Power Flow Formulation Research Articles

PowerModelsDistribution.jl: An open-source framework for exploring distribution power flow formulations

In this work we introduce PowerModelsDistribution, a free, open-source toolkit for distribution power network optimization, whose primary focus is establishing a baseline implementation of steady-state multi-conductor unbalanced distribution network optimization problems, which includes implementations of Power Flow and Optimal Power Flow problem types. Currently implemented power flow formulations for these problem types include AC (polar and rectangular), a second-order conic relaxation of the Branch Flow Model (BFM) and Bus Injection Model (BIM), a semi-definite relaxation of BFM, and several linear approximations, such as the simplified unbalanced BFM. The results of AC power flow have been validated against OpenDSS, an open-source “electric power distribution system simulator”, using IEEE distribution test feeders (13, 34, 123 bus and LVTestCase), all parsed using a built-in OpenDSS parser. This includes support for standard distribution system components as well as novel resource models such as generic energy storage (multi-period) and photovoltaic systems, with the intention to add support for additional components in the future.

On the impact of different voltage unbalance metrics in distribution system optimization

With increasing penetrations of single-phase, rooftop solar PV installations, the relative variations in per-phase loading and associated voltage unbalance are expected to increase. High voltage unbalance may increase network losses and lead to failure of three-phase equipment such as motor loads. However, solar PV panels are connected to the grid through inverters, which can provide reactive power support and may mitigate some of these negative effects. In this paper, we utilize a three-phase AC optimal power flow (OPF) formulation to minimize voltage unbalance using reactive power from solar PV inverters. When considering actions to reduce voltage unbalance, it is important to recognize that various organizations such as IEC, NEMA and IEEE provide different and partially inconsistent definitions of voltage unbalance in their power quality standards. This paper analyzes the impact of the different voltage unbalance metrics using different combinations of voltage unbalance objectives and constraints. For our analysis, the optimization scheme is tested on two unbalanced low-voltage distribution networks. We observe that minimizing voltage unbalance defined by one standard might actually increase voltage unbalance as defined by another standard, potentially resulting in equipment damage. We also observe that minimizing voltage unbalance does not always lead to lower network losses. However, considerable reduction in voltage unbalance with low network losses can be achieved by minimizing the losses while simultaneously enforcing limits on multiple definitions of voltage unbalance.

Integration of Electric Vehicles in Low-Voltage Distribution Networks Considering Voltage Management

The expected growth of the number of electric vehicles can be challenging for planning and operating power systems. In this sense, distribution networks are considered the Achilles’ heel of the process of adapting current power systems for a high presence of electric vehicles. This paper aims at deciding the maximum number of three-phase high-power charging points that can be installed in a low-voltage residential distribution grid. In order to increase the number of installed charging points, a mixed-integer formulation is proposed to model the provision of decentralized voltage support by electric vehicle chargers. This formulation is afterwards integrated into a modified AC optimal power flow formulation to characterize the steady-state operation of the distribution network during a given planning horizon. The performance of the proposed formulations have been tested in a case study based on the distribution network of La Graciosa island in Spain.

Comparison of Linear and Conic Power Flow Formulations for Unbalanced Low Voltage Network Optimization

The electrification of transport and heating systems brings new challenges for the low voltage distribution networks, which in the future might be more prone to congestion. This calls for methods to perform optimization and control actions, to ensure their correct operation. Their large extension and non-negligible degree of voltage unbalance make low voltage systems difficult to analyze. This paper features a comparison of three unbalanced formulations for power flow and OPF analyses: the full nonlinear, non-convex AC formulation, a second-order cone relaxation and a linear approximation. Accuracies and computational efforts are evaluated for different problem types, to assess whether the conic and linear formulation are suitable to perform low voltage systems calculations.

