Bus Voltage Phase Angles Research Articles

Voltage angle‐based coherency identification in power system

AbstractCoherency detection in power system is vital step for controlled‐islanding. Model‐based slow‐coherency method is the traditional method of detecting coherent areas, which suffers from computational burdens. With the implementation of a wide‐area measurement system (WAMS) in power systems, coherent areas can be detected early after a disturbance that helps to ensure quick separation or remedial action. In this study, a novel Synchrophasor‐based coherency identification method has been proposed with bus voltage angle measurement. A new algorithm is proposed based on the first and second derivatives of the bus voltage phase angle using hierarchical clustering method for coherency detection. The proposed approach leverages hierarchical clustering method and the dynamics reflected in the bus voltage phase angle behavior to achieve automatic and accurate coherency identification. Real time simulation results demonstrate the ability of the proposed algorithm to detect the coherency with three consecutive PMU data from all the system buses and showcase coherent area boundaries. Case studies show robustness of the proposed method and is least affected by the latency in communication network compared to conventional methods. The proposed algorithm has been mathematically formulated and evaluated with different benchmark systems in OPAL‐RT real‐time simulation environment with HYPERSIM platform.

Synchrophasor-Based Out-of-Step Prediction in Large Grids

Out-of-step (OOS) condition is a potential problem in the power system and uncontrolled islanding is one of the severe consequences of out-of-step condition which leads to cascaded tripping of the system. To avoid this undesired cascaded tripping, early prediction of out-of-step condition is essential before losing the synchronization of generators and between the system areas. Controlled islanding is the last emergency action that can be taken by splitting the system intentionally into coherent islands. Conventionally, OOS condition has been detected with impedance-based measurement techniques which have their own limitations. With the implementation of a wide-area measurement system with synchrophasor measurement units (PMUs) in the power system, it is now possible to measure the bus voltage angle at a much faster rate than with SCADA. In this paper, synchrophasor-based bus voltage angle measurement has been used for early prediction of OOS condition in power systems. A new algorithm has been formulated for both generator and tie lines based on the trajectory of first and second derivatives of the bus voltage phase angle for early detection of OOS condition. The prediction speed of proposed method is found to be between 5% and 70% of time to OOS, after disturbance clearance. The proposed algorithm has been mathematically formulated and evaluated with three benchmark systems in real-time simulation environment with an OPAL-RT real-time simulator and a HYPERSIM platform.

A new continuous variables and convex relaxation based formulation for reconfiguration of unbalanced mutually coupled active distribution networks

With the integration of renewable energy-based sources and demand response initiatives, the distribution network has transformed into an active network with bidirectional flows. Network Reconfiguration (NR), i.e., changing network topology, is an alternative to minimize power loss in the distribution network with sectionalizing switches (normally closed) and tie-line switches (normally open). Due to the discrete nature of switches (on or off), the NR formulation is generally a mixed-integer NP-hard nonlinear problem. This paper proposes the first NR formulation for an unbalanced active distribution network with continuous variables to indicate switches’ statuses in NR while considering mutual coupling between line impedances. The obtained solution provides the optimal real power and voltage magnitude settings of dispatchable Distributed Generators (DGs) and network topology for the minimum active power loss in the network. The proposed formulation considers all practical aspects of the distribution network, i.e., mutual coupling, load unbalances, angular unbalance, and the absence of one or two phases in a line section, which, otherwise, have not been considered together in previous formulations. Further, a novel convex relaxation for bus voltage phase angles avoids explicit voltage recovery. Compared to prior formulations, numeric results for unbalanced radial distribution networks showcase the proposed formulation’s effectiveness.

The Efficacy of Multi-Period Long-Term Power Transmission Network Expansion Model with Penetration of Renewable Sources

