Mixed Integer Quadratic Programming Research Articles

Cooperative Optimization of Networked Microgrids for Supporting Grid Flexibility Services Using Model Predictive Control

The transition towards fully renewable energy-based power systems will require to increase the number of reserves at the System Operators' (SOs) disposal to provide flexibility on the energy management process. The microgrid's ability of integrating distributed energy resources, loads and energy storage systems (ESS) appears as a powerful flexibility tool. Nevertheless, the associated control problem of microgrids increases with the number of connected devices. A structuration of the distribution grids in networks of microgrids is proposed, focusing on their ability to provide flexibility services. The complexity of the associated optimization algorithm is faced using Distributed Model Predictive Control (MPC). The algorithm is divided in two steps. The first one is applied to the cooperative participation of microgrids in the day-ahead market. The second step covers the interaction with the SO offering flexibility services in exchange for a financial benefit. The financial benefit is optimally shared between the networked microgrids to satisfy the power profile requested by the SO at the lowest cost. As the proposed control algorithm presents both continuous and binary variables, its associated optimization problem is formulated using the Mixed Logic Dynamic (MLD) framework, which results in a Mixed Integer Quadratic Programming problem (MIQP).

Convex Model for Estimation of Single-Phase Photovoltaic Impact on Existing Voltage Unbalance in Distribution Networks

Due to the increasing adoption of solar power generation, voltage unbalance estimation gets more attention in sparsely populated rural networks. This paper presents a Monte Carlo simulation augmented with convex mixed-integer quadratic programming to estimate voltage unbalance and maximum photovoltaic penetration. Additionally, voltage unbalance attenuation by proper phase allocation of photovoltaic plants is analysed. Single-phase plants are simulated in low-voltage distribution networks and voltage unbalance is evaluated as a contribution of measured background and photovoltaic-caused unbalance. Voltage unbalance is calculated in accordance with EN 50160 and takes into account 10-minute average values with 5% tolerance condition. Results of the optimization revealed substantial unbalance attenuation with optimal phase selection and increased potential of local generation hosting capacity in case of higher background unbalance.

Predictive management approach for the coordination of wind and water-based power supplies

This paper addresses the coordinated operation between wind farms and reservoir hydropower plants in networks with weak transmission capacity. The joint operation of the energy sources is captured by a Virtual Power Plant (VPP) representation. With the aim of maximising renewable energy penetration while ensuring a suitable VPP integration, a supervisory controller for the energy dispatch layer is proposed. To deal with the variable nature of both resources, the controller has been designed into a model predictive control framework including mixed-integer quadratic programming (given the hybrid considered model) for solving the related optimisation problem, and complementary techniques as constraints softening and time-varying weighting. The results indicate that the predictive control approach achieves a better Independent System Operator (ISO) reference tracking that, together with reservoir management, are the most important factors to ensure a suitable VPP incorporation to the main grid. As a case study, a sub-network of the Argentinian power system is tackled. The controller performance is quantitatively and qualitatively assessed under uncertain conditions and compared with other approaches over a nine-year period. By means of the proposed power management policy, the ISO’s effort to balance the sub-network is reduced 56% with regard to other approaches without any water spillage.

BMFO-SIG: A Novel Binary Moth Flame Optimizer Algorithm with Sigmoidal Transformation for Combinatorial Unit Commitment and Numerical Optimization Problems

To achieve paramount economy, hybrid renewable energy sources are getting importance, as renewable sources are cost less. Over past few years, wind energy incorporation drew more consideration in electrical market, as wind power acting affirmative role in energy saving as well as sinking emission pollutants. Moth-flame optimizer is recently proposed meta-heuristics search algorithm, which is encouraged by direction-finding nature of moth and their convergence towards light. Albeit, moths are having a strong ability to maintain a fixed angle with respect to the moon and possess an effective mechanism for travelling in a straight line for long distances. However, they are ensnared in a deadly/useless spiral path around imitation source of lights. In proposed research, to improve the performance of moth-flame optimizer around deadly corkscrew path around imitation source of lights, for discrete optimization problems, two binary variants has been implemented and tested for 23 benchmark functional including combinatorial unit commitment problem. This paper critically analyze per megawatt cost saving while incorporating wind energy sources along with thermal units. The searches for allocation of generators (units that participate in generation to take up load) and once units are decided allocation of generations (economic load dispatch) is done by mixed integer quadratic programming (MIQP). To verify the feasibility and efficacy of operation of proposed algorithms, small- and medium-scale power systems consisting of 5-, 6-, 10-, 20- and 40-generaing unit systems taken into consideration. Commitment and scheduling pattern has been evaluated with and without wind integration and it has been experimentally found that proposed algorithm gives superior type of solutions as compared to other recently reported meta-heuristics search algorithms.