Scheduling electric vehicle charging to minimise carbon emissions and wind curtailment

This paper presents an investigation of the potential for coordinated charging of electric vehicles to i) reduce the CO 2 emissions associated with their charging by selectively charging when grid carbon intensity (gCO2/kWh) is low and ii) absorb excess wind generation in times when it would otherwise be curtailed. A method of scheduling charge events that seeks the minimum carbon intensity of charging while respecting EV and network constraints is presented via a time-coupled linearised optimal power flow formulation, based on plugging-in periods derived from a large travel dataset. Schedules are derived using real half-hourly grid intensity data; if charging in a particular event can be done entirely through use of renewable energy that would otherwise have been curtailed, its carbon intensity is zero. It was found that if ‘dumb’ charged from the current UK mainland (GB) grid, average emissions related to electric vehicle (EV) charging are in the range 35–56 gCO2/km; this can be reduced to 28–40 gCO2/km by controlled charging – approximately 20–30% of the tailpipe emissions of an average new petrol or diesel car sold in Europe. There is potential for EVs to absorb excess wind generation; based on the modelled charging behaviour, 500,000 EVs (20% of Scotland’s current car fleet) could absorb around three quarters of curtailment at Scotland’s largest onshore wind farm. • Electric vehicle (EV) charging is scheduled for minimum carbon intensity. • EVs’ potential to absorb excess renewables by charging selectively is investigated. • Real GB grid carbon intensity (gCO 2 /kWh) and wind curtailment data used. • Scheduling can reduce EV emissions from 35 to 56 gCO 2 /km to 28–40 gCO 2 /km. • 500,000 EVs can absorb 75% of curtailment at GB’s largest onshore wind farm.

Cost-Benefit Analysis of Moving-Target Defense in Power Grids

We study moving-target defense (MTD) that actively perturbs transmission line reactances to thwart stealthy false data injection (FDI) attacks against state estimation in a power grid. Prior work on this topic lacks an analysis of the relationship between MTD's effectiveness (in detecting FDI attacks) and the associated cost of the perturbations (incurred by the grid operator). To address the issue, we present formal design criteria to select MTD reactance perturbations that are truly effective. Based on a key optimal power flow (OPF) formulation, we find that the effective MTD may incur a non-trivial operational cost. We show that MTD's detection capability and the associated cost depend on the separation between the column spaces of measurement matrices before and after the MTD perturbation. We use the metric of smallest principal angles between the subspaces to characterize the separation. We show that different degrees of the separation provide a spectrum of tradeoffs between the MTD's detection capability and its cost. Furthermore, we present closed-form expressions in the case of a two-bus system to illustrate the tradeoffs. While our analysis is primarily based on a direct current (dc) power flow model, we show that the perturbations designed using this model are also effective in detecting FDI attacks against ac power flows. Similarly, the cost-benefit tradeoff still holds under the ac power model. Extensive simulations, using the MATPOWER simulator and benchmark IEEE bus systems, verify and illustrate the proposed design approach that for the first time addresses both key aspects of cost and effectiveness of the MTD.

Quadratically Constrained Quadratic Programming Formulation of Contingency Constrained Optimal Power Flow with Photovoltaic Generation

Contingency Constrained Optimal Power Flow (CCOPF) differs from traditional Optimal Power Flow (OPF) because its generation dispatch is planned to work with state variables between constraint limits, considering a specific contingency. When it is not desired to have changes in the power dispatch after the contingency occurs, the CCOPF is studied with a preventive perspective, whereas when the contingency occurs and the power dispatch needs to change to operate the system between limits in the post-contingency state, the problem is studied with a corrective perspective. As current power system software tools mainly focus on the traditional OPF problem, having the means to solve CCOPF will benefit power systems planning and operation. This paper presents a Quadratically Constrained Quadratic Programming (QCQP) formulation built within the matpower environment as a solution strategy to the preventive CCOPF. Moreover, an extended OPF model that forces the network to meet all constraints under contingency is proposed as a strategy to find the power dispatch solution for the corrective CCOPF. Validation is made on the IEEE 14-bus test system including photovoltaic generation in one simulation case. It was found that in the QCQP formulation, the power dispatch calculated barely differs in both pre- and post-contingency scenarios while in the OPF extended power network, node voltage values in both pre- and post-contingency scenarios are equal in spite of having different power dispatch for each scenario. This suggests that both the QCQP and the extended OPF formulations proposed, could be implemented in power system software tools in order to solve CCOPF problems from a preventive or corrective perspective.

