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ime - domain simulation is an important approach for power system dynamic analysis . however , the complete system model , or interchangeably the long - term stability model , typically includes different components where each component requires several differential and algebraic equations ( dae ) to represent , at the same time , these dynamics involve different time scales from millisecond to minute . as a result , the total number of dae of a real power system can be formidably large and complex such that time domain simulation over long time intervals is expensive@xcite . these constraints are even more stringent in the context of on - line stability assessment . intense efforts have been made to accelerate the simulation of long - term stability model . one approach is to use a larger time step size to filter out the fast dynamics or use automatic adjustment of step size according to system behavior in time - domain simulation @xcite@xcite@xcite from the aspect of numerical method . another approach is to implement the quasi steady - state ( qss ) model in long - term stability analysis @xcite@xcite from the aspect of model approximation . nevertheless , the qss model suffers from numerical difficulties when the model gets close to singularities which were addressed in @xcite-@xcite . moreover , the qss model can not provide correct approximations of the long - term stability model consistently as numerical examples shown in @xcite@xcite . in addition , sufficient conditions of the qss model were developed in @xcite which pointed to a direction to improve the qss model . as a result , the qss model requires improvements in both model development and numerical implementation . this paper contributes to the latter one . in this paper , we apply pseudo - transient continuation ( @xmath0 ) which is a theoretical - based numerical method in power system long - term stability analysis . pseudo - transient continuation method can be implemented directly in the long - term stability model to accelerate simulation speed compared with conventional implicit integration method . on the other hand , the method can also be applied in the qss model to overcome possible numerical difficulties due to good stability property . this paper is organized as follows . section [ sectiondyptc ] briefly reviews general pseudo - transient continuation method in dae system . section [ sectionptcinpowersystem ] includes a introduction about power system models followed by implementation of pseudo - transient continuation method in the long - term stability model and the qss model respectively . section [ sectionnumerical ] presents three numerical examples to show the feasibility of the method . and conclusions are stated in section [ sectionconclusion ] . pseudo - transient continuation is a physically - motivated method and can be used in temporal integration . the method follows the solution of dynamical system accurately in early stages until the steady state is approaching . the time step is thereafter increased by sacrificing temporal accuracy to gain rapid convergence to steady state @xcite . if only the steady state of a dynamical system instead of intermediate trajectories is of interest , pseudo - transient continuation method is a better choice than accurate step - by - step integration . on the other hand , compared with methods that solve nonlinear equations for steady state such as line - search and trust region methods , pseudo - transient continuation method can avoid converging to nonphysical solutions or stagnating when the jacobian matrix is singular . this is particularly the case when the system has complex features such as discontinuities which exist in power system models . therefore , @xmath0 method can be regarded as a middle ground between integrating accurately and calculating the steady state directly . @xmath0 method can help reach the steady state quickly while maintain good accuracy for the intermittent trajectories . for ode dynamics , sufficient conditions for convergence of @xmath0 were given in @xcite . the results were further extended the semi - explicit index - one dae system in @xcite . we recall the basic algorithm here . we consider the following semi - explicit index - one dae system : @xmath1 with initial value @xmath2 . here @xmath3 , @xmath4 , @xmath5^t \in \re^{n_1+n_2}$ ] , and @xmath6 where @xmath7 is a nonsingular scaling matrix . we assume the initial condition for ( [ dae ] ) is consistent , i.e. @xmath8 and seek to find the equilibrium point @xmath9 such that @xmath10 and satisfies @xmath11 .    as stated before , the step - by - step integration is too time consuming if the intermediate states are not of interest . on the other hand , newton s method for @xmath12 alone usually fails as the initial condition is not sufficiently near the equilibrium point . the @xmath0 procedure is defined by the iteration : @xmath13 where @xmath14 is adjusted to efficiently find @xmath9 rather than to enforce temporal accuracy . the convergence results in @xcite@xcite assume that the time step is updated with `` switched evolution relaxation''(ser ) : @xmath15    the algorithm is shown as below :    set @xmath16 and @xmath17 . evaluate @xmath18 .    