Three-phase Unbalanced Distribution Systems Research Articles

Three-phase model of SCIG-based variable speed wind turbine for unbalanced DSLF analysis

Steady state performances of the electric power distribution system are normally assessed or evaluated based on load flow analysis. To properly carry out the analysis, a valid steady state load flow model of each distribution system component, including the wind power plant (WPP), needs to be developed. The present paper proposes a method for modeling and integrating squirrel cage induction generator (SCIG)-based variable speed WPP into a three-phase unbalanced distribution system load flow (DSLF) analysis. The proposed method is based on a single-phase T-circuit model of fixed speed WPP, which has successfully been applied to balanced electric power systems. In the present work, the single-phase T-circuit model is extended and modified to be used in steady state load flow analysis of three-phase unbalanced distribution systems embedded with SCIG-based variable speed WPP. Results of the case studies confirm the validity of the proposed method.

Multiple-node model of wind turbine generating system for unbalanced distribution system load flow analysis

This paper discusses a method to integrate a wind turbine generating system (WTGS) into a three-phase unbalanced distribution system load flow (DSLF) analysis. The proposed method is based on the single-phase multiple-node model. In the present work, the single-phase multiple-node model is extended to a three-phase multiple-node model to facilitate the load flow analysis of a three-phase unbalanced power system network. The multiple-node model (i.e., three-node model) will only modify the load flow analysis by introducing two lines and two load buses to the distribution system network where the WTGS is installed. Thus, a standard three-phase load flow program can be employed to compute the unknown quantities in the DSLF problem formulation. The proposed method is verified by incorporating the model into the load flow analysis of three-phase distribution networks. The investigation uses two representative distribution networks (i.e., 19-bus and 25-bus networks). The results of the study confirm the validity of the proposed method.

Iterative convex relaxation of unbalanced power distribution system integrated multi-energy systems

This study describes a multi-period, multi-energy operation of a three-phase unbalanced Electric Distribution System (EDS) that coordinates with a District Heating System (DHS) and a Natural Gas (NG) distribution system. The electrical subsystem problem is formulated as a bi-level programming problem, with levels 1 and 2 solving the subsystem’s linearized and relaxed nonlinear versions, respectively. Second-Order Cone Programming (SOCP) and polyhedral relaxations are availed to circumvent the non-convexities involved in the subsystems. The SOCP relaxation for an unbalanced power distribution network is inexact; also, the subsystems applying polyhedral relaxation may not generate a meaningful solution as the relaxations are not tight. Therefore, a solution recovery algorithm is developed for each subsystem to recover a feasible solution from their relaxed counterparts. The successive bound tightening algorithm employing a solution recovery procedure is proposed for each subsystem, improving solution quality and strengthening the relaxations with the desired computational efficiency. The proposed solution strategy to optimize the Multi-Energy System (MES) operation cost is verified on the three-phase IEEE-13 and IEEE-123 bus systems, each coordinating with a 30-node DHS and a 6-bus NG network. The results analyses demonstrate that the proposed solution strategy efficiently achieves an optimal solution, reducing maximum relaxation error below 0.1% for each subsystem. The proposed strategy delivered a 14.58% reduction in real power losses and a 12.73% decrease in phase voltage unbalance rate for the EDS in MES. Furthermore, a 1.96% decrease in operational cost demonstrates the techno-economic benefits of the proposed strategy.

A graph theory-based solution method for capacitor bank and voltage regulator allocation problems in an unbalanced distribution system

This paper presents a graph theory-based method to solve voltage and reactive power (voltage/var) control problems for a three-phase unbalanced distribution system (UDS). Multi-point fixed and switched capacitor bank (CB), solar distributed generation (SDG) and voltage regulator (VR) location are determined by simple and straightforward graph theory-based measures. The main contribution of this paper is – a single-stage optimisation-based voltage/var control method for UDS unlike commonly used two-stage optimisation methods; to find the location and optimal size of the CB. As the proposed graph theory method used power flow outputs to determine the most suitable nodes for CB and SDG connection, optimisation is needed only for optimal sizing at a certain location which reduces the computational burden. Secondly, optimal placement of VR is used to maintain the voltage level/voltage unbalance of the system. Multiple test cases with optimal configurations of switched CBs and optimal power factor operation-based SDG are used to demonstrate the effectiveness of proposed optimal operation techniques for overall improvement of performance parameters of UDS.