DC Optimal Power Flow With Joint Chance Constraints

Managing uncertainty and variability in power injections has become a major concern for power system operators due to increasing levels of fluctuating renewable energy connected to the grid. This work addresses this uncertainty via a joint chance-constrained formulation of the DC optimal power flow (OPF) problem, which satisfies all the constraints jointly with a pre-determined probability. The few existing approaches for solving joint chance-constrained OPF problems are typically either computationally intractable for large-scale problems or give overly conservative solutions that satisfy the constraints far more often than required, resulting in excessively costly operation. This paper proposes an algorithm for solving joint chance-constrained DC OPF problems by adopting an S$\ell _1$QP-type trust-region algorithm. This algorithm uses a sample-based approach that avoids making strong assumptions on the distribution of the uncertainties, scales favorably to large problems, and can be tuned to obtain less conservative results. We illustrate the performance of our method using several IEEE test cases. The results demonstrate the proposed algorithm's advantages in computational times and limited conservativeness of the solutions relative to other joint chance-constrained DC OPF algorithms.

A new unbalanced three-phase governor power flow formulation based on the current injections method

This paper presents a new unbalanced three-phase governor power flow formulation using the three-phase current injection equations (TCIM) expressed as a function of voltages expressed in rectangular coordinates. The proposed technique considers the representation of the primary frequency regulation for synchronous generators, as well as, voltage and frequency regulation in a droop-based control for distributed generation units (DGs). The static response of the generation governor, droop control equations and an equation to define the network reference angle for a given value are incorporated into the modeling. As the system is unbalanced, an internal additional bus, having balanced voltages, was created for each synchronous generator. As a consequence, two additional equations were included to keep internal voltages balanced. The inclusion of new equations results in an augmented formulation which is then linearized and solved using the Newton–Raphson method, in which the original structure of the Jacobian matrix from the current injections method is preserved. The proposed formulation allows the distribution of the active losses among the generating units as well as the determination of the frequency deviation due to variations, either in the total amount of generation or in the total system load. Any bus (load or generation bus) of the system can be chosen as the system reference bus. The results obtained using the unbalanced three-phase 6-bus test system were validated using the well known RTDS (real time digital simulator). The 25-bus islanded test system was also used in the simulations with the inclusion of synchronous generation (SG) units to demonstrate the effectiveness of the proposed methodology.

On the numerical analysis based on successive approximations for power flow problems in AC distribution systems

This paper proposes a new power flow formulation for alternating-current distribution networks for radial and mesh topologies. This formulation corresponds to a successive approximation based on a modification of the conventional Gauss-Seidel numerical method by using a successive approximation approach. This power flow method allows working with complex variables by reducing the number of required calculations and avoiding the transformation of the power flow model into polar coordinates. Additionally, it does not use derivatives for approximating the problem as it occurs with Taylor-based approaches. Simulation results confirm that the proposed method is faster concerning computational time, as well as in the total number of iterations required. Numerical comparisons with classical methods such as Gauss-Seidel, Newton-Raphson, Levenberg-Marquardt, graph-based methods, and linear approximations have been made and implemented in MATLAB software to demonstrate the effectiveness of the proposed approach regarding power flow solutions in radial and mesh distribution networks.

Optimal Switch Placement in Distribution Systems: A High-Accuracy MILP Formulation

A new solution method is introduced to the problem of optimally deploying manual and automatic switches in distribution systems, where the product of two continuous variables and the inverse of a continuous variable are reformulated as a linear relation. This leads to a (mixed integer linear problem) MILP power flow formulation too. The objective function includes cost and reliability. The cost term itself includes capital investment, installation, and maintenance costs (MC) as well as customer interruption cost (CIC); while the reliability term is represented by system average interruption duration index (SAIDI). The problem is formulated as a MILP, which guarantees a global optimal solution. The effectiveness of the proposed method is validated through various case studies and sensitivity analyses on the RBTS4, followed by a comprehensive discussion and analysis of results. The proposed MILP formulation prescribes fewer switches while achieving lower SAIDI, compared to that of a previous MINLP formulation.

Hidden Convexities in Decentralized Coordination for the Distribution Networks

The modern power grid is undergoing unprecedented levels of transformations due to the rising prevalence of diverse power entities, cyber-enablement of grid components and energy deregulations. In this paper, we focus on distribution networks (DNs) to enable the seamless plug-and-play coordination of actuating cyber-enabled power entities for cost-effective and feasible system operations. The proposed distributed algorithm empowers individual cyber-physical agents residing in active power nodes with the ability to iteratively compute local actuation setpoints by exchanging information with neighbouring entities. The main contribution of this work is the identification of hidden convexities in the original non-convex optimal power flow (OPF) formulation for the DN via strategic decomposition and strong duality principles. These eliminate the need for OPF relaxations/approximations. Strong convergence and feasibility results are presented via theoretical analysis and practical simulation studies conducted on realistic systems.

