big-M Method Research Articles

Stackelberg game-based optimal scheduling of integrated energy systems considering differences in heat demand across multi-functional areas

Demand-side management is very critical in China’s energy systems because of its high fossil energy consumption and low system energy efficiency. A building shape factor is introduced in describing the architectural characteristics of different functional areas, which are combined with the characteristics of the energy consumed by users to investigate the features of heating load in different functional areas. A Stackelberg game-based optimal scheduling model is proposed for electro-thermal integrated energy systems, which seeks to maximize the revenue of integrated energy operator (IEO) and minimize the cost of users. Here, IEO and users are the Stackelberg game leader and followers, respectively. The leader uses real-time energy prices to guide loads to participate in demand response, while the followers make energy plans based on price feedback. Using the Karush–Kuhn–Tucker (KKT) condition and the big-M method, the model is transformed into a mixed-integer quadratic programming (MIQP) problem, which is solved by using MATLAB and CPLEX software. The results demonstrate that the proposal manages to balance the interests of IEO and users. Furthermore, the heating loads of public and residential areas can be managed separately based on the differences in energy consumption and building shape characteristics, thereby improving the system operational flexibility and promoting renewable energy consumption.

Daily peak-shaving model of cascade hydropower serving multi-grids considering an HVDC channel shared constraint

High-voltage direct current (HVDC) is widely applied in large-scale hydropower transmission in China because of its long distance, large capacity and low power loss. However, when multiple hydropower stations send power by sharing one HVDC transmission channel, it is challenging to ensure the security of the sending-end power grid. To address this problem, a short-term peak-shaving model serving multiple power grids considering an HVDC channel shared (HCS) constraint is proposed. First, a daily peak-shaving model for cascade hydropower stations serving multiple power grids coupling with conventional HVDC constraints is established. Second, for the power transmission of different hydropower stations sharing one HVDC channel, the HCS constraint restructured by set-calculation is integrated into the model. Finally, the Big-M method is adopted to linearize the restructured HCS constraint to build an MILP model due to its flexibility in handling massive inequality constraints. The result from the case study shows that the proposed HCS constraint can be well coupled into the model to ensure power transmission security. The peak-valley difference in dry and flood season reduced by 37% and 21%, respectively, which indicates that the model can make full use of the flexibility of hydropower to achieve a satisfactory peak-shaving result.

Distributed robust optimization scheduling of a steel plant integrated energy system considering the uncertainty of byproduct coal gas

The steel plant integrated energy system (SPIES) is an important form in the steel industry. Improving the utilization efficiency of steam, electricity, coal gas and other energy flows is of great significance for both economic and environmental benefits. In this paper, a SPIES scheduling model is established according to the operation characteristics of coal gas holders, boilers and other equipment in steel plants. Meanwhile, to cope with the uncertainty of byproduct coal gas, this paper adopts an imprecise Dirichlet model (IDM) to construct a fuzzy set containing multisource coal gas production information. Then, according to duality theory and the big-M method, the original distributed robust optimization (DRO) model is transformed into a traditional mixed integer linear programming (MILP) model, which is solved by the column-and-constraint generation (CC&G) algorithm. Finally, a real steel production system is given in a case study. Case study illustrate that compared with the traditional robust method, the method proposed in this paper for a SPIES can effectively reduce the conservatism of the scheduling decision. Numerical simulation show that the proposed method can reduce total cost by 55,307.1¥, accounting for 1.91% of the total cost compared with robust optimization method and save 1,326.94 s of computational time compared with the stochastic optimization method, thus reaching balance between conservatism and computational efficiency.

Optimal operation method for load restoration of distribution networks under disturbance scenario

Extreme events can cause damages to transmission system equipment, resulting in a major power outage. In disturbance scenario, local distributed energy resources (DERs) and switches can be fully utilized to restore power loads, which is critical for enhancing the resiliency. A service restoration model for distribution network with soft open point (SOP) is proposed, in which the ZIP loads are considered. Traditional tie switch (TS) operations and the power output capabilities of local DERs are coordinated to restore loads in outage area. On this basis, SOP flexibility is explored to further improve the restoration performance. The restoration model includes nonlinear constraints and integer variables, which is usually a mixed-integer nonlinear programming (MINLP) problem, and is difficult to obtain a feasible solution within an acceptable time. In particular, the ZIP load model is not directly amenable with conic solvers. To improve solution efficiency, convex relaxation, linear approximation, and big-M methods are introduced to convert MINLP problem into a mixed-integer second-order cone programming (MISOCP) problem. Simulation tests are studied to verify the effectiveness of the proposed model.

