Non-convex Optimal Power Flow Problem Research Articles

Optimal Power Flow of Hybrid Wind/Solar/Thermal Energy Integrated Power Systems Considering Costs and Emissions via a Novel and Efficient Search Optimization Algorithm

The OPF problem has significant importance in a power system’s operation, planning, economic scheduling, and security. Today’s electricity grid is rapidly evolving, with increased penetration of renewable power sources (RPSs). Conventional optimal power flow (OPF) has non-linear constraints that make it a highly non-linear, non-convex optimization problem. This complex problem escalates further with the integration of renewable energy resource (RES), which are generally intermittent in nature. This study suggests a new and effective improved optimizer via a TFWO algorithm (turbulent flow of water-based optimization), namely the ITFWO algorithm, to solve non-linear and non-convex OPF problems in energy networks with integrated solar photovoltaic (PV) and wind turbine (WT) units (being environmentally friendly and clean in nature). OPF in the energy networks is an optimization problem proposed to discover the optimal settings of an energy network. The OPF modeling contains the forecasted electric energy of WT and PV by considering the voltage value at PV and WT buses as decision parameters. Forecasting the active energy of PV and WT units has been founded on the real-time measurements of solar irradiance and wind speed. Eight scenarios are analyzed on the IEEE 30-bus test system in order to determine a cost-effective schedule for thermal power plants with different objectives that reflect fuel cost minimization, voltage profile improvement, emission gases, power loss reduction, and fuel cost minimization with consideration of the valve point effect of generation units. In addition, a carbon tax is considered in the goal function in the examined cases in order to investigate its effect on generator scheduling. A comparison of the simulation results with other recently published algorithms for solving OPF problems is made to illustrate the effectiveness and validity of the proposed ITFWO algorithm. Simulation results show that the improved turbulent flow of water-based optimization algorithm provides an effective and robust high-quality solution of the various optimal power-flow problems. Moreover, results obtained using the proposed ITFWO algorithm are either better than, or comparable to, those obtained using other techniques reported in the literature. The utility of solar and wind energy in scheduling problems has been proposed in this work.

Voltage Stability Analysis in Medium-Voltage Distribution Networks Using a Second-Order Cone Approximation

This paper addresses the voltage stability margin calculation in medium-voltage distribution networks in the context of exact mathematical modeling. This margin calculation is performed with a second-order cone (SOCP) reformulation of the classical nonlinear non-convex optimal power flow problems. The main idea around the SOCP approximation is to guarantee the global optimal solution via convex optimization, considering as the objective function the λ-coefficient associated with the maximum possible increment of the load consumption at all the nodes. Different simulation cases are considered in one test feeder, described as follows: (i) the distribution network without penetration of distributed generation; (ii) the distribution network with penetration of distributed generation; and (iii) the distribution grid with capacitive compensation. Numerical results in the test system demonstrated the effectiveness of the proposed SOCP approximation to determine the λ-coefficient. In addition, the proposed approximation is compared with nonlinear tools available in the literature. All the simulations are carried out in the MATLAB software with the CVX package and the Gurobi solver.

A Real-Time Alternating Direction Method of Multipliers Algorithm for Nonconvex Optimal Power Flow Problem

The high penetration rate of smart devices such as storage elements brings new challenges to the optimal power flow (OPF) problem in power systems, which is generally nonconvex and difficult to be solved in real time. This article proposes a set of two fully distributed algorithms by combining the alternating direction method of multipliers and proximal alternating minimization techniques. The first one is a basic distributed algorithm for offline scheduling of 24-h ahead OPF. The other extended one is the warm-starting algorithm by using the offline scheduling solution as the initial point, and the warm-starting algorithm can get converged faster than the basic algorithm. Both algorithms aim to provide a highly feasible solution for the nonconvex OPF problem, and simulation conducted on four radial power systems with batteries has validated the performance of these two algorithms.