Tractable Data Enriched Distributionally Robust Chance-Constrained Conservation Voltage Reduction

This paper proposes a tractable distributionally robust chance-constrained conservation voltage reduction (DRCC-CVR) method with enriched data-based ambiguity set in unbalanced three-phase distribution systems. The increasing penetration of distributed renewable generation not only brings clean power but also challenges the voltage regulation and energy-saving performance of CVR by introducing high uncertainties to distribution systems. In most cases, the conventional robust optimization methods for CVR only provide conservative solutions. To better consider the impacts of load and PV generation uncertainties on CVR implementation in distribution systems and provide less conservative solutions, this paper develops a data-based DRCC-CVR model with tractable reformulation and data enrichment method. Even though the uncertainties of load and photovoltaic (PV) can be captured by data, the availability of smart meters (SMs) and micro-phasor measurement units (PMUs) is restricted by cost budget. The limited data access may hinder the performance of the proposed DRCC-CVR. Thus, we further present a data enrichment method to statistically recover the high-resolution load and PV generation data from low-resolution data with Gaussian Process Regression (GPR) and Markov Chain (MC) models, which can be used to construct a data-based moment ambiguity set of uncertainty distributions for the proposed DRCC-CVR. Finally, the nonlinear power flow and voltage dependant load models and DRCC with moment-based ambiguity set are reformulated to be computationally tractable and tested on a real distribution feeder in Midwest U. S. to validate the effectiveness and robustness of the proposed method.

A SOCP-Based ACOPF for Operational Scheduling of Three-Phase Unbalanced Distribution Systems and Coordination of PV Smart Inverters

The proliferation of distributed energy resources (DERs) imposes new challenges to distribution system operation, e.g., power quality issues. To overcome these challenges and enhance system operation, it is critical to effectively utilize all available resources and accurately characterize unbalanced distribution networks in operational tools. This paper proposes a convex second-order-cone programming (SOCP)-based AC optimal power flow (ACOPF) model for three-phase unbalanced distribution networks, including smart inverters and Volt-VAr controller (VVC) devices. Reactive power-voltage (Q-V) characteristics of smart inverters of solar photovoltaic (PV) units are also modeled. Moreover, the settings of Q-V characteristics of VVC are co-optimized within the proposed ACOPF, considering the allowable range of the IEEE 1547-2018 standard. Furthermore, dynamic analyses are conducted to verify the stability of optimal settings of VVC. The proposed models are tested on an actual 1747-node primary distribution feeder in Arizona. The results illustrate the effectiveness of the proposed ACOPF for unbalanced systems in providing an optimal solution while capturing the non-linearity and non-convexity of ACOPF. By co-optimizing settings, system operation is improved due to the flexibility of adjusting reactive power output from PV units with VVC. The time-domain simulations show that the optimal settings cause no stability issue for the distribution system.

Integration of Distributed Generation and Plug-in Electric Vehicles on Power Distribution System by Using Queuing Theory

Plug-in electric vehicles (PEVs) and distributed generation (DG) can positively and negatively impact the distribution system. Therefore, this paper presents the modeling and analysis of DG and PEVs’ penetration levels of the three-phase unbalanced radial distribution system. The study aims to optimize the distribution system’s DG sizing and PEV charging to minimize total power loss. The test system is the 4th circuit of the Nonsung service station along Thaharn Road, Muang District, Udon Thani, Thailand. According to objective function and constraints, such control variables as installation buses and rated outputs of DG and the penetration levels of PEVs were obtained to evaluate the total power loss. Significantly, the charging demand of PEVs is an uncertain load estimated by queuing theory integration with the minimization tool called the differential evaluation (DE) method. According to the result comparison of a four case simulation, the total power losses of queuing theory and DE application are minimum. Finally, total power losses conform to the regulation of the Provincial Electricity Authority (PEA), Thailand.