Multi-objective optimal siting and sizing of BESS considering transient frequency deviation and post-disturbance line overload

With the development of renewable energy sources, more frequent and severe active power disturbances emerge in power systems, and jeopardize the frequency stability and line loading security. The battery energy storage system (BESS) is able to adjust output power flexibly, and an attractive solution to improve frequency dynamics and power flow distribution. This paper proposes a multi-objective optimal siting and sizing scheme for the BESS to arrest frequency excursion and mitigate line overload under major disturbances. First, an extended average system frequency model is developed to estimate the required transient frequency regulation capability (TFRC) for frequency stability, and the post-disturbance power flows. Then, taking the required TFRC and line capacity as constraints, a multi-objective optimization model is established to minimize the life cycle cost of the BESS and generation cost. Furthermore, based on the linear weighted method, big-M method, and multi-cut generalized Benders decomposition, the intractable optimization model is transformed into a master problem for investment decision making, and a set of subproblems accounting for normal operation, frequency stability and line loading security. Additionally, the master problem and subproblems are solved iteratively to determine the location and capacity of the BESS. Case studies are conducted to validate the proposed scheme, showing superior performance in improving frequency nadir and alleviating post-disturbance line overload.

A Smoothing Method for Ramp Metering

Ramp metering offers great potential to mitigate traffic congestion and improve freeway management efficiency under traffic congestion conditions. This paper proposes an optimization program for freeway dynamic ramp metering based on Cell Transmission Model (CTM). This problem has been formulated as a discrete time optimal control problem with smooth state equations and constraints to meter traffic inflow from on-ramps. In the proposed model, the ‘min’ operators in the primal CTM are non-differentiable and thus, the corresponding optimal control problem cannot be solved directly using conventional gradient based methods. In this paper, we introduce a smooth approximation to approximate the ‘min’ operators and then a unified computational approach is developed to solve the problem. Theoretical analysis is carried out, showing that the optimal solution obtained from the approximated problem converges to the optimal solution of the primal CTM. Compared to the classical inequality relaxation method, our method can resolve the flow holding-back problem and reduce under fundamental diagram phenomenon. Compared with the Big-M method, our method has better efficiency. To achieve the desired traffic response control in real application, a series of online optimal control problems are solved using Model Predictive Control (MPC). Simulation studies show that our method can significantly improve freeway traffic management efficiency.

Real-Time Schedule of Microgrid for Maximizing Battery Energy Storage Utilization

This paper proposes a real-time schedule model of a microgrid (MG) for maximizing battery energy storage (BES) utilization. To this end, a BES life model is linearized using piece-wise linearization and big-M method to assess the BES life loss (BLL) in a real-time manner. The cost-effective schedule model of the MG with multiple energy resources aims to maximize BES utilization while ensuring its sufficient lifespan. Corresponding to the optimization model, approximate dynamic programming (ADP) for maximizing BES utilization (ADP-MBU) in the real-time schedule is proposed to optimize the system under stochastic environments. In ADP-MBU, a new value function approximation method employing the BES cumulative life loss (BCLL) is developed to improve the optimality and applicability. The proposed ADP-MBU algorithm can achieve satisfactory approximate optimality while reflecting the variation of real-time BLL. Case studies validate the applicability of the proposed MG schedule model and the advantages of the proposed ADP-MBU algorithm.

Impact of Large-scale concentrated solar power on energy and auxiliary markets

Concentrated solar power (CSP) generation has been widely regarded as a promising technology to increase the penetration of renewable energy-based generation units in power systems. Numerous research studies have been conducted in various ways to promote the participation of a CSP generator as a price-taker in electricity markets. However, it remains an open question as to how the strategic behavior of a CSP generator as a price-maker affects electricity markets. To this end, we conducted a numerical analysis to quantify the impact of a strategic large-scale CSP generator on energy and auxiliary markets. A three-stage stochastic bi-level model is proposed to describe the strategic behavior of a CSP generator as a price-maker in electricity markets. The impact of the risk-averse level of a CSP generator is also incorporated. Moreover, network constraints are considered to make our conclusion more realistic. The proposed bi-level model is recast into a mathematical programming with equilibrium constraints form firstly by using Karush-Kuhn-Tucker conditions and then a mixed-integer-liner-programming form by using the strong duality theory, and big-M method. Case studies based on an IEEE-56 bus system demonstrate that a price-maker CSP generator can decrease the price of both energy markets and reserve markets. However, an associated thermal energy storage tank can alleviate the negative impact of a CSP generator on electricity markets. Additionally, the high risk-averse level of a CSP generator will lead to an increased equilibrium price.

