Three-phase Unbalanced Distribution Networks Research Articles

Power Losses Reduction in Three-Phase Unbalanced Distribution Networks Using a Convex Optimization Model in the Complex Domain

This article presents a solution methodology to minimize power losses in three-phase unbalanced distribution networks. This approach involved an efficient complex-domain model that is categorized under mixed-integer convex optimization. The methodology employed consisted of efficient load rotation at each constant power node via a three-phase rotation matrix that allows defining each load connection to minimize the expected power imbalance at the terminals of the substation, as well as the total grid power losses, and improve voltage profile performance at each system phase. The load imbalance, expressed as a percentage, can be defined as a function of the active, reactive, or apparent power. In addition, considering the complex-domain representation of three-phase electrical networks under steady-state conditions, a mixed-integer convex model was formulated to reduce the power imbalances. With the purpose of determining the initial and final power losses of these distribution systems, the successive approximations method was employed to address the three-phase power flow problem. As a result, numerical validations in the IEEE 25-bus system and a 35-node three-phase feeder showed that the final active power losses vary depending on the objective function analyzed. Therefore, for the test feeders studied, it is necessary to evaluate each objective function, with the aim of finding the one that yields the best numerical results. Power losses reductions of about 3.8056 % and 6.8652 % were obtained for both test feeders via the proposed optimization methodology. All numerical validations were performed in the Julia programming environment, using the JuMP optimization tool and the HiGHS solver.

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.

Modified Matrix Completion-Based Detection of Stealthy Data Manipulation Attacks in Low Observable Distribution Systems

A composite detection technique against stealthy data manipulations is developed in this paper for distribution networks that are low observable. Attack detection strategies typically rely on state estimation which becomes challenging when limited measurements are available. In this paper, a modified matrix completion approach provides estimates of the system state and its error variances for the locations in the network where measurements are unavailable. Using the error statistics and their corresponding state estimates, bad data detection can be carried out using the chi-squared test. The proposed approach employs a moving target defence strategy (MTD) where the network parameters are perturbed through distributed flexible AC transmission system (D-FACTS) devices such that stealthy data manipulation attacks can be exposed in the form of bad data. Thus, the bad data detection approach developed in this paper can detect stealthy attacks using the MTD strategy. This technique is implemented on 37-bus and 123-bus three-phase unbalanced distribution networks to demonstrate the attack detection accuracy even for a low observable system.

Reactive Power Optimization of Distributed PVConnected to Three-Phase Unbalanced Distribution Network

This study presents an approach to optimize reactive power in three-phase unbalanced distribution networks using an enhanced particle swarm optimization algorithm. The study analyzes the characteristics of reactive power optimization and establishes a reactive power optimization model for the distribution network using a multi-objective optimization method. Inequality constraints are utilized to determine the reactive power optimization model constraints for three-phase unbalanced distribution networks. The study applies an improved particle swarm optimization algorithm to solve the reactive power model and achieve optimization. The experimental results demonstrate that the proposed method considerably reduces the unused power of the optimized power network within 0-500ms and achieves a maximum reactive power optimization time of fewer than 8.2s, thereby enhancing the efficiency of reactive power optimization in distribution networks.

RETRACTED ARTICLE: Distributed energy management for networked microgrids in a three-phase unbalanced distribution network

RETRACTED ARTICLE: Distributed energy management for networked microgrids in a three-phase unbalanced distribution network

Flexible and curtailable resource activation in three-phase unbalanced distribution networks

The need for flexibility and curtailable resources is crucial for ensuring the healthy operation of future distribution networks (DN). In this work, we propose a network-state driven framework that distribution system operators (DSOs) can utilize for activating flexible and curtailable resources for alleviating network voltage and thermal issues, while accounting for network voltage and current imbalances. This approach assumes the availability of dynamic network state information and uses nodal sensitivities for calculating a flexibility activation signal (FAS). The signal design is motivated by volt-Var and volt-watt inverter control, and thus bounded. The FAS also considers network voltage and current imbalances and incentivizes activation of active and reactive power flexibilities for reducing imbalance in addition to mitigating voltage and thermal imbalances in a three-phase unbalanced distribution network. The FAS design resembles optimal power flow duals, often used as locational marginal prices. The gains associated with the imbalance component of the objective function of three-phase unbalanced resource activation (TPU-RA) is performed using Pareto optimality. A numerical case study is presented showing the efficacy of the proposed framework in avoiding network issues while reducing voltage unbalance factor by more than 80%. Further, DN’s flexibility needs are quantified for location and time of day.

