IN stability studies, the full dynamic model of a power

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1 Simplified time-domain simulation of detailed long-term dynamic models Davide Fabozzi Tierry Van Cutsem, Fellow, IEEE Abstract Time-domain simulation of power system long-term dynamics involves te solution of large sparse systems of nonlinear stiff differential-algebraic equations. Simulation tools ave traditionally focused on te accuracy of te solution and, in spite of many algoritmic improvements, time simulations still require a significant computational effort. In some applications, owever, it is sufficient to ave an approximate system response of te detailed model. Te paper revisits te merits of te Backward Euler metod and proposes a strategy to control its step size, wit te objective of filtering out fast stable oscillations and focusing on te aperiodic beaviour of te system. Te proposed metod is compared to detailed simulation as well as to te quasi-steadystate approximation. Illustrative examples are given on a small but representative system, subject to long-term voltage instability. Index Terms long-term dynamics, stiff decay property, backward Euler metod, quasi-steady-state approximation, long-term voltage instability I. INTRODUCTION IN stability studies, te full dynamic model of a power system takes on te form: = g(x, y, z) () ẋ = f(x, y, z) (2) z(t + k ) = (x, y, z(t k )) (3) Beind tese simple equations, is idden a large set of nonlinear, stiff, sparse, differential-algebraic, continuous-discrete time equations [], [2], [3], [4]. Te Algebraic Equations (AEs) () deal wit te network, wose response is assumed instantaneous under te quasi sinusoidal (or pasor) approximation. Te rectangular components of bus voltages are grouped into y. Te Ordinary Differential Equations (ODEs) (2) relate to a wide variety of penomena and controls ranging from te sort-term dynamics of power plants, static var compensators, induction motors, etc., to te long-term dynamics of secondary frequency control, load self restoration, etc. Vector x includes te corresponding state variables, suc as rotor angles, flux linkages, motor slips, controller state variables, etc. Te discrete-time equations (3) capture te discrete controls and protections tat act on te system, ranging from te fast switcing of sunt compensation to long-term controls suc as generator setpoints, load tap cangers, overexcitation limiters, D. Fabozzi (fabozzi@montefiore.ulg.ac.be) is P.D. student at te Dept. of Electrical Engineering and Computer Science of te University of Liège, Sart Tilman B37, B-4 Liège, Belgium. T. Van Cutsem (t.vancutsem@ulg.ac.be) is a researc director of FNRS (Fund for Scientific Researc) and adjunct professor at te same department /9/$ IEEE etc. Te corresponding (sunt susceptance, transformer ratio, etc.) variables are grouped into z, wic undergoes step canges from z(t k ) to z(t+ k ) at some instant t k [4]. A great number of numerical integration metods as been proposed to transform te ODEs (2) into AEs at eac time step of a simulation [5], [6]. Among tem, variable-step size algoritms ave been also implemented to speed up te computation [6]. As te power systems community tecnological paradigm for integration as traditionally focused on te accuracy of te solution, integration metods for timedomain simulation wit increasing accuracy (and growing complexity) ave been proposed [7], [], [2], [8], [9], [], []. Te trapezoidal metod as been soon recommended [] and is widely used. In spite of tese improvements, running detailed simulations of te full model still requires a significant computational effort. Tis effort may be proibitive for realtime applications, suc as dynamic security assessment, for instance. On te oter and, tis ig accuracy is not needed in some applications. For instance, wen cecking long-term voltage instability, te electromecanical oscillations are generally of marginal interest, as long as no generator loses syncronism. In te searc for simplified but efficient time-domain simulation, te full model (-3) became a target for simplification. One proven tecnique is te Quasi-Steady-State (QSS) approximation, wic is based on time decomposition [4], [2]. It consists in decomposing x into a fast component x f and a slow component x s, considering te fast part as infinitely fast and replacing te corresponding ODEs by AEs. Tus, te full model is simplified into: = g(x f, x s, y, z) (4) = f f (x f, x s, y, z) (5) ẋ s = f s (x f, x s, y, z) (6) z(t + k ) = (x f, x s, y, z(t k )) (7) in wic f as been decomposed into f f and f s, corresponding to te fast and slow parts, respectively. Instead of using te equations (5) stemming from te detailed model, an ad oc reduced model can be used, as detailed in [4], [2], [3]. Te QSS approximation proved to be a good and computationally efficient alternative to full time simulation of a full model; owever, te very QSS approximation as limits: a long-term degradation of system operating conditions may result in motor stalling or loss of syncronism [4]. In tis case, a fast instability of te full model (-3) appears as an early singularity of te QSS model (4-7) tat gives little information about te nature of te problem; Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.

