Parameter-free optimization algorithm effective and precise solution of the optimal power flow problem.

The need to de‐carbonize the current energy infrastructure, and the increasing integration of renewables pose a number of difficult control and optimization problems. Among those, the optimal power flow (OPF) problem—i.e., the task to minimize power system operation costs while maintaining technical and network limitations—is key for operational planning of power systems. The influx of inherently volatile renewable energy sources calls for methods that allow to consider stochasticity directly in the OPF problem. Here, we present recent results on uncertainty quantification for OPF problems. Modeling uncertainties as second‐order continuous random variables, we will show that the OPF problem subject to stochastic uncertainties can be posed as an infinite‐dimensional L2‐problem. A tractable reformulation thereof can be obtained using polynomial chaos expansion (PCE), under mild assumptions. We will show advantageous features of PCE for OPF subject to stochastic uncertainties. For example, multivariate non‐Gaussian uncertainties can be considered easily. Finally, we comment on recent progress on a Julia package for PCE.

The optimal power flow (OPF) problem is a non-linear and non-smooth optimization problem. OPF problem is a complicated optimization problem, especially when considering the system constraints. This paper proposes a new enhanced version for the grey wolf optimization technique called Developed Grey Wolf Optimizer (DGWO) to solve the optimal power flow (OPF) problem by an efficient way. Although the GWO is an efficient technique, it may be prone to stagnate at local optima for some cases due to the insufficient diversity of wolves, hence the DGWO algorithm is proposed for improving the search capabilities of this optimizer. The DGWO is based on enhancing the exploration process by applying a random mutation to increase the diversity of population, while an exploitation process is enhanced by updating the position of populations in spiral path around the best solution. An adaptive operator is employed in DGWO to find a balance between the exploration and exploitation phases during the iterative process. The considered objective functions are quadratic fuel cost minimization, piecewise quadratic cost minimization, and quadratic fuel cost minimization considering the valve point effect. The DGWO is validated using the standard IEEE 30-bus test system. The obtained results showed the effectiveness and superiority of DGWO for solving the OPF problem compared with the other well-known meta-heuristic techniques.

The optimal power flow (OPF) problem is a well-known non-convex optimization problem that aims to minimize the cost of electric power generation subject to consumer demand, the physics of power flow, and technological constraints. To find an optimal solution to this problem, local search techniques such as interior point methods are typically used. However, due to the non-convex nature of the problem, these methods are likely to result in a sub-optimal solution. The goal of this paper is to characterize the worst-case performance of local search on the OPF problem. To accomplish this, we formulate the OPF problem as a canonical quadratically-constrained quadratic program (QCQP). Then, we study the problem of finding the worst-case local minimum of this QCQP, which is non-convex and hard to solve in general. We find a relaxation of this problem into a semidefinite program (SDP) and show that it is exact for certain cases. Using some test cases which are known to have multiple local minima, we demonstrate the effectiveness of the proposed relaxation to bound the worst-case local minimum. We compare the obtained upper bound on local minima to the lower bound provided by the standard SDP relaxation of the OPF problem to understand how much SDP outperforms local search for a given problem.

In order to solve the optimal power flow (OPF) problem, a unique algorithm based on a search and rescue method is applied in this study. For the OPF problem under three objective functions, the SAR offers a straightforward and reliable solution. The three objective functions are used to minimize the fuel cost, power loss and voltage deviation as a single objective function. The OPF problem for benchmark test system, including the IEEE-14 bus, IEEE-30 bus, and IEEE-57 bus, are solved by the Search and Rescue algorithm (SAR) under specific objective functions that are determined by the operational and economic performance indices of the power system. To demonstrate the efficacy and possibilities of the SAR algorithm, SAR is contrasted with alternative optimization techniques such as harmony search algorithm, gradient method, adaptive genetic algorithm, biogeography-based optimization, Artificial bee colony, gravitational search algorithm, particle swarm optimization, Jaya algorithm, enhanced genetic algorithm, modified shuffle frog leaping algorithm, practical swarm optimizer, Moth flam optimizer, whale and moth flam optimizer, grey wolf optimizer, cheap optimization algorithm and differential evolution algorithm. The value of minimum power losses based on SAR technique is equal to 0.459733441487247 MW for IEEE-14 bus. The value of minimum total fuel cost based on SAR technique is equal to 8051.12225602148 $/h for IEEE-14 bus. The value of minimum voltage deviation based on SAR technique is equal to 0.0357680148269292 for IEEE-14 bus. The value of minimum power losses based on SAR technique is equal to 2.71286428848434 MW for IEEE-30 bus. The value of minimum total fuel cost based on SAR technique is equal to 798.197578585806 $/h for IEEE-30 bus. The value of minimum voltage deviation based on SAR technique is equal to 0.0978069572088536 for IEEE-30 bus. The value of minimum total fuel cost based on SAR technique is equal to 38017.7691758245 $/h for IEEE-57 bus. The acquired results for the OPF compared to all competitor algorithms in every case of fitness function demonstrate the superiority of the SAR method.

