Convex Programming Model Research Articles

Two-stage robust programming model for active distribution network with a high proportion of renewable energy

To reduce operating costs and obtain the optimal location and capacity of renewable energy access, a two-stage robust planning model is studied for active distribution networks with a high proportion of renewable energy access. To build a function to minimize annual comprehensive costs, the constraints of power flow, branch power, and node voltage of the active distribution network are studied. Considering the characteristics and properties of power flow safety constraints, the mixed integer second-order cone transformation is used to construct a mixed integer convex programming model, which is convenient for efficiently obtaining the global optimal solution. To avoid the influence of uncertainty, a two-stage robust model is constructed. The explicit maximum value method is used to obtain the solution results that satisfy the planning model of all uncertain problems. Experiments show that the model can obtain the optimal access position and capacity of renewable energy. After application, it can reduce the annual comprehensive cost of the distribution network and ensure the stable and safe operation of the active distribution network.

Energy management optimization in smart railway stations with the ability to charge plug-in hybrid electric vehicles

The increasing development of traffic in the world has led to an increase in the consumption of electrical energy in electric vehicles (EVs). One of the most widely used electric vehicles is the electric train. Thus, energy optimization becomes more crucial in electrical railways to reduce consumption and total electricity bill as well as environmental issues. Consequently, electrical railway energy management must be technically and economically efficient and effective. This paper proposes an energy efficiency optimization framework for intelligent railway stations that provide plug-in electric vehicle charging parking capacity use, renewable energy sources (REs), and regenerative braking energy (RBE). The structure of the proposed convex programming (CP) model is used for fast and efficient optimization of decision variables, Equipment size, and the cost function of the station for reducing the power purchased from the Grid. Furthermore, by presenting meta-heuristic algorithms, the state variables of the energy storage system (ESS) and Plug-in hybrid electric vehicles (PHEVs) battery have been optimized. The results obtained from the proposed method prove that the operation cost of the station decreases by 61.4 %.

Multi-period scheduling problem of combination of home health care and outpatient services: based on Chinese family doctor contract services

Inspired by the service mode of family doctor contract service in China, this paper presents a multi-period problem of the combination of home health care and outpatient service over a planning horizon of multiple days. In the planning horizon, patients receive services with fixed frequencies and have different preferences for different dates. Taking the patients’ service date regulations and doctors’ working regulations into account, we propose a mixed-integer nonlinear and convex programming model, which aims to minimize the total operating costs and maximize the patients’ preference satisfaction. A hybrid tabu search algorithm with an embedded shake procedure is designed to solve the problem. Numerical experiment results imply that the hybrid tabu search algorithm performs well when compared with the exact method on small instances, and it can also provide higher quality solutions when compared with variable neighborhood search method. We also analyze problem properties and parameter sensitivities through some small instances.

A new sampling scheme combining maximum entropy and moment matching techniques for reactor physics uncertainty quantification

The sampling method is widely used for the quantification of reactor physics analysis uncertainty, while the traditional statistical sampling methods often require a huge number of samples to ensure the stability of uncertainty quantification results. For this reason, a new two-step sampling scheme based on the combination of information entropy theory and moment matching technique is proposed for the quantification of the reactor physics uncertainty. We construct a convex programming model to optimize initial samples by simultaneously maximizing information entropy and minimizing moment deviations. The optimized samples can not only characterize the statistics of the real distribution better, but also control the information entropy with a smaller sample size. Moreover, a quantitative criterion for sample selection induced by the maximum entropy and moment matching is proposed. Finally, a series of empirical studies are carried out to show the capability and efficiency of the introduced two-step sampling method.

End-to-end delay guaranteed Service Function Chain deployment: A multi-level mapping approach

Network Function Virtualization (NFV) enables service providers to maximize the business profit via resource-efficient QoS provisioning for customer requested Service Function Chains (SFCs). In recent applications, end-to-end delay is one of the crucial QoS requirements that need to be guaranteed in SFC deployment. Whereas it has been considered in the literature, accurate and comprehensive modeling of the delay in the problem of maximizing the service provider’s profit has not yet been addressed. In this paper, at first, the problem is formulated as a mixed-integer convex programming model. Then, by decomposing the model, a two-level mapping algorithm is developed, that at the first level, maps the functions of the SFCs to virtual network function instances; and in the second level, deploys the instances in the physical network. Simulation results show that the proposed approach achieves a near-optimal solution in comparison to the performance bound obtained by the optimization model, and also superiors the existing solutions.