The electrical energy demand increase does evolve rapidly due to several socioeconomic factors such as industrialisation, population growth, urbanisation and, of course, the evolution of modern technologies in this 4th industrial revolution era. Such a rapid increase in energy demand introduces a huge challenge into the power system, which has paved way for network operators to seek alternative energy resources other than the conventional fossil fuel system. Hence, the penetration of renewable energy into the electricity supply mix has evolved rapidly in the past three decades. However, the grid system has to be well planned ahead to accommodate such an increase in energy demand in the long run. Transmission Network Expansion Planning (TNEP) is a well ordered and profitable expansion of power facilities that meets the expected electric energy demand with an allowable degree of reliability. This paper proposes a DC TNEP model that minimises the capital costs of additional transmission lines, network reinforcements, generator operation costs and the costs of renewable energy penetration, while satisfying the increase in demand. The problem is formulated as a mixed integer linear programming (MILP) problem. The developed model was tested in several IEEE test systems in multi-period scenarios. We also carried out a detailed derivation of the new non-negative variables in terms of the power flow magnitudes, the bus voltage phase angles and the lines’ phase angles for proper mixed integer variable decomposition techniques. Moreover, we intend to provide additional recommendations in terms of in which particular year (within a 20 year planning period) can the network operators install new line(s), new corridor(s) and/or additional generation capacity to the respective existing power networks. This is achieved by running incremental period simulations from the base year through the planning horizon. The results show the efficacy of the developed model in solving the TNEP problem with a reduced and acceptable computation time, even for large power grid system.

Recycling Newton–Krylov algorithm for efficient solution of large scale power systems

Power flow calculations are crucial for the study of power systems, as they can be used to calculate bus voltage magnitudes and phase angles, as well as active and reactive power flows on lines. In this paper, a new approach, the Recycling Newton–Krylov (ReNK) algorithm, is proposed to solve the linear systems of equations in Newton–Raphson iterations. The proposed method uses the Generalized Conjugate Residuals with inner orthogonalization and deflated restarting (GCRO-DR) method within the Newton–Raphson algorithm and reuses the Krylov subspace information generated in previous Newton runs. We evaluate the performance of the proposed method over the traditional direct solver (LU) and iterative solvers (Generalized Minimal Residual Method (GMRES), the Biconjugate Gradient Stabilized Method (Bi-CGSTAB) and Quasi-Minimal Residual Method (QMR)) as the inner linear solver of the Newton–Raphson method. We use different test systems with a number of busses ranging from 300 to 70000 and compare the number of iterations of the inner linear solver (for iterative solvers) and the CPU times (for both direct and iterative solvers). We also test the performance of the ReNK algorithm for contingency analysis and for different load conditions to simulate optimization problems and observe possible performance gains.

Robust Topology Detection of Distribution Network

This work proposes an M-estimator of scatter matrix for data-driven topology detection in Active Distribution Networks (ADNs) using bus voltage phase angle measurements available from Micro Phasor Measurement Units ( $\mu$ PMUs). The proposed method is robust to the presence of bad measurements and non-Gaussian $\mu$ PMU noise.

Estimation of Voltage Dependent Load Models Through Power and Frequency Measurements

This letter proposes an optimization-free technique to estimate the parameters of voltage dependent loads through the measurements of only the load active and reactive power consumption and frequency deviations at the load bus. The technique is shown to be a relevant consequence of the dependency of power flow equations on the bus voltage phase angles. The WSCC 9-bus and the IEEE 14-bus systems serve to illustrate the proposed technique.

Robust Data Filtering for Estimating Electromechanical Modes of Oscillation via the Multichannel Prony Method

This paper develops a robust estimator of correlation as a data preprocessing stage to the Prony method that is able to suppress white impulsive noise. The method consists of the following steps. First, the bus voltage magnitudes and phase angles are combined to build a set of complex-valued measurements. Second, the outliers of the complex-valued data samples, which are induced by impulsive noise, are identified and suppressed using the iteratively reweighted phase–phase correlator; the latter is a robust estimator of correlation for complex-valued Gaussian processes, which has been extended here to handle outliers in both magnitude and phase angle of voltage phasor measurements. Finally, the classical Prony method is applied on the robustly estimated voltage phase angles. The good performance of the proposed method is demonstrated through simulations carried out on the two-area four-machine system, on the simplified WECC 179-bus system, as well as on real PMU data. Simulation results show that the method is very fast to compute and is compatible with real-time application requirements.

Distributed Control of Inverter-Interfaced Microgrids With Bounded Transient Line Currents

Distributed generators (DGs) in a microgrid are tightly coupled through power lines, whose dynamics should not be ignored. If not properly handled, large transient line currents may trigger false protection even under normal operating conditions. Droop-based control adjustments also unnecessarily increase frequency and voltage oscillations. Targeting at these problems, this paper presents a distributed control solution for inverter-interfaced microgrids. The objective of primary control is to realize the desired regulations of bus voltages and frequency as well as suppression of transient line currents. The objective of secondary control is to maintain fair load sharing. At secondary control level, a consensus algorithm is introduced to calculate the references for phase angles of bus voltages based on fair load sharing and dc power flow. At primary control level, a feedback linearization based control algorithm with dynamic control bounds is designed for voltage regulation and transient line current suppression. In addition to a common reference frame, the subsystem controllers only require measurements of local and neighboring subsystems. The effectiveness of the proposed control solution is demonstrated through simulations based on both simplified and detailed models.

