Transient Stability Constrained Optimal Power Flow Research Articles

Fast approach for transient stability constrained optimal power flow based on dynamic reduction method

The main challenge to solve the transient stability constrained optimal power flow (TSC-OPF) problem is its huge dimension due to numerous discretised transient variables and constraints. This problem becomes more serious when large power systems are considered. This study presents a fast approach to realise a global TSC-OPF based on dynamic reduction method, which decomposes the power system into several coherent areas and represents the original system by a reduced equivalent system. In this approach, the single transient stability constraint is obtained by simulating the reduced system instead of the full system. The new approach reduces the simulation execution time and thus increases the efficiency of the TSC-OPF. Two case studies indicate that the proposed approach can remarkably reduce the CPU time of the TSC-OPF procedure, compared with the TSC-OPF based on the full-system simulation. The new approach is very practical in solving the TSC-OPF problem in large power systems where numerous machines are coherent.

Selective transient stability-constrained optimal power flow using a SIME and trajectory sensitivity unified analysis

A novel approach based on both single machine equivalent (SIME) and trajectory sensitivity methods is proposed to formulate a transient stability-constrained optimal power flow (TSC-OPF) in the Euclidian space, where only one single stability constraint is necessary in the optimization problem to represent all dynamic and transient stability constraints of the multi-machine system. This formulation remarkably reduces the dimension of the optimization problem to one similar to a conventional OPF, resulting in a tractable approach to the preventive control of transient stability in realistic power systems. A unified framework of time domain analysis is proposed, where the transient stability, trajectory sensitivity and SIME analyses are all combined to assess the system's stability and to compute the sensitivity coefficients of the proposed transient stability constraint. Based on these sensitivity coefficients, three non-heuristic selection criteria are proposed to perform the preventive control by rescheduling only a selected number of generators, which is the commonly accepted practice followed by the system's operators. The validity and the effectiveness of the proposed method are numerically demonstrated in the 10-machine 39-bus New England system and Mexican 46-machine 190-bus system.

Contingency filtering technique for transient stability constrained optimal power flow

Transient stability constrained optimal power flow (TSCOPF) is an important and difficult problem. When multiple contingencies are considered, a reliable contingency filtering technique should be used to reduce the scale of TSCOPF problem. This study brings in the concepts of active contingency and critical contingency, and develops a novel contingency filtering strategy. Based on time-domain numerical simulations, the proposed contingency filtering strategy first screens all the considered contingencies and identifies active contingencies whose severe indices violate the pre-defined threshold of transient stability, then further finds out the critical contingencies in which some generators are most severely disturbed according to the severe indices trajectories. The severe indices can be such as maximal relative rotor angles, maximal transient generator voltage dips and so on. Taking only the critical contingencies into account, the scale of TSCOPF problem is reduced significantly. Interior point method is used to solve the reduced TSCOPF problem. Numerical results on several cases indicate that the proposed contingency filtering technique is reliable and efficient. Compared with the conventional TSCOPF approach, which involves all the contingencies, the proposed contingency filtering strategy possesses overwhelming advantages in CPU time and memory consumption, and is hopeful to solve TSCOPF problems with many contingencies.

A novel margin sensitivity based method for transient stability constrained optimal power flow

Transient stability constrained optimal power flow (TSCOPF) is still a difficult but important problem in power system planning and operation. In this paper, the challenging TSCOPF problem is divided into optimal power flow (OPF) and stability analysis processes, and solved iteratively by the interior point method (IPM). The novelty of the proposed approach is that the transient stability constraints for each contingency are incorporated into the OPF model as a single stability constraint derived from the minimum kinetic energy for normal unstable case or the minimum accelerating power distance for extreme unstable case using the time domain simulation based single machine equivalent (SIME) method. The proposed constraint is robust and scalable for large power systems as well as applicable to multi-swing unstable, normal unstable and extreme unstable cases. In addition, this stability constraint is further refined to overcome the issue of over-stabilization by guiding the solution gradually across the stability boundary. As a whole, a complete solution method capable to solve multi-contingency TSCOPF problems is presented in this paper. The effectiveness and computation performance of the proposed approach have been fully studied and extensively validated using the New England 10-generator system and the IEEE 50-generator 145-bus system.