Optimal Power Flow with Renewable Generation: A Modified NSGA-II-based Probabilistic Solution Approach

The rapid expansion of renewable generation has drastically increased the planning complexity of modern power systems as additional uncertainties, environmental concerns, and technical–economic issues should be accounted for. Within this context, the best operation performance of contemporary power system operators (SOs) depends not just on tractable realistic optimal power flow (OPF) formulations, but also on powerful optimization approaches. In this work, a tractable life-like multi-objective probabilistic OPF-based model for the SO’s medium-term operation considering high penetration of renewable resources is proposed. This model includes an explicit formulation of the operation of dispatchable and non-dispatchable generation, shunt reactive power sources, and under-load tap-changing (ULTC) transformers. The resulting model is a large-scale probabilistic multi-objective non-convex nonlinear mixed-integer programming (NLMIP) problem with continuous, discrete, and binary variables. To ensure tractability, uncertainties are modeled through a fast and efficient 2m probabilistic approach. To handle the nonlinearities and non-continuous variables that characterize the problem, a modified non-dominated sorting genetic algorithm (NSGA)-II solution approach is proposed and effectively tested.

Multi‐slack power flow for islanded microgrids with radial and meshed topologies

This study proposes a new power flow formulation for islanded microgrids. The proposed power flow is based on the effect of the superposition principle and the solution of a small non-linear subproblem to determine the frequency of the microgrid and the voltage magnitude in the angular reference node in each iteration. These variables are determined by considering the following equations in the non-linear subproblem: active and reactive power balance equations and specified phase at the angular reference node. The application of the superposition principle allowed to obtain two versions of the proposed technique: one for radial networks - based on the current summation method - and another for meshed networks - based on the Gauss-Zbus method. Therefore, the iterative framework prosed in this study expands in a simple and integrated way the two most commonly used power flow methods in conventional distribution networks for islanded microgrids. The tests in microgrids with 33, 310 and 1438 nodes showed that the proposed approach has the same accuracy as Newton-Raphson algorithm, but with significantly lower computational cost in large scale microgrids. In addition, the proposed method for island microgrid showed good accuracy and convergence for the most common load models applied in power flow studies of islanded microgrids.

Optimal Load Restoration in Active Distribution Networks Complying With Starting Transients of Induction Motors

Large horsepower induction motors play a critical role as industrial drives in production facilities. The operational safety of distribution networks during the starting transients of these motor loads is a critical concern for the operators. In this paper, an analytical and convex optimization model is derived representing the starting transients of the induction motor in a semi-static fashion. This model is used to find the optimal energization sequence of different loads (static and motor loads) following an outage in a distribution network. The optimization problem includes the optimal control of the converter-based DGs and autotransformers that are used for the induction motor starting. These models together with the semi-static model of the induction motor are integrated into a relaxed power flow formulation resulting in a Mixed-Integer Second Order Cone Programming (SOCP) problem. This formulation represents the transient operational limits that are imposed by different protection devices both in the motor side and network side. The functionality of the proposed optimization problem is evaluated in the case of a large-scale test study and under different simulation scenarios. The feasibility and accuracy of the optimization results are validated using I) off-line time-domain simulations, and II) a Power Hardware-In-the-Loop experiment.

Comparison Between Linear and Nonlinear Governor Power Flow Formulations

AbstractThe main objective of this paper is to review and compare two methodologies to solve the power flow problem considering active power generation droop control. The first methodology is based on a DC power flow model, and the second one is based on a conventional full Newton power flow formulation. The study of two test systems is used to validate the proposed analysis; the first one is the 11-bus system with two distinct areas, and the second one is the 39-bus New England power system. The results presented validate and indicate the effectiveness of the DC power flow approach in order to estimate the steady-state system frequency deviations.

