Holomorphic Embedding Load Flow Research Articles

Photovoltaic hosting capacity assessment of a distributed network based on an improved holomorphic embedding load flow method and stochastic scenario simulation

The increasing proportion of distributed energy in the distribution network poses a significant challenge to effectively absorbing distributed generation (DG). On this premise, the DG hosting capacity (HC) assessment in the future distribution network based on the secure operation boundary of the power grid is necessary. In this paper, an improved holomorphic embedding load flow method (HELM) is employed to construct the relationship between the photovoltaic HC (PVHC) and each constraint to determine the upper limit of the comprehensive PVHC. Furthermore, a screening method of PV locations in the distribution network is constructed to screen the alternative PV locations with strong voltage regulation ability and reduce the alternative multi-point schemes, paving the way for simplifying the PVHC assessment based on a stochastic scenario simulation. Finally, a PVHC assessment based on the stochastic scenario simulation is proposed to expand the applicability of the assessment based on improved HELM. The negative impact of PV uncertainty is quantified by generating stochastic load flow scenarios based on the upper limit of the PVHC from improved HELM. The PVHC assessment in the high-voltage distribution network IEEE30 is analyzed to prove the efficiency and comprehensiveness of PVHC assessment in the distribution network.

Interval Holomorphic Embedding Load Flow Method: A novel approach for interval analysis considering load and generation uncertainties

Interval Holomorphic Embedding Load Flow Method: A novel approach for interval analysis considering load and generation uncertainties

Area interchange control using the holomorphic embedding load flow method considering automatic generation control

Area interchange control using the holomorphic embedding load flow method considering automatic generation control

Optimal allocation of distributed generation in active distribution power network considering HELM-based stability index

Traditionally, assessment of static voltage stability for distribution power networks with distributed generation (DG) utilizes the continuous power flow (CPF) method by calculating load margin index (LMI) at various load levels, which suffers from slow calculation speed and low accuracy when power flow calculation becomes ill-conditioned. To overcome this issue, this paper presents a novel holomorphic embedding load flow method (HELM)-based approach that calculates voltage stability indices for various types of nodes in a distribution network. Different orders of sensitivity information are calculated using the HELM method, and the criteria for assessing voltage stability are derived, which are used as constraints to optimize the location and capacity of DG. Such formulation of the optimization problem does not need to calculate the maximum load margin using CPF. Moreover, it can be effectively solved using the Genetic Algorithm and easily adaptable to various operating conditions of DG, which improve the calculation speed of the stability index. The effectiveness of the proposed approach is verified on IEEE 33-node and IEEE 69-node distribution power network models. Results show that considering the HELM VSI, the location, and capacity of DG in the distribution network can be better optimized. For the IEEE 33-node model, the total cost is reduced by approximately 2.33%, the network loss is reduced by 1.2%, and the HELM voltage stability criteria index is increased by about 4.23%; while for the IEEE 69-node system, the total cost is reduced by about 2.98%.

Holomorphic Embedding Load Flow Method in Three-Phase Distribution Networks With ZIP Loads

The Holomorphic Embedding Load flow Method (HELM) employs analytic continuation of complex analytic functions to calculate the practical (high voltage) load flow solution and provides promising convergence guarantees. Although HELM has been applied to various types of power systems, the literature has not thoroughly investigated three-phase distribution networks with common load types/connections. In this paper, we extend HELM to three-phase distribution networks with ZIP loads, while preserving its prominent convergence promises. Specifically, we introduce a HELM formulation, which emphasizes on accurate modeling of constant-current loads (and the voltage dependence of current injections) for both wye and delta connections. We perform extensive numerical experiments on standard test systems with various equipment/connection types to demonstrate the accuracy and efficacy of the proposed HELM compared with the existing literature.