Distributed Optimal Voltage Control Considering Latency and Asynchronous Communication for Three Phase Unbalanced Distribution Systems

Integration of distributed energy resources (DERs) helps realize sustainability and reduce carbon footprint, but may cause voltage control issues in the distribution power system. The impact is mainly driven by the intermittency and variability of DERs typically connected at the edge. Traditional centralized optimal voltage control lacks scalability, and local voltage controls are not optimal. Distributed algorithms are scalable, robust to failures, and possibly require less communication resources compared to centralized. This paper presents a new online distributed voltage control algorithm with primal-dual updates and auxiliary variables for three-phase unbalanced distribution systems. The phase-specific active and reactive power of participating voltage control agents is regulated considering the power converter operational limits. The algorithm is validated to regulate voltage for both static and dynamic loading conditions for three-phase unbalanced IEEE 13 and 123 bus systems modeled in the OpenDSS. The robustness of the algorithm is validated for delayed asynchronous communication, modeling errors, and measurement noises. Simulation results and convergence analysis show the superior performance of the proposed algorithm.

A Hybrid Compensator for Unbalanced AC Distribution System With Renewable Power

A hybrid compensator with a new coordinated control is proposed for mitigating power quality aspects and integration of renewable power in an unbalanced three-phase four-wire distribution system. The proposed compensator comprises of thyristor switched capacitor (TSC) banks, delta connected thyristor controlled reactor (TCR), a 3rd and a 5th harmonic tuned passive LC filter & a low rating three-phase four-wire D-STATCOM. The TSC-TCR operates asymmetrically for the base var compensation and also eliminates negative sequence current during unbalanced loading at the point of common coupling (PCC). The D-STATCOM works in coordination with the TSC-TCR and the passive filters to compensate fundamental var, zero sequence current, voltage flicker & 7th harmonic currents at the PCC. The D-STATCOM thus serves as a supportive var compensator as well as an active filter in selective harmonic compensation mode, thereby improving the var compensation and harmonic current filtering at the PCC. A new coordination control system is proposed to control the operation of the D-STATCOM which is also employed to integrate a PV source & a doubly fed induction generator (DFIG) based wind energy conversion system (WECS) to the PCC. This scheme with grid connected and islanded mode of operation using D-STATCOM exhibits low-voltage ride-through capabilities. Further, the chance of resonance of the passive filters used with source inductance is eliminated in the proposed scheme. Simulation results show that the proposed hybrid system effectively compensates var, negative & zero sequence current, reduces voltage flicker and harmonic currents at PCC.

Log(v) 3LPF: A Linear Power Flow Formulation for Unbalanced Three-Phase Distribution Systems

In this work, we introduce Log(v) 3LPF, a linear power flow solver for unbalanced three-phase distribution systems. Log(v) 3LPF uses a logarithmic transform of the voltage phasor to linearize the AC power flow equations around the balanced case. We incorporate the modeling of ZIP loads, transformers, capacitor banks, switches and their corresponding controls and express the network equations in matrix-vector form. With scalability in mind, special attention is given to the computation of the inverse of the system admittance matrix, Ybus. We use the Sherman-Morrison-Woodbury identity for an efficient computation of the inverse of a rank-k corrected matrix and compare the performance of this method with traditional LU decomposition methods using Big- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}$</tex-math></inline-formula> notation. We showcase the solver for a variety of network sizes, ranging from tens to thousands of nodes, and compare the Log(v) 3LPF with commercial-grade software, such as OpenDSS.

Injected Current Sensitivity Based Load Flow Algorithm for Multi-Phase Distribution System in the Presence of Distributed Energy Resources

This paper presents an injected current sensitivity (ICS) based multi-phase load flow (LF) methodology with minimal Jacobian computation for the three-phase unbalanced distribution system with multiple Distributed Energy Resources (DERs). In the proposed approach, first, a generalized bus admittance and Jacobian matrix have been formulated that include various distribution system components such as lines, shunt capacitors, voltage control devices such as regulators, different transformer connections, and load models. Second, DERs are modeled as a PV bus and incorporated into the proposed LF algorithm by formulating a reactive power injection-based sensitivity matrix that iteratively computes its net power injections. Unlike the traditional Jacobian matrix-based LF methods where an update of Jacobian (size and element) is required for each PV bus, this method does not require any update, hence, making it faster and more flexible to include a large number of DERs. From the various tests performed on the standard IEEE three-phase distribution network and their derivatives, it has been observed that the proposed approach is accurate, efficient, computationally less complex, and scalable.