2 2 following a large disturbance, te system may become unstable in te sort-term period and, ence, even not enter te long-term period. Detecting tis situation requires to couple detailed and QSS simulations [3]; a more accurate simulation of long-term dynamics may require to retain some of te fast dynamics [2]. Automatic model simplification metods are to be developed. Tus, a trade-off as to be found between solution accuracy and completeness, on one and, and computational efficiency, on te oter and. Tis paper presents a new approac for simplified time domain simulation of electric power systems, a compromise between te lengty simulation of a full model and te fast simulation of a very simplified model. We believe tat, wile te wole scope of te penomena to be observed in a power system can only be tracked by means of a full simulation on a full model, tere is a range of penomena tat can be covered by a simplified simulation at a reasonable computational cost. Tus, we advocate a paradigm sift from accuracy towards computational efficiency. Te approac exploits te stiff decay property and te simplicity of te backward Euler metod to obtain approximate solutions of te full model, witout a priori simplification of te latter as in te QSS approac. Besides efficiency, te objectives of te proposed solver are to: (a) ignore fast, opefully stable, oscillations of lower concern (typically electromecanical inter-macine oscillations), but (b) track fast aperiodic canges tat may bring te system to unacceptable operating conditions, and (c) deal efficiently wit te numerous discrete events (3). Tis paper is organized as follows. Some properties of integration metods are revisited in Section II, wile te proposed approac is presented in Section III. A small test system is used in Section IV to illustrate te proposed sceme and compare it to full and QSS simulations. Te conclusion in Section V closes te paper. II. SHORT REVIEW OF INTEGRATION METHODS Given an ODE of te type: ẋ = f(x) (8) and considering a discretization of te simulation interval in time steps of lengt, te TrapezoidalMetod (TM) computes an approximation of te real solution according to: x n = x n + 2 ẋn + 2 ẋn (9) wile te Backward Euler Metod (BEM) uses: x n = x n + ẋ n () A. A-Stability Consider te simple scalar ODE, often referred to as test equation : ẋ = λx () were λ is a complex number. Te region of absolute stability of a given integration metod is te region of te λ plane A Fig.. I(λ) TM C B R(λ) A I(λ) C BEM Absolute stability regions of te TM and BEM metods B R(λ) suc tat applying te metod to (), wit λ from witin tis region, yields a solution satisfying te absolute stability requirement: x n < x n (2) Tese regions are sown saded in Fig., for te TM and BEM metods, respectively [5]. It is desirable tat, wenever te exact solution is stable, so is te simulated one. Tis property is called A-Stability [5] and olds wen te region of absolute stability of te metod contains te entire left alf plane (R(λ) < ). Tus bot TM and BEM are A-stable. Tus, for point A in Fig., were R(λ) < (and ence R(λ) < ), bot TM and BEM produce stable responses. For point B, were te system is unstable, TM produces an unstable response wile BEM still gives a stable one. Tis drawback is referred to as yper-stability [9]. For point C, te responses of bot metods will be unstable. B. Stiff decay As far as computational efficiency is concerned, te possibility of taking large steps is a very important feature. By taking large integration steps, some fast oscillations of lower concern will be ignored. However, in order to accommodate a step size comparable or larger tan te period of tese oscillations, te integration metod must be numerically robust. In tis context, te stiff decay property plays a crucial role [5]. Consider te test equation modified as follows: ẋ = λ(x g(t)) (3) were g(t) is a bounded function. An integration metod as stiff decay if, for a given t n > : x n g(t n ) as R(λ) (4) Wen TM is applied to (3), it is easily sown tat: x n g(t n )=a f (x n g(t n )) wit a f = 2+λ 2 λ (5) wile wit BEM, one as: x n g(t n )=a f (x n g(t n )) wit a f = λ (6) were a f is often called amplification factor. Wen n R(λ) tends to, a f tends to for TM and to for BEM. Hence, according to (4), TM does not ave stiff decay wile BEM as. Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.

