convergence was proved under standard Wolfe conditions and some suitable assumptions. Furthermore, the numerical experiments showed the proposed method is most robust and superior efficiency compared to some existing methods.

Economic Dispatch Optimization Strategies and Problem Formulation: A Comprehensive Review

Economic Dispatch Problems (EDP) refer to the process of determining the power output of generation units such that the electricity demand of the system is satisfied at a minimum cost while technical and operational constraints of the system are satisfied. This procedure is vital in the efficient energy management of electricity networks since it can ensure the reliable and efficient operation of power systems. As power systems transition from conventional to modern ones, new components and constraints are introduced to power systems, making the EDP increasingly complex. This highlights the importance of developing advanced optimization techniques that can efficiently handle these new complexities to ensure optimal operation and cost-effectiveness of power systems. This review paper provides a comprehensive exploration of the EDP, encompassing its mathematical formulation and the examination of commonly used problem formulation techniques, including single and multi-objective optimization methods. It also explores the progression of paradigms in economic dispatch, tracing the journey from traditional methods to contemporary strategies in power system management. The paper categorizes the commonly utilized techniques for solving EDP into four groups: conventional mathematical approaches, uncertainty modelling methods, artificial intelligence-driven techniques, and hybrid algorithms. It identifies critical research gaps, a predominant focus on single-case studies that limit the generalizability of findings, and the challenge of comparing research due to arbitrary system choices and formulation variations. The present paper calls for the implementation of standardized evaluation criteria and the inclusion of a diverse range of case studies to enhance the practicality of optimization techniques in the field.

An optimal siting and economically optimal connectivity strategy for urban green 5G BS based on distributed photovoltaic energy supply

The emergence of ultra-dense 5G networks and a large number of connected devices will bring with them significant increases in energy consumption, operating costs, and CO2 emissions. At the same time, the deployment of distributed photovoltaic (DPV) in megacities plays an important role in promoting the integration of “building-photovoltaic”, adjusting the city's clean energy supply system, and building a green, and low-carbon smart city. The development of a new “DPV-5G Base Station-Energy Storage (DPV-5G BS-ES)” coupled DC microgrid system and its pre-deployment investment costs are fundamental factors to be considered when the problem of large-scale DPV and BS deployment in cities has to be addressed. There are almost no published papers on this issue in the literature, which only consider the problem of maximizing the deployment of 5G BS service coverage or only perform optimization analyses for the performance of microgrid systems without considering the economic optimality and low-carbon green issues in the construction of urban utilities. In order to solve the problem of 5G BS deployment and economic optimization of topology in the “DPV-5G BS-ES” coupled DC microgrid system, a novel deployment strategy is proposed in this paper. The strategy is divided into: (a) The Geographic Information System (GIS) for extracting and analyzing 3D data from all buildings in the area. (b) A GIS-based formulation of a mixed integer quadratic constrained planning model (BSSCP) under discrete fixed radius coverage constraints is used to delineate building clusters within the area and determine the optimal deployment location of 5G BS under each sub-cluster. (c) An Improved Self-Organizing Mapping (ISOM) model combines GIS and microgrid performance parameters to address the economically optimal connection of grid lines for DPV in 3D space. (d) Simulation experiments were conducted for the power output and profitability of eight different PV panels and tracking systems to calculate the economically optimal type of DPV generation hardware. The experimental results show that the BSSCP model proposed in this study reduced the 5G BS deployment cost input by up to 35.8974 % compared to that of the control experimental model. Meanwhile, the ISOM model obtained an average reduction of 52.3741 % in the cost of connecting cables for the optimized 25 DPV planning and DC microgrid topology.

Design optimization method of heating and cooling full cycle storage air conditioning in integrated energy system

Heating and cooling storage air conditioning is an important part of the integrated energy system, which can play an important role in the flexible conversion of thermoelectric coupling systems. Because storage air conditioning involves the dynamic characteristics of power supply on the source side and cold and heat demand on the load side, system capacity allocation is a problem of technical and economic optimization. This paper breaks through the traditional design method based on the hourly load distribution law of typical design days and proposes a performance-oriented design method for the bidirectional full-cycle dimension storage air conditioning system based on the system’s annual operation performance, which can take into account the hourly dynamic load distribution law of every day in the full heating season and the full cooling season coupled with the bidirectional demand of cold and heat loads. A more scientific and reasonable capacity configuration scheme for the storage air conditioning system is obtained. The design is presented with a concrete case.