Hub location problem considering spoke links with incentive-dependent capacities

This study proposes a hub-and-spoke (HS) network in which the capacity of a spoke link depends on the incentive provided to the transporters. The incentives influence the capacities of the spoke links, which in turn affect the hub selections and flow assignment. A mixed integer convex programming model is developed to optimize the proposed HS network. The model minimizes total transport cost by locating p hubs, routing flows between origin–destination (OD) pairs and determining the incentives on spoke links. A customized outer-approximation algorithm is presented to provide exact solutions to the developed model, which is difficult to solve optimally within acceptable computational times for large-scale instances. An approximation algorithm based on the stepwise linearization method is then developed to provide suboptimal solutions. The developed model and algorithms are validated in computational experiments with different problem scales. The experiments demonstrate that the proposed HS network tends to route the flows on more spoke links, balance the transport costs on the potential paths between OD pairs and serve more demands with lower transport costs, when compared with the HS network with fixed capacities.

Resilience promotion of active distribution grids under high penetration of renewables using flexible controllers

Flexible voltage controllers (FVCs), such as voltage regulation transformers (VRTs) and shunt capacitor banks (SCBs), can play a significant role in enhancing the resilience of active distribution systemS (ADS) equipped with large-scale renewable distributed generation. In this paper, a comprehensive resilience response framework is presented, which provides preventive and emergency actions both before and after hurricane events. In this study, a preventive action identifies the on/off state of dispatchable distributed generation (DDG) units and discrete settings of FVCs for both before and after a hurricane event, whereas an emergency action includes the redispatch of DDG unit, load curtailment, and voltage regulation after a hurricane event. The core of the proposed framework is a trilevel max–min robust optimization (MMRO) model, which enhances the resilience of the ADS operation problem against multistage extreme N − k contingencies and renewable power generation uncertainty. The proposed MMRO model is formulated as a mixed-integer convex programming model by relaxing nonconvex AC power flow equations into mixed-integer second-order cone relaxation (MISOCR) equations. Finally, to test the effectiveness of the proposed trilevel MMRO model, we conduct experiments on a modified IEEE-33/69 bus distribution system. The numerical results show that the proposed trilevel MMRO model is an effective model for enhancing the resilience of ADSs.

Convex Formulation for Optimal Active and Reactive Power Dispatch

This paper proposes a convex programming model to solve the optimal power flow problem (OPF) related to the problems of generation active power dispatch and opportunity costs to meet the reactive power needs in energy markets. The problem is initially represented as a bilevel optimization problem where the upper level considers the minimization of opportunity costs of generating units through a second-order cone programming OPF model, and the lower level is a linear programming model that minimizes the value of active power offered by generators to the market energy. Subsequently, through the concept of strong duality, the problem is transformed into a one-level convex optimization model. Comparisons of the proposed modeling with other level formulations are presented. Results are presented for the IEEE14 and IEEE30 bus systems.

Modeling and evaluating the impact of electricity price on commute network flows of battery electric vehicles

ABSTRACT In many urban areas and urban-rural regions, the price of electricity is not only subject to time-of-use tariffs, but also dependent on the usage purpose or service sector, significantly varying across different geographic locations or land uses. Such a spatial price difference greatly affects the charging behavior of electric vehicle drivers. This paper models and analyzes the possible impact of electricity-charging price on the routing choice and flow pattern of electric vehicles in a home-workplace commute traffic network. The particular circumstance in such a network accommodating electric vehicles is that commuters driving battery electric vehicles have opportunities to charge their vehicles at both home and workplace and take into account different electricity-charging prices in their travel choices. For the illustration and evaluation purposes, we presented a small synthetic example and proposed a convex programming model for characterizing the commute network flow pattern of electric vehicles that may be charged at home and/or workplace, encapsulating a piecewise electricity-charging cost function that implies commuters’ combined home-based and workplace-based charging behaviors. Given different electricity-charging prices at home and workplace, solving this problem requires tracking all individual paths in the network. As such, a path-based solution algorithm sketched in the disaggregated simplicial decomposition framework, with some modifications in the path generation phase and an added k-shortest path search procedure, is designed for problem solutions. We implemented the modeling and solution methods for deriving the network flow pattern of electric vehicles and comparing it with that of gasoline vehicles, as well as evaluating the network performance change caused by varying electricity-charging prices with multiple network-level and link-level evaluation matrices.