3 3 x 3 Proposed Algoritm Trapezoidal Metod x 3 Proposed Algoritm Fig. 2. Small disturbance Fig. 3. Large disturbance Te advantage of integration metods wit stiff decay lies in teir ability to skip rapidly varying solution details wile maintaining a decent description of te coarse beavior of te solution. Conversely, integration metods witout stiff decay need to be solved wit small step size even if only a coarse beavior of te solution is sougt, oterwise errors get propagated and pronounced numerical oscillations are experienced [5]. C. A simple example Consider te following system of ODEs: x = x 2 (7) x 2 = 36 x.5 x 2 u(t) (8) x 3 = Mx 2 x u(t) (9) were u(t) denotes te unit step function. Te initial conditions are x () = x 2 () = and x 3 () =. Tevariable of interest is x 3.WenM =, x 3 evolves in an unstable aperiodic way, typical of a saddle-node bifurcation. Wit M, te comparatively fast oscillation produced by x and x 2 is added to tat evolution. Te disturbance stems from te step function. Its magnitude is adjusted troug te value of M. Te QSS approximation of te slow dynamics is obtained by considering x and x 2 as fast variables, wic yields: = x 2 = 36 x u(t) x 3 = x u(t) In tis particular example, te QSS approximate model does not depend on te severity of te disturbance, measured by M. Hence, even if te disturbance is so severe tat it leads to a fast instability, te QSS model will not reveal it. Figure 2 sows te effect of a small disturbance (M =). Te reference response as been obtained wit TM using a small time step of. s. Te oscillation period is in te order of s. Wen using a large time step = s, TM experiences artificial numerical oscillations due to te propagation of errors, wile BEM provides a nice coarse approximation of te reference solution. Integrating te QSS model wit te same step size gives results very close to te reference and te BEM solution. Figure 3 sows te effect of a large disturbance (M = 4) leading to a fast instability. Using te same large time step = s, BEM provides a coarse approximation of te reference solution. Since it is independent of M, teqss solution is te same as in te previous case. Compared to QSS approximation, BEM sows a better ability to track fast aperiodic canges. D. Newton iterations We consider a simultaneous sceme in wic te AEs () and te algebraized ODEs stemming from (2) are solved togeter by Newton metod. Traditionally, te oter option is to andle te two sets separately, in a partitioned way, wile performing functional iterations on x. Tis combination, owever, would not converge wen increasing te time step to te extent considered in tis work [6]. Assuming algebraized ODEs written in compact form as: f(x, y) = (2) were te dependence on z is dropped for clarity, te ν-t iteration of te Newton metod requires to solve te linear system: [ ][ ] [ gy g x y ν y ν g(x ν, y ν ] ) f y f x x ν x ν = f(x ν, y ν ) [ ] b ν y = b ν (2) x were g y is te Jacobian matrix of g wit respect to y, and similarly for te oter matrices, wile b y and b x are mismatc vectors, relative to algebraic and differential equations, respectively. For efficiency reasons, a disonest Newton metod is used, in wic te Jacobian is updated as rarely as possible []. Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.