Hierarchical eco-driving control for plug-in hybrid electric vehicles under multiple signalized intersection scenarios

With the development of vehicle-to-everything (V2X) and autonomous driving technologies, plug-in hybrid electric vehicles (PHEVs) enable to absorb surrounding information to enhance economic driving. This study proposes a dynamic inverse hierarchical optimization method, which incorporates traffic-signal phase and timing as well as road data, to plan economic velocity and improve PHEV energy consumption. The hierarchical control framework determines the desired speed and arrival time in the upper layer using the shortest path faster algorithm. The lower level accounts for multi-objective velocity planning based on particle swarm optimization and Pareto theory. The inverse layering method solves the economic velocity optimization problem. With the support of V2X and autonomous driving technologies, the method enhances energy efficiency and computational efficiency in PHEVs through dynamic inverse hierarchical optimization. Simulation results highlight that the proposed algorithm leads to 7.43% improvement in energy consumption economy and the reduced calculation time, compared to the existing solutions. The hardware-in-the-loop experiments validate the real-time applicability of the proposed algorithm.

A hybrid data-driven and model-based approach for computationally efficient stochastic unit commitment and economic dispatch under wind and solar uncertainty

Stochastic unit commitment (UC) and economic dispatch (ED) are imperative in dealing with uncertainty in renewable forecast for power system operation and planning such that the overall expected production cost is minimized over the planning horizon. However, accurate calculation of the expected production cost requires assessment of a very large number of different scenarios of uncertain renewable resources, such as solar and wind, which is practically infeasible to simulate in real time. This article proposes a hybrid data-driven and physics-based model-predictive paradigm to efficiently solve for stochastic unit commitment and economic dispatch considering uncertainty in wind and solar power forecasts. The novelty of the approach lies in decoupling the production cost estimation from the unit commitment and economic dispatch optimization problems under uncertainty without compromising on the fidelity of the solutions. A data-driven machine learning model is first developed to predict the mean optimal production cost. A physics-based inverse problem is then solved to get the stochastic UC and ED profiles from the expected cost. The presented approach considers, for the first time, solar uncertainty in UC/ED determination and enables efficient and accurate propagation of wind and solar uncertainty to estimate the statistics of the production cost. The effectiveness of the developed approach is demonstrated systematically on a stylized RTS-GMLC single-node system. The overall framework predicts the expected cost 62.5% more accurately than the existing state-of-the-art, on unforeseen days during the entire year, and yields, for the first time, the associated physically consistent UC and ED profiles. The solutions are also shown to be flexible in providing adequate daily reserves to address any statistical deviations from probabilistic power forecasts. The computational time associated with the presented method is only about 10 s compared to over 24 h needed for a conventional stochastic UC/ED determination under uncertainty on an Intel Core i9 processor with 32 GB of RAM.

Development of an Interactive Software Tool for Designing Solvent Recovery Processes.

Solvents are used in chemical and pharmaceutical industries as a reaction medium, selective dissolution and extraction media, and dilution agents. Thus, a sizable amount of solvent waste is generated due to process inefficiencies. Most common ways of handling solvent waste are on-site, off-site disposal, and incineration, which have a considerable negative environmental impact. Solvent recovery is typically not used because of potential difficulties in achieving required purity guidelines, as well as additional infrastructure and investments that are needed. To this end, this problem must be studied carefully by involving aspects from capital needs, environmental benefits, and comparison with traditional disposal methods, while achieving the required purity. Thus, we have developed a user-friendly software tool that allows engineers to easily access solvent recovery options and predict an economical and environmentally favorable strategy, given a solvent-containing waste stream. This consists of a maximal process flow diagram that encompasses multiple stages of separations and technologies within those stages. This process flow diagram develops the superstructure that provides multiple technology pathway options for any solvent waste stream. Separation technologies are placed in different stages; depending on the component, they can separate in terms of their physical and chemical properties. A comprehensive chemical database is created to store all relevant chemical and physical properties. The pathway prediction is modeled as an economic optimization problem in General Algebraic Modeling Systems (GAMS). With GAMS code as the backend, a Graphical User Interface (GUI) is created in Matlab App Designer to provide a user-friendly tool to the chemical industry. This tool can act as a guidance system to assist professional engineers and provide an easy comparative estimate in the early stages of process design.

