Transient Stability Constrained Optimal Power Flow Research Articles

Economic scheduling method for voltage‐dependent loads in a power system based on a multitime scale framework

AbstractThere are many voltage‐dependent loads (VDLs) in power systems. When the unit reserve supply is low, the transmission network adjusts the voltage of grid‐connected nodes to provide additional resources. Many reactive power devices with different regulatory performance levels and timescales must be coordinated to employ flexible VDLs to the maximum extent, and consequently, a transient stability‐constrained optimal power flow (TSCOPF) problem with mixed integers and multiperiod coupling must be solved. A multitimescale scheduling ‘day‐ahead‐intraday‐real‐time’ framework is designed for the optimal economic scheduling of VDLs. In the day‐ahead stage, the optimal scheduling period of the discrete reactive power control devices is determined. In the intraday stage, a rolling optimization model based on model predictive control is established, and the specific action times of the discrete reactive power devices are determined. In the real‐time stage, the optimal regulation strategy of the continuous reactive power control device is optimized by rolling. Then, to consider the speed and accuracy of the solution at the same time, a continuous sensitivity correction method is adopted to simplify and solve the original model. Finally, with a modified IEEE‐39 system and a modified IEEE‐118 system as examples, it is verified that the multitimescale scheduling framework maximizes the exploitation of the flexible resources of VDLs to ensure voltage security and stability and improve the economy of system operation under different scenarios.

Stability assessment of a transmission system with converted-interfaced generation by means of a transient stability-constrained optimal power flow

This paper proposes a transient-stability constrained optimal power flow (TSCOPF) algorithm that includes, together with synchronous generators, renewable power parks with nonsynchronous converter-interfaced generation (CIG) applying grid-forming and grid-following controls. The solution of the TSCOPF algorithm allows the comparison of control alternatives for CIG in terms of generation cost and contribution to power system transient stability. The algorithm is applied to a model of the Iberian Peninsula power system on which a 2021 incident consisting in the disconnection from the rest of continental Europe is analyzed. The effect of each control on the cost of maintaining the stability of the system is discussed.

Transient stability constrained optimal power flow considering fourth-order synchronous generator model and controls

This paper presents a novel mathematical programming model for the optimal power flow with transient stability constraints, considering a detailed fourth-order model for synchronous generators and their controls. Such controls are the simplified excitation system for voltage magnitude and a frequency control system formed by a governor and a turbine. Moreover, in order to manage convergence issues and to determine causes of instability, load shedding is included. The proposed model considers, in the same mathematical expression, the prefault or the initial steady stage, the fault stage, and the postfault. The differential equations that represent the rotor angle stability constraints were transformed into algebraic equations using the trapezoidal integration method. The proposed nonlinear programming model was implemented using the mathematical programming language AMPL and solved using the nonlinear solver IPOPT. The well-known WSCC 9-bus, and the IEEE 68-bus systems were used for testing accuracy and efficiency. A comparative analysis was also carried out with transient stability-constrained optimal power flow using the second-order generator model to be able to observe not only the economic differences but also to be able to compare the performance of the systems, that is, if the system configuration satisfies the requirements usually demanded by the transmission system operator. The results show the ability of the proposed model to predict the dynamic behavior of the system under contingency and to properly dispatch the generation resources to withstand these perturbations. It is concluded that a more detailed model provokes an overall improvement of the system stability, while performing economical and reliable dispatch decisions. • A new NLP mathematical programming model for the TSC-OPF problem. • The proposed model considers a detailed fourth-order model for generators. • The proposed model considers the excitation system, governor, and turbine controls. • A comparative analysis with the TSC-OPF using a second-order generator model.

Microgrid Operation Optimization Considering Transient Stability Constraints: A New Bidirectional Stochastic Adaptive Robust Approach

This article proposes a new bidirectional stochastic adaptive robust framework with transient stability constraints to optimally and securely operate microgrids (MGs). Within the proposed framework, the uncertainties of photovoltaic/wind turbine generation, load, and electricity price are modeled using adaptive robust optimization (ARO), while the uncertainty of unscheduled islanding occurrence is modeled by stochastic programming. Besides covering the uncertainties related to MG operation, the proposed approach considers transient stability uncertainties leading to a more secure MG operation scheduling framework. To the best of the authors’ knowledge, this is the first time that transient stability analysis along with stochastic adaptive robust optimization (SARO) is proposed for MG operational scheduling. In addition, a new bidirectional scenario generation method is proposed to model MG unscheduled-islanding occurrence and MG restoration to the grid-connected mode. Modeling both the grid-connected and unscheduled-islanded MG operation modes as well as their transitions further distinguishes the proposed approach from the previous ones. Furthermore, a new solution methodology is presented to solve the proposed MG scheduling problem. This methodology is capable of solving both SARO and transient stability constrained-optimal power flow problems. The proposed MG operation optimization framework and solution method are tested on the modified IEEE 33-bus network.