while @xmath19 is too large . solve @xmath20 . set @xmath21 . evaluate @xmath18 . update @xmath22 according to ( [ delta ] ) . step 2.a is a newton step which is typically solved by an iterative method which terminates on small linear residuals while it may also be solved by inexact newton iteration . note that the failure of @xmath0 usually can be well signaled by reaching the bound on the total number of iterations @xcite . the convergence of @xmath0 for smooth @xmath23 was proved in @xcite under the assumptions that the dae has index one in a certain uniform sense , that it has a global solution in time , and that the solution converges to a steady state . the result were further extended to nonsmooth @xmath23 in @xcite with @xmath24 in ( [ ptc ] ) replaced by a generalized derivative . next we explain why @xmath0 has a better stability property . firstly , conventional integration methods insist on a small norm of the linear residual at each step and will either converge , diverge to infinity , or stagnate at a point where iteration matrix is singular . however , @xmath0 method will accept an increase in the residual , responding to that increase by decreasing @xmath22@xcite . in addition , @xmath0 stems from backward euler method which is an attractive choice when stability is the desired property instead of accuracy . one may think of @xmath0 method as a predictor - corrector method where the simple predictor is from previous time step and the corrector is backward euler with newton iteration @xcite . to see this , consider the implicit euler step from @xmath25 with @xmath26 , @xmath27 thus , @xmath28 is the root of the following equation : @xmath29 the newton s method to find the root of the above equation is : @xmath30 if we take @xmath31 , then by binomial inverse theorem , the first newton iterate is : @xmath32 which is exactly @xmath0 step . as @xmath0 method has a better stability property , it can be applied in the qss model when conventional integration method fails to converge . in this section , we firstly introduce power system models in long - term stability analysis . then we apply @xmath0 method to the long - term stability model with modifications . finally , we present an algorithm of @xmath0 method in the qss model to overcome possible numerical difficulties .      the long - term stability model for calculating system dynamic response relative to a disturbance can be described as : @xmath33    equation ( [ algebraic eqn ] ) describes the transmission system and the internal static behaviors of passive devices , and ( [ fast ode ] ) describes the internal dynamics of devices such as generators , their associated control systems , certain loads , and other dynamically modeled components . @xmath34 and @xmath35 are continuous functions , and vector @xmath36 and @xmath37 are the corresponding short - term state variables and algebraic variables . besides , equations ( [ slow ode ] ) and ( [ slow dde ] ) describe long - term dynamics including exponential recovery load , turbine governor , load tap changer ( ltc ) , over excitation limiter ( oxl ) , etc . @xmath38 and @xmath39 are the continuous and discrete long - term state variables respectively , and @xmath40 is the maximum time constant among devices . note that shunt compensation switching and ltc operation are typical discrete events captured by ( [ slow dde ] ) and @xmath41 is shunt susceptance and the transformer ratio correspondingly . transitions of @xmath41 depend on system variables , thus @xmath41 change values from @xmath42 to @xmath43 at distinct times @xmath44 where @xmath45 , otherwise , these variables remain constants . since short - term dynamics have much smaller time constants compared with those of long - term dynamics , @xmath46 and @xmath41 are also termed as slow state variables , and @xmath47 are termed as fast state variables .    if we represent the long - term stability model and the qss model in @xmath48 time scale , where @xmath49 , and we denote @xmath50 as @xmath51 , then the long - term stability model of power system can be represented as : @xmath52 where the study region @xmath53 , and @xmath54 , @xmath55 , @xmath56 , @xmath57 .    at the same time , the qss model can be represented as : @xmath58 moreover , the long - term stability model ( [ complete ] ) can be regarded as two decoupled systems ( [ couple1 ] ) and ( [ couple2 ] ) showed as below when @xmath41 jump from @xmath42 to @xmath43 : @xmath59 and @xmath60 discrete variables @xmath41 are updated first and then system ( [ couple2 ] ) works with fixed parameters @xmath41 . similarly , when @xmath41 jump from @xmath42 to @xmath43 , the qss model ( [ qss ] ) can be decoupled as : @xmath61 and @xmath62      assuming @xmath63 is nonsingular , then the long - term stability model ( [ couple2 ] ) with @xmath41 fixed as parameters is a semi - explicit index-1 dae system . @xmath0 method requires initial condition to satisfy the algebraic equations , however , the discrete equation ( [ couple1 ] ) will violate this condition whenever it works . as a result , we need to modify the original @xmath0 method for its implementation in long - term stability model . in power system long - term stability model , @xmath64^t$ ] , @xmath65^t\in\re^{p+m+n}$ ] , @xmath66 , where @xmath67 is the identiy matrix of size @xmath68 . in order to make the initial condition of @xmath0 consistent , we switch back to implicit integration method whenever discrete variables jump and set the step length to be @xmath69 . moreover , @xmath0 method is implemented for the post - fault system starting from @xmath70several seconds after the contingency . in examples of this paper , @xmath70 was set to be @xmath71 . the proposed algorithm is shown as below . run the long - term stability model up to @xmath70 by implicit integration method . set the value @xmath72 at @xmath70 as the initial condition @xmath73 of @xmath0 , and set @xmath17 .    