A Linear AC Power Flow Model for Unbalanced Multi-Phase Distribution Networks Based on Current Injection Equations

Distribution networks typically have unbalanced loads; however, most works on Active Distribution Networks (ADNs) use single-phase representations. This paper proposes a linear alternate current (AC) three-phase optimal power flow (OPF) for a day-ahead operation planning of ADNs and microgrids. The proposed AC linear three-phase optimal power flow model is based on polar coordinate systems and considering some consistent approximations for linearization of power flow equations. The shunts admittances of the network and the mutual impedances between phases of the three-phase lines are considered in the model. In this paper, the proposed linear AC three-phase OPF model is used to define the optimal day-ahead operation planning of active distribution systems and microgrids with three-phase feeders and three-phase distributed generators (DG) and energy storage (ES). Since this model proposes a three-phase representation of the system, it provides more accurate results, making it an effective tool for optimal planning of the ADN operation. The proposed model is illustrated using the IEEE 34-bus test system and results show its effectiveness and excellent performance, presenting an error of less than 4% for the voltage magnitude of the buses and less than 2% for the apparent power flow of the lines, when compared with non-linear models.

It is an extra heavy burden to maintain the distribution network model since of its frequent changing and large scale. In reality, the network model may involve significant parameter errors, which makes the model not applicable. This paper presents a robust data-driven approach to linearize the three-phase power flow. A three-phase linear power flow (LPF) model for active distribution networks (ADNs) is proposed. To address the issues of measurement outliers and data collinearity, the support vector regression (SVR) algorithm is applied to the linearization regression to obtain the proposed three-phase LPF model. The training process is offline and the obtained LPF model can be used to calculate the power flow online. Numerical tests demonstrate that the proposed approach can achieve satisfactory calculation accuracy with high computational efficiency, even under data collinearity and measurement outliers. Since the proposed LPF has very high accuracy, this model can be incorporated into distribution networks optimization problems.

The power flow model is usually used to estimate power from the central calculation for the whole power system. In recent years, data-driven approach is an emerging technique to determine power flow without knowing circuit parameters. However, with the development of renewable energy, more complex power grids are placing an increasing burden on central computing. Therefore, this paper proposes a method to estimate power from a single bus only which provides a quick reference to the power of a single bus. This method is based on modified linear power flow model, the core is to solve the challenge of power estimation using only one single bus data rather than whole power system data by analyzing the electrical relationship. The proposed modified linear power flow model includes both the nonlinear physical power flow model and the linear regression model. Simulation results on the IEEE Standard bus 30 show that this method has higher accuracy than other linear power flow models in single bus power calculation.

This paper presents a method for accurately treating active power losses in linear (DC) optimal power flow models. Quadratic nodal losses are approximated by iteratively adding linear constraints. In each iteration a linear programming problem is solved, and constraints on the nodal loss functions are built as linearizations around the current system operating state. The performance of the proposed model - in terms of computational time and convergence properties - is demonstrated on the IEEE 118 bus test system.

Linear power flow formulations and optimal operation of three-phase autonomous droop-controlled Microgrids

A per-node granularity decentralized optimal power flow for radial distribution networks with PV and EV integration

Accurate line parameters are the basis for the optimal control and safety analysis of distribution networks. The lack of real-time monitoring equipment in grids has meant that data-driven identification methods have become the main tool to estimate line parameters. However, frequent network reconfigurations increase the uncertainty of distribution network topologies, creating challenges in the data-driven identification of line parameters. In this paper, a line parameter identification method compatible with an uncertain topology is proposed, which simplifies the model complexity of the joint identification of topology and line parameters by removing the unconnected branches through noise reduction. In order to improve the solving accuracy and efficiency of the identification model, a two-stage identification method is proposed. First, the initial values of the topology and line parameters are quickly obtained using a linear power flow model. Then, the identification results are modified iteratively based on the classical power flow model to achieve a more accurate estimation of the grid topology and line parameters. Finally, a simulation analysis based on IEEE 33- and 118-bus distribution systems demonstrated that the proposed method can effectively realize the estimation of topology and line parameters, and is robust with regard to both measurement errors and grid structures.

Effective post-disaster repair strategies and emergency resource dispatch schemes can accelerate power restoration and reduce outage losses. However, existing repair and resource scheduling strategies typically rely on the assumption of three-phase balanced distribution networks. In reality, distribution networks frequently operate under three-phase unbalanced conditions due to unbalanced loads, asymmetric line parameters, and non-full-phase connections of small-scale distributed energy resources. This paper introduces a linear three-phase power flow model to characterize the coupling relationships among the three phases. Using wind turbines as an example, the paper employs the probabilistic scenario method to address the uncertainty of renewable energy generation. A stochastic optimization model is established for the coordinated scheduling of repair crews, distribution network reconfiguration, and mobile generators. The proposed model is validated on a modified IEEE 37-bus system. The simulation results demonstrate that three-phase unbalanced constraints will impact resource scheduling and load restoration plans, and the developed model enhances system security during the repair process.