The name SuperOPF is used to refer several projects, problem formulations and soft-ware tools intended to extend, improve and re-define some of the standard methods of optimizing electric power systems. Our work included applying primal-dual interior point methods to standard AC optimal power flow problems of large size, as well as extensions of this problem to include co-optimization of multiple scenarios. The original SuperOPF problem formulation was based on co-optimizing a base scenario along with multiple post-contingency scenarios, where all AC power flow models and constraints are enforced for each, to find optimal energy contracts, endogenously determined locational reserves and appropriate nodal energy prices for a single period optimal power flow problem with uncertainty. This led to example non-linear programming problems on the order of 1 million constraints and half a million variables. The second generation SuperOPF formulation extends this by adding multiple periods and multiple base scenarios per period. It also incorporates additional variables and constraints to model load following reserves, ramping costs, and storage resources. A third generation of the multi-period SuperOPF, adds both integer variables and a receding horizon framework in which the problem type is more challenging (mixed integer), the size is even larger, and it must be solved more frequently, pushing the limits of currently available algorithms and solvers. The consideration of transient stability constraints in optimal power flow (OPF) problems has become increasingly important in modern power systems. Transient stability constrained OPF (TSCOPF) is a nonlinear optimization problem subject to a set of algebraic and differential equations. Solving a TSCOPF problem can be challenging due to (i) the differential-equation constraints in an optimization problem, (ii) the lack of a true analytical expression for transient stability in OPF. To handle the dynamics in TSCOPF, the set of differential equations can be approximated or converted into equivalent algebraic equations before they are included in an OPF formulation. In Chapter 4, a rigorous evaluation of using a predefined and fixed threshold for rotor angles as a mean to determine transient stability of the system is developed. TSCOPF can be modeled as a large-scale nonlinear programming problem including the constraints of differential-algebraic equations (DAE). Solving a TSCOPF problem can be challenging due to (i) the differential-equation constraints in an optimization problem, (ii) the lack of a true analytical expression for transient stability constraint in OPF. Unfortunately, even the current best TSCOPF solvers still suffer from the curse of dimensionality and unacceptable computational time, especially for large-scale power systems with multiple contingencies. In chapter 5, thse issues will be addressed and a new method to incorporate the transient stability constraints will be presented.

The feasible region plays an important role in optimal power flow (OPF) problems. However, constructing the feasible region of general optimal power flow problems is a challenging task. In this paper, a trajectory-unified (TJU) method is developed to compute the feasible region of general large-scale OPF problems. This is the first attempt, to our knowledge, to develop a numerical method to compute feasible regions of OPF problems. In addition, the projection of the computed feasible region into a desired low-dimensional sub-space is presented. By employing the proposed TJU method, we compute the feasible region of a 9-bus and the IEEE 118-bus OPF problem. It is shown that the feasible region of a power system grows in size from light-loading conditions to medium-loading conditions while it shrinks in size from medium-loading conditions to heavy-loading conditions. This discovery of a geometric property of the feasible solution asserts the observations that OPF problems are generally easy to solve during medium-loading conditions but are generally difficult to solve during heavy-loading conditions.

A memory-guided Jaya algorithm to solve multi-objective optimal power flow integrating renewable energy sources

Gradient pyramid mechanism and Nelder-Mead simplex enhanced Colony Predation Algorithm for optimal power flow problems

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.

Due to its significance in the operation of power systems, the optimal power flow (OPF) problem has attracted increasing interest with the introduction of smart grids. Optimal power flow developed as a crucial instrument for resource planning effectiveness as well as for enhancing the performance of electrical power networks. Transmission line losses, total generation costs, FACTS (flexible alternating current transmission system) costs, voltage deviations, total power transfer capability, voltage stability, emission of generation units, system security, etc., are just a few examples of objective functions related to the electric power system that can be optimized. Due to the nonlinear nature of optimal power flow problems, the classical approaches may become locked in local optimums, hence, metaheuristic optimization techniques are frequently used to solve these issues. The most recent optimization strategies used to solve optimal power flow problems are discussed in this paper as the state of the art (according to the authors, the most pertinent studies). The presented optimization techniques are grouped according to their sources of inspiration, including human-inspired algorithms (harmony search, teaching learning-based optimization, tabu search, etc.), evolutionary-inspired algorithms (differential evolution, genetic algorithms, etc.), and physics-inspired methods (particle swarm optimization, cuckoo search algorithm, firefly algorithm, ant colony optimization algorithm, etc.).