while @xmath74 is too large . if discrete variables jump at @xmath44    update @xmath41 according to ( [ couple1 ] ) . set @xmath75 , @xmath17 . solve the newton step @xmath76 . set @xmath77 . evaluate @xmath78 . otherwise    set @xmath79 . solve the newton step @xmath80 . set @xmath77 . evaluate @xmath78 . update @xmath22 according to ( [ delta ] ) . note that @xmath81 and @xmath82 depend on the specific integration method used . for instance , if implicit trapezoidal method is used , then @xmath83 @xmath84      assuming @xmath85 is nonsingular , then the qss model ( [ coupleqss2 ] ) with @xmath41 fixed as parameters is a semi - explicit index-1 dae system . @xmath0 method requires initial condition to satisfy the algebraic equations , however , the discrete equation ( [ coupleqss1 ] ) will violate this condition whenever it works . as a result , we need to modify the original @xmath0 method for its implementation in the qss model .    in the qss model , @xmath64^t$ ] , @xmath65^t\in\re^{p+m+n}$ ] , @xmath86 , where @xmath67 is the identity matrix of size @xmath87 . in order to make the initial condition of @xmath0 consistent , we switch back to implicit integration method whenever discrete variables jump and set the step length to be @xmath69 . besides , the qss model is implemented at @xmath88when short - term dynamics settle down after the contingency . usually , @xmath88 can be set as 30s . the proposed algorithm is shown as below . run the long - term stability model up to @xmath88 by implicit integration method . set the value @xmath72 at @xmath88 as the initial condition @xmath73 of the qss model , and set @xmath17 . start to run the qss model . if the qss model has a numerical difficulty by using implicit integration method , then go to step 3 , otherwise , continue with the qss model .    while @xmath74 is too large . if discrete variables jump at @xmath44    update @xmath41 according to ( [ coupleqss1 ] ) . set @xmath75 , @xmath17 . solve the newton step @xmath76 . set @xmath77 . evaluate @xmath78 . otherwise    set @xmath79 . solve the newton step @xmath80 . set @xmath77 . evaluate @xmath78 . update @xmath22 according to ( [ delta ] ) .    similarly , @xmath81 and @xmath82 depend on the specific integration method used . if implicit trapezoidal method is used , then @xmath89 @xmath90 in this section , three examples are to be presented . the first two examples were the same 145-bus system in which the qss model met numerical difficulties during simulation while the long - term stability model was stable in long - term time scale . firstly , @xmath0 method was implemented in the long - term stability model and the speed was more than 7 times faster than the trapezoidal method . secondly , when the qss model by trapezoidal method met difficulty , @xmath0 method was implemented in the qss model and provided correct approximations and the speed was still more than 5 times faster than the long - term stability model by trapezoidal method . and in the last example which was a 14-bus system , the long - term stability model was unstable and @xmath0 method successfully captured the instability which was signaled by reaching the bound of maximum iteration in the newton step . all simulations were done in psat-2.1.6@xcite . the system was a 145-bus system @xcite . there were exciters and power system stabilizers for each of generator 1 - 20 . and there were turbine governors for each of generator 30 - 40 . besides , there were three load tap changers at lines between bus 73 - 74 , bus 73 - 81 and bus 90 - 92 respectively . the contingency was a line loss between bus 1 - 6 . in this case , the post - fault system was stable after the contingency in the long - term time scale and it took 122.39s for the time domain simulation of the long - term stability model by implicit trapezoidal method . however , @xmath0 method only took 16.12s for the simulation of the long - term stability model which was about @xmath91 of the time consumed by trapezoidal method . [ tdsptc ] shows that the trajectories by @xmath0 method followed closely to the trajectories by trapezoidal method , and finally both converged to the same long - term stable equilibrium point . thus @xmath0 method provided correct approximations for the long - term stability model in terms of trajectories and stability assessment .     