4 4 Fig. 4. y y(t k ) y(t + k ) Solving a discontinuity y(t k + ) At te k-t time step, te mismatc vector b ν x be easily obtained as: b ν x = c k f(xν k, y ν k ) 2 wit c k = x k + f(x k, y k ) 2 t of TM can + xν k (22) were c k is a constant at all iterations ν performed at step k. A similar derivation for BEM yields: b ν x = c k f(x ν k, y ν k )+ xν k (23) wit c k = x k (24) E. Discontinuity andling If an external disturbance is applied to te system or a discrete transition (3) takes place at time t k, te differential variables x remain uncanged, wile te algebraic variables y undergo a discontinuity, as illustrated in Fig. 4. In principle, a re-initialization is required, in wic te new value y(t + k ) is computed and used in (2) to obtain te new derivative ẋ(t + k ). If so, a reduced system involving g y as to be built, factorized and solved. Alternatively, te parameter cange can be stretced over a very small time step. Tis also as a cost, since te drastic reduction of te step size requires to update te Jacobian in (2). Tis procedure, owever, can be skipped wenever c k does not depend on y k, because te mismatc vector does not involve any value tat is modified by te discontinuity. In tis case, tere is no need to solve for y(t + k ) and te simulation may just proceed wit te new parameters. Te above condition on c k does not old for TM, as sown by Eq. (22), wile it does for BEM, as can be seen from (24). Wen computational efficiency is sougt, te opportunity of avoiding discontinuity andling - and te resulting Jacobian updates and factorizations - is a key feature. Tis is a significant advantage in view of te numerous discrete canges tat occur in te simulation of power system long-term dynamics, as stressed in te Introduction. In tis respect, it is preferable to represent discrete controllers troug discrete cange in parameters of te type (3) rater tan by a continuous-time approximation, wic would add rows and columns to te Jacobian matrix. III. PROPOSED APPROACH We focus on getting simplified solutions of te full model (-3). In so far as accuracy requirements are relaxed, BEM appears attractive tanks to te possibility it offers to make large time steps witout experiencing numerical oscillations, and to pass troug discontinuities witout re-initializations. A. Step size control Large integration steps are desirable, for computation efficiency reasons, and made possible by te stiff decay property of BEM. However, tere are reasons for not taking excessive step sizes: ) some long-term oscillations of interest (e.g. frequency response in isolated or ydro-generation dominated systems) would be filtered out by large steps; 2) discrete events would be excessively delayed and/or syncronized. To preserve computational efficiency, te exact transition times are not identified (no interpolation is used) but rounded to te next integration step. Hence, too large a step size may delay te transition and artificially syncronize multiple events; 3) wen fast transients are triggered, smaller steps may be needed to track te system response and avoid divergence of te Newton iterations. Te proposed approac consists in: (i) integrating wit a maximum step cosen in accordance wit te first two requirements above, (ii) reducing te step size automatically wen te Newton iterations exibit convergence difficulties, and (iii) recovering to as soon as possible. Note tat tis step size control differs from te traditional Local Truncation Error (LTE) control [6]. Indeed, te latter criterion makes little sense in so far as accuracy requirements are relaxed and te empasis is put on te coarse response of te system. Te logic to control te step size, wen convergence difficulties are met (see item 3. above), consists of bounding te mismatc vector b x of te first Newton iteration. Consider again te test equation (), and assume a linear predictor (explicit Euler metod) is used to obtain a first guess of x at te k-t step, i.e. x k = x k + ẋ k = x k + (λx k ) Substituting tis expression for x k in (23) and (24), wit ν =, yields: b x = x k λx k + x k = λ2 x k (25) wic sows tat te mismatc b x varies linearly wit te step size. Tis property was found to apply pretty well to te mismatc vector of te nonlinear power system models. In tis case, te infinite norm b x (largest component magnitude) of te first mismatc vector is considered. Tus, wen a convergence problem is detected, te step size is reduced so as to bring b x near some value τ. Namely, if a previous step size k brings a mismatc norm: b x = d k k (26) ten, te step size k tat will make te new mismatc norm equal to te bound τ is given by: τ = d k k (27) since BEM is a first-order integration metod, it does not make sense to use more tan first-order Taylor expansion for prediction [6] Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.