An Improved Two-Stage Optimization Procedure for Optimal Power Flow Calculation

The paper focuses on the problem of optimal power flow. Minimized production and transmission costs were considered to design an optimal power flow search method based on the novel two-stage optimization procedure that takes into account the voltage and power constraints. As a result, the efficient optimum search method for the non-convex optimal power flow problem is proposed and implemented. The efficient L-BFGS-B optimization algorithm is employed to allow for the constraints. The stochastic modification of the global optimization algorithm of the Lipschitz function was used in the first stage to improve the L-BFGS-B method. Two illustrative examples on the IEEE 118-bus test scheme and real regional grid demonstrate the efficiency of the proposed two-stage optimization procedure. The possibility of taking into account the renewable sources in the proposed optimal power flow algorithm is briefly discussed

Heuristic Algorithm Based Optimal Power Flow Model Incorporating Stochastic Renewable Energy Sources

Today's electricity grid is rapidly evolving, with increased penetration of renewable energy sources (RES). Conventional Optimal Power Flow (OPF) has non-linear constraints that make it a highly non-linear, non-convex optimisation problem. This complex problem escalates further with the integration of RES, which are generally intermittent in nature. In this article, an optimal power flow model combines three types of energy resources, including conventional thermal power generators, solar photovoltaic generators (SPGs) and wind power generators (WPGs). Uncertain power outputs from SPGs and WPGs are forecasted with the help of lognormal and Weibull probability distribution functions, respectively. The over and underestimation output power of RES are considered in the objective function i.e. as a reserve and penalty cost, respectively. Furthermore, to reduce carbon emissions, a carbon tax is imposed while formulating the objective function. A grey wolf optimisation technique (GWO) is employed to achieve optimisation in modified IEEE-30 and IEEE-57 bus test systems to demonstrate its feasibility. Hence, novel contributions of this work include the new objective functions and associated framework for optimising generation cost while considering RES; and, secondly, computational efficiency is improved by the use of GWO to address the non-convex OPF problem. To investigate the effectiveness of the proposed GWO-based approach, it is compared in simulation to five other nature-inspired global optimisation algorithms and two well-established hybrid algorithms. For the simulation scenarios considered in this article, the GWO outperforms the other algorithms in terms of total cost minimisation and convergence time reduction.

Convex formulation of confidence level optimization of DG affine reactive power controllers in distribution grids

Volatile productions and consumptions generate a stochastic behavior of distribution grids and make its supervision difficult to achieve. Usually, the Distributed Generators reactive powers are adjusted to perform decentralized voltage control. Industrial controllers are generally equipped with a local affine feedback law, which settings are tuned at early stage using local data. A centralized and more efficient tuning method should aim to maximize the probability that all the node voltages of distribution grids remain within prescribed bounds. When the characteristics of the stochastic power forecasts are known, the centralized algorithm allows to update the settings on a regular time basis. However, the method requires to solve stochastic optimization problem. Assuming that stochastic variables have Gaussian distributions, a procedure is given which guarantees the convergence of the stochastic optimization. Convex problems drastically reduce the difficulty and the computational time required to reach the global minimum, compared to nonconvex optimal power flow problems. The linear controllers with optimized parameters are compared to traditional control laws using simulations of a real distribution grid model. The results show that the algorithm is reliable and moreover fast enough. Hence, the proposed method can be used to update periodically the control parameters.

A Novel Fast Semidefinite Programming-Based Approach for Optimal Reactive Power Dispatch

Reactive power planning problem is the key to secure and economic operation of power systems. Optimal management of existing reactive sources leads to loss minimization and economic performance of the system. Because of the nonlinear inter-relation between the physical parameters of the electric grid, this problem is a highly nonlinear and nonconvex constrained optimization problem. The application of semidefinite programming (SDP) to power system problems has recently gained considerable research attention. A recent SDP formulation uses a convex relaxation to the nonconvex optimal power flow problem under some technical conditions. This paper proposes a novel equivalent convex optimization formulation for the optimal reactive power dispatch (ORPD) problem and presents a new framework for finding the global optimum. Numerical results for the IEEE 30-bus and 118-bus test systems show that the proposed scheme obtains the optimum operation point and outperforms various state-of-the-art methods significantly.