Frequency Control of Distributed Energy Resources in Distribution Networks

Abstract This paper studies the contribution of distributed energy resources (DERs) installed in distribution systems to the frequency regulation of transmission systems. To this aim, the paper compares three strategies to retrieve the input signal of the DER frequency regulators, namely (i) measuring the frequency at the point of common coupling with the transmission system; (ii) measuring locally the frequency signals at every bus with regulation of the distribution system; and (iii) computing the average of such signals. The impact of measurement issues caused by the computation of the numerical derivative of the bus voltage phase angle, as well as of noise, delays and loss of information are also thoroughly analyzed in the case study.

Hybrid fuzzy charged system search algorithm based state estimation in distribution networks

This paper proposes a new hybrid charged system search (CSS) algorithm based state estimation in radial distribution networks in fuzzy framework. The objective of the optimization problem is to minimize the weighted square of the difference between the measured and the estimated quantity. The proposed method of state estimation considers bus voltage magnitude and phase angle as state variable along with some equality and inequality constraints for state estimation in distribution networks. A rule based fuzzy inference system has been designed to control the parameters of the CSS algorithm to achieve better balance between the exploration and exploitation capability of the algorithm. The efficiency of the proposed fuzzy adaptive charged system search (FACSS) algorithm has been tested on standard IEEE 33-bus system and Indian 85-bus practical radial distribution system. The obtained results have been compared with the conventional CSS algorithm, weighted least square (WLS) algorithm and particle swarm optimization (PSO) for feasibility of the algorithm.

A Novel Out-of-step Splitting Protection Based on the Wide Area Information

Using the wide area information obtained from phasor measurement units, a novel out-of-step splitting protection is proposed. In order to analyze the different characteristic of bus voltage phase angles, a new coordinate system is introduced. In this coordinate system, all of the converted bus voltage phasor ends are stable, while only the common beginning is moving during the out-of-step oscillation process. Its conformal property guarantees the voltage phase angle differences in the old and new coordinate systems are equal. After the coordinate transformation, the formula about the converted power network neutral point trajectory is derived. The intersection of the trajectory and the broken line which connects the bus voltage phasor ends is the out-of-step center. All of the buses are divided into two parts by the out-of-step center. Through a series of proofs, it is found that the absolute values of the bus voltage phase angles in one part are always smaller than 110°, while the voltage phase angles of the rest buses vary from −180° to 180°. These different characteristics could be used to group these buses and locate the out-of-step center. It is very simple because only the voltage phase angle information is applied, and there are no complex mathematical operations. Finally, a novel out-of-step splitting protection scheme is given. The digital simulation results prove the proposed scheme is efficient.

Real‐time calculation of power system bus voltage using a hybrid approach combining the Newton–Raphson method and dynamic programming

The calculation of the magnitudes and phase angles of the bus voltage is a challenging task in real‐time applications for power systems. Voltage profile, which denotes the present conditions of a power system, is determined by executing the traditional AC power flow program or by searching the supervisory control and data acquisition system. The AC power flow program is not suitable for several real‐time applications, such as contingency analysis and security control calculations, because of its complexity and convergence problems. Fast computation is the major concern in such applications. In this paper, a new method based on sensitivity factors, referred to as Jacobian‐based distribution factors (JBDFs), is proposed for calculating the magnitudes and phase angles of bus voltages. This method requires setting up JBDFs and deriving optimal solution paths of bus voltage for non‐swing buses through dynamic programming under base‐case loading conditions. Under real‐time conditions, the proposed method initially calculates real and reactive power line flows via JBDFs, and then computes the voltage magnitudes and phase angles of non‐swing buses through the derived optimal solution paths. The excellence of the proposed hybrid calculation method is verified by IEEE test systems. Simulation results demonstrate that the proposed method exhibits fast computation and high accuracy. Thus, the method is suitable for real‐time applications. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A novel efficient model for the power flow analysis of power systems

The overall system status calculated by power flow analysis is the most basic information used for all decisions taken by power system operators and planners. While conventional AC power flow solutions are computationally tractable, approximate DC models are employed in many applications, such as optimal power flow studies and unit commitment problems, mainly due to the linear nature of DC models. These models do not provide any information on the reactive power and voltage magnitude quantities and occasionally inaccurate results of the active power values. This paper presents an efficient power flow approach compromising both the conflicting aspects of speed and accuracy. The proposed model adopts bus voltage magnitudes and phase angles as state variables. Given the nonlinear nature of transmission system losses, an iterative method for solving the problem is proposed. Simulation results reveal that the proposed method outperforms conventional methods from an execution time viewpoint, while preserving acceptable accuracy. Different system conditions are also investigated to reveal the robustness and reliability of the proposed model.