Distribution Locational Marginal Pricing (DLMP) for Unbalanced Three-Phase Networks

This paper applies the principles of distribution locational marginal pricing (DLMP) to unbalanced three-phase distribution networks. We first propose a linear model for AC optimal power flow derived through a series of approximation and reformulation techniques. Then a scenario-generation algorithm is proposed to properly model the uncertain parameters in the linear model. Through a proposed No U-Turn sampler (NUTS) based algorithm, probability density functions (PDFs) of DLMPs are calculated. These PDFs provide statistical information about the locational and temporal price risks. By means of applications to two IEEE unbalanced test networks, the numerical results show promising performance for the proposed linear model and the NUTS-based algorithm in creating PDFs of DLMPs. DLMP price densities will be increasingly useful as distribution system operators seek flexible, low risk solutions from embedded generators and aggregators of distributed energy resources.

Flexible and curtailable resource activation in three-phase unbalanced distribution networks

The need for flexibility and curtailable resources is crucial for ensuring the healthy operation of future distribution networks (DN). In this work, we propose a network-state driven framework that distribution system operators (DSOs) can utilize for activating flexible and curtailable resources for alleviating network voltage and thermal issues, while accounting for network voltage and current imbalances. This approach assumes the availability of dynamic network state information and uses nodal sensitivities for calculating a flexibility activation signal (FAS). The signal design is motivated by volt-Var and volt-watt inverter control, and thus bounded. The FAS also considers network voltage and current imbalances and incentivizes activation of active and reactive power flexibilities for reducing imbalance in addition to mitigating voltage and thermal imbalances in a three-phase unbalanced distribution network. The FAS design resembles optimal power flow duals, often used as locational marginal prices. The gains associated with the imbalance component of the objective function of three-phase unbalanced resource activation (TPU-RA) is performed using Pareto optimality. A numerical case study is presented showing the efficacy of the proposed framework in avoiding network issues while reducing voltage unbalance factor by more than 80\%. Further, DN's flexibility needs are quantified for location and time of day.

Neumann Series Based Voltage Sensitivity Analysis for Three Phase Distribution System

In this letter, a simplified voltage sensitivity analysis technique that can provide accurate estimates of voltage change across the network for a given change in bus power injections in a three-phase unbalanced distribution network is proposed. This technique is derived from the first-order approximation of the Neumann series, which allows maintaining the accuracy of the solution while the computational effort is reduced. The proposed technique is tested on a 559-bus unbalanced distribution system with multiple distributed generation resources. The results show that the average error in the voltage estimates with the proposed method is not more than 0.3% with the execution time of similar order relative to the state-of-the-art sensitivity analysis methods.

Robust three-phase state estimation for PV-Integrated unbalanced distribution systems

As the increasing penetration of renewable energy brings uncertainty and fluctuation into the operating conditions of distribution systems, distribution system state estimation (DSSE) needs to track not only the state of the network, but also the state of the renewable energy sources robustly and under unbalanced conditions. Aiming at delivering comprehensive situational awareness of distribution systems hosting solar photovoltaic (PV) power plants, this paper proposed a joint state estimation model for unbalanced distribution systems integrated with single-phase and three-phase PV power plants, as well as an extended weighted least absolute value (EWLAV) estimation algorithm for handling simultaneous measurement errors and control input errors. The proposed state estimation model considers various loss components of single-phase and three-phase power electronic converters, and is applicable to general unbalanced conditions including unbalanced phasing and parameters of distribution lines, PV power plants, and loads. The proposed EWLAV algorithm leverages sparse l1 optimization principles and can identify and suppress the gross errors in both measurements and converter control inputs. Simulation results show that the developed model can reliably reflect the characteristics of three-phase unbalanced distribution networks and PV plants, and the EWLAV algorithm produces unbiased results under simultaneous measurement errors and control input errors, which conventional estimators such as the weighted least squares (WLS) and weighted least absolute values (WLAV) cannot achieve.

A benchmark model for low voltage distribution networks with PV systems and smart inverter control techniques

Unbalanced three-phase low-voltage distribution networks (LVDNs) modeling, optimization, and control are essential for enabling high photovoltaic (PV) penetration levels. Accordingly, a new case study is developed to show the gaps and challenges at different PV penetration levels in LVDNs. In this case study, the aim is to provide a better understanding of LVDNs’ behavior in order to support the development and validation of the models and tools. Therefore, a reduction model is proposed to decrease the simulation time by lowering the number of buses in the IEEE European LV Test Feeder, with a negligible error. In addition, an OpenDSS-Julia interface is developed to demonstrate the effects of different PV penetration levels on the inverters’ behavior, active power curtailment, and voltage level in LVDNs. Results are demonstrated concerning several limitations and challenges in using existing smart inverter control techniques, in terms of the inverters’ behavior, active power curtailment, and the voltage level. These limitations and challenges include over-voltage issues using the constant power factor technique, high active power curtailment using the volt–watt technique, and high current flows in the network assets and poor power factors using the volt–var technique. In addition, state-of-the-art system models have not taken-into-account the modeling of uncertainty effects on the performance of PV modules. Similarly, such models have largely ignored the internal and standby losses in the inverter models. These neglected issues may lead to under- or over-estimation of the impacts of PV systems on LVDNs and inaccurate estimations of the network’s ability to accommodate high PV penetration levels.

