Large-scale Mixed-integer Nonlinear Programming Research Articles

A smart energy scheduling under uncertainties of an iron ore stockyard-port system using a rolling horizon algorithm

Planning the efficient use of electricity in iron ore stockyard operations is a strategic issue due to the constant rise in energy prices nowadays and its considerable impact on production costs. This paper proposes a new large-scale mixed-integer nonlinear programming (MINLP) model for stockyard-port energy planning solved by the energy scheduling algorithm and a commercial solver to minimize power costs. The proposed nonlinear optimization problem is solved through an equivalent MILP model to minimize the flows of power and material between the stockyard-port equipment. The electrical machines are powered by different electricity energy providers, and eventually consume storage energy from batteries. The energy scheduling algorithm allows the planner to find a solution that saves electrical power costs in real time under unforeseen operational changes. Numerical results obtained through the proposed algorithm with a scheduling horizon of 24 h, show that the presence of the battery in the stockyard-port electrical grid allows for an energy cost reduction of up to 17.88% compared to the case without the battery. The energy scheduling based on rolling horizon algorithm provides feasible solutions near to the optimal solution, with an average distance of 1.78%, and it has an affordable computation time in instances where the MINLP model is not able to provide a solution.

Risk-based distributionally robust optimal dispatch for multiple cascading failures in regional integrated energy system using surrogate modeling

Regional integrated energy system (RIES) with the close coupling of multiple energies faces various cascading failure risks in its operation, and formulating the risk-based optimal dispatch scheme is important to ensure the secure and economic operation of the RIES. Considering that the occurrence probability of each subsequent failure path caused by the initial failure in the cascading failure chain varies with the change of dispatch scheme, the cooling/heating and gas pipeline dynamics and the uncertainty of renewable energy, a risk-based distributionally robust optimal dispatch model for multiple cascading failures in a RIES is proposed. A kernel ambiguity set is constructed by using the maximum mean discrepancy between the real and reference probability distributions and the kernel mean embedding technology, which can obtain the more accurate worst-case probability distribution. Moreover, a surrogate modeling method based on piecewise convex quadratic functions is proposed to learn the mapping between the cascading failure risks and the decision variables, and the original large-scale mixed integer nonlinear programming model is transformed into a small-scale mixed integer quadratic constrained programming model for efficient solution. Finally, case study on an actual RIES demonstrates the effectiveness and advantages of the proposed method. Compared to optimal dispatch without considering the cascading failure risks, the proposed method can reduce the total failure risk cost by nearly 60%.

Data-driven stochastic robust optimization of sustainable utility system

Sustainable utility systems reduce reliance on fossil fuels by using renewable energy sources. Multi-scale uncertainties associated with renewable energy and utility systems pose challenges to the modeling and optimization of sustainable utility systems. This study proposes a sustainable retrofit framework for utility systems based on a data-driven stochastic robust optimization approach. Kernel density estimation and fuzzy clustering were employed to capture the uncertainty features in a holistic framework. A life cycle assessment approach was used to calculate the global warming potential (GWP) of the sustainable utility system, and a multi-objective environmental and economic optimization model was developed. A nested decomposition-based algorithm was proposed to solve a large-scale mixed-integer nonlinear programming problem. Finally, a case study of an industrial utility system was conducted to demonstrate the effectiveness of the proposed method. The optimization results show that the proposed method reduces GWP by 9 % after introducing renewable energy and achieves a balance between economic and environmental performance.

Optimizing the monthly scheduling of crudes in a terminal-refinery system

This paper focuses on solving the optimization of crude oil operations scheduling carried out in a real system composed of a refinery and a marine terminal, over a monthly horizon. In the present article, we introduce a large-scale mixed-integer non-linear programming (MINLP) model that faithfully represents the operation and characteristics of the system. Considering the model's complexity and its non-linear and non-convex nature, the challenge lies in solving the model in a time frame that meets the user's needs. To tackle this problem, we develop a temporal decomposition method in conjunction with a linear approximation.

