Power Flow Jacobian Matrix Research Articles

Cross-variance analysis to online estimate power flow Jacobian matrix using limited PMU data

This paper proposes a data-driven approach to estimate the power flow Jacobian matrix online with only small-scale data set collected by phasor measurement unit (PMU). For data in limited amount, one of the greatest challenges to estimate the Jacobian matrix is how to quantify and treat the interference of multiple uncertainties, which are analytically intractable by traditional approaches. To tackle these uncertainties, in this work, through analyzing the cross-variance matrix between voltage measurements and load data, the joint density probability of uncertainties is modeled in matrix-level. Furthermore, a random matrix theory (RMT) based approach is proposed to shrink the singular values of the cross-variance matrix to alleviate the unfavorable influence of uncertainties. This approach is based on the property that the singular values of a random matrix are asymptotically converged to a deterministic distribution. Numerous cases prove that the proposed method is capable of obtaining more precise estimation results even with limited PMU data. Besides, this estimation would supply extra information assistant for power system monitoring, such as voltage stability assessment (VSA) and topology change detection (TCD).

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.

Analysis of Network Impacts of Frequency Containment Provided by Domestic-Scale Devices Using Matrix Factorization

This paper studies the distribution network impacts of frequency containment services derived from domestic-scale devices. Risk metrics considering the likelihood and severity of violations are proposed, given uncertainty in the location of these devices. A novel linearization approach is proposed to enable detailed simulations of large-scale networks, capturing MV–LV coupling in European-style networks of over 100 000 nodes. The approach combines a novel factorization of the power flow Jacobian matrix and an efficient linearization update step. The first of these innovations improves the scalability and practicality of the linearization, reducing the memory and computational requirements by as much as twenty times, whilst the latter reduces the median linearization error by at least 50% in all networks studied. It is demonstrated that rapid voltage change (RVC) and overvoltage constraints could limit the uptake of these devices to just a fraction of one percent of customers if the response is not managed. Sensitivity analysis demonstrates that both the likelihood and severity of constraint violations increases rapidly with the size of the devices used for frequency response.

Alleviating Fractal and Ill-Conditioning Problems of the AC Power Flow Using a Polynomial Form

Power system resilience assessment, and mitigation, which deals with high-impact, low probability events, is gaining a great deal of attention among power researchers. To perform such an analysis, the availability of tools able to analyze modern power grids under normal and extreme disturbances is imperative. One such tool is the AC power flow algorithm. The latter should exhibit a numerical stability and an excellent convergence property under a broad range of operating conditions of a power system. One major difficulty that needs to be overcome stems from the nonlinearity and non-convexity of the AC power flow equations. As shown by Thorp and Naqavi, the Newton-Raphson (NR) method suffers from a fractal behavior, and the power flow Jacobian matrix becomes singular at points located on the fractal hypersurface, preventing that algorithm to converge. Furthermore, the Jacobian matrix becomes ill-conditioned for power systems that have relatively short lines or are heavy loaded. To address these problems, we propose a transformation of the AC power flow equations from a sinusoidal form to a polynomial form, which is solved using a modified Newton's method. The new polynomial form, whose order is equal to the number of solutions, encompasses node-based equations, Pythagorean equations, and loop-based equations. Furthermore, our approach is quadratic in three cases. Simulation results performed on several power systems reveal the good performance of the proposed method.

Online Static Voltage Stability Monitoring for Power Systems Using PMU Data

A new online static voltage stability monitoring method for power systems is proposed by using phasor measurement unit (PMU) data in this paper. This approach uses the real-time power, voltage, and phase angle data collected by the PMU to estimate the power flow Jacobian matrix of the system, and then the static voltage stability is monitored via the minimum singular values (MSVs) of the power flow Jacobian matrix. The novelty of the approach lies in the fact that it only utilizes PMU data for implementing online monitoring of the power system static voltage stability, independent of the physical model and its parameters. The application results on the IEEE 57-bus test system verify the effectiveness of the proposed approach.

