Distribution System Power Flow Research Articles

Power Flow Analysis in Unbalanced Three-Phase Distribution Systems using Backward/Forward Sweep and Current Injection Methods

The electrical power distribution system is a part of the power system that distributes electricity from the transmission network to customers. In the distribution system, imbalances often occur due to the varying load profiles in each phase. This can cause voltage imbalances in the distribution system. This study aims to compare two power flow analysis methods, Backward/Forward Sweep and Current Injection. The study analyses the voltage and power loss conditions on each phase at each bus and line in the three-phase distribution system under unbalanced conditions. Simulations were conducted on two IEEE test buses, IEEE 19-Bus and IEEE 33-Bus with radial configurations. The power flow calculation results using the Backward and Forward Sweep method showed that in the IEEE 19-Bus system, the highest voltage drop percentage occurred on phase b at bus 19, at 3.14%, the highest voltage imbalance percentage occurred at bus 19, at 0.1409%, and the total active and reactive power losses were 7.352 kW and 3.164 kVAR. In the IEEE 33-Bus system, the highest voltage drop percentage occurred on phase c at bus 18, at 5.85%, the highest imbalance percentage occurred at bus 15, at 0.2077%, and the total active and reactive power losses were 19.107 kW and 8.22 kVAR. The percentage difference between the two methods used is less than one percent, indicating that both methods are sufficiently accurate in analyzing power flow in an unbalanced distribution system.

Improved Newton-Raphson method with simplified Jacobian matrix and optimized iteration rate for power flow calculation of power system

With high penetration of renewable energy and novel loads connected to the distribution network, the voltage fluctuation becomes more severe and frequent, which may cause over- and under-voltage. The distribution system operator should calculate the power flow and validate the state to optimize the distribution network. Power flow calculation is the solution to the multivariate nonlinear problem, and the Newton-Raphson method is an effective algorithm for solving nonlinear problems. However, calculating the Jacobian matrix is a crucial process of the Newton-Raphson method, which is time-consuming. Therefore, this paper proposed an improved Newton-Raphson method, which simplifies and decreases the iterations of the calculation process of the Jacobian matrix to improve the calculation rate. To verify the effectiveness of the proposed method, the power flow of the IEEE 33-node power distribution system is calculated by the improved Newton-Raphson method and the conventional Newton-Raphson method.

A Fully-Decentralized Transactive Energy Management Under Distribution Network Constraints via Peer-to-Peer Energy Trading

Peer-to-peer (P2P) energy trading is seen as a promising method for distributed transactive energy (TE) management. Since distribution network is used for electricity transfer, physical constraints should be imposed during the P2P trading. In this paper, we propose a P2P-based distributed TE management approach considering distribution network constraints and product differentiation for P2P energy trading between different buses. A multi-period AC optimal power flow (OPF) model is used to monitor power flow in distribution system, and the exactness of second-order cone relaxation is recovered by the convex-concave procedure, making sure that P2P transactions do not violate network constraints. Moreover, we propose a fully-decentralized variant of alternating direction method of multipliers for clearing P2P transactions in a fully-decentralized manner under AC power flow constraints. Our proposed variant does not need any interactions with the central supervisor in comparison with the traditional one. Finally, numerical results on IEEE 33-bus and 141-bus distribution systems verify the effectiveness of our proposed method in ensuring physical feasibility, lowering data communication and protecting data privacy during P2P energy trading.

Simultaneous Robust State Estimation, Topology Error Processing, and Outage Detection for Unbalanced Distribution Systems

This paper proposes an efficient algorithmic approach that overcomes the critical challenges in real-time unbalanced distribution system state estimation, topology error processing, and outage identification simultaneously. These challenges include (1) limited locations of measurement devices and unsynchronized measurement data as well as missing and bad data, (2) complicated mixed-phase switch actions and mutual impedances and shunt admittances, and (3) the nonlinear nature of unbalanced distribution system power flow with distributed energy resources (DERs). A single snap-shot mixed-integer quadratic programming (MIQP) optimization framework is proposed to cope with these challenges, which simultaneously identifies real-time network topology, estimates system state, and detects the outages via analytical constraints. An AC optimal power flow (ACOPF) approach is proposed to accurately model unbalanced distribution systems. In the proposed MIQP formulation, nonlinearities due to the complicated mixed-phase switch operations and the ACOPF approach are linearized. The effectiveness of the proposed approach is verified on an actual distribution feeder in Arizona. The results illustrate that the proposed model is robust, accurate, and computationally efficient for implementation in a distribution management system (DMS).