Emergency sources pre-positioning for resilient restoration of distribution network

In recent years, the large-scale power outages caused by extreme natural disasters have caused considerable economic losses. In this paper, an optimization decision-making method is proposed to allocate emergency power resources in a distribution system ahead of an extreme natural disaster. A resource allocation optimization model is proposed to allocate different resources among microgrids and critical loads in the distribution system. The factors including capacity, rental prices, energy reserves at each stage, and so on are all considered. A heuristic method is proposed to solve the model. By transforming the optimization model into a mixed-integer quadratic program, the resource allocation plan can be obtained efficiently. A case based on the IEEE 123-bus system is performed. The results validate the effectiveness and efficiency of our method.

Linear Quadratic Regulator of Discrete-Time Switched Linear Systems

This brief studies the linear quadratic regulation problem of discrete-time switched linear systems via dynamic programming. Discrete decision variables (referred as switch inputs) are introduced to represent the subsystem selection. This formulation leads to a mixed integer quadratic programming problem, which is known to be computationally hard in general. Instead of deriving suboptimal policies with analytical bounds on its optimality, the optimal switching sequence is computed in a computationally efficient manner when the input matrix is restricted as a vector. The unique contribution of this brief is an analytical expression of both the optimal switching condition (determining the subsystem selection) and the optimal control law. Proof of optimality is presented by fractional optimization and the practical merits of our approach is validated through simulations on a second-order system in comparison with recent pruning methods.

Two novel locally ideal three-period unit commitment formulations in power systems

The thermal unit commitment problem has historically been formulated as a mixed integer quadratic programming problem, which is difficult to solve efficiently, especially for large-scale systems. The tighter characteristic reduces the search space; therefore, as a natural consequence, it significantly reduces the computational burden. In the literature, many tightened formulations for a single unit with parts of constraints were reported without a clear derivation process. In this paper, a systematic approach is developed to create tight formulations. The idea is to use new variables in high-dimensional space to capture all the states of a single unit within three periods and then use these state variables systematically to derive three-period locally ideal expressions for a subset of the constraints in unit commitment. Meanwhile, the linear dependence of those new state variables is leveraged to keep the compactness of the obtained formulations. Based on this approach, we propose two tight models. The proposed models and other four state-of-the-art models are tested on 56 instances over a scheduling period of 24 h for systems ranging from 10 to 1080 generating units. The simulation results show that our proposed unit commitment formulations are tighter and more efficient (Increased by 13.6%) than other state-of-the-art models. After transforming our models into mixed integer linear programming formulations, our models are still tighter and more efficient (Increased by 67.3%) than other models.

Optimal Generator Start-Up Sequence for Bulk System Restoration With Active Distribution Networks

This paper proposes a new scheme for bulk system restoration considering the available black-start resources (BSRs) in the distribution system (DS). The DS assisted generator start-up sequence (D-GSS) scheme is realized in a decentralized way, so that it does not only benefit the bulk system GSS by utilizing available BSRs in the DS, but also respects the independent operation of the transmission system operator (TSO) and distribution system operators (DSOs). First, the decentralized D-GSS scheme is presented including the interaction of the TSO and DSOs and the corresponding compact models. The models of the decentralized D-GSS scheme are built with the bulk system GSS modeled as a mixed-integer quadratic program (MIQP), the DS multi-step operation modeled as mixed-integer second-order conic programs (MISOCPs) and reliable power supply assessments of renewable energy sources as linear programs (LPs). Finally, a projection function based analytical target cascading (P-ATC) method is developed to iteratively solve the decentralized models with model complexity reduction. The P-ATC algorithm can handle integer variables and improve the computational efficiency of the iterative calculation process. The effectiveness of the proposed method is validated using the small-scale T6D2 system and large-scale T118D5 system, showing good GSS performance and computation efficiency.