Sizing battery energy storage and PV system in an extreme fast charging station considering uncertainties and battery degradation

This paper presents mixed integer linear programming (MILP) formulations to obtain optimal sizing for a battery energy storage system (BESS) and solar generation system in an extreme fast charging station (XFCS) to reduce the annualized total cost. The proposed model characterizes a typical year with eight representative scenarios and obtains the optimal energy management for the station and BESS operation to exploit the energy arbitrage for each scenario. Contrasting extant literature, this paper proposes a constant power constant voltage (CPCV) based improved probabilistic approach to model the XFCS charging demand for weekdays and weekends. This paper also accounts for the monthly and annual demand charges based on realistic utility tariffs. Furthermore, BESS life degradation is considered in the model to ensure no replacement is needed during the considered planning horizon. Different from the literature, this paper offers pragmatic MILP formulations to tally BESS charge/discharge cycles using the cumulative charge/discharge energy concept. McCormick relaxations and the Big-M method are utilized to relax the bi-linear terms in the BESS operational constraints. Finally, a robust optimization-based MILP model is proposed and leveraged to account for uncertainties in electricity price, solar generation, and XFCS demand. Case studies were performed to signify the efficacy of the proposed formulations.

Distribution system planning considering peak shaving of energy station

Energy stations (ESs) connected in a distribution system (DS) may lead to great impacts on the planning scheme of DS. In this context, this paper carries out a long-term DS planning model considering the peak shaving of ES, which is achieved by scheduling the input energy of ES. By regarding DS and ES as different stakeholders, a decentralized framework is devised to shave the electric peak loads in the DS planning, where the coupling relationship between the TOU price and exchanged power (e.g., the input power of ES) is clearly expressed. Specifically, an explicit adjustment formula is developed to represent the coupling relationship based on the concept of elasticity. In addition, an easily reformulated solution method is developed to address the probability distribution (PD) uncertainty of electric and cooling loads in the uncertainty-moment-based distributionally robust optimization (DRO) planning model. The chance-constrained power balance is expressed in a second order conic (SOC) format based on the conditional value at risk (CVaR) method and duality theory. Then, the SOC constraints are linearized according to the polyhedral linearization method. Furthermore, the bilinear terms of the planning model are respectively linearized by the McCormick and big-M methods. Finally, the proposed planning model is tested on a modified IEEE 33-node DS with an ES and a practical 99-node DS with an ES. Numerical results show that the proposed planning model is effective in managing PD uncertainties of loads as well as reducing costs of DS.

Data-driven adjustable robust Day-ahead economic dispatch strategy considering uncertainties of wind power generation and electric vehicles

Considering the intermittent features of wind power generation and electric vehicles, it is important for microgrid to formulate available dispatch strategy while ensuring the system economy. A two-stage data-driven adjustable robust optimization (ARO) model is presented in this paper to realize an optimal day-ahead economic dispatch strategy for the microgrid considering the uncertainties of wind power generation and electric vehicles. In the optimization model, an imprecise-Dirichlet-model-based nonparametric ambiguity set is constructed without any presumption about the probability distributions of the uncertainties. Further, a polyhedron uncertainty set obtained from the ambiguity set is driven by historical data, and this becomes narrower with an increase in the amount of the historical data used to construct the ambiguity set. The data-driven ARO problem is converted into a traditional two-stage ARO model with the obtained polyhedron uncertainty set. Further, this can be decomposed into a mixed integer linear programming (MILP) master problem and a sub problem with a max–min structure. Then, duality theory, Big-M method, and column and constraint generation (C&CG) algorithm are applied to achieve the optimized day-ahead economic dispatch strategy for the microgrid. Case studies illustrate the solution robustness, economic benefit, flexible adjustment capability and uncertainty depiction ability of the presented method.

