Intermediate variables for the calculation of the transient voltage in the d-axis when using the RK4 method ABS,FW , ABS,FW Active and reactive power absorbed by a flywheel energy storage system [p.u.].

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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.INDEX TERMS numerical methods, optimal power flow, power system stability, transient stability,Number of synchronous generators and buses, respectively. c iCost coefficient of the synchronous generator at bus i (in e M W h ). P g,i,0 , Q g,i,0Steady-state active and reactive power generated by the synchronous generator at bus i (in p.u.). P g,i,t , Q g,i,tActive and reactive power generated by the synchronous generator at bus i and time t (in p.u.).Active and reactive load at bus i and time t (in p.u.). P e,i,t , Q e,i,tElectric output active and reactive power in the rotor of the synchronous generator at bus i and time t (in p.u.). P m,iMechanical input power of the synchronous generator at bus i (in p.u.).V i,0 , α i,0 Steady state voltage magnitude and angle at bus i (in p.u. and rad, respectively). V i,t , α i,tVoltage magnitude and angle at bus i and time t (in p.u. and rad, respectively). I brSteady state current of branch br (in p.u.). Y i,j , θ i,jMagnitude and angle of the element (i, j) in the bus admittance matrix Y (in p.u. and rad, respectively). Y L,br,j , θ L,br,j Magnitude and angle of the element (br, j) in the line admittance matrix Y L (in p.u. and rad, respectively). δ i,tRotor angle of synchronous generator i at time t in synchronously rotating reference frame (in rad). δ COI,tThe rotor angle corresponds to the center of inertia (COI) at time t (in rad). ω s Synchronous rotor speed (in rad s ). ∆ω i,0Steady-state rotor speed deviation of the