A Robust Method for Distribution System State Estimation with Predator-Prey Brain Storm Optimization

In this paper, a new method is proposed for distribution system static state estimation with low redundant measurements. It is different from transmission system static state estimation because the distribution system static state estimation has low redundancy of measurements while the transmission system static state estimation has enough redundancy. This paper makes use of the nest structure of DistFlow of the distribution system power flow calculation to transform the formulation into evaluating the state variable vector at distribution substations. In other words, the distribution state estimation problem may be expressed as an optimization problem. As the optimization method, PPBSO (Predator-Prey Brain Storm Optimization) of evolutionary computation is proposed to solve the problem. The proposed method is successfully applied to a sample system.

Three Solution Approaches to Stochastic Multi-Period AC Optimal Power Flow in Active Distribution Systems

This paper proposes three novel solution approaches (A1, A2, A3) to solve stochastic multi-period AC optimal power flow (S-MP-OPF) for day-ahead flexibility procurement from distributed energy resources (DER) in active distribution systems (ADSs). The S-MP-OPF is a mixed-integer nonlinear programming (MINLP) problem due to the binary variables modeling the operation of storage devices and flexible loads. The proposed three approaches have a shared common first step, which resorts to a new mixed-integer linear programming (MILP) model approximation of the S-MP-OPF problem. The MILP model employs second-order Taylor series expansion of trigonometric terms and formulates the linear approximations relying on variables such as square of voltage magnitude and voltage angle difference. This first step serves also the purpose of fixing the binary variables to the values computed by the MILP problem. Then, the approach A1 only checks the AC feasibility of MILP solution while approaches A2 and A3 further optimize continuous variables. Specifically, the sophisticated heuristic approach A2 employs sequential linear programming and AC power flow while the approach A3 models and solves directly the remaining nonlinear programming (NLP) problem. The performances of these approaches, a benchmark MINLP solver, and a state-of-the-art method are thoroughly compared in three radial or weakly meshed ADSs of 34, 31, and 191 nodes, respectively. The numerical results indicate that, albeit the approach A2 performs best overall, these approaches present distinct accuracy vs speed trade-offs, which make them suitable to problems of different sizes and diverse accuracy or speed requirements.

Multi-period Two-stage Robust Optimization of Radial Distribution System with Cables Considering Time-of-use Price

In the existing multi-period robust optimization methods for the optimal power flow in radial distribution systems, the capability of distributed generators (DGs) to regulate the reactive power, the operation costs of the regulation equipment, and the current of the shunt capacitor of the cables are not considered. In this paper, a multi-period two-stage robust scheduling strategy that aims to minimize the total cost of the power supply is developed. This strategy considers the time-of-use price, the capability of the DGs to regulate the active and reactive power, the action costs of the regulation equipment, and the current of the shunt capacitors of the cables in a radial distribution system. Furthermore, the numbers of variables and constraints in the first-stage model remain constant during the iteration to enhance the computation efficiency. To solve the second-stage model, only the model of each period needs to be solved. Then, their objective values are accumulated, revealing that the computation rate using the proposed method is much higher than that of existing methods. The effectiveness of the proposed method is validated by actual 4-bus, IEEE 33-bus, and PG 69-bus distribution systems.