The optimal power flow (OPF) problem is a relevant subject for the secure and economic power systems operation. For instance, OPF can be used to reduce the power system technical losses. A reduction in 0.1% in the losses accounts for near 50 billion USD in cost savings. Likewise, OPF can be used along with power production forecast tools, for renewable energy sources, to evaluate their impact in the grid security operation planning. In an OPF problem, an objective function related with demand supply generation cost, power line losses, or violation limits is optimized subject to several system security constraints. These constraints are related with branch power flow limits, voltages limits, power injection limits, among others. The exact model of the objective function and constraints of the OPF problem is neither linear nor convex. In this paper it is presented the OPF problem in a Quadratically Constrained Quadratic Program (QCQP) approach. Objective function is based on a quadratic function of the bus voltages and constraints are formulated as Quadratic forms. A two-bus system is used to demonstrates the non-convexity of the OPF problem. Also, it is presented the Rank- 1 convex relaxation of the OPF problem which relaxes the QCQP model into a positive Semidefinite Programming (SDP) model. Once the OPF problem is relaxed into a SDP convex form, a global optimal solution can be obtained. An application example for the OPF problem is presented for the IEEE 14 system. The QCQP and the SDP results are compared and discussed.

This paper proposes a recursive conic approximation methodology to deal with the optimal power flow (OPF) problem in unbalanced bipolar DC networks. The OPF problem is formulated through a nonlinear programming (NLP) representation, where the objective function corresponds to the minimization of the expected grid power losses for a particular load scenario. The NLP formulation has a non-convex structure due to the hyperbolic equality constraints that define the current injection/absorption in the constant power terminals as a function of the powers and voltages. To obtain an approximate convex model that represents the OPF problem in bipolar asymmetric distribution networks, the conic relation associated with the product of two positive variables is applied to all nodes with constant power loads. In the case of nodes with dispersed generation, a direct replacement of the voltage variables for their expected operating point is used. An iterative solution procedure is implemented in order to minimize the error introduced by the voltage linearization in the dispersed generation sources. The 21-bus grid is employed for all numerical validations. To validate the effectiveness of the proposed conic model, the power flow problem is solved, considering that the neutral wire is floating and grounded, and obtaining the same numerical results as the traditional power flow methods (successive approximations, triangular-based, and Taylor-based approaches): expected power losses of 95.4237 and 91.2701 kW, respectively. To validate the effectiveness of the proposed convex model for solving the OPF problem, three combinatorial optimization methods are implemented: the sine-cosine algorithm (SCA), the black-hole optimizer (BHO), and the vortex search algorithm (VSA). Numerical results show that the proposed convex model finds the global optimal solution with a value of 22.985 kW, followed by the VSA with a value of 22.986 kW. At the same time, the BHO and SCA are stuck in locally optimal solutions (23.066 and 23.054 kW, respectively). All simulations were carried out in a MATLAB programming environment.

Investigation of optimal power flow solution techniques considering stochastic renewable energy sources: Review and analysis

In this paper, we consider the Optimal Power Flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consists in a spatial branch-and-bound algorithm based on a compact quadratically-constrained convex relaxation. It is computed by solving the semidefinite rank relaxation of OPF once at the root node of the algorithm. An important result is that the optimal value of our compact relaxation is equal to the rank relaxation value. Then, at every sub-nodes of our branch-and-bound, the lower bound is obtained by solving a quadratic convex problem instead of an SDP. Another contribution is that we add only O(n+m) variables that model the squares of the initial variables, where n is the number of buses in the power system and m the number of transmission lines, to construct our relaxation. Then, since the relations between the initial and auxiliary variables are non-convex, we relax them to get a quadratic convex relaxation. Finally, in our branch-and-bound algorithm, we only have to force a reduced number of equalities to prove global optimality. This quadratic convex relaxation approach is here tailored to the OPF problem, but it can address any application whose formulation is a quadratic optimization problem subject to quadratic equalities and ring constraints. Our first experiments on instances of the OPF problem show that our new algorithm Compact OPF (COPF) is more efficient than the standard solvers and other quadratic convex relaxation based methods we compare it with.

This study proposes an improved particle swarm optimization method to find optimal power flow by using the power transmission loss as an objective function. In the literature, PSO is a well — known intelligent search method to handle the solution of optimal power flow problem. A novel scheme which is called Improved PSO (IPSO) is defined by modifying the initialization step of PSO algorithm and choosing the load bus voltages, generator active and reactive powers, line flow capacities as penalty functions in the objective function. PSO and IPSO-based optimal power flow solutions are compared with each other on IEEE 118 and 300 bus systems. According to the test results, the power loss obtained by IPSO-based solution has less power loss than PSO-based solution. This proposed method can be used to obtain faster desirable solutions and better power loss results for the optimal power flow problem in case of power loss minimization.