Holomorphic Embedded Analysis of Unified Power Quality Conditioner Compensated Power Distribution System

Unified power quality conditioner (UPQC) has been used to stabilize the voltage profile with reduced harmonics. Thus, it is important to do the impact study of UPQC over distribution system. But iterative load flow method sometime fails to converge with rise in system complexity. Thus, recursive holomorphic embedding load flow is best suited over here. This article proposes a recursive mathematical modeling of UPQC using holomorphic embedding method. This holomorphic embedded model of UPQC is based on series and shunt converters operation determined by phasor diagram. Different operating modes of UPQC are also modeled and tested. The proposed models have been tested over IEEE 13, 123 bus distribution test systems and a 19 bus practical Indian distribution grid. A rigorous comparative study with existing iterative load flow methods has been done. On analyzing the results, it shows great efficacy and reliability in terms of voltage sag mitigation and total harmonic distortion compared to conventional methods.

A Holomorphic Embedding Power Flow Algorithm for Islanded Hybrid AC/DC Microgrids

This paper proposes a novel power flow algorithm for islanded hybrid AC/DC microgrids. The algorithm is based on a recursive (non-iterative) method called the holomorphic embedding load flow method. The convergence properties of this method are more favorable than other iterative-based techniques such as Newton Raphson. The proposed power flow algorithm accounts for the special characteristics of hybrid microgrids by developing new holomorphically embedded versions for the DG droop characteristics on both the AC and DC microgrid as well as the interlinking AC/DC converter droop characteristics. The proposed approach is tested on two systems and in the presence of multiple interlinking systems. For each case, the results from the proposed algorithm are compared to detailed time-domain simulations performed using PSCAD/EMTDC to validate the accuracy of the proposed algorithm.

GPU-accelerated N-1 static security analysis based on fine-grained parallelism HELM

With the increment of scale of power system, the accuracy and efficiency requirements of N-1 static security analysis (SSA) keep increasing. In order to satisfy these requirements, this paper proposed a GPU-accelerated N-1 SSA method to reduce computation burden and ensure the accuracy of alternating current power flow (ACPF) and connectivity test simultaneously. First, in the algorithm, ACPF is performed by holomorphic embedding load flow method (HELM). The construction and solution of sparse linear system both adopts fine-grained parallelism. According to the mathematical lemma of combining linear equations, this paper performs fine-grained parallelism Padé approximation of HELM. Second, considering the high time complexity of the original search-based algorithm and adjacency matrix method which is not suitable for sparse graph, this paper proposes a GPU-accelerated Union Set algorithm. Its main advantage is lower time complexity compared to other graph theory methods. And most importantly, it is very suitable for parallelization. Shared memory can be used in this algorithm well. Therefore, it is not complicated to perform a block-level parallelism in GPU-accelerated Union Set. Finally, the overall framework of GPU-accelerated N-1 SSA method is shown in this paper. It has been tested on large-scale power systems of up to 2869 buses. The computation results are compared with MATPOWER and the execution times are compared with other mainstream method to show the improvement in performance.

HELMpy, Open Source Package of Power Flow Solvers, Including the Holomorphic Embedding Load Flow Method (HELM), Developed on Python 3

HELMpy, Open Source Package of Power Flow Solvers, Including the Holomorphic Embedding Load Flow Method (HELM), Developed on Python 3

HELMpy, open source package of power flow solvers, including the Holomorphic Embedding Load Flow Method (HELM), developed on Python 3

HELMpy, open source package of power flow solvers, including the Holomorphic Embedding Load Flow Method (HELM), developed on Python 3

Stratified Control Applied to a Three-Phase Unbalanced Low Voltage Distribution Grid in a Local Peer-to-Peer Energy Community

This paper presents control relationships between the low voltage distribution grid and flexibilities in a peer-to-peer local energy community using a stratified control strategy. With the increase in a diverse set of distributed energy resources and the next generation of loads such as electric storage, vehicles and heat pumps, it is paramount to maintain them optimally to guarantee grid security and supply continuity. Local energy communities are being introduced and gaining traction in recent years to drive the local production, distribution, consumption and trading of energy. The control scheme presented in this paper involves a stratified controller with grid and flexibility layers. The grid controller consists of a three-phase unbalanced optimal power flow using the holomorphic embedding load flow method wrapped around a genetic algorithm and various flexibility controllers, using three-phase unbalanced model predictive control. The control scheme generates active and reactive power set-points at points of common couplings where flexibilities are connected. The grid controller’s optimal power flow can introduce additional grid support functionalities to further increase grid stability. Flexibility controllers are recommended to actively track the obtained set-points from the grid controller, to ensure system-level optimization. Blockchain enables this control scheme by providing appropriate data exchange between the layers. This scheme is applied to a real low voltage rural grid in Austria, and the result analysis is presented.

