Mixed-integer Second-order Cone Programming Model Research Articles

Reliability-Oriented Reconfiguration of Power Distribution Systems Considering Load and RES Production Scenarios

The ever-increasing integration of non-dispatchable distributed generation, i.e., renewable energy sources (RES), arises new challenges in the field of power system's reliability. Distribution network reconfiguration (DNR) is a cost-effective approach for the distribution system operator (DSO) that wishes to enhance system's reliability without infrastructure upgrades. This paper introduces a novel path-based mixed-integer second-order cone programming model to optimally solve the reliability-oriented DNR problem. The DSO's objectives that are optimized are: a) improvement of distribution system's reliability indices and b) minimization of power losses. The proposed model is enriched with a scenario-based stochastic programming formulation that considers multiple levels of load and RES production. The standard 33-nodes distribution system and a real-world 83-nodes distribution system are employed to prove the efficiency and applicability of the model. Firstly, the multi-objective nature of the reliability-oriented DNR problem is investigated by conducting a sensitivity analysis, which reveals a trade-off region between reliability indices and power losses. Moreover, the obtained results show different global optimal solutions when the variability of load and RES production is considered. This highlights the importance of considering a scenario-based approach for load and RES production when solving the reliability-oriented DNR problem.

An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy

In order to adjust to the change of the large-scale deployment of photovoltaic (PV) power generation and fully exploit the potentialities of an integrated energy distribution system (IEDS) in solar energy accommodation, an evaluation method on maximum hosting capacity of solar energy in IEDS based on convex relaxation optimization algorithm is proposed in this paper. Firstly, an evaluation model of maximum hosting capacity of solar energy for IEDS considering the electrical-thermal comprehensive utilization of solar energy is proposed, in which the maximization of PV capacity and solar collector (SC) capacity are fully considered. Secondly, IEDS’s potential in electricity, heat, and gas energy coordinated optimization is fully exploited to enhance the hosting capacity of solar energy in which the electric distribution network, heating network, and natural gas network constraints are fully modeled. Then, an enhanced second-order cone programming (SOCP)-based method is employed to solve the proposed maximum hosting capacity model. Through SOCP relaxation and linearization, the original nonconvex nonlinear programming model is converted into the mixed-integer second-order cone programming model. Meanwhile, to ensure the exactness of SOCP relaxation and improve the computation efficiency, increasingly tight linear cuts of distribution system and natural gas system are added to the SOCP relaxation. Finally, an example is given to verify the effectiveness of the proposed method. The analysis results show that the maximum hosting capacity of solar energy can be improved significantly by realizing the coordination of an integrated multi-energy system and the optimal utilization of electricity, heat, and gas energy. By applying SOCP relaxation, linearization, and adding increasingly tight linear cuts of distribution system and natural gas system to the SOCP relaxation, the proposed model can be solved accurately and efficiently.

A convex-relaxation based method for optimal water-power flow

This paper proposes a convex reformulation for the non-linear optimal water-power flow (OWPF) problem to optimize the operation cost of the integrated electricity–water network (IEWN). Due to the significant challenge in seeking the optimal solution of non-convex problems, the original OWPF problem is reformulated as a tractable mixed-integer second-order cone programming (MISOCP) problem. For the power distribution network, a second-order cone (SOC) relaxation is employed to address the non-linear branch power flow equation. For the water distribution network, a big-M trick is introduced to handle the unknown direction of water flow, then SOC relaxations and convex envelopes are employed to deal with quadratic and bilinear terms in non-convex constraints, respectively. To enhance the approximation accuracy of the proposed MISOCP model, a convex combination method and a sequential convexification approach are developed. Numerical results demonstrate that the proposed method outperforms the original non-linear formulation and linearized reformulation in accuracy, efficiency, and robustness.