Online Distributed Routing Problem of Electric Vehicles

Considering the penetration of numerous electric vehicles (EV) into the transportation sector, the EV routing problem that jointly optimizes the charging and routing process of EVs is becoming increasingly popular, which should be solved in online fashion for the practical requirements. First, we start with an offline EVs routing and charging (EVRP) problem which is a large-scale mixed integer nonlinear program (MINLP). Considering the NP hardness of MINLP, solving the offline EVRP directly is a time-consuming task that is not suitable for the online setting. Hence, a Benders decomposition based method is proposed to decompose the offline EVRP into a master problem and a set of sub-problems which allows for distributed implementation. To further speed up the computation, we relax the mixed integer master problem to one that can be equivalently solved as a linear program. Moreover, a novel kind of valid cut is added to the relaxed master problem to further reduce the number of iterations. In order to adapt the offline EVRP to the online setting, by introducing the virtual depot, we utilize the rolling-horizon framework to tackle the uncertainty of future information, where the offline EVRP is solved in real-time repeatedly. Finally, simulations using real road map in Belgium are performed. Besides, numerical results validate that the computation speed of proposed algorithm is faster than the state-of-the-art algorithms by several orders of magnitude, and showcase the capability of proposed algorithm to solve large size instances up to 350 nodes with 35 EVs within 100 seconds.

Spatial–Temporal Energy Management of Base Stations in Cellular Networks

The operations of base stations (BSs) contribute most of the energy consumption in the cellular wireless networks. Powering BSs by distributed energy resources (DERs), such as photovoltaic (PV) and energy storage, is an effective way to reduce on-grid power consumption and build green wireless networks. Optimal energy management of BSs helps to reduce electricity bills for the wireless network and provides flexibility to the power networks. This article proposes the concept of spatial–temporal energy management (ST-EM) for the energy management of BSs. On the one hand, the BSs manage their power consumption according to the real-time prices; on the other hand, the BSs adjust the user associations and change their power consumption according to the price differences among different BSs. The ST-EM of BSs is formulated as a large-scale mixed-integer nonlinear programming (MINLP) problem, which is proven to be NP-hard. We propose a heuristic approach to search for one suboptimal solution by decomposing the original problem into an energy network optimization subproblem and a communication network optimization subproblem. The two subproblems are solved alternatively until convergence. Numerical experiments are conducted to verify the effectiveness of our proposed method.

Multidisciplinary Design Optimization Approach to Integrated Space Mission Planning and Spacecraft Design

Space mission planning and spacecraft design are tightly coupled and need to be considered together for optimal performance; however, this integrated optimization problem results in a large-scale mixed-integer nonlinear programming (MINLP) problem, which is challenging to solve. In response to this challenge, this paper proposes a new solution approach to this problem based on decomposition-based optimization via augmented Lagrangian coordination. The proposed approach leverages the unique structure of the problem that enables its decomposition into a set of coupled subproblems of different types: a mixed-integer quadratic programming (MIQP) subproblem for mission planning, and one or more nonlinear programming (NLP) subproblem(s) for spacecraft design. Because specialized MIQP or NLP solvers can be applied to each subproblem, the proposed approach can efficiently solve the otherwise intractable integrated MINLP problem. An automatic and effective method to find an initial solution for this iterative approach is also proposed so that the optimization can be performed without a user-defined initial guess. The demonstration case study shows that, compared to the state-of-the-art method, the proposed formulation converges substantially faster and the converged solution is at least the same or better given the same computational time limit.

Monthly schedule of crude refinery: Multi-scale strategy and multi-criteria objective

The optimization of crude refinery schedule, a subject of extensive research, frequently results in large-scale mixed-integer nonlinear programming problems. This paper optimizes a realistic industrial refinery's schedule over a one-month horizon. To achieve a rapid solution time, we developed a novel multi-scale strategy that treats the optimization model at two temporal (longer horizon divided into shorter periods) and two spatial scales (the refinery and its coarse-grained version). Problems at the shorter temporal and coarser spatial scales are solved quickly and contribute to the efficient solution of the original problem. Among other novelties, the model includes the frequently overlooked residue tanks downstream of the CDU, which have a significant influence on the solution property. Additionally, an exact mixed-integer linear formulation of the crude swing penalty was proposed, allowing optimization of the distillation sequence as part of a multi-criteria objective. Several examples demonstrate the improvement in computing time and solution quality.

Evaluate the capacity of electricity-driven water facilities in small communities as virtual energy storage

In small communities, such as small islands, campuses of universities/industrial parks/lab facilities, and remote villages/cities, the Water-Energy Nexus (WEN) operators generally need to purchase electricity from the power grid to meet a portion of their power demand. The WEN of these communities creates new opportunities of providing virtual energy storage (VES) and virtual power plant (VPP) services to the power grid by properly controlling their electricity-driven water facilities (EDWF). This paper investigates the capacity of EDWFs in small communities as VES and VPP. Small communities in four different types of areas, e.g., island, desert, urban, and rural, are studied based on their geographical specifications, to evaluate and compare the capabilities of their WENs as VES and VPP respectively. WENs with high penetration of photovoltaic resources are considered, for which battery energy storage systems (BESS) are needed. An optimization approach of operating the EDWFs as VES is proposed to reduce the required size of BESSs. The state-of-the-art convex technologies are adopted to reduce the computational burden of the developed optimization models which are large-scale mixed-integer nonlinear programming problems. Besides, the WEN of small communities is modeled as a VPP to provide spinning reserve energy for the power grid in emergency conditions. This paper also describes how the EDWFs can be controlled to serve as a virtual system and provides substantial benefits for the small communities. Numerical results validate that it is economically beneficial to operate the WENs of small communities as VES or VPP for renewable energy management of power grids.