On the robustness and scalability of semidefinite relaxation for optimal power flow problems

Semidefinite relaxation techniques have shown great promise for nonconvex optimal power flow problems. However, a number of independent numerical experiments have led to concerns about scalability and robustness of existing SDP solvers. To address these concerns, we investigate some numerical aspects of the problem and compare different state-of-the-art solvers. Our results demonstrate that semidefinite relaxations of large problem instances with on the order of 10,000 buses can be solved reliably and to reasonable accuracy within minutes. Furthermore, the semidefinite relaxation of a test case with 25,000 buses can be solved reliably within half an hour; the largest test case with 82,000 buses is solved within eight hours. We also compare the lower bound obtained via semidefinite relaxation to locally optimal solutions obtained with nonlinear optimization methods and calculate the optimality gap.

Distributed optimal power flow of DC microgrids: A penalty based ADMM approach

The popularity of direct current (DC) networks have made their optimal power flow (OPF) problem a hot topic. With the proliferation of distributed generation, the many problems of centralized optimization methods, such as single point failure and slow response speed, have led to utilization of measures such as distributed OPF methods. The OPF problem is non-convex, which makes it difficult to obtain an optimal solution. The second-order cone programming (SOCP) relaxation method is widely utilized to make the OPF problem convex. It is difficult to guarantee its exactness, especially when line constraints are considered. This paper proposes a penalty based ADMM approach using difference-of-convex programming (DCP) to solve the non-convex OPF problem in a distributed manner. The algorithm is composed of distributed x iteration, z iteration and λ, μ iteration. Specifically, in the distributed z iteration, the active power flow injection equation of each line is formulated as a difference of two convex functions, and then the SOCP relaxation is given in a different form. If the SOCP relaxation is inexact, a penalty item is added to drive the solution to be feasible. Then, an optimal solution can be obtained using a local nonlinear programming method. Finally, simulations on a 14-bus system and the IEEE 123-bus system validate the effectiveness of the proposed approach.

Modified adaptive differential evolution based optimal operation and security of AC-DC microgrid systems

The recent advancement in power electronic systems has enabled the formation of AC-DC microgrids by integration of DC microgrids into conventional AC grid. In this paper, an approach for optimal power flow (OPF) problem formulation in AC-DC microgrid networks is proposed. In the formulated AC-DC microgrid OPF problem, the objective functions considered for minimization are the total cost of generation in both the AC and DC networks and the cost of active power transfer from the AC grid to the DC microgrid. The OPF problem is subject to the limits of voltage magnitudes, power flows, and the AC-DC power electronic converter's limits. This paper also presents an approach to reformulate the nonlinear AC-DC OPF problem as an equivalent traditional AC OPF problem. A modified adaptive differential evolution algorithm is proposed to solve the non-convex AC OPF problem, Simulation studies are performed on an IEEE 30-bus and IEEE 39-bus test system connected to 6-bus and 9-bus DC microgrids under both normal operation and network contingency conditions. The results obtained on the test systems by the proposed approach confirm the validity of the developed approach.

On the efficiency of metaheuristics for solving the optimal power flow

In this article, various nonlinear and non-convex optimal power flow (OPF) objective functions are optimized using eight different optimization algorithms, i.e., moth-flame optimization algorithm (MFO), gray wolf optimizer (GWO), dragonfly algorithm (DA), sine–cosine algorithm (SCA), antlion optimizer (ALO), multi-verse optimizer (MVO), grasshopper algorithm (GOA) and ion motion algorithm (IMO) for different test systems, i.e., IEEE 57-bus test system and IEEE 118-bus test system. As per no free lunch algorithm, in general, no algorithm can be considered better than other, and hence, the performance of algorithm relies heavily upon system under consideration. Applications through 22 different case studies are pursued to assess efficiency of algorithms under consideration. An attempt has been made to evaluate algorithms based on the best solution, average solution, average simulation time and trend of convergence. Different statistical tests are performed to identify the efficiency of algorithms under consideration. Test results suggest that MFO performs better as compared to rest of the algorithms for most test cases proving its mettle in dealing with nonlinear and non-convex OPF problem.