A Comparison of machine learning regression models for critical bus voltage and load mapping with regards to max reactive power in PV buses

The aim of this paper is to compare voltage and system loading mapping capabilities of a variety of regression algorithms, such as Adaptive Network based Fuzzy Inference System (ANFIS), Artificial Neural Networks (ANN), K-Nearest Neighbors (KNN), Support Vector Regression (SVR), and Decision Tree (DT). A voltage sensitivity matrix is generated from the power flow Jacobian matrix for a loading scenario near the unstable point. Principal Component Analysis (PCA) is used to separate the system, close to the critical point, in order to group the buses into coherent voltage controlling areas. For different reactive power injection scenarios, we have different bus voltages that can be mapped by the aforementioned regression algorithms. The algorithms are trained with limited amounts of data, in order to establish a fair comparison between them. The present work shows that ANFIS and KNN have a better performance in critical voltage and load prediction when compared to the rest. The academic IEEE 14 and 118 bus systems are employed with all its limits considered, so the results may be reproduced.

An Optimal Power-Flow Approach to Improve Power System Voltage Stability Using Demand Response

The increasing penetration of renewables has driven power systems to operate closer to their stability boundaries, increasing the risk of instability. We propose a multiperiod optimal power-flow approach that uses demand responsive loads to improve the steady-state voltage stability, which is measured by the smallest singular value (SSV) of the power-flow Jacobian matrix. In contrast to past work that employs load shedding to improve the stability, our approach improves the SSV by decreasing and increasing individual loads while keeping the total loading constant to avoid the fluctuation of the system frequency. Additionally, an energy payback period maintains the total energy consumption of each load at its nominal value. The objective function balances SSV improvements against generation costs in the energy payback period. We develop an iterative linear programming algorithm using singular value sensitivities to obtain an ac-feasible solution. We demonstrate its performance on two IEEE test systems. The results show that demand response actions can improve the static voltage stability, in some cases, more cost effectively than generation actions. We also compare our algorithm's performance to that of an iterative nonlinear programming algorithm from the literature. We find that our approach is approximately six times faster when applied to the IEEE 9-bus system, allowing us to demonstrate its performance on the IEEE 118-bus system.

Efficient method to compute all the type‐1 low‐voltage power flow solutions in distribution systems

The power flow equations usually have multiple solutions including a high-voltage solution and many low-voltage solutions, among which only the type-1 solutions (where the power flow Jacobian matrix has only one positive real-part eigenvalue) are closely related to the voltage instability phenomenon. This study proposes an efficient method to compute all the type-1 low-voltage power flow solutions in distribution systems. First, the geometric properties of the power flow solution space of distribution systems have been studied. Second, the propositions which can guarantee to locate all the type-1 power flow solutions have been suggested and proved. Finally, the conventional implicit Z-bus method is modified to compute all the type-1 germ solutions, based on the suggested propositions; and then the Newton–Raphson method is utilised to trace all the type-1 low-voltage power flow solution branches which originate from the known type-1 germ solutions. The 7-node, 33-node, and 69-node systems are used to validate and demonstrate the proposed method.

Voltage stability constrained multi‐objective optimisation model for long‐term expansion planning of large‐scale wind farms

This study proposes a new model for long-term planning of large-scale wind farms (WFs) considering voltage stability constraints. The proposed model is a multi-objective optimisation problem which aims to maximise the profit of WF investor from wind energy procurement and to minimise the overall power generation cost. By performing modal analysis on the reduced power flow Jacobian matrix, proper locations for installation of WFs are determined. Moreover, the impact of voltage stability constraints on the capacity of WFs is investigated. In comparison with other studies, the main contributions of this study are: (i) to propose a new methodology for determination of the best locations for WFs, (ii) to study the impact of voltage stability as an important security constraint in long-term planning of wind energy. The proposed voltage stability constrained multi-objective wind power planning (VSC-MOWPP) model is solved via ɛ -constraint method in General Algebraic Modelling System optimisation package environment. The compromise solution of Pareto optimal set is selected using fuzzy satisfying approach. The proposed VSC-MOWPP model is examined on the IEEE 39-bus standard test system. The numerical investigations substantiate the suitability of the proposed model for multi-year planning of WFs at the presence of long-term voltage stability constraints.