Distributionally robust location–allocation models of distribution centers for fresh products with uncertain demands

This paper studies the location–allocation problem of distribution centers (DCs) for fresh products, in which it requires to determine the number and locations of DCs and design the allocation scheme for fresh products. The main challenge in optimizing this problem is to handle the uncertain demands with only partially known probability distribution information. To address this challenge, this paper first characterizes the ambiguous probability distributions of fresh products demands by an ambiguity set. Then, from different perspectives, this paper proposes two distributionally robust (DR) location–allocation models to find the optimal decisions that perform best in view of the worst-case distribution within the ambiguity set. The first model is with ambiguous expectation, which is transformed into a mixed-integer linear programming (MILP) model via employing the linear decision rule technique. The second model is with ambiguous chance constraints. This model is a semi-infinite chance constraint programming problem, which is transformed into a mixed-integer second-order cone programming (MISOCP) model by deriving safe approximations of the ambiguous chance constraints. To verify the feasibility and effectiveness of the proposed models, a case study about Hema fresh chain stores in Beijing is addressed. The experimental results demonstrate that the optimal decision provided by the proposed optimization methods can immunize against ambiguous probability distributions of demands with a small cost price.

Minimization of the distribution operating costs with D-STATCOMS: A mixed-integer conic model

This paper addresses the problem of determining the optimal location and operation of distribution static synchronous compensators (D-STATCOMs) in power distribution networks. Metaheuristic algorithms have been conventionally used to solve this mixed-integer nonlinear programming problem. In contrast, we propose a mixed-integer second-order cone programming (MI-SOCP) model that guarantees global optimum and fast convergence of the algorithms in optimization solvers. The model seeks to minimize the annual installation cost and the operating cost, subject to active and reactive power balance constraints, voltage regulation bounds, devices capabilities, and the number of D-STATCOMs available to be connected. The multiobjective nature of the problem is also analyzed and solved using the proposed MI-SOCP model. An extensive set of simulations on the IEEE-33 nodes test system demonstrate the advantages of this approach compared to conventional heuristic algorithms and with solutions given by algebraic modeling software.

Increasing RES Hosting Capacity in Distribution Networks Through Closed-Loop Reconfiguration and Volt/VAr Control

This article presents a novel mixed-integer second-order cone programming model to increase the photovoltaic (PV) hosting capacity and optimize the operation of distribution networks. The problem considers voltage and reactive (Volt/VAr) control through the optimal operation of capacitors banks, substations’ on-load tap changers, voltage regulators, and network reconfiguration with radial and closed-loop operation topologies. The proposed formulation considers voltage-dependent models for loads and capacitor banks. The objective function maximizes the PV hosting capacity of the network. Numerical experiments are carried out using the 33-node and the 85-node networks. Results demonstrate the effectiveness of the proposed formulation to increase the penetration of PV sources, especially when the closed-loop operation is allowed, together with network reconfiguration and Volt/VAr control.

A mixed-integer second-order cone model for optimal siting and sizing of dynamic reactive power compensators in distribution grids

The problem of the optimal placement and sizing of dynamic reactive power compensators in AC distribution networks is addressed in this paper from convex optimization. The exact mixed-integer nonlinear programming (MINLP) model is transformed into a mixed-integer second-order cone programming (MISOCP) model. The main advantage of the MISOCP formulation is the possibility of finding a global optimum with branch & cut combined with interior-point method due to the convex structure of the continuous part of the problem, i.e., the multi-period branch optimal power flow. The dynamic reactive power compensators are sized and dimensioned considering daily load curves and variable reactive power injections. Numerical validations are tested in the 33- and 69-bus test feeders using the CVX tool available for MATLAB with the MOSEK solver. These simulations demonstrate the effectiveness and robustness of the MISOCP approach when compared with the solution of the exact MINLP obtained in the GAMS software.

On the Conic Convex Approximation to Locate and Size Fixed-Step Capacitor Banks in Distribution Networks

The problem of the optimal siting and sizing of fixed-step capacitor banks is studied in this research from the standpoint of convex optimization. This problem is formulated through a mixed-integer nonlinear programming (MINLP) model, in which its binary/integer variables are related to the nodes where the capacitors will be installed. Simultaneously, the continuous variables are mainly associated with the power flow solution. The main contribution of this research is the reformulation of the exact MINLP model through a mixed-integer second-order cone programming model (MI-SOCP). This mixed-integer conic model maintains the nonlinearities of the original MINLP model; however, it can be solved efficiently with the branch & bound method combined with the interior point method adapted for conic programming models. The main advantage of the proposed MI-SOCP model is the possibility of finding the global optimum based on the convex nature of the power flow problem for each binary/integer variable combination in the branch & bound search tree. The numerical results in the IEEE 33- and IEEE 69-bus systems demonstrate the effectiveness and robustness of the proposed MI-SOCP model compared to different metaheuristic approaches. The MI-SOCP model finds the final power losses of the IEEE 33- and IEEE 69-bus systems of 138.416kW and 145.397kW, which improves the best literature results reached with the flower pollination algorithm, i.e., 139.075 kW, and 145.860kW, respectively. The simulations are carried out in MATLAB software using its convex optimizer tool known as CVX with the Gurobi solver.