Chordal Conversion Based Convex Iteration Algorithm for Three-Phase Optimal Power Flow Problems

The three-phase optimal power flow (OPF) problem has recently attracted a lot of research interests due to the need to coordinate the operations of large-scale and heterogeneous distributed energy resources in unbalanced electric power distribution systems. The nonconvexity of the three-phase OPF problem is much stronger than that of the single-phase OPF problem. Instead of applying the semidefinite programming relaxation technique, this paper advocates a convex iteration algorithm to solve the nonconvex three-phase OPF problem. To make the convex iteration algorithm computationally efficient for large-scale distribution networks, the chordal conversion based technique is embedded in the convex iteration framework. By synergistically combining the convex iteration method and the chordal based conversion technique, the proposed three-phase OPF algorithm is not only computationally efficient but also guarantees global optimality when the trace of the regularization term becomes zero. At last, to further improve the computational performance, a greedy grid partitioning algorithm is proposed to decompose a single large matrix representing a distribution network to many smaller matrices. The simulation results using standard IEEE test feeders show that the proposed algorithm is computationally efficient, scalable, and yields global optimal solutions while resolving the rank conundrum.

Dual <inline-formula> <tex-math notation="LaTeX">${M}$ </tex-math> </inline-formula>-Convex Variable Subsets Family and Extremum Analysis for the OPF Problem

The optimal power flow (OPF) model is a central optimization problem in power system network. In this paper, we propose a novel approach to solve the OPF problem that has a convex objective function and non-convex feasible domain due to the constraints. Based on the concept of abstract convex analysis, we construct the dual ${M}$ -convex subsets family of original variables by using the variable separation method, followed by the analysis of the extremum according to the infimum base of the ${M}$ -convex subsets. It is challenging to obtain an explicit mapping function among the separated variables due to the nonlinear equality constraints. We, therefore, use the theorem of implicit function and the differential function to do duality analysis on separated variables. Based on the min–max principle of the primal-dual problem of OPF, we derive the condition of the complementary factor which leads the Lagrange dual problem to maximum, and make the variable separation that results in a Minkowski-type dual ${M}$ -convex subset. We then can obtain a local minimum using the principle of abstract convex optimization, which will be the global optimal solution under the Karush–Kuhn–Tucker conditions. We evaluate the proposed approach on several IEEE systems. The simulation results indicate that the approach is feasible and effective to deal with the non-convex OPF problem with a non-convex feasible domain.

An Efficient Chaotic Cuckoo Search Framework for Solving Non-Convex Optimal Power Flow Problem

The optimal power flow (OPF) problem is a very complicated task in power systems. OPF problem has a set of equality and inequality constraints. This paper looks at a chaotic cuckoo search (CCS) algorithm for solving non-convex OPF problem. The proposed CCS is a bio-inspired optimization calculation that is inspired by the behaviour of cuckoos people in nature. The chaotic guide is a variation of qualities combined with CS. A sinusoidal chaotic is integrated with CS algorithm and tested on standard IEEE 30-bus test system to the point of improving its global speed of convergence and enhancing its performance. The elitism scheme is also serves to save the best cuckoo during amid the procedure when updating the cuckoo. The results show clearly the superiority of CCS in searching for the best function values results when compared with well-known metaheuristic search algorithms.

Optimal Power Flow With Power Flow Routers

Power flow routing is an emerging control paradigm for the dynamic control of electric power flows. In this paper, we propose a generic model of a power flow router (PFR) and incorporate it into the optimal power flow (OPF) problem. First, a generic PFR architecture is proposed to encapsulate the desired functions of PFRs. Then, the load flow model of PFRs is developed and incorporated into the OPF framework. To pursue global optimality of the non-convex PFR-incorporated OPF (PFR-OPF) problem, we develop a semidefinite programming (SDP) relaxation of PFR-OPF. By introducing the regularization terms that favor a low-rank solution and tuning the penalty coefficients, a rank-1 solution can be obtained and used for recovering an optimal or near-optimal solution of the PFR-OPF and the results are verified in numerical tests. The efficacy of the PFR-OPF framework allows us to investigate the impact of PFR integration. With the system loadability as an example, the numerical results show that remarkable enhancement can be achieved by installing PFRs at certain critical buses of the network.