Parallel augment Lagrangian relaxation method for transient stability constrained unit commitment

This paper presents a transient stability constrained unit commitment (TSCUC) model which achieves the objective of maintaining both transient stability and economical operation. In the TSCUC model, transient stability constraints are incorporated into the framework of unit commitment. In order to solve TSCUC problem, augmented Lagrangian relaxation (ALR) combined with variable duplication techniques and the auxiliary problem principle (APP) is used to decompose the TSCUC problem into two sub-problems: one sub-problem is a traditional unit commitment (UC) problem with prevailing constraints; another sub-problem is modeled as a transient stability constrained optimal power flow (TSCOPF) problem. The first sub-problem is solved by dynamic programming, while the second sub-problem is solved using a reduced-space interior point method. In ALR, an efficient hybrid sub-gradient method is developed to update all Lagrangian multipliers. The iterative process continues until the duality gap is sufficiently small. Finally, case studies show that the proposed methodology is very efficient for solving TSCUC problems within a parallel computing framework.

Transient stability constrained optimal power flow using independent dynamic simulation

Transient stability constrained optimal power flow (TSC-OPF) is originally a non-linear optimisation problem with variables and constraints in time domain, which is not easy to deal with directly because of its huge dimension. This study presents an efficient approach to realise TSC-OPF by introducing an independent dynamics simulation algorithm into the optimisation procedure. In the new approach, the simulation algorithm is used to realise the dynamics constraints and to deduce the transient stability constraint, whereas the optimisation algorithm verifies the steady state and the transient stability constraints together. The new TSC-OPF has just one more constraint than that of a conventional OPF and can be solved by a conventional OPF algorithm with small modification. Moreover, the new approach makes it easy to improve the accuracy and efficiency of TSC-OPF, because of its flexibility in choosing machine models and simulation methods, which is important for large power systems. In the study, the proposed approach is tested with two small power systems, where the two-axis machine model and a mixed time step simulation method are used to assess system transient stability with a 1-ms time resolution.

A global approach to transient stability constrained optimal power flow using a machine detailed model

Transient stability constrained optimal power flow (TSC-OPF) is originally a nonlinear optimization problem with variables and constraints in time domain, which is not easy to deal with because of its huge dimension, especially for systems with detailed machine models. This paper presents an efficient approach to realize TSC-OPF by introducing an independent dynamics simulation algorithm into the optimization procedure. In the new approach, the simulation algorithm is used to realize the dynamic constraints and to deduce the transient stability constraint, while the optimization algorithm verifies the steady state and the transient stability constraints together. The new TSC-OPF has just one more constraint than that of a conventional OPF and can be solved by a conventional OPF algorithm with small modification. In the new approach, there is no limitation for the machine model and the simulation method. The nonlinearity of the power system is taken fully into account. In the paper, the proposed approach is verified with a small three-machines system. The simulation results show the machine model influences greatly the system transient stability and the TSC-OPF results. The widely used machine classical model in the TSC-OPF over-estimates the system transient stability and under-estimates the TSC-OPF costs.

Solution of transient stability-constrained optimal power flow using artificial bee colony algorithm

The transient stability constraint should be taken into consideration for the solution of the optimization problems in power systems. This paper presents a solution for the transient stability-constrained optimal power flow (TSCOPF) by a novel approach based on the artificial bee colony algorithm. The formulas of TSCOPF are derived through the addition of rotor angle inequality constraints into optimal power flow relationships. In this nonlinear optimization problem, the objective function is taken into consideration as the fuel cost of the system. The proposed approach is tested on both a WSCC 3-generator, 9-bus system and a IEEE 30-bus system, and the test results obtained are compared to prove the efficiency and effectiveness of the approach with other approaches in the literature.

Two approaches to transient stability-constrained optimal power flow

Formulation of transient stability-constrained optimal power flow (TSC-OPF) and finding a practical solution for the problem have gained much attention recently. In this paper, two approaches to include transient stability constraints in the OPF problem considering detailed dynamic models for generators and their controls are introduced. The first method is based on the maximum relative rotor angle deviation (MRRAD) of generators which suits systems that have specific requirements on MRRAD. The second method represents the transient stability margin of the system based on generators output power (GOP) and hence does not rely on MRRADs. The transient stability boundary can be represented by a nonlinear function of GOP. The Artificial Neural Network (ANN) curve-fitting tool is used to derive a mathematical formulation for the transient stability boundary (TSB). The closed form representation of the TSB is then inserted in the OPF problem as a new constraint. The proposed method is examined using the WSCC 9-bus, the New England 39-bus and the IEEE 300-bus test systems. The results indicate that the proposed methods lead to lower computational time in solving TSC-OPF which has been a serious challenge for this problem.