Solving economic optimization problems by mathematical programming methods

Solving economic optimization problems by mathematical programming methods

A real-time balancing market optimization with personalized prices: From bilevel to convex

This paper studies the static economic optimization problem of a system with a single aggregator and multiple prosumers in a Real-Time Balancing Market (RTBM). The aggregator, as the agent responsible for portfolio balancing, needs to minimize the cost for imbalance satisfaction in real-time by proposing a set of optimal personalized prices to the prosumers. On the other hand, the prosumers, as price taker and self-interested agents, want to maximize their profit by changing their supplies or demands and providing flexibility based on the proposed personalized prices. We model this problem as a bilevel optimization problem. We first show that the optimal solution of this bilevel optimization problem can be found by solving an equivalent convex problem. In contrast to the state-of-the-art Mixed-Integer Programming (MIP)-based approach to solve bilevel problems, this convex equivalent has very low computation time and is appropriate for real-time applications. Next, we compare the optimal solutions of the proposed personalized scheme and a uniform pricing scheme. We prove that, under the personalized pricing scheme, more prosumers contribute to the RTBM and the aggregator’s cost is less. Finally, we verify the analytical results of this work by means of numerical case studies and simulations.

Batch-to-batch optimization for economic performance improvement in batch processes by rational disturbances utilization

Batch processes usually adopt a two-layer architecture control strategy to reduce the calculation cost while ensuring economic performance. The upper layer in the strategy obtains the nominal optimal reference trajectory by solving an economic optimization problem, while the lower MPC one possesses the task of tracking this trajectory as closely as possible. The disadvantage of this two-layer strategy lies in the lower layer resisting all the disturbance for satisfactory tracking results, regardless of the fact that part of the disturbance will be beneficial to economic performance. In order to realize the economic promotion in batch processes, a batch-to-batch optimization method is proposed in this paper. As an alternative to the traditional lower-layer tracking MPC, the proposed algorithm uses Gaussian process regression to establish a more accurate linear time-varying model and learns the information from the previous batches to obtain a more precise state estimation, thus continuously moving up economics by solving the economic optimization problem. Convergence of the proposed algorithm is proved and simulation is performed to verify the effectiveness of the proposed algorithm on economic performance improvement in comparison with the traditional tracking one.

Fuel Economy Optimization for Platooning Vehicle Swarms via Distributed Economic Model Predictive Control

Cooperation among multiple connected vehicles (CVs) brings swarm intelligence to our transportation systems and helps improve their performance. This study proposes a fuel economy optimization approach for a platooning vehicle swarm via distributed economic model predictive control (DEMPC). Within the DEMPC framework, each CV shares its assumed trajectory in the predictive horizon with its neighboring CVs at each control loop. With its neighbors’ and its own assumed trajectories, each CV first solves an open-loop control optimization problem for platoon formation, and then solves an open-loop economic optimization problem for direct fuel economy improvement. In particular, the optimal cost of the former optimization problem is used in the latter one to build an upperbound constraint for stability guarantee. The asymptotic convergence of assumed terminal states is given in the analysis, and the recursive feasibility of the two optimization problems are proved with an explicit constraint on the weight matrices in the open-loop control optimization problem. Based on these analyses, the asymptotic stability of the closed-loop system is finally proved through Lyapunov analysis. Numerical simulation results validate the effectiveness of the proposed approach in terms of closed-loop stability and fuel economy improvement. Note to Practitioners—This work aims to optimize the fuel economy for a swarm of vehicles running in a platoon. Existing approaches on platoon control mainly focus on the accurate tracking of the following vehicles, but this may cause aggressive control and hence lead to poor fuel economy. Therefore, this paper proposes a distributed economic model predictive control approach to explicitly optimize the fuel consumption rate of each vehicle. The proposed method can improve fuel economy and reduce communication requirements and burdens for platooning vehicle swarms. The paper assumes no communication time delay and packet drops, so the impact of unreliable communication will be studied in the future work.

