Optimal Power Flow Formulation Research Articles

Bi-Level Volt-VAR Optimization to Coordinate Smart Inverters With Voltage Control Devices

Conservation voltage reduction (CVR) uses Volt-VAR optimization (VVO) methods to reduce customer power demand by controlling feeder's voltage control devices. The objective of this paper is to present a VVO approach that controls system's legacy voltage control devices and coordinates their operation with smart inverter control. An optimal power flow (OPF) formulation is proposed by developing linear and nonlinear power flow approximations for a three-phase unbalanced electric power distribution system. A bi-level VVO approach is proposed, where Level 1 optimizes the control of legacy devices and smart inverters using a linear approximate three-phase power flow. In Level 2, the control parameters for smart inverters are adjusted to obtain an optimal and feasible solution by solving the approximate nonlinear OPF model. Level 1 is modeled as a mixed integer linear program (MILP) while Level 2 as a nonlinear program with linear objective and quadratic constraints. The proposed approach is validated using 13-bus and 123-bus three-phase IEEE test feeders and a 329-bus three-phase PNNL taxonomy feeder. The results demonstrate the applicability of the framework in achieving the CVR objective. It is demonstrated that the proposed coordinated control approach help us to reduce the feeder's power demand by reducing the bus voltages; the proposed approach maintains an average feeder voltage of 0.96 p.u. A higher energy saving is reported during the minimum load conditions. The results and approximation steps are thoroughly validated using OpenDSS.

Line-Wise Optimal Power Flow Using Successive Linear Optimization Technique

Optimal power flow (OPF) is an important power systems optimization tool. However, due to the inherent nonlinearity of the solution space, OPF methods may yield a local optimal solution, leading to economic loss. For this reason, an OPF remains a major research topic. Changing power balance equations can change the solution space, alter solution time, and create a chance of reaching a better solution. In this paper, a new simple line-wise optimal power flow (LWOPF) formulation is proposed using line-wise power balance equations and solved using the successive linear programming (SLP) technique. The study results on nine different test systems, up to a 9 241-bus real power system, show that the proposed method is accurate, provides monotonic convergence, scales well for large systems and is consistently faster, up to twice the speed of MATPOWER and other proven bus-wise SLP approaches to solve the OPF.

Incorporating Unbalanced Operation Constraints of Three-Phase Distributed Generation

In this letter, we provide a convex-concave procedure based model to approximate the unbalance operation constraints of three-phase fossil-fuel distributed generation. This model transfers the non-convex unbalanced operation constraints into an iterative convex optimization procedure. The simplicity and convexity make it easy to integrate into a broad range of convex distribution optimal power flow formulations.

Voltage stability constrained line‐wise optimal power flow

The purpose of optimal power flow (OPF) is to optimise an objective function subject to a set of operating constraints. It remains an active research area because of using the bus-wise power balance equations, which leads to a non-linear solution space, resulting in OPF yielding local optimal solutions, thereby causing significant economic loss. In this study, first, a new line-wise OPF (LWOPF) formulation is proposed. Thereafter, a maximum loadability factor, as a voltage collapse indicator, is derived and combined with LWOPF constraints to form a voltage stability constrained LWOPF (VSCLWOPF) model. As the line-wise power balance equations are based upon the square of voltage magnitudes, it results in significant improvement in the solution space and lower-order terms in all computational steps. The LWOPF and VSCLWOPF formulations, are solved using non-linear optimisation technique, tested on several benchmark and real power systems. Results show that the proposed LWOPF is accurate, provides monotonic convergence, and scales well for large systems. It provides a better solution and is consistently faster, up to twice the speed of MATPOWER, due to reduced computational needs. Results of VSCLWOPF show that, for the same voltage stability level, the solution costs less than that obtained by classical bus-wise OPF.

Convex relaxation of Sparse Tableau Formulation for the AC optimal power flow

Optimal power flow (OPF) approaches employing such methods as semi-definite programming (SDP) have garnered considerable interest in the literature of the last decade. The OPF formulations for these approaches have almost universally relied on Ybus admittance matrix representations, which derive from nodal analysis, and restrict allowable network elements to be voltage-controlled only. Limitations of nodal analysis long been recognized, and to overcome these, commercial power system software often employs modified nodal analysis (MNA). Here, we consider a novel general power system modeling approach based on multi-port representation of individual components with Sparse Tableau Formulation (STF) of network constraints, which is more versatile than MNA for OPF requiring many monitored links with constrained flows. In STF, one is better able to exploit the fact that the vast majority of power grid network elements have voltage-current behavior that is well-modeled as linear. This opens the door to simple, engineering-based convex relaxations. We discuss two relaxations, admittance-based and current-based. The tightness of the relaxation is shown to improve when angle constraints and narrow bounds for active power generation are provided. Sequential bound tightening and reduced spatial branch-and-bound are discussed to obtain stronger relaxation solution. We conduct case studies to show the effectiveness of our relaxations with standard test cases.