A robust optimization approach for multi-product inventory management in a dual-channel warehouse under demand uncertainties

This study explores the multi-product inventory problem of a dual-channel warehouse to determine the optimal continuous review inventory policies. The warehouse is divided into two areas, one for fulfilling online orders and the other for storing products and fulfilling offline orders. Uncertainties are involved in the demands of the products during lead time in the two areas with their means and variances as the only known demand information. Using a distributionally robust optimization approach, the problem is formulated as a multi-product inventory model with an individual chance constraint and a multi-product inventory model with a joint chance constraint for the warehouse capacity to minimize the annual total expected cost. Two types of service levels are considered to ensure an adequate customer satisfaction. Through mathematical manipulations, the developed distributionally robust multi-product inventory models are transformed into convex programming models which can be solved efficiently, and the corresponding solution algorithms are developed. In particular, the closed-form solution of the order quantities is derived for the model with the individual chance constraint. Numerical experiments are performed to verify the effectiveness and practicality of the proposed models and the solution approaches in dealing with demand uncertainties and to draw specific managerial insights. Effects of important problem parameters on inventory policies and cost performance are also analyzed through numerical studies. Recommendations on the warehouse structure are given for business firms engaged in both online and offline sales. The continuous review inventory policies obtained by the proposed approach are robust and are flexible in making decisions for the operations of dual-channel warehouses and supply chains with only limited demand information. The proposed algorithms are proved to be very efficient through computational experiments.

Оптимізаційні задачі модернізації пропускних здатностей дуг відмовостійких мереж

Розглянуто математичні моделі двох класів задач модернізації пропускних здатностей дуг відмовостійких орієнтованих мереж. Відмовостійкою вважається мережа, для якої можна задовольнити всі вимоги на передачу потоків, якщо матиме місце одна, будь-яка відмова, з усіх можливих одиничних відмов мережі. У першому класі задач (задача A) для передачі потоків можуть використовуватись всі можливі шляхи в мережі. У другому класі задач (задача P) для передачі потоків задіяно тільки шляхи із напередзаданої множини шляхів. Математичні моделі представлено задачами лінійного, булевого та нелінійного програмування з блочною структурою матриці обмежень. Матеріал статті представлено в п’яти розділах. У розд. 1 описано поняття одиничної відмови та сценарію відмов мережі, наведено зміст оптимізаційних задач A та P для модернізації пропускних здатностей дуг відмовостійкої мережі, описано тестову мережу (6 вершин та 19 дуг) для перевірки алгоритмів розв’язання задач модернізації відмовостійких мереж. У розд. 2 описано базові моделі задач лінійного програмування для знаходження пропускних здатностей дуг відмовостійкої фізичної структури мережі (задача A) та відмовостійкої логічної структури мережі (задача P), розглянуто їх властивості. У розд. 3 описано задачі A та P у формі моделей змішаного булевого лінійного програмування. Наведено оптимальні розв’язки задачі A для різних сценаріїв відмов на прикладі тестової мережі. Розв’язки знайдено за допомогою програми Gurobi з NEOS-сервера, де математичну модель задачі A описано мовою моделювання AMPL. У розд. 4 описано нелінійні моделі опуклого програмування для задач A та P, призначені для знаходження оптимальних за вибраним критерієм пропускних здатностей дуг відмовостійких мереж, та описано декомпозиційний алгоритм їх розв’язання. У розд. 5 наведено опис програмного забезпечення мовою програмування ФОРТРАН для декомпозиційного алгоритму на основі ефективних реалізацій r-алгоритмів Шора. Проведено порівняння декомпозиційного алгоритму з програмою IPOPT на основі результатів розв’язання тестових задач.

Home health care routing and scheduling problem with the consideration of outpatient services

In China, family doctor contract service can provide not only the home health care service for the elderly or patients with mobility difficulties at their homes but also the outpatient service mainly for ordinary patients in the community care center. This paper presents a home health care routing and scheduling problem with the consideration of outpatient services. By considering the constraints about time windows, skill requirements, and working regulations, the problem is formulated as a mixed-integer nonlinear and convex programming model to minimize the total travel costs of the door-to-door service and the total waiting penalties of out-patients, and maximize the total benefit of patients’ preference satisfaction. We adopt an outer-approximation method to obtain its global ε-optimal solutions for the small scale problem and develop a hybrid genetic algorithm to solve the large problem. A small instance is set up to analyze the problem properties and the performance of the outer-approximation method. The results of large scale examples show that the proposed hybrid genetic algorithm can provide high-quality solutions with short computing times.

