Stability-constrained Optimal Power Flow Research Articles

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.

Preventive–corrective stability constrained OPF in microgrids for accommodating DG plug-and-play

To accommodate unpredictable plug-and-play (PnP) of distributed generators (DGs) in droop controlled microgrids, this paper proposes a preventive–corrective stability constrained optimal power flow (SC-OPF). Firstly, considering that DG droops are restricted by load sharing, we introduce the battery energy storage system (BESS) with adjustable droops, which acts as a new control resource. Secondly, we propose a bi-level optimization problem, where the upper level ensures minimizing the preventive–corrective coordinated cost and accommodating DG PnPs without violating stability constraints and the lower level focuses on the stability index given by semi-definite program (SDP) based on the Lyapunov equation. With the previously proposed analytical stability sensitivity, the lower level problem is reformulated by a series of linear constraints through the Benders decomposition method, and hence the entire bilevel problem can be solved efficiently and accurately. We apply the modified IEEE 33-bus system to validate the proposed SC-OPF model. From simulation results, the proposed model significantly improves the economic cost, stability, and computational efficiency compared to other SC-OPF models without considering BESS droop control or PnPs.

Small signal stability constrained optimal power flow model based on trajectory optimization

The essence of traditional power system’s small signal stability analysis model using the eigenvalue analysis method is to analyze the time-invariant system obtained after the approximate linearization of the system, because it only considers the equilibrium state and therefore cannot consider the power system nonlinearity. In contrast, time-domain simulation can fully consider the system nonlinearity from the whole small signal dynamic period, because it considers not only the equilibrium state but also the state after the equilibrium point. On the basis of the time-domain simulation idea, this paper proposes an SSSC-OPF (small signal stability constrained-optimal power flow) based on optimizing the rotor angle trajectory. The objective function and the SSS constraint (small signal stability constraint) of the model are extracted from rotor angle curves of each generator, and the system stability is ensured by both in concert. Compared with the general SSSC-OPF model using the SSS constraint based on the eigenvalues, the proposed model can consider nonlinearity while avoiding the errors caused by the approximate linearization in the general SSSC-OPF model, and has a higher degree of generalizability. Finally, this paper performs the simulations in three test systems, the IEEE 9-bus test system, the IEEE 39-bus test system, and the IEEE 118-bus test system, and verifies the proposed model’s effectiveness by comparing the simulation results.

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.

Small-Signal Stability-Constrained Optimal Power Flow for Inverter Dominant Autonomous Microgrids

This article details and solves a small-signal stability-constrained optimal power flow (SSSC-OPF) for inverter-based ac microgrids. To ensure a sufficient stability margin during optimal generation, a small-signal stability constraint is embedded into the conventional OPF formulation. This condition is enforced using a Lyapunov stability equation. A reduced-order model of the microgrid is adopted to alleviate the computational burden involved in solving the resulting SSSC-OPF. Even then, the resulting stability conditions are highly nonlinear and cannot be handled using the existing methods. To tackle the nonconvexity in the SSSC-OPF due to the presence of the nonlinear stability constraint, two distinct convex relaxation approaches, namely semidefinite programming and parabolic relaxations, are developed. A heuristic penalty function is added to the objective function of the relaxed SSSC-OPF, which is solved sequentially to obtain a feasible point. While off-the-shelf tools fail to produce any feasible point within hours, the proposed approach enables us to solve the SSSC-OPF in near real time. The efficacy of the proposed SSSC-OPF is evaluated by performing numerical studies on multiple benchmarks as well as real-time studies on a microgrid system built in a controller/hardware-in-the-loop setup.

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.

Optimal Power Flow Considering Global Voltage Stability Based on a Hybrid Modern Heuristic Technique

Integrating large renewable energy to grid complicates power systems’ steady-state operation which leads to the high risk of voltage instability due to its intermittent and stochastic properties. The steady-state voltage stability margin (VSM) can be measured by the smallest singular value (SSV) of the load flow Jacobian matrix. Unlike other voltage stability indices such as L-index which shows the probability of voltage collapse for a particular bus, namely, local stability index, SSV represents the global voltage stability and the smaller the value is, the riskier the whole system's stability is. Voltage stability constrained-optimal power flow (VSC-OPF) is an effective tool to stabilize system voltage. Yet VSC-OPF is a highly non-linear and non-convex problem because of AC power flow and implicit SSV constraints. Such challenge is tackled, in this work, by a novel hybrid heuristic method, differential evolutionary particle swarm optimization (DEEPSO) which can easily formulate explicit constraints and objective functions to obtain near-optimal solutions efficiently. Several case studies are conducted on the IEEE 30-bus system, and they demonstrate the effectiveness of the hybrid technique and presents some insights on voltage profiles under various system operating condition.