Container slot allocation and dynamic pricing of time-sensitive cargoes considering port congestion and uncertain demand

This paper studies the container slot allocation problem for time-sensitive cargoes with consideration of dynamic pricing and port congestion under uncertain demand. Time-sensitive cargoes call for express delivery. However, port congestion is a non-negligible factor to affect the delivery time. Hence, a new freight rate function with respect to the delivery time considering port congestion is proposed for time-sensitive cargoes, and two different slot allocation strategies are proposed: Loyal Strategy and Expansive Strategy. Accordingly, we formulate the problem as a one-stage container slot allocation model and a two-stage container slot allocation model, respectively. Both of the two models are stochastic mixed-integer quadratic programming models with chance constraints due to the involvement of dynamic pricing, port congestion, and uncertain demand. To solve the proposed models, a tailored algorithm that combines the Sample Average Approximation approach and the Reformulation Linearization Technique (SAA-RLT) is developed in this paper. Finally, numerical experiments are carried out to verify the applicability and effectiveness of the proposed models and the solution algorithm.

Fast mixed integer optimization (FMIO) for high dose rate brachytherapy

The purpose of this work was to develop an efficient quadratic mixed integer programming algorithm for high dose rate (HDR) brachytherapy treatment planning problems and integrate the algorithm into an open-source Monte Carlo based treatment planning software, RapidBrachyMCTPS. The mixed-integer algorithm yields a globally optimum solution to the dose volume histogram (DVH) based problem and, unlike other methods, is not susceptible to local minimum trapping. A hybrid linear-quadratic penalty model coupled to a mixed integer programming model was used to optimize treatment plans for 10 prostate cancer patients. Dose distributions for each dwell position were calculated with RapidBrachyMCTPS with type A uncertainties less than 0.2% in voxels within the planning target volume (PTV). The optimization process was divided into two parts. First, the data was preprocessed, in which the problem size was reduced by eliminating voxels that had negligible impact on the solution (e.g. far from the dwell position). Second, the best combination of dwell times to obtain a plan with the highest score was found. The dwell positions and dose volume constraints were used as input to a commercial mixed integer optimizer (Gurobi Optimization, Inc.). A penalty-based criterion was adopted for the scoring. The voxel-reduction technique successfully reduced the problem size by an average of 91%, without loss of quality. The preprocessing of the optimization process required on average 4 s and solving for the global maximum required on average 33 s. The total optimization time averaged 37 s, which is a substantial improvement over the ∼15 min optimization time reported in published literature. The plan quality was evaluated by evaluating dose volume metrics, including PTV D 90, rectum and bladder D 1cc and urethra D 0.1cc . In conclusion, fast mixed integer optimization is an order of magnitude faster than current mixed-integer approaches for solving HDR brachytherapy treatment planning problems with DVH based metrics.

Integer Programming for Learning Directed Acyclic Graphs from Continuous Data.

Learning directed acyclic graphs (DAGs) from data is a challenging task both in theory and in practice, because the number of possible DAGs scales superexponentially with the number of nodes. In this paper, we study the problem of learning an optimal DAG from continuous observational data. We cast this problem in the form of a mathematical programming model that can naturally incorporate a superstructure to reduce the set of possible candidate DAGs. We use a negative log-likelihood score function with both and penalties and propose a new mixed-integer quadratic program, referred to as a layered network (LN) formulation. The LN formulation is a compact model that enjoys as tight an optimal continuous relaxation value as the stronger but larger formulations under a mild condition. Computational results indicate that the proposed formulation outperforms existing mathematical formulations and scales better than available algorithms that can solve the same problem with only regularization. In particular, the LN formulation clearly outperforms existing methods in terms of computational time needed to find an optimal DAG in the presence of a sparse superstructure.

Parameter Identification Method Based on Mixed-Integer Quadratic Programming and Edge Computing in Power Internet of Things

With the rapid development of power Internet of Things, its scale is becoming larger and larger. Many advanced applications depend on the accuracy of network model and state estimation, and the accuracy of network model and state estimation largely depends on network parameter error. Therefore, a parameter identification and estimation method based on mixed-integer quadratic programming (MIQP) and edge computing is proposed. Firstly, a “cloud-tube-edge-end” architecture of power Internet of Things is proposed, and the edge computing layer collects terminal data and conducts data analysis, which greatly reduces the computing pressure of cloud center. In this architecture, the local state estimation is used to limit the branch with error data in a specific range to prevent the measurement errors in other ranges from affecting the local estimation process. Then, the parameter identification model is transformed into MIQP model, and a penalty factor is introduced into the optimization model to identify the parameter error and measurement error in the process of minimizing the objective function. Finally, data encryption, identity authentication, and other methods are used in edge computing to achieve network security protection, so as to avoid network attacks and information leakage in the process of data transmission. The proposed method is tested and analyzed in IEEE 14-bus test system. The results show that the proposed method can accurately determine and identify the error data in a certain probability in the actual operation of the power grid, which is convenient for the controller to find out the wrong data in time and determine the source of the error data, so as to set a reasonable data value.

