Multi-objective Mixed-integer Nonlinear Optimization Research Articles

A hierarchical optimization approach to maximize hosting capacity for electric vehicles and renewable energy sources through demand response and transmission expansion planning

Within the scope of sustainable development, integrating electric vehicles (EVs) and renewable energy sources (RESs) into power grids offers a number of benefits. These include reducing greenhouse gas emissions, diversifying energy sources, and promoting the use of green energy. Although the literature on hosting capacity (HC) models has grown, there is still a noticeable gap in the discussion of models that successfully handle transmission expansion planning (TEP), demand response (DR), and HC objectives simultaneously. Combining TEP, DR, and HC objectives in one model optimizes resource use, enhances grid stability, supports renewable and EV integration, and aligns with regulatory and market demands, resulting in a more efficient, reliable, and sustainable power system. This research presents an innovative two-layer HC model, including considerations for TEP and DR. The model determines the highest degree of load shifting appropriate for incorporation into power networks in the first layer. Meanwhile, the second layer focuses on augmenting the RES and EVs’ hosting capability and modernizing the network infrastructure. System operators can choose the best scenario to increase the penetration level of EVs and RESs with the aid of the proposed model. The proposed model, which is formulated as a multi-objective mixed-integer nonlinear optimization problem, uses a hierarchical optimization technique to identify effective solutions by combining the particle swarm optimization algorithm and the crayfish optimizer. When compared to traditional methods, the results obtained from implementing the proposed hierarchical optimization algorithm on the Garver network and the IEEE 24-bus system indicated how effective it is at solving the presented HC model. The case studies demonstrated that integrating DR into the HC problem reduced peak load by 10.4–23.25%. The findings also highlighted that DR did not impact the total energy consumed by EVs throughout the day, but it did reshape the timing of EV charging, creating more opportunities for integration during periods of high demand. Implementing DR reduced the number of projects needed and, in some cases, led to cost savings of up to 12.3%.

A branch-and-bound algorithm based on NSGAII for multi-objective mixed integer nonlinear optimization problems

Solving Multi-Objective Mixed Integer NonLinear Programming (MO-MINLP) problems is a point of interest for many researchers as they appear in several real-world applications, especially in the mechanical engineering field. Many researchers have proposed using hybrids of metaheuristics with mono-objective branch and bound. Others have suggested using heuristics with Multi-Criteria Branch and Bound (MCBB). A general hybrid approach is proposed based on MCBB and Non-dominated Sorting Genetic Algorithm 2 (NSGAII) to enhance the approximated Pareto front of MO-MINLP problems. A computational experiment based on statistical assessment is presented to compare the performance of the proposed algorithm (BnB-NSGAII) with NSGAII using well-known metrics from the literature. To evaluate the computational efficiency, a new metric, the Investment Ratio (IR), is proposed that relates the quality of solution to the consumed effort. Experimental results on five real-world mechanical engineering problems and two mathematical ones indicate that BnB-NSGAII could be a competitive alternative for solving MO-MINLP problems.

Optimal Smart Inverter Control for PV and BESS to Improve PV Hosting Capacity of Distribution Networks Using Slime Mould Algorithm

In this study, an optimal reactive power (Volt/VAr) control of smart inverters for photovoltaic (PV) and battery energy storage systems (BESSs) to improve the PV hosting capacity (PVHC) of distribution networks is proposed. The primary objective of the proposed method is to improve the PVHC of a distribution network by determining the optimal oversize, dispatch, and control setting of the Volt/VAr functions of the smart inverters for both PVs and BESSs. Concurrently, the optimal locations of the PVs and BESSs are determined. The problem is formulated as a multi-objective mixed-integer nonlinear optimization to maximize the PVHC and minimize the voltage deviation simultaneously. A bio-inspired metaheuristic optimization method, i.e., the slime mould algorithm (SMA), is employed to solve the optimization problem. To assess the efficacy of the proposed PVHC improvement method, extensive simulations are conducted on an IEEE 33-node system using MATLAB software. The simulation results verify that the proposed method improves the PVHC of the distribution network compared to different cases and the default Volt/VAr control settings of the smart inverters. Furthermore, the SMA optimization method provides superior performance in finding the optimal PVHC of a distribution network compared to the conventional metaheuristic optimization methods.