Optimal Scheduling for Flexible Regional Integrated Energy System with Soft Open Point

The Regional Integrated Energy System (RIES) has brought new modes of development, utilization, conversion, storage of energy. The introduction of Soft Open Point (SOP) and the application of Power to Gas (P2G) technology will greatly deepen the coupling of the electricity-gas integrated energy system, improve the flexibility and safety of the operation of the power system, and bring a deal of benefits to the power system. On this background, an optimal dispatch model of RIES combined cold, heat, gas and electricity with SOP is proposed. Firstly, RIES architecture with SOP and P2G is designed and its mathematical model also is built. Secondly, on the basis of considering the optimal scheduling of combined cold, heat, gas and electricity, the optimal scheduling model for RIES was established. After that, the original model is transformed into a mixed-integer second-order cone programming model by using linearization and second-order cone relaxation techniques, and the CPLEX solver is invoked to solve the optimization problem. Finally, the modified IEEE 33-bus system is used to analyze the benefits of SOP, P2G technology and lithium bromide absorption chillers in reducing system network loss and cost, as well as improving the system’s ability to absorb wind and solar and operating safety.

An integrated integer programming model with a simulated annealing heuristic for the carrier vehicle traveling salesman problem

We investigate a carrier vehicle traveling salesman problem (CVTSP) to visit n-targets in Euclidean space. Each target is visited by a smaller vehicle (e.g., UAV) consecutively, launched from and landed to a carrier. The smaller vehicle and the carrier must synchronize during both launching and landing. We present a mixed-integer second-order cone programming model with TSP-like constraints to determine the best target visiting sequence and the associated launch and land coordinates. A simulated annealing algorithm is implemented to shorten the computational time. The computational experiments indicate the simulated annealing algorithm calculates good CVTSP solutions within a reasonable amount of time.

Resilience-Oriented Distributed Load Restoration Method for Integrated Power Distribution and Natural Gas Systems

This study proposes a distributed load restoration method to effectively model and solve the load restoration problem of integrated power distribution and natural gas systems (IDGSs) after faults for resilient operation. The IDGS load restoration problem is devised as a mixed-integer second-order cone programming (MISOCP) model to minimize the total load loss cost of IDGS. This model considers distribution network reconfiguration and the coupling effect between the power distribution system (PDS) and natural gas system (NGS). Moreover, a bidirectional gas flow model is used to offer more flexibility for load restoration when the NGS is faulty. By considering the independent management and information privacy of each energy subsystem, the proposed model is solved using an alternating direction method of multipliers (ADMM) algorithm in a distributed manner. Furthermore, an adaptive-ADMM algorithm is used for convergence acceleration. Finally, numerical studies are conducted to validate the proposed methods effectiveness.

Economic dispatch for a regionally integrated heat and electricity system with variable pipeline flow direction

Traditional models of heating networks are not able to take into account the changes in the value and the direction of pipeline mass flow in a multi-heat-source district heating network or the time delay characteristics of the network with constant temperature and variable flow operation. A day-ahead dispatch model for regional integrated heating and electricity system based on improved second-order cone programming method is therefore proposed. We first construct a heating network transmission model with variable value and direction of pipeline mass flow, and then use a convex relaxation to transform this nonlinear model into a mixed-integer second-order cone programming model. A hydraulic time-delay correction model is then constructed to reflect the dynamic characteristics of the heating network under variable mass flow conditions. Finally, we establish a model of an electric boiler with thermal storage to improve the flexibility of the distribution network. Case studies are used to show that the optimised results are more economical in terms of both the variable value and direction of mass flow. The proposed programming method retains the variability in the flow direction and achieves precise relaxation without the need to solve for the initial value. We also verify that the hydraulic time delay characteristics have a strong influence on the optimised results of the regionally integrated heat–electricity system under conditions of constant temperature and variable flow. In addition, an electric boiler with thermal storage can reduce renewable energy curtailment and effectively improve operational flexibility.