Data-driven distributionally robust unbalanced operation of distribution networks with high penetration of photovoltaic generation and electric vehicles

The uncertainties associated with the large-scale integration of electric vehicles (EV) and renewable energy resources introduce new challenges to operating the unbalanced three-phase distribution networks. In this paper, a data-driven distributionally robust optimization framework is proposed for the operation of the distribution network considering the uncertainties associated with the interconnected EV fleets and solar photovoltaic (PV) generation. The proposed framework leverages the column-and-constraint generation (C&CG) approach to minimize the operation cost considering the worst-case probability distributions of PV generation, the available energy in EV fleets, the arrival and departure times of EV fleets as well as their minimum and maximum energy capacities. The proposed approach is applied to the modified three-phase unbalanced IEEE 34-bus and IEEE 123-bus networks. To evaluate the performance of the proposed distributionally robust optimization framework, the results are compared to those procured by solving scenario-based stochastic programming and robust optimization problems. Furthermore, the impact of vehicle-to-grid capability on the operation of the distribution network is investigated and the in-sample and out-of-sample performances of the proposed framework are evaluated.

Three-Phase Unbalanced Distribution Network Dynamic Reconfiguration: A Distributionally Robust Approach

The active distribution network has witnessed an increasing penetration of distributed generation (DG) while the stochasticity and variability arising from DGs also impose significant challenges on system operation. To mightily accommodate the uncertainty of DG, we introduce a distributionally robust chance-constrained dynamic reconfiguration approach for a three-phase unbalanced distribution network. The proposed framework optimizes the switching cost and the expected power supply cost from upstream grid, and stipulates that the chance constraints hold under the worst-case distribution within a novel ambiguity set, which incorporates the Wasserstein distance and the first-order moment. Then we develop tractable and scalable solution methods to tackle the expected objective function and chance constraints. As a result, the proposed model is reduced to a mixed-integer linear programming problem that can readily be implemented. Numerical experiments are carried out on the IEEE 34-bus and 123-bus test systems to demonstrate the effectiveness and efficiency of the suggested approach.

Solar photovoltaic dispatch margins with stochastic unbalanced demand in distribution networks

The increase in the generation capacity of the variable photovoltaic units introduces new challenges to the operation of the unbalanced three-phase distribution networks. In this paper, a two-stage optimization problem is formulated to identify the feasible dispatch margins of photovoltaic generation considering the distribution network operation constraints. The proposed problem is solved using the column-and-constraint generation approach. The distribution network constraints are formulated as second-order cone constraints. The uncertainty in the forecasted demand and maximum photovoltaic generation as well as the unbalanced operation of the distribution network is considered in the proposed approach. The dispatch margins of photovoltaic generation are quantified considering the worst-case realization of demand in the distribution network. The impacts of energy storage and the ramping limits of the dispatchable generation resources on the dispatch margins of photovoltaic generation are addressed in this network. The dispatch margins of photovoltaic generation are quantified in the modified IEEE 13-bus system. It is shown that enforcing the ramping rates for the dispatchable units will increase the lower dispatch margins of photovoltaic generation, and leveraging energy storage increases the difference between the lower and upper photovoltaic dispatch margins.

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.

A Linear AC-OPF Formulation for Unbalanced Distribution Networks

Linear optimal power flow (OPF) formulations are powerful tools applied to a large number of problems in power systems, e.g., economic dispatch, expansion planning, state estimation, congestion management, electricity markets, among others. This article proposes a novel mixed-integer linear programming formulation for the ac-OPF of three-phase unbalanced distribution networks. The model aims to minimize the total energy production cost while guaranteeing the network's voltage and current magnitude operational limits. New approximations of the Euclidean norm, which is present in the calculation of nodal voltage and branch current magnitudes, are introduced by applying a linear transformation of weighted norms and a set of intersecting planes. The accuracy, optimality, feasibility, and scalability of the proposed linearizations are compared with common linear approximations in the literature using two unbalanced distribution test systems. The obtained results show that the proposed formulation is computationally more efficient (almost twice) while being as accurate and more conservative than the benchmarked approaches with maximum errors lower than 0.1%. Thus, its potential application in a variety of distribution systems operation and planning optimization problems is endorsed.