Fitness-distance balance based artificial ecosystem optimisation to solve transient stability constrained optimal power flow problem

ABSTRACT The Transient Stability Constrained Optimal Power Flow (TSCOPF) has become an important tool for power systems today. TSCOPF is a nonlinear optimisation problem, making its solution difficult, especially for small power systems. This paper presents a new optimisation method that incorporates Fitness-Distance Balance (FDB) with the Artificial Ecosystem Optimisation (AEO) algorithm to improve the solution quality in multi-dimensional and nonlinear optimisation problems. The proposed method, named the Fitness-Distance Balance Artificial Ecosystem Optimisation (FDBAEO), also has the capacity to solve the TSCOPF problem efficiently. In order to evaluate the proposed algorithm, it was tested on IEEE CEC benchmarks and on an IEEE 30-bus test system for the TSCOPF problem. Simulation results were compared with the basic AEO algorithm and other current meta-heuristic methods reported in the literature. The results showed that the proposed method was more effective in converging at the global optimum point in solving the TSCOPF problem compared to the other algorithms. This situation indicates that the design changes made in the decomposition phase of the AEO were more suitable for simulating the operation of the algorithm in the real world. The FDBAEO has exhibited a promising performance in solving both single-objective optimisation and constrained real-world engineering design problems.

Transient stability constrained optimal power flow solution using ant colony optimization for continuous domains (ACO R )

This paper aims to improve transient stability using the Ant Colony Optimization for Continuous Domains (ACOR). This improvement is obtained by solving the Transient Stability Constrained Optimal Power Flow (TSCOPF) problem and extracting the sensitivity coefficients. The presented costs minimization approach requires less execution time to manage energy resources efficiently and compared to other conventional methods, it also outperforms based on statistical indicators such as mean and standard deviation. Furthermore, the sensitivity analysis of the suggested method considers the active power generation of the generating units along with their terminal voltage, which notably decreases the operation cost and reduces risk to the power system. On the other hand, the application of ACOR algorithm can reduce the fuel cost of power system operation to 60,928.36 $/h with a decrease of 15.33 $/h compared to the conventional method with a standard deviation equal to 2.51 $/h and an execution time of 7.66 s. This strategy could be used in power system dispatching. This method is implemented on the New England 39 bus system, and the results demonstrate the method's efficiency compared to other conventional methods.

Stability Improvement of a Transmission Grid With High Share of Renewable Energy Using TSCOPF and Inertia Emulation

This paper proposes a Transient Stability Constrained Optimal Power Flow (TSCOPF) formulation that models non-synchronous renewable generation equipped with synthetic inertia. The proposed optimization problem calculates the optimal operating point of the system, accommodating high shares of non-synchronous renewable generation while ensuring transient stability in the event of critical incidents. Synthetic inertia controllers are used to improve the dynamic stability of the system in cases of very high share of renewable generation. The proposed tool is tested in the North-West Spanish system, a network with a high penetration of wind energy that causes a reduction in the total system inertia. The results of the study show that 1) synthetic inertia in renewable power plants can diminish electromechanical oscillations after a severe contingency, reducing the cost of ensuring transient stability; 2) using synthetic inertia the system becomes more stable when conventional generation is decommissioned following de-carbonization and renewable promotion policies; and 3) the proposed model can be used to calculate the parameters of the synthetic inertia control.