Distributed energy management for active distribution network considering aggregated office buildings

This paper proposes a distributed energy management for active distribution network (ADN) considering aggregated office buildings. It aims to dispatch the flexible resources of office buildings to actively participate in the optimization of the ADN. Firstly, by considering the comfortable indoor temperature range and travel habits of workers, an optimal operation model for the office building with integrated heating ventilation and air conditioning (HVAC) systems and electric vehicle (EV) charging piles is constructed. Secondly, based on the piecewise linearization and second-order cone relaxation (SOCR), an ADN optimization model with aggregated office buildings is formulated and converted into a mixed-integer second-order cone programming (MISOCP) model. Then, considering the private information protection, the alternating direction method of multipliers (ADMM) is used to solve the ADN optimization model to realize the distributed energy management of the ADN with aggregated office buildings. Finally, the scheduling results of the ADN optimization model under different comfortable indoor temperatures and control methods of EVs are analyzed in the winter heating scenario. Numerical results show that the proposed method can increase the penetration of distributed photovoltaic (PV) generation and decrease losses of the distribution network while ensuring the temperature and travel demands of workers in office buildings. The information privacy of each district can also be protected with the proposed method.

Optimal Restoration of Distribution Systems Considering Temporary Closed-Loop Operation

This article presents a new mathematical model to solve the restoration problem in balanced distribution systems with distributed generators (DGs) considering closed-loop topology operation during the restorative state. The restorative state is comprised of the interval of time since the permanent fault has been isolated until the time at which the faulted zone is repaired. During this interval of time, switching operations are performed to minimize the negative effects resulting from the occurrence of a permanent fault in the network. In this way, the two main objective functions of the restoration problem are to minimize the amount of load curtailment in the restored system and to minimize the number of switching operations. Conventionally, the network topology is maintained radial throughout the restorative state. In this article, the possibility of forming loops is considered for improving both objective functions. As such, a new mixed-integer second-order cone programming model is proposed, considering the temporary formation of operational loops in the restorative state, and both connected and islanded operation of the DGs. Several tests are carried out using a 53-node test system and a 2313-node system for single and multiple fault scenarios. The results obtained with the proposed model outperform the solutions achieved when only open-loop configurations are considered for the restoration problem. Moreover, it is verified that the islanded operation of the DGs provides more flexibility to the network, allowing more load to be restored.

Distributed generator-based distribution system service restoration strategy and model-free control methods

The rapid growth of distributed generator (DG) capacities has introduced additional controllable assets to improve the performance of distribution systems in terms of service restoration. Renewable DGs are of particular interest to utility companies, but the stochastic nature of intermittent renewable DGs could have a negative impact on the electric grid if they are not properly handled. In this study, we investigate distribution system service restoration using DGs as the primary power source, and we develop an effective approach to handle the uncertainty of renewable DGs under extreme conditions. The distribution system service restoration problem can be described as a mixed-integer second-order cone programming model by modifying the radial topology constraints and power flow equations. The uncertainty of renewable DGs will be modeled using a chance-constrained approach. Furthermore, the forecast errors and noises in real-time operation are solved using a novel model-free control algorithm that can automatically track the trajectory of real-time DG output. The proposed service restoration strategy and model-free control algorithm are validated using an IEEE 123-bus test system.

Optimal Restoration of Active Distribution Systems With Voltage Control and Closed-Loop Operation

This article presents a new mixed-integer second-order cone programming model for solving the restoration problem in active distribution systems considering the optimal control of distributed generators (DGs), voltage regulators (VRs), on-load tap changers (OLTCs), and capacitor banks (CBs). In contrast to most of the works found in the literature, temporary loops may be formed in the network during the restorative operation state. This operating condition allows restoring the service to more loads. In this way, the objective function of the problem minimizes (i) the total load not supplied, (ii) the number of switching operations, (iii) the changes in the statuses of the voltage control devices and in the dispatch of the DGs, and (iv) the number of basic loops formed in the system. Several tests are carried out using a 53-node system for single-fault and multiple-fault scenarios. The results obtained with the proposed approach outperform the solutions achieved when only radial configurations are allowed in the problem. Moreover, it is also verified that the voltage control allows for more efficient restoration schemes.