Optimal Selection of Conductor Sizes in Three-Phase Asymmetric Distribution Networks Considering Optimal Phase-Balancing: An Application of the Salp Swarm Algorithm

This paper presents a new methodology to simultaneously solve the optimal conductor selection and optimal phase-balancing problems in unbalanced three-phase distribution systems. Both problems were represented by means of a mathematical model known as the Mixed-Integer Nonlinear Programming (MINLP) model, and the objective function was the minimization of the total annual operating costs. The latter included the costs associated with energy losses, investment in conductors per network segment, and phase reconfiguration at each node in the system. To solve the problem addressed in this study, a master–slave methodology was implemented. The master stage employs a discrete version of the Salp Swarm Algorithm (SSA) to determine the set of conductors to be installed in each line, as well as the set of connections per phase at each of the nodes that compose the system. Afterward, the slave stage uses the three-phase version of the backward/forward sweep power flow method to determine the value of the fitness function of each individual provided by the master stage. Compared to those of the Hurricane-based Optimization Algorithm (HOA) and the Sine Cosine Algorithm (SCA), the numerical results obtained by the proposed solution methodology in the IEEE 8- and 25-node test systems demonstrate its applicability and effectiveness. All the numerical validations were performed in MATLAB.

Distribution Systems AC State Estimation via Sparse AMI Data Using Graph Signal Processing

This work establishes and validates a Grid Graph Signal Processing (G-GSP) framework for estimating the state vector of a radial distribution feeder. One of the key insights from GSP is the generalization of Shannon’s sampling theorem for signals defined over the irregular support of a graph, such as the power grid. Using a GSP interpretation of Ohm’s law, we show that the system state can be well approximated with relatively few components that correspond to low-pass Graph Fourier Transform (GFT) frequencies. The target application of this theory is the formulation of a three-phase unbalanced Distribution System State Estimation (DSSE) formulation that recovers the GFT approximation of the system state vector from sparse Advanced Metering Infrastructure (AMI) measurements. To ensure convergence of G-GSP for DSSE, the proposed solution relies on a convex relaxation technique. Furthermore, we propose an optimal sensor placement algorithm for AMI measurements. Numerical results demonstrate the efficacy of the proposed method.

Adjustable robust approach to increase DG hosting capacity in active distribution systems

We propose a robust coordinated dynamic hosting capacity (HC) evaluation method considering active network management (ANM) techniques in three-phase distribution systems. Our proposed method extends the literature in two main ways. Firstly, by applying the adjustable robust counterpart methodology, we allow fast-acting devices, such as inverters, to take live recourse actions in response to demand and generation uncertainties while not making the unrealistic assumption that they have real-time communication with a central entity. Secondly, we propose a linear uncertainty model capable of handling the strong spatial correlation between PV outputs. We show that our proposed linear model, compared to the existing linearised ellipsoidal uncertainty model, significantly reduces the computational burden while introducing a minor error. We validate the optimality and robustness of our proposed methods via numerical tests on the IEEE 37-node unbalanced three-phase distribution system.

Three-Phase Unbalance Improvement for Distribution Systems Based on the Particle Swarm Current Injection Algorithm

The aim of this study is to improve the three-phase unbalanced voltage at the secondary side of a distribution transformer. The proposed method involves compensation sources injecting three different single-phase currents into the connected point of a grid. The computations of optimal single-phase currents are performed using the circuit analysis method and particle swarm optimization algorithm. An unbalanced three-phase power distribution system model is constructed, including a transformer Δ–Δ connection, V–V connection, load balance, load unbalance combination, and three single-phase compensation current sources. The results show that the voltage unbalance rate of the electricity user side is improved to less than 1%, and the three-phase total compensation apparent power is approximately 0 VA. In the future, the application of the model as an auxiliary service could be achieved by adding an energy storage system.

Optimal allocation of DGs for non-linear objective function modeling in a three-phase unbalanced distribution system using crow search optimization algorithm

The rapid growth of consumers, industrial development and global economic expansion inspired a lot of improvement in reliable power supplies. The integration of distributed generations (DGs) in unbalanced radial distribution systems (RDS) is one of the solutions to handle all above issues. This paper proposes an optimal planning for unbalanced RDS integrated with DGs. The objective function is designed as a non-linear function, which is framed as the combination of active power losses, voltage deviation and load balancing index of RDS. The Crow Search Optimization Algorithm (CSOA) is used for identifying the optimal size, location, power factor for DG and distribution system phase in order to reduce the proposed objective function by satisfying all system constraints. The proposed model is tested on 33-bus unbalanced RDS. There is reduction of annual cost of energy loss (ACEL) due to proper integration of DG. The obtained results prove that the proposed CSOA can effectively solve the non-linear objective function and can help to obtain optimized results for unbalanced RDS.