Protection system planning for distribution networks: A probabilistic approach

This paper presents a multi-objective optimization approach to perform the allocation, sizing and coordination of control and protective devices in distribution networks with different types of distributed generation. The model consists of three objective functions that consider investment costs, the system’s non-supplied energy, and average interruption duration index. We use a probabilistic short-circuit routine to evaluate the currents values, taking into account the uncertainties of renewable power generation and the system’s loading, as well as the fault’s parameters randomness. The probabilistic approach allows the correct planning of the protection system and enables the user to admit some risk of not coordinated operation, aiming to reduce the solutions’ costs. This multi-objective mixed-integer non-linear optimization problem considers the possibility of load transfers from neighboring feeders and intentional islanding. In this paper, we use the Non-dominated Sorting Genetic Algorithm to solve this optimization problem. The algorithm is tested for a 135-bus feeder.

Joint Sampling Time and Resource Allocation for Power Efficiency in Industrial Cyber–Physical Systems

Cyber-physical systems (CPSs) form from convergence of computation, networking, and control of physical processes. In this article, the focus is on an industrial CPS consisting of many control plants and rate constrained (RC) users communicating over a single-cell orthogonal frequency-division multiple access network. We formulate a multiobjective mixed integer nonlinear optimization problem that seeks to determine the next maximum acceptable sampling moment of control plants and to minimize the energy usage in the uplinks and downlinks while taking into account the dynamics and intended performance of control plants, the quality of service of RC users, and power and subcarrier constraints. To address this problem, a decoupling approach is taken, leading to a Pareto solution. Our simulations show that the proposed scheme decreases the total power consumption in the CPS to a significant degree.

Planning the Schedule for the Disposal of the Spent Nuclear Fuel with Interactive Multiobjective Optimization

Several countries utilize nuclear power and face the problem of what to do with the spent nuclear fuel. One possibility, which is under the scope in this paper, is to dispose of the fuel assemblies in the disposal facility. Before the assemblies can be disposed of, they must cool down their decay heat power in the interim storage. Next, they are loaded into canisters in the encapsulation facility, and finally, the canisters are placed in the disposal facility. In this paper, we model this process as a nonsmooth multiobjective mixed-integer nonlinear optimization problem with the minimization of nine objectives: the maximum number of assemblies in the storage, maximum storage time, average storage time, total number of canisters, end time of the encapsulation, operation time of the encapsulation facility, the lengths of disposal and central tunnels, and total costs. As a result, we obtain the disposal schedule i.e., amount of canisters disposed of periodically. We introduce the interactive multiobjective optimization method using the two-slope parameterized achievement scalarizing functions which enables us to obtain systematically several different Pareto optimal solutions from the same preference information. Finally, a case study adapting the disposal in Finland is given. The results obtained are analyzed in terms of the objective values and disposal schedules.

Two-Timescale Multi-Objective Coordinated Volt/Var Optimization for Active Distribution Networks

This paper presents a two-timescale multi-objective coordinated volt/var optimization system for active distribution networks with renewable energy resources (RESs). In the hourly timescale, power loss and the number of control actions of on-load tap changers and (switchable) shunt capacitors for 24 h are minimized by coordinating on-load tap changers, (switchable) shunt capacitors, and reactive power outputs of RESs. A three-step method consisting of global search, user preference, and local search is adopted to solve the corresponding multi-objective mixed-integer nonlinear optimization problem. A rolling volt/var optimization is executed every 6 h, considering the day-ahead RES forecast errors. In the 15-min timescale, the RES reactive powers are optimized with RES uncertainties explicitly taken into consideration to minimize the pilot-bus voltage deviation under all representative scenarios. The pilot buses are those buses whose voltage magnitudes capture the network voltage profile. In this paper, a linear method is applied to quickly identify a set of pilot buses, and the impact of the identified pilot bus on the entire network voltage profile is evaluated. The IEEE 33-bus distribution network is used to evaluate the proposed two-timescale volt/var optimization formulations and methods with promising results.

Privacy-preserving based task allocation with mobile edge clouds

With the rapid development of mobile crowdsensing applications, task allocation has emerged as a new problem to be solved. Many task allocation strategies have been proposed to select the proper participants to complete tasks. Those methods need participants to submit their real locations to the platform in order to realize the optimal task assignment. However, those traditional task allocation strategies have two weaknesses. First, centralized task allocations result in high computing and communications loads. Second, the exposure of real locations increases participants’ concerns regarding location privacy. To address these problems, in this paper, we propose an optimal geo-indistinguishable task allocation (GITA) mechanism using mobile edge clouds. First, the new task that is received by the remote cloud is sent to the mobile edge cloud that is nearest to the task location. Then, the mobile edge clouds serve as distributed controllers to allocate the assigned tasks to the proper candidates. To protect the candidates’ real locations, we utilize a geo-indistinguishable mechanism based on differential privacy to preserve location privacy. Specifically, we obfuscate the participants’ real locations as disturbed locations, and realize the optimal task allocation based on these disturbed locations. Furthermore, we apply multiobjective mixed integer nonlinear optimization to solve this problem. Finally, extensive experimental results show that, compared with the traditional Laplace mechanism and another privacy-preserving task allocation strategy, the GITA mechanism that is proposed in this paper can decrease users’ moving distances and raise the task completion rate.