Improved models for operation modes of complex compressor stations

We study combinatorial structures in large-scale mixed-integer (nonlinear) programming problems arising in gas network optimization. We propose a preprocessing strategy exploiting the observation that a large part of the combinatorial complexity arises in certain subnetworks. Our approach analyzes these subnetworks and the combinatorial structure of the flows within these subnetworks in order to provide alternative models with a stronger combinatorial structure that can be exploited by off-the-shelve solvers. In particular, we consider the modeling of operation modes for complex compressor stations (i.e., ones with several in- or outlets) in gas networks. We propose a refined model that allows to precompute tighter bounds for each operation mode and a number of model variants based on the refined model exploiting these tighter bounds. We provide a procedure to obtain the refined model from the input data for the original model. This procedure is based on a nontrivial reduction of the graph representing the gas flow through the compressor station in an operation mode. We evaluate our model variants on reference benchmark data, showing that they reduce the average running time between 10% for easy instances and 46% for hard instances. Moreover, for three of four considered networks, the average number of search tree nodes is at least halved, showing the effectivity of our model variants to guide the solver’s search.

Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation

We consider the problem of designing a cloak for waves described by the Helmholtz equation from an integer programming point of view. The problem can be modeled as a PDE-constrained optimization problem with integer-valued control inputs that are distributed in the computational domain. A first-discretize-then-optimize approach results in a large-scale mixed-integer nonlinear program that is in general intractable because of the large number of integer variables that arise from the discretization of the domain. Instead, we propose an efficient algorithm that is able to approximate the local infima of the underlying nonconvex infinite-dimensional problem arbitrarily close without the need to solve the discretized finite-dimensional integer programs to optimality. We optimize only the continuous relaxations of the approximations for local minima and then apply the sum-up rounding methodology to obtain integer-valued controls. If the solutions of the discretized continuous relaxations converge to a local minimizer of the continuous relaxation, then the resulting discrete-valued control sequence converges weakly $$^*$$ in $$L^\infty$$ to the same local minimizer. These approximation properties follow under suitable refinements of the involved discretization grids. Our results use familiar concepts arising from the analytical properties of the underlying PDE and complement previous results, derived from a topology optimization point of view.

Stochastic modelling of sandstorms affecting the optimal operation and cleaning scheduling of air coolers in concentrated solar power plants

The operation performance of air-coolers in concentrated solar power plants decays due to particulate deposition on heat transfer surfaces. The deposition process can be seen as a stochastic phenomenon. A modelling approach is proposed to capture the uncertainty and the effect of extreme events, such as sandstorms, affecting the performance of plants located in dry places through dust or sand deposition on the air coolers. A case study of a concentrated solar power plant located in Dubai is analysed. Sandstorms generate acute and drastic fouling of the air coolers, and this is modelled as a stochastic process using historical aerosol dispersion data. Ten scenarios are generated by sampling the probability distribution of sandstorms occurrence and intensity. The optimal operation (cleaning schedule and airflow profiles) of the air coolers is established using Benders decomposition to solve the resulting large-scale mixed integer non-linear programming problem. The results of the stochastic scenarios demonstrate that substantial savings of $ 0.6 M − $ 2.7 M per year are achieved by the optimal operation. Cost is minimized by a combined reactive and proactive cleaning policy which accounts for the frequency, intensity and seasonal variability of sandstorms, in addition to the variability on local radiation and weather conditions.