An improved micro-particle swarm optimization algorithm and its application in transient stability constrained optimal power flow

SUMMARY In this paper, an improved micro-particle swarm optimization (μPSO) algorithm is proposed and applied to the development of a computational method for solving Transient Stability Constrained Optimal Power Flow (TSCOPF) problem. The improvement includes two novel intelligent strategies: the dynamic space adjustment and the selective re-initialization. The first strategy is designed to accelerate the convergence speed by confining the search in a reduced space, while the second strategy is to reduce the possible overabundance by re-initializing the particles only in a non-overlapping space. The simulation results demonstrate the superiority of the improved μPSO algorithm over both the original μPSO algorithm and the standard PSO algorithm. The proposed method is validated through case studies for solving TSCOPF problems on IEEE 39-bus system and IEEE 162-bus system. The results show that the algorithm can utilize a small population to obtain an equally good or better optimization outcomes compared to those of the original μPSO algorithm and the standard PSO algorithm. Copyright © 2012 John Wiley & Sons, Ltd.

A Two-Level Parallel Decomposition Approach for Transient Stability Constrained Optimal Power Flow

Transient stability constrained optimal power flow (TSCOPF) is able to reduce costs while keeping the operation point away from the stability boundary. While especially useful in modern power system operations, TSCOPF problems are practically very hard to solve; unacceptable computational time is considered to be one of the largest barriers in applying TSCOPF-based solutions. Based on the reduced-space interior point method (RIPM)-which takes advantage of the relatively few degrees of freedom and shows promising potential for solving large-scale TSCOPF problems-this paper introduces a parallel RIPM algorithm with high computing efficiency for multi-contingency TSCOPF problems. A two-level parallelism is developed to fully utilize the computing power of a Beowulf cluster equipped with multi-core CPUs. First, several compute-intensive steps of the TSCOPF algorithm are decomposed according to different contingencies with mathematical equivalent transformations, the corresponding computing tasks are assigned, stored, and processed on different nodes. Second, the distributed computing task is accelerated using elemental decomposition on Jacobian matrices, and then high performance multithreaded mathematical libraries are employed to fully exploit the multi-core CPUs on each node. The effectiveness of the proposed parallel algorithm is benchmarked on a Beowulf cluster with 16 computing nodes with 128 CPU cores using a series of test cases including up to 2746 buses and 16 contingencies. The results of the case studies indicate that the proposed parallel decomposition approach inherits the optimal solution and convergence properties of the original serial interior point method (IPM) approach and shows great capacity in accelerating TSCOPF solution.

A Hybrid Method for Transient Stability-Constrained Optimal Power Flow Computation

A hybrid method combining classical deterministic programming technique and evolutionary algorithm (EA)-enhanced stochastic optimization strategy is proposed for more effective and flexible computation of transient stability-constrained optimal power flow (TSC-OPF). The method consists of searching the maximum TSC-feasible solution region with EA and deterministically optimizing an operating point in the found region via a conventional OPF solution. Stability constraints are exactly treated by a rigorous transient stability assessment (TSA) procedure, where complex system model and multi-contingency can be readily considered and stability margin is applied as the transient stability index (TSI) to avoid over-stabilizing. Benefiting from both the EA and the deterministic programming, the proposed method can continuously approach the global optima in a robust, reliable and fast-convergent way. The method is verified on the New England test system and a dynamic equivalent system of a real-world large power grid. Comparing with existing methods, it has provided more economic solutions and better adaptability to multi-swing instability and multi-contingency stabilizing. It can also be a unified approach for solving other stability constrained-OPFs.