A distributed approach to the OPF problem

This paper presents a distributed approach to optimal power flow (OPF) in an electrical network, suitable for application in a future smart grid scenario where access to resource and control is decentralized. The non-convex OPF problem is solved by an augmented Lagrangian method, similar to the widely known ADMM algorithm, with the key distinction that penalty parameters are constantly increased. A (weak) assumption on local solver reliability is required to always ensure convergence. A certificate of convergence to a local optimum is available in the case of bounded penalty parameters. For moderate sized networks (up to 300 nodes, and even in the presence of a severe partition of the network), the approach guarantees a performance very close to the optimum, with an appreciably fast convergence speed. The generality of the approach makes it applicable to any (convex or non-convex) distributed optimization problem in networked form. In the comparison with the literature, mostly focused on convex SDP approximations, the chosen approach guarantees adherence to the reference problem, and it also requires a smaller local computational complexity effort.

Solving non-linear, non-smooth and non-convex optimal power flow problems using chaotic invasive weed optimization algorithms based on chaos

Invasive Weed Optimization (IWO) algorithm is a simple but powerful algorithm which is capable of solving general multi-dimensional, linear and nonlinear optimization problems with appreciable efficiency. Recently IWO algorithm is being used in several engineering design owing to its superior performance in comparison with many other existing algorithms. This paper presents a Chaotic IWO (CIWO) algorithms based on chaos, and investigates its performance for optimal settings of Optimal Power Flow (OPF) control variables of OPF problem with non-smooth and non-convex generator fuel cost curves (non-smooth and non-convex OPF). The performance of CIWO algorithms are studied and evaluated on the standard IEEE 30-bus test system with different objective functions. The experimental results suggest that IWO algorithm holds immense promise to appear as an efficient and powerful algorithm for optimization in the power system.

A current-based SCOPF with enhanced particle swarm optimization

A current-based security-constrained optimal power flow (SCOPF) is presented in this paper and an Enhanced particle swarm optimization (EPSO) is developed to solve the non-convex optimal power flow problem. The SCOPF can be divided into three steps involving security analysis, severest event selection, and a preventive algorithm. Firstly, novel security analysis is conducted before a fault occurs in the system using the current-based power flow technique. Secondly, a ranking method is used to highlight the most severe events caused by a specific facility. Finally, a preventive algorithm makes use of the contingency information, and can maintain the operator system security, avoiding congestion when a fault occurs. In addition, this method not only enhances the neighborhood search, but also searches for the optimum solution quickly to advance the convergence.

A Distributed Approach for the Optimal Power-Flow Problem Based on ADMM and Sequential Convex Approximations

The optimal power-flow (OPF) problem, which plays a central role in operating electrical networks, is considered. The problem is nonconvex and is, in fact, NP hard. Therefore, designing efficient algorithms of practical relevance is crucial, though their global optimality is not guaranteed. Existing semidefinite programming relaxation-based approaches are restricted to OPF problems where zero duality holds. In this paper, an efficient novel method to address the general nonconvex OPF problem is investigated. The proposed method is based on an alternating direction method of multipliers combined with sequential convex approximations. The global OPF problem is decomposed into smaller problems associated with each bus of the network, the solutions of which are coordinated via a light communication protocol. Therefore, the proposed method is highly scalable. The convergence properties of the proposed algorithm are mathematically substantiated. Finally, the proposed algorithm is evaluated on a number of test examples, where the convergence properties of the proposed algorithm are numerically substantiated, and the performance is compared with a global optimal method.

A semidefinite programming method with graph partitioning technique for optimal power flow problems

A new semidefinite programming (SDP) method with graph partitioning technique to solve optimal power flow (OPF) problems is presented in this paper. The non-convex OPF problem is converted into its convex SDP model at first, and then according to the characters of power system network, the matrix variable of SDP is re-arranged using the chordal extension of its aggregate sparsity pattern by the graph partitioning technique. A new SDP-OPF model is reformulated with the re-arranged matrix variable, and can be solved by the interior point method (IPM) for SDP. This method can reduce the consumption of computer memory and improve the computing performance significantly. Extensive numerical simulations on seven test systems with sizes up to 542 buses have shown that this new method of SDP-OPF can guarantee the global optimal solutions within the polynomial time same as the original SDP-OPF, but less CPU times and memory.