5 5 Fig. 5. x x(t + ) st guess x(t) (no prediction) st guess (prediction) A case in wic prediction is not useful t G 4 3 M 2 G Dividing (27) by (26) yields: τ k = k b (28) x As convergence of te Newton sequence improves over successive time steps, te same formula yields increasing values of k. Once te latter reaces, te step size is locked to tis value until a new convergence problem, if any, calls for a new reduction. Te two parameters of te proposed step size control are tus and τ. Teir coice is somewat system dependent and may require some trial-and-error adjustments, as wit any numerical integration procedure. B. Dealing wit prediction Prediction is present in many numerical integration metods, eiter to speed up convergence or as part of te LTE estimation. Wit te step size control presented in te previous section, prediction was found to be beneficial most of te time but detrimental on some occasions were system state variables evolve rapidly. Indeed, wen large steps are made, te predicted value of x may fall far from te solution point, causing te Newton iterations to diverge. As sketced in Fig. 5, it may be more advantageous not to use prediction in tose cases. Tis was already observed in [] in te context of detailed time simulation. Te decision to use prediction or not is taken as follows. As te solution is not known wen starting te iterations, te initial guess closest to te solution cannot be identified. Instead, te norm b x of te mismatc vector is used again. Wen a first-order prediction is used: x k = x k + f(x k, y k ) (29) and te mismatc vector is given by (24), were ν =: b x = x k f(x k, y k )+ x k wile wen no prediction is used: (3) x k = x k (3) and te mismatc vector becomes: b x = x k f(x k, y k )+ x k = ẋ k (32) Tus, wit no prediction, te mismatc vector amounts to minus te vector of state variable derivatives at te previous Fig. 6. One-line diagram of te test system time step. From tere on, prediction will be enabled if b x computed from (3) is smaller tan ẋ k. It sould be empasized tat tis test comes at te negligible cost of evaluating te time derivatives (if (3) is selected) or at no cost (if (29) is selected). IV. APPLICATION TO A SMALL TEST SYSTEM A. Test system We present ereafter simulation results obtained on a 4-bus test system, wose one-line diagram is sown in Fig. 6. Tis is a variant of te system considered in [4] to illustrate long-term voltage instability mecanisms. In tis system, a large load is fed by an external system troug a long double-circuit line, as well as by a local generator controlling te voltage at te close bus 2. Te long-term dynamics comes from te Load Tap Canger (LTC) controlling te voltage of te distribution bus 3, and te OverExcitation Limiter (OEL) protecting te generator at bus 2. Te LTC delayis2sontefirst tap cange and s on subsequent canges. Te OEL as an inverse-time caracteristic [4]. Wit respect to te model detailed in [4], te Tévenin equivalent at bus as been replaced by a large equivalent generator beind its step-up transformer, wile bot generators are driven by a steam turbine wit speed governor. Te purpose of tis modification is to ave te slower, common frequency mode of oscillation in te system responses. Te disturbance considered is te tripping of one of te two circuits, at t = s. Tis outage, togeter wit te reactive power limitation of te generator at bus 2, causes te maximum power deliverable to te load to decrease below te level tat te LTC tries to restore indirectly. Hence, long-term voltage instability results. All pasors are projected on reference axes rotating at te speed of te Center Of Inertia (COI) and rotor angles are defined wit respect to tis COI reference. Tis is essential to avoid pasors to rotate and rotor angles to increase wit time, wen te system settles at a new frequency. Tis may cause divergence of Newton iterations wen making large steps, or at least it may require to update te Jacobian frequently. B. Simulation metods Te objective is to compare te time responses provided by respectively: Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.

6 6.5 V 4 Proposed Algoritm.5 V 4 Proposed Algoritm Fig. 7. Case. Transmission side voltage Fig. 8. Case 2. Transmission side voltage ) a reference metod. Tis is full simulation of te full model (-3). For accuracy purposes, TM as been used wit variable step size controlled by LTE; 2) te proposed algoritm. Te default step size as been taken as s. Te mismatces are monitored over successive iterations to early detect possible divergence. In te latter case, te iterations are stopped, and te step size is reduced as described in Section III-A; 3) te QSS approximation of te type (4-7), using a constant step size = s. Te retained dynamics (6) relate to te non-windup integrator tat determines te instant of OEL activation as well as te steam turbine reeat time constant. Tese ODEs ave been integrated by BEM. Te relevant Jacobian matrices are evaluated numerically, using finite differences. Tey are updated at te beginning of an iteration if te decrease of te mismatces is not large enoug. An update also takes place in case a state variable reaces or leaves a limit. C. Case : pure long-term voltage instability In tis first scenario, te wole load is represented wit an exponential model. Te time evolutions of te voltage at bus 4 computed by te tree metods are sown in Fig. 7. Te electromecanical oscillations tat follow te line tripping die out rater fast, indicating tat te sort-term dynamics are stable. Te LTC starts responding at t =2s, wile te OEL is activated at t 7 s. From tere on, te LTC fails restoring te distribution voltage and depresses te transmission one. Tis degradation stops wen te LTC its its limit. Bot te proposed approac and te QSS approximation provide evolutions almost undistinguisable from te bencmark metod. D. Case 2: long-term instability causing loss of syncronism Tis scenario differs from Case by a larger active power generation at bus 2. Figure 8 sows tat te voltage starts evolving as in te previous case, wit an OEL activation at.4.2 δ 2 (rad) Proposed Algoritm Fig. 9. Case 2. Rotor angle of generator at bus 2 t 5 s, but eventually plunges down at about t 274 s. Tis is caused by a loss of syncronism, as seen from te rotor angle in Fig. 9 (generator at bus 2, referred to COI). Te proposed metod provides a system evolution ardly distinguisable from te bencmark, tereby sowing its ability to track fast aperiodic canges. Te QSS simulation, on te oter and, reaces a singularity at t = 224 s, identified by dots in Figs. 8 and 9. Altoug long-term voltage instability can be diagnosed from te QSS response, te resulting fast instability cannot be identified by tis metod. E. Case 3: long-term instability causing motor stalling Tis scenario differs from Case by te load model. Part of te load consists of an equivalent induction motor supposed to represent multiple motor loads. Figure sows tat te voltage evolves under te effect of rotor oscillations, two tap canges, and te OEL activation at t 35 s. Immediately after, it falls sarply. As can be seen from te slip curve in Fig., tis is caused by te stalling of te motor (wic decelerates up to standstill; no tripping Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.