Fast Flexible Direct Power Flow for Unbalanced and Balanced Distribution Systems

The study proposes a fast flexible direct power flow solution for radial distribution systems and a fast flexible direct weakly meshed power flow solution for weakly meshed distribution systems. The algorithm is based on the direct forward sweep power flow solution, which is an updated version of the backward/forward sweep solution. The fast flexible direct power flow uses a unique conversion matrix (CM) to rapidly determine the power flow solution. The inverted conversion matrix and its slide-modified matrix are used to solve the power flow problem in a single forward sweep step, which is a novel feature of this work. To ensure the invertibility of the conversion matrix, it is constructed to have a small condition number and a determinant of minus one, and all of its eigenvalues must be equal to that of minus one. Additionally, by modifying the conversion matrix to accommodate the loop branch using the break-point idea, a new weakly meshed conversion matrix (WMCM) is generated with the same following modification as for the radial network and employed in the fast flexible direct weakly meshed power flow (FFDWMPF) solution for the weakly meshed distribution network. The usage of a single matrix in the power flow solution and advanced direct techniques decreases the number of iterations and CPU execution time when MATLAB programming is executed. Furthermore, the proposed method is flexible enough to incorporate any changes in the radial or weakly meshed distribution system just by incorporating the changes in the CM and WMCM for any radial or weakly meshed system. Moreover, the robustness of FFDPF and FFDWMPF is evaluated under various loading scenarios on balanced radial and weakly meshed distribution networks. Finally, to validate the proposed algorithm, the proposed strategy is applied to numerous balanced and unbalanced distribution systems.

Operation Method of On-Load Tap Changer on Main Transformer Considering Reverse Power Flow in Distribution System Connected with High Penetration on Photovoltaic System

The increasing use of photovoltaics (PVs) in distribution systems owing to the low-carbon policy has given rise to the need for various technological changes. In particular, the operation of on-load tap changers (OLTCs) has attracted attention. In traditional distribution systems, the OLTC operates via a line-drop compensator (LDC), which focuses on the load to solve the low-voltage problem; however, the problem of over-voltage caused by PVs persists. Currently, a method for operating an OLTC using the measured voltage is being researched; however, solving the voltage problem for several feeders connected to a main transformer (MT) is not viable. Therefore, this study proposes an OLTC operation method to address the feeder with the largest voltage problem depending on the direction of power flow. The proposed method selects a point where the OLTC operates using the difference between the measured and reference voltages. Setting the reference voltage can solve the problem that occurs due to the direction of power flow. Finally, the effectiveness of the proposed method is verified via case studies. Based on the results, we can conclude that the proposed method effectively solves the voltage problem, and an increase in hosting capacity can be expected.

Probabilistic optimal siting and sizing of distributed generation and shunt capacitors considering feeder flow control units using a novel distribution power flow

A modern energy system should be reliable, economically viable, and environment-friendly. This paper addresses the above issue by proposing a probabilistic approach of optimally placing and sizing distributed generation and shunt capacitors in a distribution network. Objectives are to simultaneously reduce the cost, emission, network loss, improve the system voltage profile, and avoid line overloads. A feeder flow control generator is considered so that the distribution network draws a constant amount of active and reactive power from the substation. The operation is modeled using a Zero-PQVδ bus pair in the power flow. A novel distribution system power flow is proposed to embody the operation philosophy. Uncertainties of renewable power sources and load demand are modeled using ‘Hong's 2m point estimate method’. A fuzzy multi-objective satisfaction criterion is used to realize multiple objectives. Simulation studies on a thirty-three node distribution network reveal the efficacy of the proposed method.

A fixed-point current injection power flow for electric distribution systems using Laurent series

This paper proposes a new power flow (PF) formulation for electrical distribution systems using the current injection method and applying the Laurent series expansion. Two solution algorithms are proposed: a Newton-like iterative procedure and a fixed-point iteration based on the successive approximation method (SAM). The convergence analysis of the SAM is proven via the Banach fixed-point theorem, ensuring numerical stability, the uniqueness of the solution, and independence on the initializing point. Numerical results are obtained for both proposed algorithms and compared to well-known PF formulations considering their rate of convergence, computational time, and numerical stability. Tests are performed for different branch R/X ratios, loading conditions, and initialization points in balanced and unbalanced networks with radial and weakly-meshed topologies. Results show that the SAM is computationally more efficient than the compared PFs, being more than ten times faster than the backward–forward sweep algorithm.