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.

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.

Gas-to-electricity investment planning for power system stability improvement and environmental sustainability in Nigeria

Power outage is a prominent feature of the current Nigerian power system. However, a properly planned energy sector can help the nations quest for energy sustainability and economic development. Techno-economic assessment of the Nigerian energy facilities for efficient gas-to-grid power integration is presented in this paper using the particle swarm optimization algorithm (PSO) for solving a voltage stability-constrained optimal power flow model on Matlab environment. Investment in gas-fired DG technology can be an economic and sustainable approach for reducing the detrimental effeccts of gas-flaring practices of the petroleum industries on the environment. The technical benefits such as voltage profile improvement and voltage stability enhancement are the main focus of the technical analysis carried out in this study.

SDP-Based Optimal Power Flow With Steady-State Voltage Stability Constraints

This paper proposes a voltage stability-constrained optimal power flow (VSC-OPF) model based on semidefinite programming (SDP) relaxation. The minimum singular value of the power flow Jacobian is used as a steady-state voltage stability index, which is incorporated into the SDP formulation. To model an ADP constraint for voltage stability, an auxiliary matrix based on the power flow Jacobian is constructed, and this auxiliary matrix can be reformulated as a function of the semidefinite variable matrix defined for SDP relaxation. The resulting SDP-based VSC-OPF model is formulated and solved via the solver SDPT3 and the toolbox YALMIP. Extensive simulations on IEEE test systems validate the effectiveness of the proposed model.

Inverse Stability Problem and Applications to Renewables Integration

In modern power systems, the operating point, at which the demand and supply are balanced, may take different values due to changes in loads and renewable generation levels. Understanding the dynamics of stressed power systems with a range of operating points would be essential to assuring their reliable operation, and possibly allow higher integration of renewable resources. This letter introduces a non-traditional way to think about the stability assessment problem of power systems. Instead of estimating the set of initial states leading to a given operating condition, we characterize the set of operating conditions that a power grid converges to from a given initial state under changes in power injections and lines. We term this problem as "inverse stability", a problem which is rarely addressed in the control and systems literature, and hence, poorly understood. Exploiting quadratic approximations of the system's energy function, we introduce an estimate of the inverse stability region. Also, we briefly describe three important applications of the inverse stability notion: (i) robust stability assessment of power systems w.r.t. different renewable generation levels, (ii) stability-constrained optimal power flow (sOPF), and (iii) stability-guaranteed corrective action design.

A New Voltage Stability-Constrained Optimal Power-Flow Model: Sufficient Condition, SOCP Representation, and Relaxation

A simple characterization of the solvability of power flow equations is of great importance in the monitoring, control, and protection of power systems. In this paper, we introduce a sufficient condition for power flow Jacobian nonsingularity. We show that this condition is second-order conic representable when load powers are fixed. Through the incorporation of the sufficient condition, we propose a voltage stability-constrained optimal power flow (VSC-OPF) formulation as a second-order cone program (SOCP). An approximate model is introduced to improve the scalability of the formulation to larger systems. Extensive computation results on Matpower and NESTA instances confirm the effectiveness and efficiency of the formulation.

Optimal Curtailment of Non-Synchronous Renewable Generation on the Island of Tenerife Considering Steady State and Transient Stability Constraints

The increasing penetration of non-synchronous, renewable energy in modern power systems displaces synchronous generation and affects transient stability. This is just one of the factors that has led to preventive curtailment of renewable energy sources in an increasing number of electrical grids. Transient stability constrained optimal power flow (OPF) techniques provide a tool to optimize the dispatch of power systems while ensuring a secure operation. This work proposes a transient stability-constrained OPF model that includes non-synchronous generation with fault ride-through capability and reactive support during voltage dips. The model is applied it to the IEEE 39 Bus benchmark test case and to the power system of the Spanish island of Tenerife, and solved using the open-source library IPOPT that implements a primal-dual interior point algorithm. The solution of the model makes it possible to optimize the dispatch of conventional plants and the curtailment of non-synchronous generation, as well as to explore methods to reduce generation cost. Fault ride-through capability, synchronous inertia and fault clearing times are identified as useful tools to reduce the curtailment of non-synchronous generation, especially during periods of low load and high availability of renewable energy sources.