A modified holomorphic embedding method based hybrid AC-DC microgrid load flow

Load flow study is a steady state analytical tool for power system network analysis where mostly iterative numerical methods are used to solve the load flow equations. But with the rise in complexity of the power networks due to amalgamation of both AC and DC power system within a network, the convergence time of these iterative load flow algorithms has been increased quite high. To resolve this problem a recursive load flow method has been developed called Holomorphic Embedding Load Flow Method (HELM) which reduces time complexity and also increase the probability of existence of the load flow solution. Here a new holomorphic embedding load flow problem formulation for a hybrid AC-DC microgrid system using line impedance values has been proposed. By using Padẽ approximant, we solve this holomorphic embedding load flow functions which guarantees the solution of load flow problem. Comparative study with well-known load flow methods over different test cases and a practical Indian distribution grid shows the efficiency and applicability of this method over the hybrid power networks.

Distributed slack bus model formulation for the holomorphic embedding load flow method

Distributed slack bus model formulation for the holomorphic embedding load flow method

Remote Voltage Control Using the Holomorphic Embedding Load Flow Method

This paper proposes a new remote voltage control approach based on the non-iterative holomorphic embedding load flow method (HELM). Unlike traditional power-flow-based methods, this method can guarantee the convergence to the stable upper branch solution if it exists and does not depend on an initial guess of the solution. Bus type modifications are set up for remote voltage control. A participation factor matrix is integrated into the HELM to distribute reactive power injections among multiple remote reactive power resources so that the approach can remotely control the voltage magnitudes of desired buses. The proposed approach is compared with a conventional Newton–Raphson (N–R) approach by study cases on the IEEE New England 39-bus system and the IEEE 118-bus system under different conditions. The results show that the proposed approach is superior to the N–R approach in terms of tractability and convergence performance.

Holomorphic embedding load flow for unbalanced radial distribution networks with DFIG and tap‐changer modelling

In this study, the holomorphic-embedding method as a linear and non-iterative method is developed and implemented to solve the load flow of the three-phase unbalanced radial distribution networks. Previously, this method was used to solve the load flow problem of balanced transmission networks. The proposed methodology could be applied for a variety of unbalanced radial distribution networks with any number of buses and branches. The characteristics of unbalanced radial distribution networks including line coupling effects, unbalanced loads and unsymmetrical phases are considered in this study. The proposed method uses an unbalanced network structure and modifies the calculation approach of the admittance matrix. In this manner, an unbalanced network is modelled the same as the balanced networks. The tap-changer, doubly-fed induction generator (DFIG), and photovoltaic system are modelled in the admittance matrix or current injection vector. The performance of the proposed method is validated on the 19-bus unbalanced radial distribution network. Finally, the effectiveness of the proposed method in solving the load flow problem is investigated.

Three-Phase Unbalanced Optimal Power Flow Using Holomorphic Embedding Load Flow Method

Distribution networks are typically unbalanced due to loads being unevenly distributed over the three phases and untransposed lines. Additionally, unbalance is further increased with high penetration of single-phased distributed generators. Load and optimal power flows, when applied to distribution networks, use models developed for transmission grids with limited modification. The performance of optimal power flow depends on external factors such as ambient temperature and irradiation, since they have strong influence on loads and distributed energy resources such as photo voltaic systems. To help mitigate the issues mentioned above, the authors present a novel class of optimal power flow algorithm which is applied to low-voltage distribution networks. It involves the use of a novel three-phase unbalanced holomorphic embedding load flow method in conjunction with a non-convex optimization method to obtain the optimal set-points based on a suitable objective function. This novel three-phase load flow method is benchmarked against the well-known power factory Newton-Raphson algorithm for various test networks. Mann-Whitney U test is performed for the voltage magnitude data generated by both methods and null hypothesis is accepted. A use case involving a real network in Austria and a method to generate optimal schedules for various controllable buses is provided.