Edge Learning With Unmanned Ground Vehicle: Joint Path, Energy, and Sample Size Planning

Edge learning (EL), which uses edge computing as a platform to execute machine learning algorithms, is able to fully exploit the massive sensing data generated by Internet of Things (IoT). However, due to the limited transmit power at IoT devices, collecting the sensing data in EL systems is a challenging task. To address this challenge, this article proposes to integrate unmanned ground vehicle (UGV) with EL. With such a scheme, the UGV could improve the communication quality by approaching various IoT devices. However, different devices may transmit different data for different machine learning jobs and a fundamental question is how to jointly plan the UGV path, the devices’ energy consumption, and the number of samples for different jobs? This article further proposes a graph-based path planning model, a network energy consumption model, and a sample size planning model that characterizes F-measure as a function of the minority class sample size. With these models, the joint path, energy and sample size planning (JPESP) problem is formulated as a large-scale mixed-integer nonlinear programming (MINLP) problem, which is nontrivial to solve due to the high-dimensional discontinuous variables related to UGV movement. To this end, it is proved that each IoT device should be served only once along the path, thus the problem dimension is significantly reduced. Furthermore, to handle the discontinuous variables, a tabu search (TS)-based algorithm is derived, which converges in expectation to the optimal solution to the JPESP problem. Simulation results under different task scenarios show that our optimization schemes outperform the fixed EL and the full path EL schemes.

Solving the Problem of Large-Scale Optimal Scheduling of Distributed Energy Resources in Smart Grids Using an Improved Variable Neighborhood Search

Since the last decade, power systems have been evolving dynamically due to smart grid technologies. In this context, energy management and optimal scheduling of different resources are very important. The main objective of this paper is to study the optimal scheduling of distributed energy resources (OSDER) problem. This problem is a challenging, complex and very large-scale mixed-integer non-linear programming (MINLP) problem. Its complexity escalates with incorporation of uncertain and intermittent renewable sources, electric vehicles, variable loads and markets which makes it hard to be solved using traditional optimization algorithms and solvers. However, it can be handled efficiently and without approximation or modification of the original formulation using modern optimization algorithms such as metaheuristics. In this paper, an improved version of the variable neighborhood search (IVNS) algorithm is proposed to solve the OSDER problem. The proposed algorithm was tested on two large-scale centralized day-ahead energy resource scenarios. In the first scenario, the 12.66 kV, 33-bus test system with a total of 49,920 design variables is used whilst in the second scenario, the 30 kV, 180-bus test system is used with a total of 154,800 design variables. The optimization results using the proposed algorithm were compared with five existing optimization algorithms, i.e., chaotic biogeography-based optimization (CBBO), cross-entropy method and evolutionary PSO (CEEPSO), chaotic differential evolution with PSO (Chaotic-DEEPSO), Levy differential evolution with PSO (Levy-DEEPSO), and the variable neighborhood search (VNS). For the first test system, the IVNS has achieved a score of −5598.89 while for the second test system it has achieved a score of −3180.15. A comparative study of the results has shown that the proposed IVNS algorithm performs better than the remaining algorithms for both cases.

Coyote Optimization Algorithm-Based Approach for Strategic Planning of Photovoltaic Distributed Generation

The optimal planning for distributed generations (DGs) associated with photovoltaics (PVs) in the utility-owned distribution system is crucial for increasing high penetration of renewables while against practical system operation constraints. Such PV-DG planning is categorized as a complicated mixed-integer nonlinear programming (MINLP) problem and is extremely difficult to solve by using conventional methods. In recent years, several bio-inspired metaheuristic algorithms have been proposed to tackle various complicated real-parameter optimization problems. This paper proposes a two-stage approach including a new bio-inspired algorithm, Coyote Optimization Algorithm (COA), to solve the large-scale MINLP PV-DG sizing problem considering different load levels. The objective function terms under consideration include the total system power loss and voltage regulator tap changes at different load levels while against limits of rms bus voltages, tap changes, and PV-DG constraints at each candidate bus. The proposed method is tested using the IEEE 123-bus unbalanced benchmark system and an actual utility distribution network. Results obtained are then compared with those obtained by a classic MINLP solver-based and four other bio-inspired methods. Moreover, results also show that the proposed method leads to lower loss, a minimum number of regulator tap changes, and higher PV penetration capacity among the compared methods and is suitable for solving the large-scale PV-DG planning problem in distribution systems.

A Benchmark for Joint Channel Allocation and User Scheduling in Flexible Heterogeneous Networks

Flexible duplexing is a promising technique to improve the spectral efficiency of future cellular networks, which has been proposed mainly to provision asymmetric uplink (UL) and downlink (DL) traffic scenarios, through flexible channel allocations. However, this flexibility in the channel allocation process, which is responsible for allocating the underlying channel to different base stations in a heterogeneous network (HetNet), has brought new technical challenges due to the introduction of complex UL-to-DL and DL-to-UL interference scenarios. This paper analyzes the joint channel allocation (CA) and user scheduling (US) process for orthogonal frequency-division multiple access-based flexible HetNets, while considering exact inter-cell/intra-cell interferences. The resulting joint problem is a large-scale mixed-integer nonlinear programming problem that is computationally intractable, therefore, it has been re-formulated into a convex upper bound problem to find benchmark solutions for CA in flexible HetNets. Since, no new CA scheme has been proposed yet for the HetNets employing flexible duplexing techniques, we discuss the efficacy of existing schemes under various UL and DL traffic scenarios.