A Mixed-Integer Convex Model for the Optimal Placement and Sizing of Distributed Generators in Power Distribution Networks

The optimal placement and sizing of distributed generators is a classical problem in power distribution networks that is usually solved using heuristic algorithms due to its high complexity. This paper proposes a different approach based on a mixed-integer second-order cone programming (MI-SOCP) model that ensures the global optimum of the relaxed optimization model. Second-order cone programming (SOCP) has demonstrated to be an efficient alternative to cope with the non-convexity of the power flow equations in power distribution networks. Of relatively new interest to the power systems community is the extension to MI-SOCP models. The proposed model is an approximation. However, numerical validations in the IEEE 33-bus and IEEE 69-bus test systems for unity and variable power factor confirm that the proposed MI-SOCP finds the best solutions reported in the literature. Being an exact technique, the proposed model allows minimum processing times and zero standard deviation, i.e., the same optimum is guaranteed at each time that the MI-SOCP model is solved (a significant advantage in comparison to metaheuristics). Additionally, load and photovoltaic generation curves for the IEEE 69-node test system are included to demonstrate the applicability of the proposed MI-SOCP to solve the problem of the optimal location and sizing of renewable generators using the multi-period optimal power flow formulation. Therefore, the proposed MI-SOCP also guarantees the global optimum finding, in contrast to local solutions achieved with mixed-integer nonlinear programming solvers available in the GAMS optimization software. All the simulations were carried out via MATLAB software with the CVX package and Gurobi solver.

Coordinated scheduling of integrated power and gas grids in consideration of gas flow dynamics

Natural gas, hydrogen, methane, or their mixture play important roles in the energy systems, and the coordinated optimal scheduling for the integrated energy systems are requisite in consideration of complicated system characteristics. This paper focuses on mixed-integer second-order cone programming-based optimal scheduling for the electric grids with gas flow dynamics in consideration of AC power flow, unit commitment, and line switching. For the gas grids, differential continuity equations and differential momentum equations, representing gas flow dynamics, are discretized to a group of algebraic equations with the implicit trapezoidal rules. In the algebraic equations, nonconvex bilinear terms with integer variables, representing gas flow directions, are transformed into linear inequality constraints with McCormick envelopes, and quadratic terms are relaxed by the second-order cone (SOC) approach. For the electric grids, SOC relaxation-based AC power flow models are employed, and the line switching problem is modeled by the improved SOC relaxation approach with McCormick envelopes. The entire problem is established as a mixed-integer second-order cone programming model, and the impacts of gas flow dynamics on the feasible region compared to the steady-state region are discussed.

Reconfiguration of Radial Distribution Systems: An Efficient Mathematical Model

For a few decades, researchers try to reach a better solution by proposing new models or methods for distribution system reconfiguration (DSR) problem. This paper presents an efficient model for reconfiguration of radial distribution systems, aiming loss minimization. The proposed model can be easily solved by every commercial solvers and software. The proposed mixed-integer second-order cone programming (MISOCP) model was tested on 16-, 33-, 69-, 70-, 119-, and 136-bus test systems as well as Taiwan power company network (TPC). The results show high efficiency of the proposed model for reconfiguration of radial distribution systems.

Distribution Network-Constrained Optimization of Peer-to-Peer Transactive Energy Trading Among Multi-Microgrids

This article proposes a two-level network-constrained peer-to-peer (P2P) transactive energy for multi-microgrids (MGs), which guarantees the distribution power network security and allows MGs to trade energy with each other flexibly. At the lower level, a P2P transactive energy is employed for multi-MGs to trade energy with each other. A multi-leader multi-follower (MLMF) Stackelberg game approach is utilized to model the energy trading process among MGs. We prove the existence and the uniqueness of the Stackelberg equilibrium (SE) and provide the closed-form expression for SE. For privacy concerns, we provide several distributed algorithms to obtain SE. At the upper level, the distribution system operator (DSO) reconfigures the distribution network based on the P2P transactive energy trading results by applying the AC optimal power flow considering the distribution network reconfiguration. If there are any network violations, DSO requests trading adjustments at the lower level for network security. We reformulate the DSO operation problem in a mixed-integer second-order cone programming (MISOCP) model and ensure its solution accuracy. Numerical results for a 4-Microgrid system, a modified IEEE 33-bus and 123-bus distribution power system show the effectiveness of the proposed transactive model and its solution technique.