Parametric Transient Stability Constrained Optimal Power Flow Solved by Polynomial Approximation Based on the Stochastic Collocation Method

To better respond to the impact of power system-uncertain parameters on transient stability, a novel model named the parametric transient stability constrained optimal power flow (parametric TSCOPF) is proposed. It seeks the optimal control scheme of transient stability constrained optimal power flow (TSCOPF) expressed by the function of uncertain parameters in power systems. The key difficulty to solve this model lies in that the relationship between the parametric TSCOPF solution and uncertain parameters is implicit, which is hard to derive generally. To this end, this paper approximates the optimal solution of parametric TSCOPF by polynomial expressions of uncertain parameters based on the stochastic collocation method. First, the parametric TSCOPF model includes both uncertain parameters and transient stability constraints, in which the transient stability constraint is constructed as a set of polynomial expressions using the SCM. Then, to derive the relationship between the parametric TSCOPF solution and uncertain parameters, the SCM is applied to the parametric Karush–Kuhn–Tucker (KKT) conditions of the parametric TSCOPF model, so that the optimal solution of the parametric TSCOPF is approximated by using polynomial expressions with respect to uncertain parameters. The proposed parametric TSCOPF model has been tested on a 3-machine, 9-bus system, and the IEEE 145-bus system, which verifies the effectiveness of the proposed method.

Transient stability constrained optimal power flow applied to distribution systems with synchronous generators

This paper proposes a Transient Stability Constrained Optimal Power Flow (TSC-OPF) formulation in order to calculate the optimum operating point of synchronous generators in distribution networks, in terms of dispatch of their active and reactive powers. However, the problem is complex due to the high number of constraints and variables, as well as, the presence of nonlinear constraints. To circumvent these adversities, an optimization problem is proposed in this paper with the application of a set of mathematical approximations in the constraints of the active and reactive power balance equations. In addition, an approximation is explored in the swing equation of synchronous generators belonging to a group of coherent machines, particularly regarding the active power injection from the generators. The proposed TSC-OPF is tested in a 31-bus radial distribution system with two and four generators. The results show that the running time to solve the proposed optimization problem with approximations becomes smaller when compared to the time to solve it without approximations.

Evaluation of Numerical Methods for TSCOPF in a Large Interconnected System

Transient stability-constrained optimal power flow (TSCOPF) models comprehensively analyze the security and economic operation of power systems. However, they require a high computational effort and can suffer from convergence problems when applied to large systems. This study analyzes the performance of eleven numerical integration algorithms applied to ordinary differential equations that represent power system dynamics in a TSCOPF model. The analyzed algorithms cover a range of explicit and implicit methods, including the recently published semi-explicit and semi-implicit Adams-Bashforth-Moulton formulas, together with several initialization techniques. The integration methods are applied to a model of the Iberian Peninsula power system, and their performance is discussed in terms of convergence, accuracy, and computational effort. The results show that most implicit methods converge to the solution, even for large time steps. In particular, the Adams-Moulton method of order two and Simpson’s rule, both initialized with RK4, outperform the trapezoidal rule, which is the default method in TSCOPF models.

The Voltage Security Region Calculation Method of Receiving-End Power System Based on the Equivalence of Transient Process

In order to ensure the voltage security of the receiving-end power grid under both steady state and N-1 faults, it is necessary to conduct an accurate dynamic reactive power reserve assessment to ensure its normal voltage operation range at steady state. In other words, it’s to determine the power grid voltage security region which is a transient stability constrained optimal power flow (TSCOPF) problem essentially. However, it needs to consider a large amount operational snapshots and corresponding transient constraints under each snapshot, which increases the computational burden heavily. To overcome this issue, this paper proposes a calculation method for the voltage security region of receiving-end power grid through the equivalence of the transient process. Firstly, based on the power grid partition and the matrix of graph theory, the different snapshots are selected to obtain typical snapshots, and then for the typical snapshots sought, the transient stability constraints under the fault are transformed into steady-state constraints. In other words, the form is consistent with the steady-state equation but the impedance matrix parameters are equivalently adjusted. In essence, it is equally to replace the constraints of the optimization problem under the premise of ensuring that the consistency of the boundary conditions, so as to greatly accelerate the solution speed under the premise of ensuring the accuracy of the solution, and the calculation cases made in the modified IEEE9 system and the modified IEEE39 system prove the effectiveness and reliability of the proposed method in this paper.