Optimal power management of dependent microgrid considering distribution market and unused power capacity

This study presents an optimal power management for a microgrid (MG) in distribution market environment. The MG operator is able to have interactions with the distribution market operator (DMO) and adjacent MG (AMG) operator to supply its local loads. The DMO regulates the electricity market, and assigns the electricity price and power profile for the MG. The motivation behind the use of The DMO is that it works as an entity between MG and independent system operator (ISO) in order to guarantee a flexible operation by reducing power fluctuation and reduce the unintentional peak loads in low market price hours. The unused power capacity in AMG is used during the islanded hours through an additional interconnection point connected to the MG. Using this method reduces the need for installing new generation resources, which will be a practical and economical solution for MG developer. This MG which is able to provide power from distribution market and the AMG with two interconnection points is named dependent MG (DMG). A market-based stochastic model containing distribution market constraints and AC power flow formulations employs conditional value at risk (CVaR) methodology to capture the loads and wind uncertainties.The effectiveness of the presented model is evaluated on a 20 kV test system using different case studies. In this test system, the operator is the owner of all generation units. The numerical analysis explicate that presented model reduces the operation cost of DMG with the aim of responsive loads, unused power capacity, energy storage system (ESS) and power generation units. The market-based scheduling also provides operational flexibility for distribution system by adjusting flexibility limit of market constraints. It is also shown that changing risk preferences level changes the power generation pattern in ESS and causes a costly operation of resources in risk-averse strategy. The competence of this model is significant when the preventive maintenance (PM) program is carried out, and the DMG should rely on its flexible resources and AMG’s available power capacity, which could reduce the load shedding.

An MISOCP-Based Solution Approach to the Reactive Optimal Power Flow Problem

In this letter, we present an alternative mixed-integer non-liner programming formulation of the reactive optimal power flow (ROPF) problem. We utilize a mixed-integer second-order cone programming (MISOCP) based approach to find global optimal solutions of the proposed ROPF problem formulation. We strengthen the MISOCP relaxation via the addition of convex envelopes and cutting planes. Computational experiments on challenging test cases show that the MISOCP-based approach yields promising results compared to a semidefinite programming based approach from the literature.

Storage investment and network expansion in distribution networks: The impact of regulatory frameworks

In this paper we propose a bi-level equilibrium model that allows to analyze the impact of different regulatory frameworks on storage and network investment in distribution networks. In our model, a regulated distribution system operator decides on network investment and operation while he anticipates the decisions of private agents on storage investment and operation. Since, especially in distribution networks, voltage stability and network losses have a decisive influence on network expansion and operation, we use a linearized AC power flow formulation to adequately account for these aspects. As adjustments of the current regulatory framework, we consider curtailment of renewable production, the introduction of a network fee based on the maximum renewable feed-in, and a subsidy scheme for storage investment. The performance of the different alternative frameworks is compared to the performance under rules that are commonly applied in various countries today, as well as to a system-optimal (first-best) benchmark. To illustrate the economic effects, we calibrate our model with data from the field project Smart Grid Solar. Our results reveal that curtailment and a redesign of network fees both have the potential to significantly reduce total system costs. On the contrary, investment subsidization of storage capacity has only a limited impact as long as the distribution system operator is not allowed to intervene in storage operation.

A Multi-Step Reconfiguration Model for Active Distribution Network Restoration Integrating DG Start-Up Sequences

The ever-increasing penetration of Distributed Generators (DGs) in distribution networks suggests to enable their potentials in better fulfilling the restoration objective. The objective of the restoration problem is to resupply the maximum energy of loads considering their priorities using minimum switching operations. Basically, it is desired to provide a unique configuration that is valid regarding the load and generation profiles along the entire restorative period. However, this unique configuration may not satisfy at the same time: I) the DG start-up requirements at the beginning of the restoration plan and II) the topological conditions that would allow the DG to provide later on the most efficient support for the supply of loads. Therefore, it is proposed in this paper to allow a limited number of reconfiguration steps according to the DG start-up requirements. In addition, this paper presents a novel formulation for the reconfiguration problem that accounts for partial restoration scenarios where the whole unsupplied area cannot be restored. The decision variables of the proposed multi-step restoration problem are: I) the line switching actions at each step of the reconfiguration process, II) the load switching actions during the whole restorative period and, III) the active/reactive power dispatch of DGs during the whole restorative period. A relaxed AC power flow formulation is integrated to the optimization problem in order to ensure the feasibility of the solution concerning the operational safety constraints. The overall model is formulated in terms of a mixed-integer second-order cone programming. Two simulation scenarios are studied in order to illustrate different features of the proposed strategy and to demonstrate its effectiveness particularly in the case of large-scale outages in distribution networks.