Optimal Water–Power Flow-Problem: Formulation and Distributed Optimal Solution

This paper formalizes an optimal water–power flow (OWPF) problem to optimize the use of controllable assets across power and water systems while accounting for the couplings between the two infrastructures. Tanks and pumps are optimally managed to satisfy water demand while improving power grid operations; for the power network, an ac optimal power-flow formulation is augmented to accommodate the controllability of water pumps. Unfortunately, the physics governing the operation of the two infrastructures and coupling constraints leads to a nonconvex (and, in fact, NP-hard) problem; however, after reformulating OWPF as a nonconvex, quadratically constrained quadratic problem, a feasible point pursuit-successive convex approximation approach is used to identify feasible and optimal solutions. In addition, a distributed solver based on the alternating direction method of multipliers enables water and power operators to pursue individual objectives while respecting the couplings between the two networks. The merits of the proposed approach are demonstrated for the case of a distribution feeder coupled with a municipal water distribution network.

Optimal Power Flow of a Power System Incorporating Stochastic Wind Power Based on Modified Moth Swarm Algorithm

The combined heat and power (CHP) generator not only generates electrical power but also generates heat energy in a single process, which decreases the emission level significantly. The integration of wind power into the power system will lead to an impact on the economic operation of the system as well as the bus voltage and transmission losses. In this paper, a formulation of optimal power flow (OPF) problem of a power system incorporating stochastic wind power is presented. To solve this problem, a modified version of the moth swarm algorithm (MMSA) is proposed. Three objective functions, which are the minimization of operating cost, the minimization of transmission power loss, and the voltage profile improvement, are considered in this paper. To minimize the operating cost, the direct, overestimation, and underestimation costs of wind power units are considered. Two test systems are considered to prove the effectiveness and the superiority of the proposed MMSA in comparison with other methods. The comparison with other methods proves the efficiency and the superiority of the proposed MMSA.

A Solution to the Optimal Power Flow Problem Considering WT and PV Generation

This paper proposes a combination of phasor particle swarm optimization (PPSO) and a gravitational search algorithm, namely a hybrid PPSOGSA algorithm, for optimal power flow (OPF) in power systems with an integrated wind turbine (WT) and solar photovoltaic (PV) generators. The OPF formulation includes the forecasted active power generation of WT and PV as dependent variables, whereas the voltage magnitude at WT and PV buses is considered as control (decision) variables. Forecasting the output power of WT and PV generators is based on the real-time measurements and the probabilistic models of wind speed and solar irradiance. The proposed OPF approach and the solution method are verified on the IEEE 30-bus test system. The robustness and efficiency of the proposed PPSOGSA algorithm in solving the OPF problem are evaluated by comparing with 20 well-established metaheuristic optimization methods under the same system data, control variables, and constraints. The statistical features of the OPF results are estimated by using the Monte Carlo method.

Hynet: An Optimal Power Flow Framework for Hybrid AC/DC Power Systems

High-voltage direct current (HVDC) systems are increasingly incorporated into today's AC power grids, necessitating optimal power flow (OPF) tools for the analysis, planning, and operation of such hybrid systems. To this end, we introduce hynet, a Python-based open-source OPF framework for hybrid AC/DC grids with point-to-point and radial multi-terminal HVDC systems. hynet's design promotes ease of use and extensibility, which is supported by the particular mathematical model and software design presented in this paper. The system model features a unified representation of AC and DC subgrids as well as a concise and flexible converter model, which enable the compact description of a hybrid AC/DC power system and its OPF problem. To support convex relaxation based OPF solution techniques, a state space relaxation is introduced to obtain a unified OPF formulation that is analogous to the OPF of AC power systems. This enables the direct generalization of relaxation-related results for AC grids to hybrid AC/DC grids, which is shown for the semidefinite and second-order cone relaxation as well as associated results on exactness and locational marginal prices. Finally, hynet's object-oriented software design is discussed, which provides extensibility via inheritance and standard design patterns, and its robust and competitive performance is illustrated with case studies.

Piecewise Linearization of Quadratic Branch Flow Limits by Irregular Polygon

This letter addresses the issue of linearization of quadratic thermal limits of transmission lines for linear optimal power flow formulation. A new irregular polygon based linearization is proposed. The approach is purely based on geometrical concepts and does not introduce any optimization problem. Comparison of the number of constraints is given with different errors for different branch capabilities and systems. Test case analysis is also presented to validate the irregular polygon linearization strategy with optimal value and computational time results.