Green Large-Scale Fog Computing Resource Allocation Using Joint Benders Decomposition, Dinkelbach Algorithm, ADMM, and Branch-and-Bound

With the increasing demands for large-scale computing in Internet of Things network, fog computing emerges as a potential solution. However, the time and energy costs are the bottlenecks for developing fog computing. In this paper, we investigate the green fog computing by maximizing the network utility function considering energy efficiency with the constraints of power and interference. The proposed problem is a large-scale mixed integer nonlinear programming. To deal with such kind of problems, we design an algorithm framework to solve the problem in a distributed and parallel manner. The outer loop of the problem is based on the Benders decomposition to divide the integer variables and continuous variables into the master problems and subproblems, respectively. In the subproblem, we use the Dinkelbach algorithm to transform the fractional programming into an equivalent solvable form. In the inner loop, the large-scale problem with only continuous variables is handled by the alternating direction method of multipliers algorithm. For the master problem, we propose a centralized branch-and-bound algorithm to deal with the complexity. We also discuss the properties and performances of our algorithm. Finally, the simulation results indicate that our proposed algorithm is energy-efficient and time-saving.

An optimization model for siting and sizing of vehicle-to-grid facilities in a microgrid

In this paper, we propose a new optimization model for investment planning for Vehicle-to-Grid (V2G) technology in a microgrid. The V2G technology allows the microgrid to take advantage of temporal arbitrage in the electricity market, and hence, improve economic viability of electric vehicles. We present both a mathematical model and a custom solution methodology for this problem. The proposed stochastic model is a very large-scale mixed-integer nonlinear programming problem, and thereby, we develop an algorithm that provides high-quality solutions. A case study for 14-node and 37-node microgrid test systems for a planning horizon of five years is presented, and a thorough sensitivity analysis for effects of uncertainty, investors preferred payback period, market price fluctuations and electric vehicles arrival and departure rates is discussed. The simulation results show that investing in the V2G technology can considerably improve the long-term economics of the microgrid, but the obtained profit and the payback period can vary from one plan to another.

Efficient real time optimization using a data-driven piecewise affine model

A data-driven real time optimization (RTO) scheme is proposed for complex processes under uncertainties. The objective is to optimize the economic performance of a plant when its first-principle model is difficult to obtain. To this end, a piecewise affine (PWA) model is built by fitting the operating data to predict the production of a process. The trust region of the PWA model is also characterized to ensure reliable predictions. Then, the RTO is formulated as a mixed-integer nonlinear program (MINLP) based on the PWA model. To alleviate the computational demands, we also develop a dynamic programming (DP) scheme that solves the RTO to a near-optimal solution. Two case studies show that the PWA model can accurately forecast the real productions when the process is operated within the trust region, and the proposed DP scheme takes less time to find a better solution than the state-of-the-art solver for the large-scale MINLP.

Two-Stage Optimal Economic Scheduling for Commercial Building Multi-Energy System Through Internet of Things

In this study, an advanced Internet of Things (IoT) technology is applied to the energy management of an intelligent combined cooling, heating, and power (CCHP) commercial building system. Based on the framework of a smart energy management system (SEMS) using IoT technology, a two-stage optimal scheduling model is proposed to determine the most economic CCHP commercial building system integrated with a three-way valve. In the day-ahead scheduling stage, the schedule is planned using the lowest operating costs of the system. In the real-time correction stage, the correction strategy employs minimum adjustment of the output of each unit. Moreover, the schedule plan is corrected to smooth out fluctuations in the loads and renewable energy resources (RES) in a timely manner to better absorb green energy. The day-ahead scheduling model is a large-scale mixed integer nonlinear programming (MINLP) problem solved through a linearization method proposed in this study and the mixed integer linear programming method. The real-time correction optimization model is a nonlinear programming problem solved by the quantum genetic algorithm (QGA). A case study is employed to demonstrate that the IoT-based SEMS improves the system automation energy management level and user comfort. Furthermore, the proposed system structure can significantly reduce system operating costs and improve the utilization of waste heat from the internal combustion engine. In conclusion, the economic and environmental superiority of the two-stage optimal dispatch model is verified.