Optimal dispatching of power grid integrating wind-hydrogen systems

In power grid integrating wind-hydrogena systems, the power balance constraints may result in large-scale wind curtailment during the valley-load period. Additionally, the classic two-stage dispatching strategy cannot evaluate the real-time hydrogen-production efficiency loss in the day-ahead stage, while the one-stage dispatching method will result in computation difficulty or a non-optimal result with or without the full-scale prediction horizon. To address this problem, a novel dispatching strategy is presented in this paper. Considering the electricity and heat demand for hydrogen production, a molten salt heat storage system (HSS) is used to improve the operational flexibility. The external characteristics of hydrogen production are modelled. On this basis, an efficiency-decision method that determines the optimal hydrogen production power is presented; thus, the decision-making variable number can be notably reduced. Considering the real-time power fluctuation, a novel energy dispatching model is proposed in the day-ahead stage to optimize efficiency decisions and thermal power units. Using linearization techniques, this model converted a classic mixed-integer quadratic programming (MIQP) problem and solved it efficiently. Simulation studies on an IEEE-30-node power grid integrating wind-hydrogen systems indicate that the proposed dispatching strategy can reduce the operational cost of the power grid by 4.4%. The results confirm the effectiveness of the proposed dispatching strategy.

Privacy preservation method for MIQP-based energy management problem: A cloud-edge framework

The increasing development of information and communication technologies in power grid makes it possible to optimize the energy management of numerous physical users with cloud-based service, while users’ concerns about privacy security have attracted increasingly more attention. To attack this challenge, this paper proposes the information masking (IM) method for the energy management problem in the form of mixed-integer quadratic programming (MIQP). Additionally, the feasibility and optimality of the recovered solution in the IM of MIQP is proven and given in the form of two lemmas. Next, the implementation mechanism of IM is designed based on a cloud-edge framework. Furthermore, the general requirements of the IM of MIQP are elaborated with respect to privacy security and computational cost to better accommodate practical applications. Finally, the optimal schedule of microgrid with on-site generators and flexible demand resources is modelled and simulated to demonstrate the feasibility and effectiveness of the proposed methodology.

Integration of Joint Power-Heat Flexibility of Oil Refinery Industries to Uncertain Energy Markets

This paper proposes a novel approach to optimize the main energy consumptions of heavy oil refining industries (ORI) in response to electricity price uncertainties. The whole industrial sub-processes of the ORI are modeled mathematically to investigate the joint power-heat flexibility potentials of the industry. To model the refinery processes, an input/output flow-based model is proposed for five main refining units. Moreover, the role of storage tanks capacity in the power system flexibility is investigated. To hedge against the electricity price uncertainty, an uncertain bound for the wholesale electricity price is addressed. To optimize the industrial processes, a dual robust mixed-integer quadratic program (R-MIQP) is adopted; therefore, the ORI’s operational strategies are determined under the worst-case realization of the electricity price uncertainty. Finally, the suggested approach is implemented in the south-west sector of the Iran Energy Market that suffers from a lack of electricity in hot days of summer. The simulation results confirm that the proposed framework ensures industrial demand flexibility to the external grids when a power shortage occurs. The approach not only provides demand flexibility to the power system, but also minimizes the operation cost of the industries.

Bidding Strategies in Energy and Reserve Markets for an Aggregator of Multiple EV Fast Charging Stations With Battery Storage

Adopting extreme fast charging for electric vehicles will significantly reduce the charging time for electric vehicle owners, which will improve the public acceptance of electric vehicle. However, under the conditions of wide spread fast charging stations, large charging power of fast charging stations will bring nonnegligible impacts to the power system. For an aggregator that owns multiple fast charging stations, installing battery storage systems within the fast charging stations can reduce the impacts and give more flexibility. It will bring extra benefit if the aggregator participates in electricity markets by utilizing the flexibility of the storage. In order to deal with the operation and market participation problem for EV fast charging stations, this paper proposes bidding strategies in both energy and reserve markets for an aggregator of multiple fast charging stations with energy storage systems. Conditional Value at Risk (CVaR) based mixed integer quadratic programming formulation is built to hedge the risks of random charging demands and volatile market prices. The proposed formulation is converted into a linear programming problem and an iterative solution method is designed to reduce the computation time. Case studies based on the trajectory data of taxis in Beijing have been carried out. Simulation results show that the aggregator can achieve higher economic benefits while satisfying the fast charging demand of electric vehicles.