A New Practical Approach to Transient Stability-Constrained Optimal Power Flow

This paper presents a new, significantly improved, approach to formulate a global transient stability-constrained optimal power flow (TSC-OPF), where the sets of dynamic and transient stability constraints to be considered in the optimization process are reduced to one single stability constraint. This constraint is derived from dynamic information provided by the SIngle Machine Equivalent (SIME) method and is only expressed in terms of steady-state variables, which allows us to diminish the length of the time-domain simulation to be included into the global TSC-OPF to a single (initial) time step. In this way, the size of the resulting optimization problem is reduced to one very similar to that of a conventional OPF, overcoming the main drawback of global TSC-OPF techniques (its huge dimension) while maintaining its accuracy and improving its practical application to real power networks. Effectiveness of the proposal is demonstrated by numerical examples on the WSCC three-machine, nine-bus system and the Mexican 46-machine, 190-bus system.

Transient stability-constrained optimal power flow for online dispatch and nodal price evaluation in power systems with flexible AC transmission system devices

This study develops a new method for economic dispatch together with nodal price calculations which includes transient stability constraints and, at the same time, optimises the reference inputs to the flexible AC transmission system (FACTS) devices for enhancing system stability and reducing nodal prices. The method draws on the sensitivity analysis of time-domain transient stability simulation results to derive a set of linearised stability constraints expressed in terms of generator active-powers and FACTS devices input references. The optimal power flow (OPF) formulation based on the Newton's algorithm augmented with the set of stability inequality constraints is applied for forming economic dispatch solution and nodal prices. The solutions obtained also include the Lagrange multipliers associated with all of the constraints. Based on the Lagrange multipliers, individual components, including those associated with stability constraints, of the nodal prices are directly identified for assisting the market participants to make investment decisions. The sensitivity technique for forming the set of transient stability constraints lends itself to parallel-computing implementation. Therefore the additional computing-time requirement is minimal, given that the transient stability simulations can be performed independently of one another and outside the OPF loop. Representative studies and results are presented here.

An Enhanced Numerical Discretization Method for Transient Stability Constrained Optimal Power Flow

Although many efforts have been made in past years, transient stability constrained optimal power flow (TSOPF) remains one of the most difficult problems in power system. A popular approach to deal with transient stability constraints is the numerical discretization method, in which TSOPF is converted to a generalized large-scale nonlinear programming problem, and interior point method is preferred to solve it. In the numerical discretization and interior point method based TSOPF, more than 80%-90% CPU seconds are used in solving the primal-dual linear system. In order to improve the computational efficiency of numerical discretization TSOPF, an enhanced numerical discretization method is proposed in this paper. The key enhancement of the proposed approach is: considering the truncation error of specific numerical integration algorithm, the transient differential equations are discretized to inequality constraints instead of equality constraints. This key enhancement reduces nearly 50% of the primal-dual linear system's dimension and greatly improves the computational efficiency of interior point based TSOPF algorithm. Case studies on several test cases up to 678-bus system indicate that the enhanced approach is much more computationally efficient than the conventional numerical discretization method and is promising to solve larger TSOPF problems.

A Reduced-Space Interior Point Method for Transient Stability Constrained Optimal Power Flow

Transient stability constrained optimal power flow (TSOPF) is a big challenge in the field of power system operation because of its computational complexity. At present, the combination of numerical discretization method with interior point method (IPM) is considered as one of the best algorithms for large-scale TSOPF problems. However, it still suffers from the curse of dimensionality as well as unacceptable computational time and memory consumption. Although the TSOPF problem after numerical discretization is a very large nonlinear programming problem (NLP), its degrees of freedom is relatively small, which makes it very suitable for solution by a reduced-space technique. Considering this characteristic of TSOPF problems, the combination of a reduced-space technique with predictor-corrector IPM is presented in this paper as a way of relieving the computational burden of numerical discretization-based TSOPF algorithms. Several key steps of the reduced-space approach, including building basis matrices, partition strategy, computation of cross term and reduced Hessian, are discussed in detail. Detailed case studies indicate that the proposed reduced-space approach can remarkably reduce both CPU time and memory usage compared to the full-space approach, and therefore is a very promising method for solving large-scale TSOPF problems.