PQ assessment of EDS with DER penetration by DVSI-based DSTATCOM using an ALMS control algorithm

This paper demonstrates a two-level dual voltage source inverter (DVSI) and its operation using two isolated DC links for optimal multidirectional power flow in the electrical distribution system (EDS). In the suggested configuration, distributed energy resource (DER)-supported voltage source inverter marked as VSI-1 and other marked as VSI-2 which is integrated by a self-supported capacitor. An Adaptive Least-Mean-Square (ALMS) controller has been employed for the DVSI-based distribution static compensator (DSTATCOM) on various aspects such as previous weight, normalising weight, learning rate, better dynamic response, reduced computational burden and better estimation speed. The suggested control algorithm is implemented using MATLAB/Simulink and processes through the FPGA SPARTAN-6 control board to perform the comparison between the DVSI and VSI. Among these two topologies, the experimental performance of DVSI shows better voltage regulation, source current harmonic reduction, voltage balancing and power factor correction as per the IEEE standard guidelines.

Comparison of Radial and Zonal DC Distribution System for Hybrid-Powered Trimaran Vessels Power Flow Analysis

Two types of power grid structures have been applied to the shipboard, namely the radial and the zonal networks. These two distribution systems have different power characteristics transmitted to the bus panel. The power characteristics affect the power density of each bus at each level. This paper explores the power characteristics of the radial type and the zonal type power grid structures in a DC distribution system. The power flow of both distribution system types has been determined by the Adaptive Newton-Raphson method and simulated based on ship operation conditions using Electrical Power System Analysis Software (ETAP). In determining the power flow for the DC distribution system, one-line diagram modelling is developed in radial type and zonal type. The simulation parameters are adjusted to the operating conditions of the ship, namely sailing, manoeuvring, loading-unloading, and at the port. After that, a power flow simulation has been carried out using ETAP and has showed what had successfully implemented the power flow in the DC distribution system for radial and zonal types. The capacity of Power generation is adequate all through a ship operation. The characteristics of the transmitted power show that the zonal type has a higher efficiency rate than the radial type. The zonal type has a backup bus so that the electrical system's reliability is higher than the radial type. However, the power distributed to the equipment bus from the load bus, both the radial type and the zone type, has the exact power requirements.

Coordinated scheduling of integrated transmission and distribution systems using an improved Lipschitz dynamic programming approach

In this study, we propose an improved Lipschitz dynamic programming-based decentralized algorithm for the coordinated scheduling of integrated transmission and distribution (T&D) systems, considering unit commitments in a transmission system and the AC power flow and discrete control devices in distribution systems. In this algorithm, reverse norm cuts are introduced iteratively to approximate the value functions of subproblems of distribution systems, which are formulated as mixed-integer linear programming models. Based on the characteristics of subproblems of distribution systems, two strategies for introducing the reverse norm cuts in the subproblem of the transmission system are developed to improve the convergence. Furthermore, to further accelerate the proposed decentralized algorithm, a curve-fitting approach is designed to obtain the pre-defined approximate lower bounds of the value functions of distribution systems at each scheduling time period, which are formulated as a linear constraint set with integer variables. Finally, case studies are conducted on a constructed integrated T&D system and a practical integrated T&D system to verify the accuracy and effectiveness of the proposed algorithm.

An interval power flow for unbalanced distribution systems based on the Three-Phase Current Injection Method

Interval arithmetic has been widely used for power systems analysis considering uncertainties associated with load and generation data. However, little work can be found within the context of three-phase distribution systems due to their peculiarities such as radial topology and unbalanced loads. In this paper, a novel methodology is proposed for determining interval results of a power system load flow for three-phase unbalanced distribution networks based on the Three-Phase Current Injection Method (TPCIM) in which state variables are considered in rectangular coordinates. In the proposed approach, active and reactive powers at each load bus are modelled as interval values to represent their inherent uncertainties and the Krawczyk operator is applied into the power flow equations in order to provide reliable interval three-phase results. Additionally, the use of interval extensions and an angular rotation technique are proposed to overcome overestimation problems associated with the interval solutions. Computational simulations are carried out using IEEE 13, 33 and 69-bus test systems. The main contribution of this work is the proposition of two methods based on traditional interval arithmetic to reduce the diameter of the solution and achieve similar results when compared with Affine Arithmetic and Monte Carlo Simulations. Additionally, the computational time associated with the algorithm is extremely advantageous. The method is useful for power distribution systems operation and planning studies, representing a viable and useful tool for calculating the impact of uncertain input data on the power flow results.