Price-based low-carbon demand response considering the conduction of carbon emission costs in smart grids

The rapid development of the Industrial Internet-of-Things extends demand response (DR) research to the aspect of low-carbon emission in smart grids. This study proposed the concept of low-carbon DR (LCDR) in the electricity market as well as the price-based LCDR mechanism and its model. First, carbon cost conduction from the generation side to the demand side was analyzed, and then conduction function was quantifiably deduced. Second, the mechanism and model of price-based LCDR were proposed by considering three DR signals, namely, the electricity price, carbon price, and carbon emission intensity of the demand side, based on the traditional price-based DR (PBDR) mechanism. Third, the proposed LCDR mechanism was applied to the environmental–economic dispatch optimization problem. At last, case studies on the modified IEEE 39-bus system verified that the LCDR mechanism can reduce carbon emissions while maintaining the function of the traditional PBDR. Meanwhile, the applicability of LCDR was illustrated based on carbon emission sensitivity to LCDR model parameters. The proposed mechanism can guide participants in the electricity market in reducing electricity carbon emissions.

Distinguish between the economic optimal and lowest distribution temperatures for heat-prosumer-based district heating systems with short-term thermal energy storage

Decreasing distribution temperature is the key driving force for energy-efficient and economic-competitive district heating (DH) systems, powering the transition from the 1st generation high-temperature steam-based DH systems to the 4th generation low-temperature water-based DH systems. However, for heat prosumers integrated with thermal energy storages (TESs), the decreasing distribution temperature may lower the peak load shaving capability of TESs and raise the peak-load-related heating costs, due to decreased charging temperature. This study, therefore, aimed to investigate the overall impacts of distribution temperature and to identify the economic optimal distribution temperature for heat-prosumer-based DH systems integrated with short-term TES. Firstly, an economic optimization problem was developed. Afterwards, distribution temperatures were optimized within the feasible regions of individual scenarios, including three benchmark scenarios representing the 2nd, 3rd, and 4th generation DH and an improved scenario featured high feasibility in distribution temperature. A campus DH system in Norway was used as the case study. Results revealed that the improved scenario's economic optimal distribution temperature was significantly distinct from the 4th generation DH scenario's lowest distribution temperature, both in terms of distribution range and annual average level. Finally, broad conclusions were reached by discussing the impacts of crucial factors on economic optimal distribution temperatures.

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.

Optimization control of hydrometallurgical leaching process based on IDE-MPC

• A dynamic model is built for the hydrometallurgical leaching process based on reaction mechanism analysis. • The steady-state economic optimization model of the leaching process is established. • An improved PSO algorithm is proposed to solve the economic optimization problem of the leaching process. • A new control method combining the IDE algorithm and MPC is used to control the leaching process. The utilization rate of raw materials during the hydrometallurgical leaching process has a great influence on the whole economic benefits of the hydrometallurgy plant, so it is necessary for the leaching process to improve the utilization of materials using optimization control methods. In this paper, the dynamic model for the hydrometallurgical leaching process of a gold hydrometallurgy plant is first built based on the reaction mechanism of the process. Then, the model parameters are identified using least-squares fitting. Thereafter, with the maximum economic benefit as the objective function, the steady-state economic optimization model of the leaching process is established, and an improved particle swarm optimization algorithm is used to solve the model. Taking the optimization results as the control objective, a model predictive control method based on an improved differential evolution algorithm is proposed to control the leaching process, so as to improve the intractability and anti-disturbance performance of the controller for the leaching process. The simulation results show that the proposed optimization and control methods achieve satisfactory effects.