An approach to probabilistic hesitant fuzzy risky multiattribute decision making with unknown probability information

As a useful tool, probabilistic hesitant fuzzy set is an enhanced version for hesitant fuzzy set. It could be used to model the uncertainty very effectively. However, in probabilistic hesitant fuzzy risky multiple attribute decision making problems, the occurrence probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status are often difficult to obtain by subjective evaluation of a decision maker. This paper aims to propose two nonlinear programming models for calculating the probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status respectively. First, a nonlinear programming model using maximum entropy principle is established for determining the probabilities of elements in a probabilistic hesitant fuzzy element. Second, by introducing the water-filling theory, we put forward its extension and design a novel mathematical programming model to determine the probability of risk status. Moreover, we have proved that both the two mathematical programming models are convex programming models and their global optimal solutions can be found. Thirdly, the collective overall expected values of alternatives are calculated and the ranking order can be derived. Then, the selection of investment project is investigated, and comparison analysis shows the superiority of the presented approach.

Planning of Reserve Branches to Increase Reconfiguration Capability in Distribution Systems: A Scenario-Based Convex Programming Approach

Distribution networks are usually designed with a fixed tree-shape topology, which limits its adapting capability against failures and excessive variation of load/generation. Reconfiguration is an important tool to optimize the steady-state operating point of the distribution system under normal operating conditions, as well as increasing the network's responsiveness under interruptions by minimizing non-supplied demand. This paper studies how to increase the capabilities of reconfiguration in distribution networks through the optimal installation of reserve branches even if few branches are switchable. For this purpose, a scenario-based convex programming model is proposed to minimize the operation cost under normal operating conditions along with the energy not supplied related to network interruptions. Instead of conventional simulation adopted by previous approaches, the formulation is based on a set of linear logical constraints that represent the post-fault network reconfiguration process. Hence, classical optimization techniques can be used to solve the proposed model, which provides a suitable framework for the attainment of global optimality using efficient off-the-shelf software. Results show a decrease in the cost of energy production (2.85% and 1.08% for the 33-node and 69-node test systems) along with a decrease in the cost of energy not supplied (58.17% and 72.42% for the 33-node and 69-node test systems) due to the greater restoration capacity of the system using reserve branches.

A distributionally robust optimization approach for multi-product inventory decisions with budget constraint and demand and yield uncertainties

This study develops a distributionally robust optimization approach for inventory decisions for a retailer with limited budget ordering multiple products from multiple suppliers. The demand of a product is assumed to be dependent on the current inventory level. Uncertainties are involved in the demands and yields of the products with their means and standard deviations as the only known information. Using the distributionally robust optimization approach, the problem is formulated as a worst-case expected profit maximization model with a budget constraint. Through mathematical deduction, the developed model is transformed into a tractable convex programming model which can be solved efficiently. The closed-form solution for the order quantity using a Lagrange multiplier is proposed and the corresponding algorithm is presented in the case with one reliable or unreliable supplier. The retailer’s ordering decisions are investigated when there are multiple identical or different suppliers, and the effects of yield uncertainties are also assessed. Numerical experiments are performed to illustrate the effectiveness and practicality of the proposed models and the solution approaches in dealing with demand and yield uncertainties. Furthermore, the impacts of parameters such as the budget, means and standard deviations of yields and demands, and the correlations between yields on retailer’s ordering policies and performance are analyzed. Managerial insights in selecting suppliers are also provided. In particular, a bootstrapping method is used to estimate the means and standard deviations of the demand and yield distributions in a practical application, and the out-of-sample performance of the resulting optimal inventory policy is evaluated.

A convex optimization and iterative solution based method for optimal power-gas flow considering power and gas losses

Optimal power-gas flow (OPGF) is an important problem for the coordinated operation of an integrated electricity and natural gas system (IEGS). However, it is a challenge to solve the OPGF problem due to its non-convexity. This paper proposes a new OPGF method considering both power losses and gas losses based on convex optimization to balance modeling accuracy and tractability. As for power system modeling, a DC power flow model with power losses embedded as equivalent loads is used to approximate the nonlinear AC power flow model. As for gas system modeling, (1) a second-order cone (SOC) relaxation method is proposed to address the non-convex Weymouth equations, and the tightness of the SOC relaxation is guaranteed by introducing a penalty term and an iterative tightening procedure based on the convex envelope. (2) Gas losses of compressors are also modeled as equivalent loads and two more penalty terms on compressors are introduced to improve the accuracy of gas loss modeling. The OPGF problem is finally formulated as a mixed-integer convex programming model with unknown equivalent loads, and an iterative solution approach is then developed to solve the problem. Numerical results validate the effectiveness of the proposed OPGF method.