Two-port network model of wind turbine generator for three-phase unbalanced distribution system load flow analysis

An unbalanced electric power system can occur due to unequal lines and/or loads in each phase of the system. For a system with a high level of unbalance, the assumption that the system is balanced becomes incorrect. For a case like this, the system must be analyzed using three-phase approaches because single-phase analysis techniques can no longer be used to obtain accurate results. Therefore, the development of three-phase models of all components of the electric power system, including the wind turbine generating system (WTGS), is very important so that the system can properly be analyzed and its performance can correctly be evaluated. This paper proposes a method to incorporate WTGS in a three-phase load flow analysis of an unbalanced distribution system. The proposed method is based on the previously published single-phase two-port network model. In this paper, the single-phase two-port network model is modified and extended so that it can be applied to unbalanced systems. The model is then included in the three-phase distribution system load flow (DSLF) analysis. The proposed method is validated using 12-node and 19-node distribution systems.

State and Topology Estimation for Unobservable Distribution Systems using Deep Neural Networks.

Time-synchronized state estimation for reconfigurable distribution networks is challenging because of limited real-time observability. This paper addresses this challenge by formulating a deep learning (DL)-based approach for topology identification (TI) and unbalanced three-phase distribution system state estimation (DSSE). Two deep neural networks (DNNs) are trained for time-synchronized DNN-based TI and DSSE, respectively, for systems that are incompletely observed by synchrophasor measurement devices (SMDs) in real-time. A data-driven approach for judicious SMD placement to facilitate reliable TI and DSSE is also provided. Robustness of the proposed methodology is demonstrated by considering non-Gaussian noise in the SMD measurements. A comparison of the DNN-based DSSE with more conventional approaches indicates that the DL-based approach gives better accuracy with smaller number of SMDs.

Tie‐line planning for resilience enhancement in unbalanced distribution networks

Over the past decades, there has been a dramatic increase in the frequency of natural disasters, which are the leading causes of large-scale power outages. This paper, therefore, assesses the significance and role of optimal tie-line construction in improving the service restoration performance of unbalanced power distribution systems in the aftermath of high-impact low-probability incidents. In doing so, a restoration process aware stochastic mixed-integer linear programming model is developed to find the optimal locations for new tie-line construction in unbalanced three-phase distribution systems. In particular, the restoration process of distribution systems, including the fault isolation and system reconfiguration, is contemplated to place tie-lines in the most proper locations to enhance the manoeuvring capability of distribution systems and reducing the customer interruption time. Furthermore, the model is a stochastic one wherein uncertainty associated with potential damages alongside demand uncertainty is captured via a set of likely scenarios. To validate the effectiveness of the proposed stochastic mixed-integer linear programming model, it is tested and verified on the IEEE 13- and 123-bus test systems.

Distributed Voltage Control for Three-Phase Unbalanced Distribution Systems With DERs and Practical Constraints

Traditional voltage control schemes in the power distribution system are typically centralized or local. Local control often gives nonoptimal solutions utilizing local measurements and lacks global coordination. Centralized control gives optimal solutions but lack scalability and robustness, especially with the increasing number of distributed energy resources (DERs). Recent advancement in power electronics with improved automation and digitalization in power grid control provides an opportunity to enable distributed optimization approaches for voltage control. This article proposes an optimal distributed voltage control algorithm based on augmented Lagrangian multiplier theory and primal-dual gradients for three-phase unbalanced distribution systems. In the developed algorithm, each node uses its own local measurements and neighboring nodes measurements to compute optimal reactive power setpoints, while meeting reactive power injection constraints for DERs. The proposed algorithm also considers practical constraints, such as communication delays, measurement noises, modeling errors, limited control devices, and infrequent switching required for field implementation. The proposed algorithm is validated for voltage control in an unbalanced, three-phase IEEE 123-bus feeder modeled in OpenDSS to demonstrate superior performance.