Deep Belief Network Enabled Surrogate Modeling for Fast Preventive Control of Power System Transient Stability

The widely used transient stability-constrained optimal power flow (TSC-OPF) method for power system preventive control is very time-consuming and thus not applicable for large-scale systems. This article proposes a new deep learning-enabled surrogate model that can significantly improve computational efficiency while maintaining high accuracy. To achieve that, the deep belief network (DBN) is strategically integrated with the reference-point-based nondominated sorting genetic algorithm (NSGA-III) to develop a new preventive control framework. The DBN allows us to identify the mapping relationship between the transient stability index and system operational features. The identified functional mapping relationship is further used as the surrogate to connect the DBN results with TSC-OPF for preventive control. The integrated NSGA-III and surrogate model enable the multiobjective optimization to consider various constraints and objectives, such as minimization of costs of generation dispatch cost and load shedding while maintaining the system stability. Extensive simulation results on several IEEE test systems show that the proposed method can achieve highly efficient control solutions and outperform other alternatives in terms of computational efficiency and economic benefits.

A Preventive Transient Stability Control Strategy Through Individual Machine Equal Area Criterion Framework

Power system transient stability assessment (TSA) and control is a complex problem. TSA from the perspective of individual machine Kimbark curves forms a distinctive approach. On the other hand, transient stability constrained optimal power flow (TSC-OPF) has been adopted widely for determining preventive control actions against transient instabilities under a given contingency. This paper presents a strategy for transient stability control, which mainly integrates the concepts of individual machine methods based TSA and global TSC-OPF. The proposed workflow is mainly based on the most severely disturbed machines (MDM) and their critical trajectories. These are utilized in deciding the number of TSC-OPF transient stability constraints, the minimum upper time limit for TSC-OPF solution interval, and security-based transient stability constraint of individual machines. Further, the application of the proposed method is explained with respect to stabilization of three different transient instability scenarios viz., transient instability originates from first swing instability, transient instability originates from multi swing instability but leads to first swing instability as fault duration increases, and transient instability originates from multi swing instability, but never leads to first swing instability as fault duration increases. IEEE 39 bus and 118 bus test system are considered for testing the proposed control strategy. The proposed approach, which expresses the transient stability constraints in global TSC-OPF in terms of some selective machine (i.e MDMs) critical trajectories not only forms a distinctive approach but also provide a smooth reconciliation between economy of operation and security level while ensuring transient stability under a given contingency.

Transient Stability Constrained Optimal Power Flow in Mixed AC/Multi‐Terminal VSC HVDC System

This paper proposes a hierarchical transient stability constrained optimal power flow (TSCOPF) analysis in a power system with a multi‐terminal voltage‐source converter (VSC) high‐voltage direct current (HVDC) system and large‐scale wind farms. This aims at determining the reinforcement and operation of the power system using the multi‐terminal VSC HVDC (MTDC) system. By the method, the wind farms can be installed without any transient instability issues. On the AC grid side, transient rotor‐angle stability and transient AC voltage stability are included as additional constraints. On top of that, the constraints of DC bus voltage fluctuations and DC transmission line current deviations on the DC grid side are also taken into account. Time‐domain simulation approach is applied to investigate these transient stability constraints. The MTDC system reinforcement is designed considering wind power transmission and economic operation of the system model. The influences of the transient stability constraints on the system operation costs with high and low wind power output patterns are discussed. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Information‐gap decision theory based transient stability constrained optimal power flow considering the uncertainties of wind energy resources

Studies on transient stability constrained optimal power flow (TSCOPF) have become crucial for power systems to guarantee their dynamic securities against credible contingencies, while their optimum operations are to be continuously projected under changing conditions. However, the current approach to the TSCOPF problem is not sufficient to meet the expectations of a modern power system because it suffers from uncertainties mainly due to the rapid and large integration of distributed energy sources. This study proposes a novel method using the information-gap decision theory (IGDT) technique to solve the TSCOPF problem in the presence of uncertainties due to the penetration of wind farms. The IGDT is a non-probabilistic decision-making method that can be easily implemented to handle uncertainty in optimisation problems. While presenting applicable strategies, it does not require any information about the historical data, probability density function or membership function of the uncertain parameters. The proposed method offers an analysis for the economic dispatch in a power system with wind energy resources while providing robustness against transient instabilities and uncertainties in power generation. To demonstrate the effectiveness of the proposed method, it is implemented on New England 39-bus and IEEE 118-bus test systems.