QV interaction evaluation and pilot voltage‐reactive power coupling area partitioning in bulk power systems

This study presents a novel methodology to evaluate the QV interactions among buses and to partition the pilot voltage-reactive power coupling areas (VRPCAs) using relative gain (RG). According to the concept of a multi-input multi-output system, the QV coupling RG is first calculated based on the QV matrix, which is extracted from power flow Jacobian matrix, to evaluate the QV interactions among different buses and then to determine the VRPCAs. The voltage stability critical buses are first identified through a modified loading margin. For each critical bus, the other buses that have strong QV coupling are detected via the cross RG and are clustered into a VRPCA piloted by the corresponding critical bus. New England 39-bus system and Polish power system are used to test the performance of the proposed approach. Simulation results verify the effectiveness of the proposed approach in evaluating the QV interactions and partitioning the VRPCAs.

Measurement-Based Estimation of the Power Flow Jacobian Matrix

In this paper, we propose a measurement-based method to compute the power flow Jacobian matrix, from which we can infer pertinent information about the system topology in near real-time. A salient feature of our approach is that it readily adapts to changes in system operating point and topology; this is desirable as it provides power system operators with a way to update, as the system evolves, the models used in many reliability analysis tools. The method uses high-speed synchronized voltage and current phasor data collected from phasor measurement units to estimate entries of the Jacobian matrix through linear total least-squares (TLS) estimation. In addition to centralized TLS-based algorithms, we provide distributed alternatives aimed at reducing computational burden. Through numerical case studies, we illustrate the effectiveness of our proposed Jacobian-matrix estimation approach as compared with the conventional model-based one.

Development of the Decoupled Discreet-Time Jacobian Eigenvalue Approximation for Situational Awareness Utilizing Open PDC

With the increased number of PMUs in the power grid, effective high speed, realtime methods to ascertain relevant data for situational awareness are needed. Several techniques have used data from PMUs in conjunction with state estimation to assess system stability and event detection. However, these techniques require system topology and a large computational time. This paper presents a novel approach that uses real-time PMU data streams without the need of system connectivity or additional state estimation. The new development is based on the approximation of the eigenvalues related to the decoupled discreet-time power flow Jacobian matrix using direct openPDC data in real-time. Results are compared with other methods, such as Prony’s method, which can be too slow to handle big data. The newly developed Discreet-Time Jacobian Eigenvalue Approximation (DDJEA) method not only proves its accuracy, but also shows its effectiveness with minimal computational time: an essential element when considering situational awareness.

A Power Flow Method Using a New Bus Type for Computing Steady-State Voltage Stability Margins

In steady-state voltage stability analysis, it is well-known that as the load is increased toward the maximum loading condition, the conventional Newton-Raphson power flow Jacobian matrix becomes increasingly ill-conditioned. As a result, the power flow fails to converge before reaching the maximum loading condition. To circumvent this singularity problem, continuation power flow methods have been developed. In these methods, the size of the Jacobian matrix is increased by one, and the Jacobian matrix becomes non-singular with a suitable choice of the continuation parameter. In this paper, we propose a new method to directly eliminate the singularity by reformulating the power flow problem. The central idea is to introduce an AQ bus in which the bus angle and the reactive power consumption of a load bus are specified. For steady-state voltage stability analysis, the voltage angle at the load bus can be varied to control power transfer to the load, rather than specifying the load power itself. For an AQ bus, the power flow formulation consists of only the reactive power equation, thus reducing the size of the Jacobian matrix by one. This reduced Jacobian matrix is nonsingular at the critical voltage point. We illustrate the method and its application to steady-state voltage stability using two example systems.

Determination of Minimum Singular Value Indices at Voltage Collapse by Means of Neural Network

The minimum singular value of the power flow jacobian matrix has been used as a static voltage stability index representing system security in power flow based system models by the scalar magnitude of a voltage stability index may be a very difficult task. VSIs at voltage collapse points are difficult to predict when the reactive power generation limits are taken into account. Here a fast method to calculate the minimum singular value and the corresponding left and right singular vectors is presented. The main advantages of the developed algorithm are the small amount of computation time needed. In order to determine voltage collapse points for different patterns of load and generation increase, an optimal power flow approach for solving the maximum loading problem was used with these points the MSV is calculated and used for training and testing the NNs.