Directional Derivative-Based Transient Stability-Constrained Optimal Power Flow

This paper proposes a novel active power redispatch sequential approach to solve the transient stability-constrained optimal power flow (TSC-OPF) problem for the preventive control of transient stability. Based on an independent transient stability assessment of the power system for a given contingency scenario, two proposed power redispatch constraints are formulated and embedded into a conventional OPF formulation. Since these two new constraints only depend on steady-state variables, the dimension of the resulting TSC-OPF model is similar to that of a conventional OPF model and can be solved by standard optimization methods to perform a nonheuristic active power redispatch. The stabilization process is performed by sequentially solving the optimization and the transient stability problems until a suitable generation dispatch that provides a transiently stable equilibrium point is assessed. The validity and the effectiveness of the proposed method are numerically demonstrated in the WSCC three-machine, nine-bus system and an equivalent model of the Mexican power system.

Solution techniques for transient stability‐constrained optimal power flow – Part II

Transient stability-constrained optimal power flow is an important emerging problem with power systems pushed to the limits for economic benefits, dense and larger interconnected systems, and reduced inertia due to expected proliferation of renewable energy resources. In this study, two more approaches: single machine equivalent and computational intelligence are presented. Also discussed are various application areas, and future directions in this research area. A comprehensive resource for the available literature, publicly available test systems, and relevant numerical libraries is also provided.

Robust Transient Stability-Constrained Optimal Power Flow With Uncertain Dynamic Loads

Load dynamics has a substantial impact on transient stability of a power system, but has not been properly treated in transient stability-constrained operational planning. This paper proposes a robust transient stability-constrained optimal power flow (OPF) model considering dynamic load models and uncertain variations in the model parameters (e.g., load composition). The uncertainty is modeled by selecting a small yet representative number of deterministic scenarios that approximate the whole uncertainty space. A decomposition-based solution approach is then developed, which iterates between a master problem corresponding to the ordinary OPF, and a set of subproblems corresponding to transient stability assessment and stabilization constraint generation. The proposed model and solution approach is validated on the New England 39-bus system. The obtained solutions can well immunize against random realizations of uncertain load model parameters while maintaining transient stability.

Parallel Transient Stability-Constrained Optimal Power Flow Using GPU as Coprocessor

Graphics processing unit (GPU) has significantly increased the computing capacity of state-of-the-art high performance computing systems. This paper presents a novel GPU acceleration approach for transient stability-constrained optimal power flow (TSCOPF), which is one of the most computational challenging tasks in large-scale power system applications. Enabled by the revealed two-level decomposition parallelism in reduced-space interior point method, GPUs serve as plug-and-play coprocessors for time-consuming linear algebra operations in TSCOPF solving. Enhanced by multi-GPU processing and mixed-precision iterative refinement technique, the efficiency of solving TSCOPF is greatly improved without redesigning and reimplementing the existing algorithm framework. Numerical studies based on a series of test cases with up to 12 951 buses indicate the effectiveness of the proposed GPU-based approach for large-scale TSCOPF problems.

An Approach to Solve Transient Stability-Constrained Optimal Power Flow Problem Using Support Vector Machines

The Transient Stability-Constrained Optimal Power Flow (TSC-OPF) is a challenging optimization problem, and is the subject of several recent researches. This paper proposes a novel approach to solve TSC-OPF. In the proposed framework, Support Vector Machines (SVMs) are used to classify whether an operating condition satisfies predefined transient contingencies. A novel classification strategy is proposed to ensure the optimal solution satisfies all considered contingencies with certain security margin. Besides, the weight coefficients of the SVM are used as sensitivity measures in order to help optimization solver find solutions more effectively. The proposed approach is demonstrated for the New England system and the IEEE 300 bus system.