Distributionally Robust Chance-Constrained Approximate AC-OPF With Wasserstein Metric

Chance constrained optimal power flow (OPF) has been recognized as a promising framework to manage the risk from variable renewable energy (VRE). In the presence of VRE uncertainties, this paper discusses a distributionally robust chance constrained approximate ac-OPF. The power flow model employed in the proposed OPF formulation combines an exact ac power flow model at the nominal operation point and an approximate linear power flow model to reflect the system response under uncertainties. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed OPF model minimizes the expectation of the quadratic cost function w.r.t. the worst-case probability distribution and guarantees the chance constraints satisfied for any distribution in the ambiguity set. The whole method is data-driven in the sense that the ambiguity set is constructed from historical data without any presumption on the type of the probability distribution, and more data leads to smaller ambiguity set and less conservative strategy. Moreover, special problem structures of the proposed problem formulation are exploited to develop an efficient and scalable solution approach. Case studies are carried out on the IEEE 14 and 118 bus systems to show the accuracy and necessity of the approximate ac model and the attractive features of the distributionally robust optimization approach compared with other methods to deal with uncertainties.

Can Attackers With Limited Information Exploit Historical Data to Mount Successful False Data Injection Attacks on Power Systems?

This paper studies physical consequences of unobservable false data injection (FDI) attacks designed only with information inside a sub-network of the power system. The goal of this attack is to overload a chosen target line without being detected via measurements. To overcome the limited information, a multiple linear regression model is developed to learn the relationship between the external network and the attack sub-network from historical data. The worst possible consequences of such FDI attacks are evaluated by solving a bi-level optimization problem wherein the first level models the limited attack resources, while the second level formulates the system response to such attacks via DC optimal power flow (OPF). The attack model with limited information is reflected in the DC OPF formulation that only takes into account the system information for the attack sub-network. The vulnerability of this attack model is illustrated on the IEEE 24-bus RTS and IEEE 118-bus systems.

Steady-State Power Flow Model of Energy Router Embedded AC Network and Its Application in Optimizing Power System Operation

The energy router is an emerging device concept that is based on an advanced power electronic technique. It is able to realize flexible and dynamic electric power distribution in power systems analogous to the function of information routers in the Internet. It is of great interest to investigate how the energy router can be used to optimize power system operation. This paper formulates the steady-state power flow model of the energy router embedded system network and the related optimal power flow formulation. The role of the energy router in providing extra flexibility to optimize the system operation is studied. Case studies are carried out on a modified IEEE RTS-79 system and a modified IEEE 118 bus system with the energy router. The results show that the energy router is able to optimize the operation of the power system through controlling the power injections and voltage of ports of the energy router. Operating objective such as adjusting branch power flow, improving bus voltage, and reducing active power losses of the grid can be reached under different objective functions.

A Coordinated Droop Control Strategy for Overvoltage Mitigation in Active Distribution Networks

In this paper, a centralized control strategy for low-voltage networks is proposed, aiming to coordinate the droop characteristics of the distributed renewable energy sources. The main objectives are the overvoltage mitigation and the optimal network operation in terms of maximum power injection and uniform power curtailment. The distinct features of the proposed method are the incorporation of droop-based control techniques into the conventional optimal power flow formulation and the use of the sensitivity theory as a means to significantly decrease the computational burden. The validity of the developed method is justified using a genetic algorithm, while extensive comparisons with conventional centralized and decentralized control strategies indicate its superior performance in the operation of low-voltage networks.

Quasi-reflection based symbiotic organisms search algorithm for solving static optimal power flow problem

This paper offers a novel variant to the existing symbiotic organisms search (SOS) algorithm, to address optimal power flow (OPF) problems considering effects of valve-point loading (VE) and prohibited zones (POZ). Problem formulation includes minimization of cost, loss, voltage stability index (VSI) and voltage deviation (VD) and simultaneous minimization of their combinations. Quadratic cost function, effects of VE and effects of both VE and POZ have been considered. OPF formulation considering effects of both VE and POZ are not yet available in the literature. Efficacy of SOS in resolving OPF is recognized in the literature. An opposition based learning technique named quasi-reflection, is merged into existing SOS to enhance its prospects of getting nearer to superior quality solution. The proposed algorithm, named quasi-reflected symbiotic organisms search (QRSOS), is assessed for IEEE 30 and IEEE 118 bus test systems. It shows promising results in reducing the objective function values of both the systems by large margins (78.98 % in case of VD when compared to SOS and NSGA-II and 46.06 % in case of loss as compared to QOTLBO in IEEE 30 and IEEE 118 bus respectively). QRSOS also outperformed its predecessors, in terms of convergence speed and global search ability.