Accurate and Efficient Derivative-Free Three-Phase Power Flow Method for Unbalanced Distribution Networks

The power flow problem in three-phase unbalanced distribution networks is addressed in this research using a derivative-free numerical method based on the upper-triangular matrix. The upper-triangular matrix is obtained from the topological connection among nodes of the network (i.e., through a graph-based method). The main advantage of the proposed three-phase power flow method is the possibility of working with single-, two-, and three-phase loads, including Δ- and Y-connections. The Banach fixed-point theorem for loads with Y-connection helps ensure the convergence of the upper-triangular power flow method based an impedance-like equivalent matrix. Numerical results in three-phase systems with 8, 25, and 37 nodes demonstrate the effectiveness and computational efficiency of the proposed three-phase power flow formulation compared to the classical three-phase backward/forward method and the implementation of the power flow problem in the DigSILENT software. Comparisons with the backward/forward method demonstrate that the proposed approach is 47.01%, 47.98%, and 36.96% faster in terms of processing times by employing the same number of iterations as when evaluated in the 8-, 25-, and 37-bus systems, respectively. An application of the Chu-Beasley genetic algorithm using a leader–follower optimization approach is applied to the phase-balancing problem utilizing the proposed power flow in the follower stage. Numerical results present optimal solutions with processing times lower than 5 s, which confirms its applicability in large-scale optimization problems employing embedding master–slave optimization structures.

An Efficient Backward/Forward Sweep Algorithm for Power Flow Analysis through a Novel Tree-Like Structure for Unbalanced Distribution Networks

The increase of distributed energy resources (DERs) in low voltage (LV) distribution networks requires the ability to perform an accurate power flow analysis (PFA) in unbalanced systems. The characteristics of a well performing power flow algorithm are the production of accurate results, robustness and quick convergence. The current study proposes an improvement to an already used backward-forward sweep (BFS) power flow algorithm for unbalanced three-phase distribution networks. The proposed power flow algorithm can be implemented in large systems producing accurate results in a small amount of time using as little computational resources as possible. In this version of the algorithm, the network is represented in a tree-like structure, instead of an incidence matrix, avoiding the use of redundant computations and the storing of unnecessary data. An implementation of the method was developed in Python programming language and tested for 3 IEEE feeder test cases (the 4 bus feeder, the 13 bus feeder and the European Low Voltage test feeder), ranging from a low (4) to a very high (907) buses number, while including a wide variety of components witnessed in LV distribution networks.

Optimised TSO–DSO interaction in unbalanced networks through frequency‐responsive EV clusters in virtual power plants

The increased penetration of distributed energy resources (DERs) installed in distribution networks via power electronic converters reduces the overall rotating inertia of the power system, causing faster frequency dynamics after a large disturbance. However, these DERs coupled with energy storage systems (ESSs) can be managed to provide valuable grid support functions such as fast frequency response (FFR) and assist the transmission system operator (TSO) in frequency control. In this study, single- and three-phase clusters of electric vehicles (EVs), acting as mobile-ESSs, are grouped together as virtual power plants (VPPs) to centralise the provision of FFR for the distribution system operator (DSO) whilst considering network unbalance. Furthermore, the parameters of the EV clusters within each VPP are optimised to minimise the inertia-weighed maximum frequency deviation following a disturbance whilst adhering to network security and power quality constraints. To this aim, a variant of the metaheuristic optimisation algorithm called improved harmony search is used for the optimisation process. Finally, the merits of the proposed methodology are shown with an illustrative example of an unbalanced three-phase transmission and distribution network modelled in DIgSILENT PowerFactory. The results show that clusters of EVs grouped as VPPs effectively improve frequency support via the TSO–DSO interaction.

Conic Optimization for Electric Vehicle Station Smart Charging With Battery Voltage Constraints

This article proposes a new convex optimization strategy for coordinating electric vehicle charging, which accounts for battery voltage rise and the associated limits on maximum charging power. Optimization strategies for coordinating electric vehicle charging commonly neglect the increase in battery voltage, which occurs as the battery is charged. However, battery voltage rise is an important consideration since it imposes limits on the maximum charging power. This is particularly relevant for dc fast charging, where the maximum charging power may be severely limited, even at the moderate state of charge levels. First, a reduced-order battery circuit model is developed, which retains the nonlinear relationship between the state of charge and maximum charging power. Using this model, limits on the battery output voltage and battery charging power are formulated as the second-order cone constraints. These constraints are integrated with a linearized power flow model for three-phase unbalanced distribution networks. This provides a new multiperiod optimization strategy for electric vehicle smart charging. The resulting optimization is a second-order cone program and, thus, can be solved in polynomial time by standard solvers. A receding horizon implementation allows the charging schedule to be updated online, without requiring prior information about when vehicles will arrive.