Bi-layer optimal planning of rural distribution network based on KKT condition and Big-M method

With the increasing seriousness of environmental and energy shortage problems, new technologies such as photovoltaic distributed generations and energy storage devices will be the development trend of future rural distribution network. In this paper, a bi-layer optimal planning model of rural distribution network is established, where the optimal planning problem on the long time scale is solved in the upper-layer model, and the optimal operation problem on the short time scale is solved in the lower-layer model. Then, the lower-layer optimal operating model is converted and added into the constraints of the upper-layer model according to the Karush–Kuhn–Tucker (KKT) conditions, and the Big-M method is used to linearize the nonlinear problems in the constraints of the proposed model. Finally, the case study of a rural distribution network in Zhejiang, China is demonstrated for verifying the effectiveness and advantages of the proposed bi-layer model on alleviating the pressure of power supply and increase profits from electricity sales.

Coordinated Operation Strategy for a Virtual Power Plant With Multiple DER Aggregators

This article proposes a coordinated operation strategy for a virtual power plant (VPP) with multiple DER aggregators under the wholesale energy and regulation service market. Considering the market clearing process, VPP operation profit, and DER aggregators’ interests, a tri-layer hierarchical framework for VPP is developed to formulate the bidding plans to the superordinate market operator and the dynamic price incentive curves for the subordinate DER owners cooperatively. By using the discrete price quota curves, big-M method, KKT optimal conditions, and strong duality theorem, the tri-layer problem is transformed into an equivalent and tractable mixed integer linear programming problem. Case study with the 141-bus system is conducted to validate the effectiveness of the proposed approach. The advantages of lower cost with operation security guarantee and higher flexibility are demonstrated by comparing with other models.

Emergency reserve constrained optimal allocation of energy storage in a novel honeycomb‐like microgrid cluster with volatile renewable energy resources

With the increasing penetration of large-scale distributed renewable energies, the electricity distribution network is growing more complex with challenges brought by the intermittency and uncertainty of renewable energies. Microgrid (MG), one of feasible solutions to integrating renewable energies, has shortages because of limited capacity. Therefore, MG cluster (MGC) was proposed as an improved solution. To efficiently and friendly integrate renewable energies, a novel honeycomb-like MG cluster (H-MGC) is proposed in this paper. Considering the uncertainty of renewable energies, a robust optimisation method of the siting and sizing of energy storage system (ESS) constrained by emergency reserve is proposed. Combining the column-and-constraint generation algorithm and big-M method, the proposed optimisation model is decomposed into mixed integer second-order cone programming master problem and sub-problems, which are alternately solved. Based on the optimal results of study case, the siting and sizing results of ESS are analysed, as well as the operating characteristics of H-MGC. Compared to existing MGC system, the proposed H-MGC has a more flexible operation mode. The related costs of ESS are reduced, and the operational life is relatively extended, under the condition that the emergency reserve of system is satisfied.

Assessment of distributed photovoltaic hosting capacity in integrated electricity and heat systems considering uncertainty

Distributed photovoltaics (DPVs) have been widely integrated into power systems due to their abundance, renewability and low cost, while the stochastic nature of DPVs imposes significant influence on the hosting capacity (HC) of DPVs. Here, the integrated electricity and heat system (IEHS) is explored to increase the deployment of DPVs in comparison to the power system by exploiting the interaction between electric and thermal energy. An MILP-based HC assessment model for DPVs of the IEHS is proposed to efficiently promote the penetration level of DPV generation in distribution networks considering the uncertainty of DPV generation. To reduce the computational burden for HC assessment, the linearisation method combined with the incremental formulations of the big-M method is developed to simplify the complex non-linear model of energy devices and networks in IEHS. A scenario-based uncertainty modelling approach is applied to characterise DPV uncertainty, which improves the accuracy of HC assessment for DPVs. Comprehensive case studies according to a 33-bus electric network and a 6-node heat network validate the superiority of the proposed assessment model.

A virtual resource-based method for formulating the safety headway of opposing trains arriving at the same station on a single-track railway line

ABSTRACT In train rescheduling, the arrival and departure times of trains at stations are normally constrained by many operations rules that must be respected. Therefore, different methods were proposed to describe train operations for preventing train conflicts, especially concerning the safety headways. This paper develops a virtual resource-based method in place of the commonly used big-M method and ‘either-or’ constraints for reformulating the safety headway of opposing trains arriving at the same station on a single-track rail. This new formulation can provide an efficient decomposition mechanism by decomposing the original complex problem into several single train-based subproblems. A resource-oriented mixed-integer linear programming model based on a time-space network is developed to minimize the total train delay time with cumulative flow variables used to describe time-space resources occupation. We also present a set of numerical experiments based on a single-track railway line in China to evaluate the validity of the proposed method.