AC OPF for Smart Distribution Networks: An Efficient and Robust Quadratic Approach

Smart grid schemes in which multiple network elements and participants are managed for the benefit of the distribution network (e.g., energy loss reduction, restoration, etc.) require sophisticated algorithms to control them in the most suitable manner while catering for network constraints. Such a complex decision making process can be solved by tailoring the ac optimal power flow (OPF) problem to the corresponding smart grid scheme. Non-linear programming ac OPF formulations, however, can suffer from scalability and robustness issues which in turn might limit their adoption. Here, a novel quadratic programming formulation is proposed and compared against the non-linear, quadratically constrained, and linearized approaches. Two cases are carried out to assess their performance: 1) management of distributed generation units to maximize renewable energy harvesting (continuous control variables) and 2) control of capacitors to minimize energy losses (discrete control variables). The results demonstrate that the proposed quadratic approach significantly outperforms the more conventional formulations in both computational efficiency and robustness. This makes it a suitable alternative to be at the heart of the decision making of complex, real-time schemes to be adopted by future smart distribution networks.

Optimal power flow using hybrid differential evolution and harmony search algorithm

In this paper, Optimal Power Flow (OPF) with non-convex and non-smooth generator fuel cost characteristics is proposed using Hybrid Differential Evolution and Harmony Search (Hybrid DE-HS) algorithm. The proposed OPF formulation includes active and reactive power constraints; prohibited zones, and valve point loading effects of generators. In the problem formulation, transformer tap settings and reactive power compensating devices settings are also considered as the control variables. Therefore, OPF is a complicated optimization problem, hence there is a need to solve this problem with an accurate algorithm. The OPF solution is obtained by considering generator fuel cost, transmission loss and voltage stability index as objective functions. The effectiveness of the proposed hybrid algorithm is validated on IEEE 30, 118 and 300 bus test systems, and the results obtained with proposed Hybrid DE-HS algorithm are compared with other optimization techniques reported in the literature. For example, the quadratic fuel cost obtained by the hybrid DE-HS algorithm for IEEE 30 bus system is 799.0514 $/h, saving 0.31% of the cost obtained by the General Algebraic Modeling System (GAMS) software. This benefit increases further with the size of the system.

Optimal Power Flow for Unbalanced Bipolar DC Distribution Grids

The emergence of distributed energy resources can lead to congestion in distribution grids. DC distribution grids are becoming more relevant as more sources and loads connected to the low voltage grid use dc. Bipolar dc distribution grids with asymmetric loading can experience partial congestion resulting in a nodal price difference between the two polarities if a respective market model is applied. In order to take into account this price difference, this paper presents an optimal power flow (OPF) model formulated in terms of voltage and current. In the case of bipolar dc distribution grids, the single line approximation is no longer valid because current can flow in the neutral conductors as well. Moreover, loads and sources can be connected between any two nodes in the network. The proposed exact OPF formulation includes bilinear equations. The locational marginal prices (LMP) are derived by linearizing the problem at the optimal solution. Example cases show the various phenomena that can appear under asymmetric loading, such as pole-to-pole connections combined with pole-to-neutral connections, parallel sources, meshed grids, and their effect on the LMP.

A New Voltage Stability-Constrained Optimal Power-Flow Model: Sufficient Condition, SOCP Representation, and Relaxation

A simple characterization of the solvability of power flow equations is of great importance in the monitoring, control, and protection of power systems. In this paper, we introduce a sufficient condition for power flow Jacobian nonsingularity. We show that this condition is second-order conic representable when load powers are fixed. Through the incorporation of the sufficient condition, we propose a voltage stability-constrained optimal power flow (VSC-OPF) formulation as a second-order cone program (SOCP). An approximate model is introduced to improve the scalability of the formulation to larger systems. Extensive computation results on Matpower and NESTA instances confirm the effectiveness and efficiency of the formulation.

Flujo de potencia óptimo para redes radiales y enmalladas empleando programación semidefinida

Este artículo presenta una formulación convexa para el problema de flujo de potencia óptimo (Optimal Power Flow-OPF) en sistemas de potencia radiales y enmallados, a través de un modelo matemático basado en programación semidefinida (Semidefinite Programming-SDP), el cual trasforma el modelo cuadrático no convexo del OPF en un modelo convexo relajado, que puede ser solucionado de forma mucho más eficiente. El modelo es implementado en MATLAB usando el paquete de optimización convexa CVX. Los resultados obtenidos son comparados con el modelo no lineal del problema implementado en GAMS y MATPOWER usando cuatro sistemas típicos de la literatura especializada, dos de tipo radial y dos enmallado. Las soluciones encontradas por esta aproximación convexa, son muy cercanas a las respuestas presentadas en la literatura especializada para el modelo no lineal. Este tipo de aproximación permite obtener soluciones de muy buena calidad, muy cercanas a la solución óptima del problema en tiempos computacionales menores y susceptibles de aplicarse en problemas de operación y control en tiempo real.