Distributed Optimal Power Flow for Distribution System Using Second Order Cone Programming and Consensus Alternating Direction Method of Multipliers

Distributed Optimal Power Flow for Distribution System Using Second Order Cone Programming and Consensus Alternating Direction Method of Multipliers

Voltage Stability Constrained Optimal Power Flow for Unbalanced Distribution System Based on Semidefinite Programming

This paper proposes a voltage stability constrained optimal power flow (VSC-OPF) for an unbalanced distribution system with distributed generators (DGs) based on semidefinite programming (SDP). The AC optimal power flow (ACOPF) for unbalanced distribution systems is formulated as a chordal relaxation-based SDP model. The minimal singular value (MSV) of the power flow Jacobian matrix is adopted to indicate the voltage stability margin. The Jacobian matrix can be explicitly expressed by ACOPF state variables. The nonlinear constraint on the Jacobian MSV is then replaced with its maximal convex subset using linear matrix inequality (LMI), which can be incorporated in the SDP-based ACOPF formulation. A penalty technique is leveraged to improve the exactness of the SDP relaxation. Case studies performed on several IEEE test systems validate the effectiveness of the proposed method.

Modified soccer game optimization and its application on power flow and distribution generation placement problems of distribution systems

Modified soccer game optimization and its application on power flow and distribution generation placement problems of distribution systems

Parallel GPF solution: A GPU‐CPU‐based vectorization parallelization and sparse technique for NR implementation

As an essential simulation tool, global power flow (GPF) can approximately model mutual impacts of interactions between transmission system (TS) and distribution system (DS). However, sequential GPF lacks timely solution performance required in several applications built on top of GPF, for example, contingency analysis. To address this issue, graphic processing unit (GPU) with excellent floating-point arithmetic operations is introduced and in turn a Master-Slave-Splitting (MSS) based parallel GPF approach is proposed via vectorization parallelization under GPU-CPU heterogeneous architecture. First, the root cause behind sparsity property of GPF and its impacts on key steps of NR-based GPF approach are quantitatively analysed. Considering the studies carried out in the authors' previous work of the application of GPU and sparse techniques on TS power flow (TS-PF), its applications on parallel implementations of corresponding steps engaged in three-phase imbalanced DS power flow (DS-PF) is mainly focused on. Numerical studies validate the effects of various custom-designed parallel schemes on acceleration enhancement of individual steps in the proposed GPU-CPU parallel GPF approach.

A resilience-oriented two-stage recovery method for power distribution system considering transportation network

In recent years, extreme weather events have resulted in a number of severe power outages and economic losses. It is becoming urgent to study approaches to enhance power distribution system resilience. This paper presents a two-stage optimization method for improving resilience of power distribution systems. The two-stage method is influenced by a transportation network, which considers the interdependence between traffic roads and overhead distribution lines. In the first stage, based on the Bayesian network and the Monte Carlo method, the locations of emergency stations, e.g., charging stations and repair stations are pre-positioned to improve the recovery speed of interrupted loads after disasters. Then the second stage is to co-optimize the dispatch of mobile energy storage systems and repair crews on transportation network to minimize the shedding of critical loads while considering operation constraints of power, voltage and power flow for power distribution system. The dispatch of mobile energy storage systems and repair crews are affected by the locations of emergency stations in the first stage. The route of mobile energy storage systems and repair crews on transportation network can be obtained from the results in the second stage. The proposed method is tested on a modified IEEE 33-bus distribution test system with a corresponding transportation network. Simulation results demonstrate the effectiveness of the proposed method in enhancing power distribution system resilience by pre-positioning emergency stations and co-dispatching mobile energy storage systems and repair crews.