Distributed Online Monitoring of Quasi-Static Voltage Collapse in Multi-Area Power Systems

This paper presents a novel performance index for distributed monitoring of quasi-static voltage collapse in multi-area interconnected power systems. Based on the sensitivity of the smallest singular value of the power flow Jacobian matrix with respect to the regional load vector, the performance index applies overlapping decomposition in order to monitor the entire interconnected system with minimal area-level information exchange. Theoretical justification and practical implementation of distributed voltage stability monitoring are presented. By comparing the value of the performance index with a numerically robust threshold at each administrative control area, each area will be able to monitor quasi-static voltage instability at the entire interconnection level. Numerical simulations in the IEEE 300-bus system illustrate the effectiveness of the proposed performance index and the threshold.

LINEAR INDEX DEFINITION TO EVALUATE THE RISK OF VOLTAGE COLLAPSE IN A COMPLEX NETWORK

When dealing with electric systems, planning and operation, it is necessary to focus the attention on the problem of security. This is crucial especially when the electricity markets force towards a higher and higher exploitation of the network facilities to improve the business. In particular, the contingency analysis is a fundamental part of the security assessment: several techniques have been implemented in the past to study the impact of such perturbations. In this article, second order information derived from the singular value analysis of the power flow (PF) Jacobian matrix is used to obtain an effective ranking of contingencies with particular respect to the voltage/reactive problems. Besides, a quantification of the risk associated to each contingency is the output of a procedure that evaluates quasi-linear indices useful in both planning and operation.

Non-iterative load-flow method as a tool for voltage stability studies

Here the use of a non-iterative (NI) method for load-flow solutions is investigated. The method, previously proposed in the literature, presents some advantages in comparison with the iterative approaches usually employed. There, because the Taylor expansion is used, the power flow Jacobian matrix is not updated along the process. Here, some improvements in the implementation are executed, and few control actions are incorporated into the formulation. The method is then applied for voltage stability studies, aiming to reduce the computational time associated with.

Using redispatch generators to reduce the standing phase angle during system restoration

A method is presented that reduces the standing phase angle by using active power redispatch. The reduction of the phase angle results in dynamic problems being avoided. The method seeks a suboptimal solution, however, the lower computational load involved in the proposed technique does not come at the price of a deminished accuracy. The proposed method is based on an augmentation to the power flow Jacobian matrix and a sensitivity technique to identify the generators to be used for the redispatch process. The system during and after the redispatch process is studied using an analysis of the system eigenvalues and time-domain simulation. The results are compared with ones obtained using an optimal power flow technique, whose objective function is the minimisation of the active power deviation.

Analysis of ill-conditioned power-flow problems using voltage stability methodology

Ill-conditioned power-flow problems have been widely investigated and reported in the literatun:. A typical approach develops enhanced solution algorithms when a power-flow case is found divergent with the conventional Newton method. It is known that a genuine ill-conditioned problem is caused by the presence of a large condition number in the power-flow Jacobian matrix. Since a large condition number is associated with small singular values or eigenvalues of a matrix and the voltage collapse is also related to small eigenvalues, it is therefore postulated that an ill- conditioned power-flow problem is actually a voltage collapse problem. The objective of ths paper is to investigate the relationship between power-flow ill-conditioning and voltage instability. The findings confirm that power-flow ill-conditioning only occurs at the voltage collapse point. As a result, developing improved algorithms to solve the problem is an unprofitable strategy. The well- known voltage stability assessment techniques such as the PV curve method are sufficient for the problem. This conclusion is supported with case studies on five widely known ill-conditioned power- flow problems and rigorous mathematical analysis.

Investigation of the relationship between ill-conditioned power flow and voltage collapse

Ill-conditioned power flow problems have been widely investigated and discussed in the literature. Typical approaches are to develop enhanced solution algorithms when a power flow case is found to diverge using the conventional Newton method. It is known that a genuine ill-conditioned problem is caused by the presence of a large condition number in the power flow Jacobian matrix. Since a large condition number is associated with small singular values and the voltage collapse is related to small eigenvalues, we therefore speculate that an ill-conditioned power flow problem is actually a voltage collapse problem. The objective of this letter is to demonstrate the relationship between power flow ill conditioning and voltage instability. Case studies of five classical ill-conditioned cases have confirmed that the ill conditioning of power flow indeed occurs-and only occurs-at the voltage collapse point.