A time-slot based signal scheme model for fixed-time control at isolated intersections

Fixed-time signal control is widely used in the real world due to its low cost of implementation. Repeated signal cycles with the same phase structure are applied in the existing fixed-time signal control schemes. However, such signal schemes do not always achieve the best performance, especially when traffic demand is unbalanced. The inclusion of sub-cycles in one cycle provides a potential solution in this situation. This study proposes a time-slot based signal scheme model with a generalized cycle structure for fixed-time signal control at isolated intersections. The proposed signal scheme can generate sub-cycles with different phase structures. It optimizes the number of sub-cycles in one cycle as well as the phase sequence, cycle length, and green splits in each sub-cycle. Certain phases may be skipped in partial sub-cycles. Nonlinear programming models are formulated to minimize average vehicle delay and maximize intersection capacity for under- and over-saturated demand. Solution algorithms for the two nonlinear models are developed by solving a series of mixed-integer-quadratic-programming (MIQP) models and mixed-integer-linear-programming (MILP) models, respectively. The numerical studies validate the advantages of proposed signal scheme in terms of intersection capacity and average vehicle delay. The sensitivity analyses show that: 1) a long maximum green time can enable the conventional dual-ring, eight-phase structure to handle unbalanced demand from the perspective of intersection capacity; 2) the time-slot number of 6 or 7 is suggested considering both the solution optimality and the model complexity.

A MIQP model for optimal location and sizing of dispatchable DGs in DC networks

The allocation and dimensioning of distributed generators (DGs) in direct current (DC) power grids were addressed in this study by using a mixed-integer quadratic programming (MIQP) formulation. The MIQP model corresponded to an approximation of the mixed-integer nonlinear programming (MINLP) model that represents this problem correctly. The proposed MIQP had, for its objective function, the minimization of the power losses; as constraints, it had power balance, voltage regulation, distributed generation capacity, and the number of DGs available, among others. The general algebraic modeling system (GAMS) was employed for solving the proposed MIQP as well as the MINLP formulation. Simulation results for one DC network with 21 nodes and another with 69 revealed that the proposed MIQP model obtains high-quality results regarding the locations of the generators, the objective function, and the power dispatch in comparison to the exact MINLP model and metaheuristic techniques recently reported in specialized literature.

Moth flame optimizer-based solution approach for unit commitment and generation scheduling problem of electric power system

Abstract This paper proposes a new approach based on the moth flame optimizer algorithm. Moth flame optimizer simulates the natural fervent navigation technique adopted by moths looking for a source of light. The proposed method is further improved by priority list-based ordering; the unit commitment problem (UCP) is a non-linear, non-convex, and combinatorial complex optimization problem. It contains both continuous and discrete variables. This further increases its complexity. Moth flame optimizer is very good at obtaining a commitment pattern: allocation of power on the committed units obtained by mixed-integer quadratic programming method. Heuristic search strategies are used to adopt for the repair of minimum up and downtime, and spinning reserve constraints. MFO effectiveness is tested on the standard UCP reference IEEE model buses 14 and 30, and 10 and 20 units. The efficiency of the projected algorithms is compared to classical PSO, PSOLR, HPSO, PSOSQP, hybrid MPSO, IBPSO, LCA-PSO, NPSO, PSO-GWO, and various other evolutionary algorithms. The comparison result shows that MFO can lead to all methods reported earlier in literature.

Leveraging owners’ flexibility in smart charge/discharge scheduling of electric vehicles to support renewable energy integration

High integration of intermittent renewable energy sources (RES), predominantly wind power, has created complexities in power system operations. In addition, large fleets of Electric Vehicles (EVs) increase the uncertainty on electricity consumption significantly. Thus, an uncoordinated charging process will further complicate the grid scheduling. In this work, we propose a coordinated scheduling algorithm for charge/discharge of aggregated EV fleets to maximize the integration of wind generation and minimize the charging cost for EV owners in a vehicle-to-grid (V2G) setup. Challenges for people participation in V2G, such as battery degradation and insecurity about unexpected events, are also addressed.The real-time charge/discharge scheduling problem is formulated as a multi-objective mixed-integer quadratic programming problem that the aggregator takes advantage of real-time communication in smart grid to frequently update the EV schedule with updated real-time information about EV availabilities, wind generation forecast, and electricity price. Simulations demonstrate significant increase in wind utilization and reduction in charging cost and battery degradation compared to the uncontrolled charging scenario.