A Padé-Weierstrass technique for the rigorous enforcement of control limits in power flow studies

A new technique is presented for solving the problem of enforcing control limits in power flow studies. As an added benefit, it greatly increases the achievable precision at nose points. The method is exemplified for the case of Mvar limits in generators regulating voltage on both local and remote buses. Based on the framework of the Holomorphic Embedding Loadflow Method (HELM), it provides a rigorous solution to this fundamental problem by framing it in terms of optimization. A novel Lagrangian formulation of power-flow, which is exact for lossless networks, leads to a natural physics-based minimization criterion that yields what is plausibly the best solution. For networks with small losses, as is the case in transmission, the ac power flow problem cannot be framed exactly in terms of optimization, but the method still retains its ability to select a solution. This foundation then provides a way to design a HELM scheme to solve for the minimizing solution. Although the use of barrier functions evokes interior point optimization, this method, like HELM, is based on the analytic continuation of a germ (of a particular branch) of the algebraic curve representing the solutions of the system. In this case, since the inequality constraints result in an unavoidable singularity at s=1, direct analytic continuation by means of standard Padé approximation exhibits numerical instabilities. This has been overcome by means of a new analytic continuation procedure, denominated Padé-Weierstrass, that exploits the covariant nature of the power flow equations under certain changes of variables. A side benefit of this procedure is that it can also be used when limits are not being enforced, in order to increase the achievable numerical precision in highly stressed cases.

Theoretical convergence guarantees versus numerical convergence behavior of the holomorphically embedded power flow method

The holomorphic embedding load flow method (HELM) is an application for solving the power-flow problem based on a novel method developed by Dr. Trias. The advantage of the method is that it comes with a theoretical guarantee of convergence to the high-voltage (operable) solution, if it exists, provided the equations are suitably framed. While theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not; it depends on the analytic continuation algorithm chosen. Since the holomorphic embedding method (HEM) has begun to find a broader range of applications (it has been applied to nonlinear structure-preserving network reduction, weak node identification and saddle-node bifurcation point determination), examining which algorithms provide the best numerical convergence properties, which do not, why some work and not others, and what can be done to improve these methods, has become important. The numerical Achilles heel of HEM is the calculation of the Padé approximant, which is needed to provide both the theoretical convergence guarantee and accelerated numerical convergence. In the past, only two ways of obtaining Padé approximants applied to the power series resulting from power-system-type problems have been discussed in detail: the matrix method and the Viskovatov method. This paper explores several methods of accelerating the convergence of these power series and/or providing analytic continuation and distinguishes between those that are backed by the theoretical convergence guarantee of Stahl’s theorem (i.e., those computing Pade approximants), and those that are not. For methods that are consistent with Stahl’s theoretical convergence guarantee, we identify which methods are computationally less expensive, which have better numerical performance and what remedies exist when these methods fail to converge numerically.

The Holomorphic Embedding Loadflow Method for DC Power Systems and Nonlinear DC Circuits

The Holomorphic Embedding Loadflow Method is extended here from AC to DC-based systems. Through an appropriate embedding technique, the method is shown to extend naturally to DC power transmission systems, preserving all the constructive and deterministic properties that allow it to obtain the white branch solution in an unequivocal way. Its applications extend to nascent meshed HVDC networks and also to power distribution systems in more-electric vehicles, ships, aircraft, and spacecraft. In these latter areas, it is shown how the method can cleanly accommodate the higher-order nonlinearities that characterize the I-V curves of many devices. The case of a photovoltaic array feeding a constant-power load is given as an example. The extension to the general problem of finding DC operating points in electronics is also discussed, and exemplified on the diode model.