Global Transient Stability-Constrained Optimal Power Flow Using an OMIB Reference Trajectory

This paper presents a new approach to transient stability control using global transient stability-constrained optimal power flow (TSC-OPF) methods. Its novelty consists in using the single machine equivalent (SIME) method to perform (and improve) two main important functions of global TSC-OPF approaches: first, SIME is used to efficiently perform the power system transient stability analysis; second, SIME determines a stable one machine infinite bus equivalent rotor angular trajectory that is used as the reference stability constraint, at one specific integration step. In this way, the stability constraint is adjusted by SIME, at each iteration of the TSC-OPF method, in order to accurately reflect power system dynamic behavior. The prowess and main characteristics of the proposed approach are shown by numerical examples.

Adaptive Evolutionary Programming Incorporating Neural Network for Transient Stability Constrained Optimal Power Flow

This paper presents an adaptive evolutionary programming incorporating neural network for solving transient stability constrained optimal power flow (TSCOPF). The proposed AEP method is an evolutionary programming (EP)-based algorithm, which adjusts its population size automatically during an optimization process. The artificial neural network, which classifies the AEP individual based on its stability degrees, is embedded into the search template to reduce the computational load caused by transient stability constraints. The fuel cost minimization is selected as the objective function of TSCOPF. The proposed method is tested on the IEEE 30-bus system with two types of the fuel cost functions, i.e. the conventional quadratic function and the quadratic function superimposed by sine component to model the cost curves without and with valve-point loading effect respectively. The numerical examples show that AEP is more effective than conventional EP in terms of computational speed, and when the neural network is incorporated into AEP, it can significantly reduce the computational time of TSCOPF. A study of the architecture of the neural network is also conducted and discussed. In addition, the effectiveness of the proposed method for solving TSCOPF with the consideration of multiple contingencies is manifested.

Total Transfer Capability Assessment Incorporating Corrective Controls for Transient Stability using TSCOPF

Advancements have been made towards inclusion of both static and dynamic security into transfer capability calculation. However, to the authors' knowledge, work on considering corrective controls into the calculation has not been reported yet. Therefore, we propose a Total Transfer Capability (TTC) assessment considering transient stability corrective controls. The method is based on the Newton interior point method for nonlinear programming and transfer capability is approached as a maximization of power transfer with both static and transient stability constraints are incorporated into our Transient Stability Constrained Optimal Power Flow (TSCOPF) formulation. An interconnected power system is simulated to be subjected to a severe unbalanced 3-phase 4-line to ground fault and following the fault, generator and load are shed in a pre-defined sequence to mimic actual corrective controls. In a deregulated electricity market, both generator companies and large load customers are encouraged to actively participate in maintaining power system stability as corrective controls upon agreement of compensation for being shed following a disturbance. Implementation of this proposal on the actual power system operation should be carried out through combining it with the existing transient stabilization controller system. Utilization of these corrective controls results in increasing TTC as suggested in our numerical simulation. As Lagrange multipliers can also describe sensitivity of both inequality and equality constraints to the objective function, then selection of which generator or load to be shed can be carried out on the basis of values of Lagrange multipliers of its respective generator's rotor angle stability and active power balance equation. Hence, the proposal in this paper can be utilized by system operator to assess the maximum TTC for specific loads and network conditions.

Application of Differential Evolution Algorithm for Transient Stability Constrained Optimal Power Flow

Consideration of transient stability constraints in optimal power flow (OPF) problems is increasingly important because modern power systems tend to operate closer to stability boundaries due to the rapid increase of electricity demand and the deregulation of electricity markets. Transient stability constrained OPF (TSCOPF) is however a nonlinear optimization problem with both algebraic and differential equations, which is difficult to be solved even for small power systems. This paper develops a robust and efficient method for solving TSCOPF problems based on differential evolution (DE), which is a new branch of evolutionary algorithms with strong ability in searching global optimal solutions of highly nonlinear and nonconvex problems. Due to the flexible properties of DE mechanism, the hybrid method for transient stability assessment, which combines time-domain simulation and transient energy function method, can be employed in DE so that the detailed dynamic models of the system can be incorporated. To reduce the computational burden, several strategies are proposed for the initialization, assessment and selection of solution individuals in evolution process of DE. Numerical tests on the WSCC three-generator, nine-bus system and New England ten-generator, 39-bus system have demonstrated the robustness and effectiveness of the proposed approach. Finally, in order to deal with the large-scale system and speed up the computation, DE is parallelized and implemented on a Beowulf PC-cluster. The effectiveness of the parallel DE approach is demonstrated by simulations on the 17-generator, 162-bus system.