A comprehensive multi-objective optimization model for the vessel scheduling problem in liner shipping

The international seaborne trade volumes have been continuously increasing over the last years. Many liner shipping companies started consolidation to form stronger alliances and attract new customers. In order to remain competitive and avoid potential monetary losses, liner shipping companies have to improve efficiency of their vessel schedules. Some of the decisions that have to be made by liner shipping companies are conflicting in their nature. However, the existing vessel scheduling models generally combine the conflicting objectives into one objective function, which aims to minimize the total route service cost. Such approach imposes limitations for liner shipping companies in the analysis of tradeoffs between the conflicting objectives. To avoid the latter drawback, this study proposes a multi-objective mixed integer nonlinear optimization model for the vessel scheduling problem, which accounts for all major route service cost components reported in the literature and separates them in two conflicting groups. The original mixed integer nonlinear model is linearized by discretizing the vessel sailing speed reciprocal, and the Global Multi-Objective Optimization Algorithm is developed to solve the linearized model. A set of numerical experiments are conducted for the Asia-Mediterranean Express Service liner shipping route. Results demonstrate that negotiation of both vessel service time windows and handling rates between liner shipping companies and marine container terminal operators may significantly reduce the total route service cost components. Furthermore, vessel schedules are found to be more sensitive to the unit fuel cost as compared to the unit carbon dioxide emission cost.

Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

In this paper, we study the multiobjective co-design problem of optimal valve placement and operation in water distribution networks, addressing the minimization of average pressure and pressure variability indices. The presented formulation considers nodal pressures, pipe flows and valve locations as decision variables, where binary variables are used to model the placement of control valves. The resulting optimization problem is a multiobjective mixed integer nonlinear optimization problem. As conflicting objectives, average zone pressure and pressure variability can not be simultaneously optimized. Therefore, we present the concept of Pareto optima sets to investigate the trade-offs between the two conflicting objectives and evaluate the best compromise. We focus on the approximation of the Pareto front, the image of the Pareto optima set through the objective functions, using the weighted sum, normal boundary intersection and normalized normal constraint scalarization techniques. Each of the three methods relies on the solution of a series of single-objective optimization problems, which are mixed integer nonlinear programs (MINLPs) in our case. For the solution of each single-objective optimization problem, we implement a relaxation method that solves a sequence of nonlinear programs (NLPs) whose stationary points converge to a stationary point of the original MINLP. The relaxed NLPs have a sparse structure that come from the sparse water network graph constraints. In solving the large number of relaxed NLPs, sparsity is exploited by tailored techniques to improve the performance of the algorithms further and render the approaches scalable for large scale networks. The features of the proposed scalarization approaches are evaluated using a published benchmarking network model.

A tuned NSGA-II to optimize the total cost and service level for a just-in-time distribution network

Distribution network planning has attracted the attention of many studies during last decades. Just-in-time (JIT) distribution has a key role in efficient delivery of products within distribution networks. In modeling of JIT distribution networks, the most frequently applied objectives are related to cost and service level. However, evaluating the impact of simultaneously minimizing total costs and balance between distribution network entities in different echelons still rarely complies with the current literature. To remedy this shortcoming and model reality more accurately, this paper develops a multi-objective mixed-integer nonlinear optimization model for a JIT distribution in three-echelon distribution network. The aims are minimization of total logistics cost along with maximization of capacity utilization balance for distribution centers and manufacturing plants. A non-dominated sorting genetic algorithm-II (NSGA-II) with three different mutation operators namely swap, reversion and insertion is employed to provide a set of near-optimal Pareto solutions. Then, the provided solutions are verified with non-dominated ranked genetic algorithm (NRGA) as well. The Taguchi method in design of experiments tunes the parameters of both algorithms, and their performances are then compared in terms of some multi-objective performance measures. In addition, a genetic algorithm is used to assess Pareto optimal solutions of NSGA-II. Different problems with different sizes are considered to compare the performance of the suggested algorithms. The results show that the proposed solution approach performs efficiently. Finally, the conclusion and some directions for future research are proposed.