Solution to security constrained environmental/economic pumped-storage hydraulic unit scheduling problem by modified subgradient algorithm based on feasible values and pseudo water price

ObjectiveA security constrained environmental/economic power dispatch problem for a lossy electric power system area including a pumped-storage (p-s) hydraulic unit is formulated. An iterative solution method, which is proposed by us and based on modified subgradient algorithm operating on feasible values (F-MSG) and pseudo water price for the p-s hydraulic unit, is used to solve it. MethodThe cost function is made up of weighted sum of total fuel cost and total emission cost of the thermal units in an operation cycle. In the proposed solution method, the F-MSG algorithm is used to solve the dispatch problem of each subinterval, while the pseudo water price is employed to adjust the net amount of water spent by the p-s hydraulic unit during the considered operation period. Since all equality and inequality constraints in our nonlinear optimization model are functions of bus voltage magnitudes and phase angles, the off-nominal tap settings of tap changing transformers and susceptances values of svar systems, they are taken as independent variables. Load flow equations are added into the model as equality constraints. Therefore, the actual transmission loss is used in solution of the considered dispatch problem. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints and svar system susceptance value constraints are added into the optimization problem as inequality constraints. Since the F-MSG algorithm requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by a method, which does not add any extra independent variable into the model, before application of the F-MSG algorithm to the optimization problem. Since the method does not add any extra independent variable into the model, the solution time of the optimization problem is reduced further. ResultsThe proposed dispatch technique is demonstrated on an example power system. Pareto optimal solutions for the power system without any p-s unit are calculated first. Later on, the same Pareto optimal solutions for the power system with the p-s unit are recalculated, and the obtained savings in the sum of optimal total fuel cost and total emission cost, due to the employment of the p-s unit, are presented. We also applied the F-MSG method to the dispatch problem with p-s unit directly. We demonstrated that the proposed solution method, where the F-MSG method is employed to solve an interval’s dispatch problem, gives less solution time than the one obtained from the direct application of the F-MSG method to the dispatch problem with the p-s unit although both methods give very close sum of total fuel and emission cost values.

Solution to Security-constrained Non-convex Pumped-Storage Hydraulic Unit Scheduling Problem by Modified Subgradient Algorithm Based on Feasible Values and Pseudo Water Price

A security-constrained power dispatch problem with a non-convex cost function for a lossy electric power system area including a pumped-storage hydraulic unit is formulated. Then, an iterative solution method based on a modified subgradient algorithm operating on feasible values and pseudo water price for the pumped-storage hydraulic unit is used to solve it. In the iterative proposed solution method, the modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the pseudo water price for the pumped-storage hydraulic unit is employed to adjust the net amount of water used by the pumped-storage hydraulic unit during the considered operation period. Since all equality and inequality constraints in the proposed non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles, the off-nominal tap settings, and susceptances values of SVAR systems, they are taken as independent variables. Load flow equations are added into the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints, and SVAR system susceptance value constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method, which does not add any extra independent variable into the model, before its application to the optimization model. Since the method does not add any extra independent variable into the model, the solution time is reduced further. The proposed dispatch technique is tested on a sample power system that has 12 buses with 5 thermal units and a pumped-storage hydraulic unit. Optimal total cost value for the power system without any pumped-storage unit is calculated first. Later, the same optimal total cost value for the power system with the pumped-storage unit is recalculated, and the obtained saving in the optimal total cost value, due to the employment of the pumped-storage unit, is presented. The solution times for the dispatch problems with and without the pumped-storage unit is also presented. At the end, the modified subgradient algorithm based on feasible values is applied directly to the dispatch problem with a pumped-storage unit. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem with the pumped-storage unit.