The complexity and computational burden of nonlinear power flow (PF) models have motivated the introduction of various approximations. While these approximations provide tractability and reduced computational burden in PF, for optimal power flow (OPF) and control purpose, the accuracy of these models becomes a major concern in the presence of high renewable energy penetration/increased demand in distribution feeders. This paper compares and analyzes, for the first time to our knowledge, the performance of commonly used PF models in OPF formulations for single-phase and unbalanced three-phase distribution systems. Simulations performed on the phase-coupled and phase-decoupled versions of the IEEE 123-node and a reduced model extracted from the IEEE 8500-node feeder with varying levels of negative and positive net loads show that, ignoring mutual coupling impedances, as in the phase-decoupled PF models, is the largest source of error on the voltage profiles. Also, the performance of the linear PF models is acceptable only for a small range around the nominal operating point; hence, they can lead to large errors if used in OPF/control framework that seeks solutions on wider operating range.

Decentralized saddle-point dynamics solution for optimal power flow of distribution systems with multi-microgrids

With large-scale distributed energy resources (DERs) being integrated into distribution networks (DNs), the coordinated optimal power flow (OPF) of high voltage (HV) and medium voltage (MV) DNs becomes challenging due to enhanced uncertainties and nonlinearity. This paper presents a hierarchical distributed chance-constrained OPF model for HV and MV DNs based on a nonlinear adaptive data-driven method. The HV master problem aims to minimize the overall operational costs, and is transformed into mixed-integer convex problem via second order cone programming. A model-free chance-constrained optimization sub-problem is established to tackle inaccurate model parameters issues and DER uncertainties in MV DNs and realize rapid response to control schemes from HV DN. The nonlinearity of power flow is approximately depicted by a data-driven incomplete dimension lifting power flow (IDL-PF) constraint, which is integrated into the MV DNs optimization model. A novel multi-parameter programming (MPP)-based distributed algorithm is proposed to solve the coordinated OPF model, which can significantly accelerate the iterative speed between the master and sub-problems. Finally, the convergence, efficiency and optimality of the proposed method are validated by numerous simulations in a practical DNs in China.

In this paper, we investigate the problem of estimating a complex-valued Laplacian matrix with a focus on its application in the estimation of admittance matrices in power systems. The proposed approach is based on a constrained maximum likelihood estimator (CMLE) of the complex-valued Laplacian, which is formulated as an optimization problem with Laplacian and sparsity constraints. The complex-valued Laplacian is a symmetric, non-Hermitian matrix that exhibits a joint sparsity pattern between its real and imaginary parts. Thus, we present a group-sparse-based penalized log-likelihood approach for the Laplacian estimation. Leveraging the mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell_{2,1}$</tex-math></inline-formula> norm relaxation of the joint sparsity constraint, we develop a new alternating direction method of multipliers (ADMM) estimation algorithm for the implementation of the CMLE of the Laplacian matrix under a linear Gaussian model. Next, we apply the proposed ADMM algorithms for the problem of estimating the admittance matrix under three commonly-used measurement models that stem from Kirchhoff's and Ohm's laws, each with different assumptions and simplifications: 1) the nonlinear alternating current (AC) model; 2) the decoupled linear power flow (DLPF) model; and 3) the direct current (DC) model. The performance of the ADMM algorithm is evaluated using data from the IEEE <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$33$</tex-math></inline-formula>-bus power system data under different settings. The numerical experiments demonstrate that the proposed algorithm outperforms existing methods in terms of mean-squared-error (MSE) and F-score, thus providing a more accurate recovery of the admittance matrix.

With high-level variable renewable energy integrating into power systems, the uncertainty significant challenges to the real-time dispatch (RTD). Meanwhile, the influence of voltage magnitude and reactive power on RTD always were ignored. Based on effective steady-state security regions of power systems and linear power flow model, a real-time dispatching method considering voltage magnitude and reactive power is proposed in this paper. This model has two goals, one is the security goal, the other is the economic goal, and priority of security objectives is higher than economy. Unlike previous real-time dispatch methods, all of the operation base point, reactive power output and participation factors of automatic generation control (AGC) generators are chosen as decision variables in the model. In order to ensure computational efficiency, the model is finally transformed into a deterministic linear programming (LP) problem by robust optimization and other methods. The accuracy and computational efficiency of the method are illustrated by modified IEEE 9-bus system and IEEE 118-bus system.

With the increasing proportion of distributed energy resources, optimisation in the active distribution network becomes highly complex and challenging. To protect user privacy and perform efficient calculations, this paper proposes an online distributed optimisation between the user and the operator. To overcome the constraint non‐linearity, a new linear power flow model is adopted, which can be updated online. In addition, to solve the time‐coupled conundrum of energy storage, the Lyapunov drift plus penalty method is employed to transform the long‐time scale optimisation problem into an instantaneous problem. Furthermore, a novel proportional integral derivative (PID)–Lagrange algorithm is proposed to improve the efficiency, where the distributed optimisation algorithm is considered as a control process, and a PID controller is used to control it. The proposed PID–Lagrange algorithm has an explicit parameter tuning strategy and significantly improves the efficiency by 54.23% in the case study.

Hybrid data-driven method for distribution network topology and line parameters joint estimation under small data sets