An exact multi-objective mixed integer nonlinear optimization approach for aircraft conflict resolution

An exact mixed integer nonlinear optimization (MINO) model is presented for tackling the aircraft conflict detection and resolution problem in air traffic management. Given a set of flights and their configurations, the aim of the problem was to provide new configurations such that all conflict situations are avoided, with conflict situation understood to mean an event in which two or more aircraft violate the minimum safety distances that must be maintained in flight. The proposed model solves the problem using both horizontal (velocity and angle turn change) and vertical (altitude level change) maneuvers. As a special case, another model is presented in which only horizontal maneuvers are used. The models proposed are based on a geometric construction which involves trigonometric functions, so the constraint system is included via a large set of trigonometric and nonconvex inequalities. A multicriteria approach is presented to provide useful information to air traffic control officers about the maneuvers to be performed. The main results of an extensive computational experiment are reported in which the performance of the state-of-the-art nonconvex MINO solver Minotaur is studied.

Voltage Stability Constrained Optimal Power Flow Using NSGA-II

Voltage stability has become an important issue in planning and operation of many power systems. This work includes multi-objective evolutionary algorithm techniques such as Genetic Algorithm (GA) and Non-dominated Sorting Genetic Algorithm II (NSGA II) approach for solving Voltage Stability Constrained-Optimal Power Flow (VSC-OPF). Base case generator power output, voltage magnitude of generator buses are taken as the control variables and maximum L-index of load buses is used to specify the voltage stability level of the system. Multi-Objective OPF, formulated as a multi-objective mixed integer nonlinear optimization problem, minimizes fuel cost and minimizes emission of gases, as well as improvement of voltage profile in the system. NSGA-II based OPF-case 1-Two objective-Min Fuel cost and Voltage stability index; case 2-Three objective-Min Fuel cost, Min Emission cost and Voltage stability index. The above method is tested on standard IEEE 30-bus test system and simulation results are done for base case and the two severe contingency cases and also on loaded conditions.

Multiobjective Optimal Reactive Power Flow Using Elitist Nondominated Sorting Genetic Algorithm: Comparison and Improvement

Elitist nondominated sorting genetic algorithm (NSGA-II) is adopted and improved for multiobjective optimal reactive power flow (ORPF) problem. Multiobjective ORPF, formulated as a multiobjective mixed integer nonlinear optimization problem, minimizes real power loss and improves voltage profile of power grid by determining reactive power control variables. NSGA-II-based ORPF is tested on standard IEEE 30-bus test system and compared with four other state-of-the-art multiobjective evolutionary algorithms (MOEAs). Pareto front and outer solutions achieved by the five MOEAs are analyzed and compared. NSGA-II obtains the best control strategy for ORPF, but it suffers from the lower convergence speed at the early stage of the optimization. Several problem-specific local search strategies (LSSs) are incorporated into NSGA-II to promote algorithm's exploiting capability and then to speed up its convergence. This enhanced version of NSGA-II (ENSGA) is examined on IEEE 30 system. Experimental results show that the use of LSSs clearly improved the performance of NSGA-II. ENSGA shows the best search efficiency and is proved to be one of the efficient potential candidates in solving reactive power optimization in the real-time operation systems.

Analyzing the interaction of design and control—2. reactor-separator-recycle system

Part 1 presented a systematic procedure to analyze the interaction of design and control at the process synthesis stage by incorporating both steady-state economic and open-loop controllability measures within a multiobjective mixed-integer nonlinear optimization problem. Part 2 applies this framework to study the design of a system consisting of an isothermal reactor followed by a distillation column, whose distillate stream recycles back to the reactor. For this particular flowsheet, the resulting optimization can be written as a nonlinear programming (NLP) problem. A comparison of alternative designs shows that leaving the reactor and recycle stream compositions as variables in the NLP significantly reduces the cost compared with the cost of designs when these compositions are fixed. Economic considerations drive the optimal solution toward a larger reactor with lower reactant composition and toward a stripper column with lower boilup. The derivation of steady-state gain expressions relating the product compositions in the column to manipulated and disturbance variables allows the incorporation of controllability measures within the NLP. Generation of the noninferior solution sets shows the quantitative trade-off between the steady-state economic and controllability objectives. In general, process controllability measures improve as the reactor holdup decreases and as the number of trays in the column increases, but the opposite changes improve the disturbance controllability measures. Closed-loop dynamic simulations compare the responses of several designs identified in the optimization.