Optimal Placement of TCSC and SVC Using PSO

Flexible AC Transmission System (FACTS) devices play a vital role in improving the static as well as dynamic performance of the power system. They serve to make the otherwise rigid, conventional AC systems flexible ones as they make it possible to adjust the bus voltage magnitudes, phase angles and line impedances at faster rates in the required directions which will result in the improvement of stability and bus voltage profiles and reduction of real and reactive power losses. In order to tap maximum benefits out of their installation, the location and ratings of the FACTS devices are to be chosen optimally. In this paper a PSO based technique has been presented for solving the problem of device placement and an objective function comprising the weighted sum of the indicators of line loadings, bus voltage deviations and device cost functions has been formulated . The objective function is solved through the application of PSO technique and testing has been done using IEEE 14, 30 and 57 bus systems Keywords: FACTS devices, Thyristor Controlled Series Capacitor (TCSC), Static Var Compensator (SVC), Particle Swarm Optimization (PSO), Security enhancement.

A solution to security constrained non-convex economic dispatch problem by modified subgradient algorithm based on feasible values

A security constrained power dispatch problem with non-convex total cost rate function for a lossy electric power system is formulated. Then, an iterative solution method proposed by us and based on modified subgradient algorithm operating on feasible values (F-MSG) is used to solve it.Since all equality and inequality constraints in our nonlinear optimization model are functions of bus voltage magnitudes and phase angles, off-nominal tap settings and susceptance values of svar systems, they are taken as independent variables. Load flow equations are added to the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints and svar system susceptance value constraints are added into the optimization problem as inequality constraints. Since F-MSG algorithm requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method, which does not add any extra independent variable into the model and reducing the solution time because of it, before application of it to the optimization model.The proposed technique is tested on IEEE 30-bus and IEEE 57 bus test systems. The minimum total cost rates and the solution times obtained from F-MSG algorithm and from the other techniques are compared, and the outperformance of the F-MSG algorithm with respect to the other methods in each test system is demonstrated.

A Security-constrained Economic Power Dispatch Technique Using Modified Subgradient Algorithm Based on Feasible Values and Pseudo Scaling Factor for a Power System Area Including Limited Energy Supply Thermal Units

A security-constrained power dispatch problem with non-convex cost function for a lossy electric power system area including limited energy supply thermal units fueled under a take-or-pay agreement is formulated. Then, an iterative solution method based on a modified subgradient algorithm based on feasible values and common pseudo scaling factor for limited energy supply thermal units is proposed and used to solve the problem. In the iterative proposed solution method, a modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the common pseudo scaling factor for limited energy supply thermal units is employed to adjust the amount of fuel consumed by the limited energy supply units during the considered operation period. Since all equality and inequality constraints in the non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles (the off-nominal tap settings), they are taken as independent variables except for those belonging to the reference bus. Load flow equations are added to the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, and off-nominal tap setting constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method given in this article before application of the modified subgradient algorithm based on feasible values to the optimization model. The proposed technique is tested on an IEEE 30-bus test system with a non-convex total cost rate curve. A portion of the dispatch problem considered in this article, where only a loading of the system in a subinterval is considered, was also solved by some other recently reported dispatch techniques. First, minimum total fuel cost rates and solution times obtained from the modified subgradient algorithm based on feasible values versus other techniques are compared, and the outperformance of the modified subgradient algorithm based on feasible values in terms of both total cost rate and solution time is demonstrated. After that, the optimal total fuel cost value and solution time of the system for the operation period for the cases where the fuel constraint is ignored and not ignored are presented, and it is shown that adding the fuel constraint into the dispatch problem further decreases the optimal total fuel cost for the considered dispatch problem. Finally, the modified subgradient algorithm based on feasible values was applied to the fuel constrained dispatch problem directly. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem.

Design of State Estimator Measurements Based on Condition Indicators

In contemporary power systems, consideration may be given to added measurements for the purpose of improving the performance of an existing state estimator. The considered addition may be a measurement of a state (e.g., bus voltage phase angle or bus voltage magnitude). In this article, the condition number and other algebraic indicators of state estimator performance are studied with regard to added state measurements in an existing power system state estimator. The concept of a condition indicator is introduced, and this includes a study of certain elements of the singular spectrum of the state estimator process matrix. Eigenvalue sensitivity analysis is used to improve the condition indicators. Examples are shown to illustrate the concept. The analysis also shows that the concept of using condition indicators for measurement placement is consistent with redundancy analysis. The purpose of a study of condition indicators is to supplement measurement placement algorithms.