Minimax-Regret Robust Co-Optimization for Enhancing the Resilience of Integrated Power Distribution and Natural Gas Systems

Natural gas system that supplies flexible and distributed energy resources can play a significant role in enhancing the resilience of the power distribution system. In this paper, a minimax-regret robust resilience-constrained unit commitment model is proposed for enhancing the resilience of the integrated power distribution and natural gas system (IDGS). First, a tri-level robust cooperation optimization problem is formulated based on the minimax-regret robust criterion to enhance the distribution system resilience against worst N-k contingencies. The multi-stage extreme weather model is considered to obtain the spatial dynamics of extreme weather. Then, the proposed robust model is reformulated as a mixed-integer convex programming model by relaxing nonconvex power and natural gas flow equations into second-order conic (SOC) constraints. The SOC-based column-and-constraint generation algorithm is employed to solve the proposed two-stage robust optimization problem. The effectiveness of proposed robust optimization model is validated using several case studies applied to the 33-bus-6-node and 123-bus-20-node IDGSs.

Joint Wireless Source Management and Task Offloading in Ultra-Dense Network

The ultra-dense network (UDN) based on mobile edge computing (MEC) is an important technology, which can achieve the low-latency of 5G communications and enhance the quality of user experience. However, how to improve the task offloading efficiency is a hot topic of UDN under the constraint on the limited wireless resources. In this article, we propose a heuristic task offloading algorithm HTOA to optimize the delay and energy consumption of offloading tasks in UDN. Firstly, a convex programming model for MEC resource allocation is established, which aims to obtain the optimal allocation set of resources for offloading tasks, and optimize the execution delay of offloading tasks. Followed by, the problem of joint channel allocation and user upload power control is solved by the greedy strategy and golden section method, which aims to optimization the delay and energy consumption of task upload data. Compared with the random task offloading algorithm, numerical simulations show that the algorithm HTOA can effectively reduce the delay and energy consumption of task offloading, and perform better as the number of users increases.

Application of the Nested Convex Programming to the Optimal Power Flow in MT-HVDC Grids.

This paper deals with an application of the nested convex programming to the optimal power flow (OPF) in multi-terminal high-voltage direct-current grids (MT-HVDC). The real-world optimization problem under consideration is non-convex. This fact implies some possible inconsistencies of the conventional numerical minimization algorithms (such as interior point method). Moreover, the constructive numerical treatment of this problem is usually based on some approximative approaches, namely, on the suitable linearizations and problem relaxations. The resulting convex programming model constitutes an approximated model and can naturally involve the significant (approximation) errors. In difference to the strongly approximate computational approaches mentioned above, the numerical scheme we propose takes into account the specific bi-linear structure of the problem and operates with the originally given non-convex formulation of the problem. We implement the proposed nested optimization approach and study the numerical consistency of the resulting optimal design. The Python based numerical experiments demonstrate the imlementability of the proposed methodology. Optimization problem of the modified version of the CIGRE MT-HVDC is next used as a benchmark test for the approach we developed.

A Mixed-Integer Convex Programming Algorithm for Security-Constrained Unit Commitment of Power System with 110-kV Network and Pumped-Storage Hydro Units

The secure operation of 110-kV networks should be considered in the optimal generation dispatch of regional power grids in large central cities. However, since 110-kV lines do not satisfy the premise of R << X in the direct current power flow (DCPF) model, the DCPF, which is mostly applied in the security-constrained unit commitment (SCUC) problem of high-voltage power grids, is no longer suitable for describing the active power flow of regional power grids in large central cities. Hence, the quadratic active power flow (QAPF) model considering the resistance of lines is proposed to describe the network security constraints, and an SCUC model for power system with 110-kV network and pumped-storage hydro (PSH) units is established. The analytical expressions of the spinning reserve (SR) capacity of PSH units are given considering different operational modes, and the SR capacity of PSH units is included in the constraint of the SR capacity requirement of the system. The QAPF is a set of quadratic equality constraints, making the SCUC model a mixed-integer nonlinear non-convex programming (MINNP) model. To reduce the computational complexity of solving the model when applied in actual large-scale regional networks, the QAPF model is relaxed by its convex hull, and the SCUC model is transformed into a mixed-integer convex programming (MICP) model, which can be solved to obtain the global optimal solution efficiently and reliably by the mature commercial solver GUROBI (24.3.3, GAMS Development Corporation, Guangzhou, China). Test results on the IEEE-9 bus system, the PEGASE 89 bus system and the Shenzhen city power grid including the 110-kV network demonstrate that the relaxed QAPF model has good calculation accuracy and efficiency, and it is suitable for solving the SCUC problem in large-scale regional networks.