method . @xmath0 method provided correct approximations.,width=172 ]     method . @xmath0 method provided correct approximations.,width=172 ]     method . @xmath0 method provided correct approximations.,width=172 ]     method . @xmath0 method provided correct approximations.,width=172 ]      in this example , the system was the same as the last case . however , the qss model met numerical difficulties at 40s when implicit trapezoidal method was used , thus @xmath0 method was implemented in the qss model starting from 40s . [ tdsptcqss ] shows that the trajectories by @xmath0 method converged to the long - term stable equilibrium point which the long - term stability model converged to , and also provided good accuracy for the intermittent trajectories . it took 21.75s for @xmath0 method which was about @xmath92 of the time consumed by the long - term stability model using implicit trapezoidal method .     method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ]     method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ]     method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ]     method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ]      in this case , the 14-bus system was long - term unstable due to wild oscillations of fast variables . the system was modified based on the 14-bus test system in psat-2.1.6@xcite . there were three exponential recovery loads at bus 9 , 10 and 14 respectively and two turbine governors at generator 1 and 3 . besides , there were over excitation limiters at all generators and three load tap changers at lines between bus 4 - 9 , bus 12 - 13 and bus 2 - 4 . the system suffered from long - term instabilities and simulation by implicit trapezoidal method could not continue after 101.22s . @xmath0 method also stopped at 103.34s when the bound on the total number of iterations for the newton step was reached . thus @xmath0 method was able to capture instabilities of the long - term stability model .     method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ]     method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ]     method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ]     method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ] in this paper , modified @xmath0 methods for the long - term stability model and the qss model are given for power system long - term stability analysis with illustrative numerical examples . we make use of the fast asymptotic convergence of @xmath0 method in the long - term stability model to achieve fast simulation speed . on the other hand , we take advantage of good stability property of @xmath0 method in the qss model to overcome numerical difficulties . numerical examples show that @xmath0 can successfully provide correct stability assessment for the long - term stability model and overcome numerical difficulties in the qss model , as well as offer good accuracy for the intermediate trajectories . @xmath0 can be regarded as a good in - between method with respect to integration and steady state calculation , thus serves as an alternative method in power system long - term stability analysis . this work was supported by the consortium for electric reliability technology solutions provided by u.s . department no . de - fc26 - 09nt43321 . kurita , a. , h. okubo , k. oki , s. agematsu , d. b. klapper , n. w. miller , w. w. price jr , j. j. sanchez - gasca , k. a. wirgau , and t. d. younkins , _ multiple time - scale power system dynamic simulation _ , ieee transactions on power systems , vol 216 - 223 , 1993 . t. v. cutsem , m. e. grenier , d. lefebvre , _ combined detailed and quasi steady - state time simulations for large - disturbance analysis_. international journal of electrical power and energy systems , vol . 28 , issue 9 , pp . 634 - 642 , november 2006 . x. z. wang , h. d. chiang , _ numerical investigations on quasi steady - state model for voltage stability : limitations and nonlinear analysis_. submitted to international transactions on electrical energy systems . x. z. wang , h. d. chiang , _ analytical studies of quasi steady - state model in power system long - term stability analysis_. to appear in ieee transactions on circuits and systems i : regular papers . doi 10.1109/tcsi.2013.2284171 . c. t. kelley , d. e. keyes _ convergence analysis of pseudo - transient continuation_. siam journal on numerical analysis , vol . 508 - 523 , april 1998 . v. vittal , d. martin , r. chu , j. fish , j. c. giri , c. k. tang , f. eugenio villaseca , and r. g. farmer . _ transient stability test systems for direct stability methods_. ieee transactions on power systems , vol . 7 , no . 37 - 43 , feb . 1992
in this paper , pseudo - transient continuation method has been modified and implemented in power system long - term stability analysis . 
 this method is a middle ground between integration and steady state calculation , thus is a good compromise between accuracy and efficiency . 
 pseudo - transient continuation method can be applied in the long - term stability model directly to accelerate simulation speed and can also be implemented in the qss model to overcome numerical difficulties . 
 numerical examples show that pseudo - transient continuation method can provide correct approximations for the long - term stability model in terms of trajectories and stability assessment . 
 pseudo - transient continuation , long - term stability model , quasi steady - state model , long - term stability .