Renewable Generation and Transmission Expansion Planning Coordination with Energy Storage System: A Flexibility Point of View

This paper presents a method for coordinated network expansion planning (CNEP) in which the difference between the total cost and the flexibility benefit is minimized. In the proposed method, the generation expansion planning (GEP) of wind farms is coordinated with the transmission expansion planning (TEP) problem by using energy storage systems (ESSs) to improve network flexibility. To consider the impact of the reactive power in the CNEP problem, the AC power flow model is used. The CNEP constraints include the AC power flow equations, planning constraints of the different equipment, and the system operating limits. Therefore, this model imposes hard nonlinearity onto the problem, which is linearized by the use of first-order Taylor’s series and the big-M method as well as the linearization of the circular plane. The uncertainty of loads, the energy price, and the wind farm generation are modeled by scenario-based stochastic programming (SBSP). To determine the effectiveness of the proposed solution approach, it is tested on the IEEE 6-bus and 24-bus test systems using GAMS software.

Resilient operation planning of integrated electrical and natural gas systems in the presence of natural gas storages

In this paper, a new model is proposed for emergency response of Power Grid Systems (PGS) to severe events in the presence of Natural Gas (NG) storages. The PGS’s transmission lines outages as well as NG pipelines outages are integrated in this model. The optimal Network Reserve Requirement (NRR) allocation and NG storage scheduling are studied as emergency response strategies to relief sever events repercussion in PGS. The proposed method is linearized using the combination of piecewise linearization approach and big-M method. The effectiveness of the proposed MILP model to improve system resilience are verified through four case-studies on modified 24-bus IEEE test system integrated to 20-node NG network.

Big-M based MILP method for SCUC considering allowable wind power output interval and its adjustable conservativeness

In contrast to most existing works on robust unit commitment (UC), this study proposes a novel big-M-based mixed-integer linear programming (MILP) method to solve security-constrained UC problems considering the allowable wind power output interval and its adjustable conservativeness. The wind power accommodation capability is usually limited by spinning reserve requirements and transmission line capacity in power systems with large-scale wind power integration. Therefore, by employing the big-M method and adding auxiliary 0-1 binary variables to describe the allowable wind power output interval, a bilinear programming problem meeting the security constraints of system operation is presented. Furthermore, an adjustable confidence level was introduced into the proposed robust optimization model to decrease the level of conservatism of the robust solutions. This can establish a trade-off between economy and security. To develop an MILP problem that can be solved by commercial solvers such as CPLEX, the big-M method is utilized again to represent the bilinear formulation as a series of linear inequality constraints and approximately address the nonlinear formulation caused by the adjustable conservativeness. Simulation studies on a modified IEEE 26-generator reliability test system connected to wind farms were performed to confirm the effectiveness and advantages of the proposed method.

Planning strategies of power-to-gas based on cooperative game and symbiosis cooperation

Power-to-gas (PtG), as a promising technology proposed to store surplus renewable energy (RE), can hardly be commercialized for its low profitability. In this paper, three approaches are proposed in this paper to enhance the profitability of the PtG. Firstly, a cooperative union containing PtG is proposed and its sustainability analysis is undertaken based on Shapley Value method. Secondly, the PtG reaction heat, as an essential by-product of PtG which is valuable and therefore requires further study, is fully exploited for district heating in the operation of regional integrated energy system, which is solved by an improved SOCP method. Thirdly, a symbiosis cooperation mode is designed for wind power and PtG to enhance the benefit of PtG through optimization-based trading strategy, which is a MINLP model and solved by Big-M method. The results show that the daily profit of PtG is significantly increased with the cooperative union as the symbiosis cooperation mode can produce a 15.1% profit lift, meanwhile, exploitation of reaction heat can produce an 8.6% profit lift. Finally, our study reveals the conflict of interest between wind power and the cogeneration. A sensitivity study on the proportion of reaction heat used for district heating is performed to verify the mutually beneficial relation between PtG and the cogeneration. The findings of this paper can guide the commercialization of PtG.