Transient stability constrained optimal power flow based on trajectory sensitivity for power dispatch of distributed synchronous generators

This paper is focused on the transient stability constrained optimal power flow (TSC-OPF) problem and its application for determining the optimal dispatch of active and reactive powers of distributed synchronous generators. In order to deal with uncertainties related to the actual active and reactive load values and the network topology configuration, this paper proposes a TSC- OPF formulation that incorporates a set of mathematical constraints based on the trajectory sensitivity analysis (TSA) that represents the impact of different load scenarios and network topologies on the transient stability of the system. The TSC-OPF is solved by an iterative algorithm based on the primal-dual interior point method and the concept of rotor angle first swing (FS) stability. The proposed TSC-OPF is applied for a distribution network constituted by nine buses and two distributed synchronous generators.

Transient Stability-Constrained Optimal Power Flow Calculation With Extremely Unstable Conditions Using Energy Sensitivity Method

In this paper, a transient stability margin is proposed in terms of the kinetic energy of power systems in extremely unstable conditions. A unified energy-based transient stability constraint is formed for both normal and extremely unstable conditions in the proposed transient stability-constrained optimal power flow (TSCOPF) model. A divide-and-conquer approach is presented to solve TSCOPF by decomposing it into optimal power flow and transient stability constraint formation subproblems. The former is solved by an interior point method and the latter is derived by an energy sensitivity technique. Furthermore, an accuracy-based perturbation strategy is proposed to address the system over-stabilization issue, and a parallel calculation technique is implemented to speed up the TSCOPF solution. The effectiveness of the proposed approach is investigated and the results are validated using the New England 10-generator and IEEE 50-generator systems under extremely unstable conditions.

Analysis of Numerical Methods to Include Dynamic Constraints in an Optimal Power Flow Model

The optimization of the operation of power systems including steady state and dynamic constraints is efficiently solved by Transient Stability Constrained Optimal Power Flow (TSCOPF) models. TSCOPF studies extend well-known optimal power flow models by introducing the electromechanical oscillations of synchronous machines. One of the main approaches in TSCOPF studies includes the discretized differential equations that represent the dynamics of the system in the optimization model. This paper analyzes the impact of different implicit and explicit numerical integration methods on the solution of a TSCOPF model and the effect of the integration time step. In particular, it studies the effect on the power dispatch, the total cost of generation, the accuracy of the calculation of electromechanical oscillations between machines, the size of the optimization problem and the computational time.

Parallel solution of transient stability constrained optimal power flow by exact optimality condition decomposition

Power systems are required to achieve an optimally economic operating point while maintaining security and stability in the presence of credible contingencies. Transient stability constrained optimal power flow (TSCOPF) is a tool to bridge steady-state optimal power flow (OPF) with transient processes under a predefined set of simulated contingencies to guarantee post-fault rotor angle stability in a simulation time window. A parallel solution of TSCOPF using exact optimality condition (OC) decomposition is proposed, where generator swing equations are utilised recursively by exploring the structure of OCs of TSCOPF from the end of simulation time window to its beginning to derive an exact explicit expression consisting of generator-dynamics-related variables in terms of the steady-state variables. The OCs of the TSCOPF model are then decomposed into the OC of OPF, along with a parallel evaluation of this expression for each contingency. Multi-core processing units are applied to accelerate the evaluation process. Case studies with up to 1047 buses over 16 contingences demonstrate an 8× improvement in the computation for realistically sized power systems using the proposed decomposition strategy.

Application of chaotic whale optimisation algorithm for transient stability constrained optimal power flow

Chaotic whale optimisation algorithm (CWOA) is a metaheuristic real-parameter optimisation algorithm. This study appears to be well capable of providing solution to the transient stability constrained optimal power flow (OPF) problem of power system. Basically, transient stability constrained OPF (TSCOPF) problem is the extended study of conventional OPF problem while additionally, considering transient stability constraints along with other previously considered equality and inequality constraints of the conventional OPF problem. Here, CWOA algorithm is validated by choosing two test power systems viz . (a) New England 10-generator, 39-bus and (b) 17-generator, 162-bus test systems. Considering multiple contingency cases, the main objective of the proposed algorithm is minimisation of the total fuel cost of these two test systems. Simulation test results, as obtained from the proposed CWOA, are compared to the results offered by some other evolutionary optimisation techniques surfaced in the recent state-of-the-art literature. The results presented in this study indicate that the proposed algorithm shows its efficacy over other recently originated popular optimisation techniques (including basic WOA) in terms of extending potential of offering higher quality solutions, effectiveness and faster convergence speed.