7 7 V Proposed Algoritm Fig s Case 3. Transmission side voltage Proposed Algoritm Fig.. Case 3. Motor slip Fig. 2. Fig. 3. Case 3. Evolution of step size for 45 t 55 s 35 bx (s).2.4 Example of variation of b x wit te step size TABLE I COMPUTATIONAL EFFORT OF VARIOUS METHODS Case simulated TM wit proposed QSS No time (s) LTE control metod simulation JU NI JU NI JU NI final divergence final divergence by a protection as been assumed), wic itself results from te lack of voltage support and reactive power supply after te OEL activation. Once again, te proposed metod provides a response quite close to te bencmark, and sows its ability to track fast aperiodic canges. Te QSS simulation, on te oter and, reaces a singularity, identified by dots in Figs. and. Tis takes place at t =36s, wen te field current limit is enforced, and at a significantly iger voltage tan in Case 2. Complementary analysis of tat singularity is needed for a satisfactory diagnosis. F. Step size control in proposed metod Figure 2 sows te evolution of te step size in Case 3. Up to t =46s, te default step size =sas been successfully used. At t =46s, convergence difficulties are detected, most likely due to te fast cange in motor slip (see Fig. ). Hence, te step starts being adjusted according to Eq. (28), were τ as been set to 2. At t 52 s, is set back to for te remaining of te simulation. Te variation of b x wit te step size is sown in Fig. 3. Te latter relates to te first iteration of te first time step after te line is tripped, in Case, at t =s. Te variation is not far from linear, as for te test equation (). Tis olds true even for relatively large steps. G. Computational efforts Table I compares te computational effort of respectively te reference, te proposed and te QSS metod in te tree cases presented so far, and for different simulation orizons. Te numbers of Newton iterations ( NI in te table) and Jacobian updates ( JU in te table) are provided as measures of tis effort. Te size of te Jacobian is te same for te tree metods. As can be seen, te proposed metod is muc less demanding tan TM. It even approaces te ig computational efficiency of te QSS approac. Te additional effort wit respect to te latter is largely spent in computing te fast canges tat te QSS approximation cannot reproduce. Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.

8 Fig. 4. V 4 Proposed Algoritm Case 4. Transmission side voltage H. Case 4: oscillatory instability Te last case illustrates te yper-stability drawback of BEM, already mentioned in Section II-A. Te gain of te AVR of te generator at bus 2 as been increased so tat after te line outage te generator lacks damping torque. Te resulting oscillatory unstable response is sown wit solid line in Fig. 4. However, bot te QSS approximation and BEM ignore tis unstable oscillation, and provide a trajectory wic is close to te underlying unstable equilibrium of te system. Seeking an analogy wit te test equation (), one can say tat te simulation corresponds to point B in Fig., were te numerical response is stable but te real response is not. Te TM, on te oter and, would sow an unstable response. Several remedies could be tougt of [9]: ) escape te BEM stability region by reducing te step size, wic corresponds to moving from point B to point C in Fig.. Coming back to te proposed approac, it would be required to reduce te value of. Alternatively, te step size could be controlled by Eq. (28) wit a significantly lower value of τ; 2) switc to a iger-order stiff-decay integration metod, less prone to yper-stability, and preserving te interesting features of BEM; 3) switc to anoter solver, suc as TM, not prone to yperstability. Te cange in simulation sceme outlined could be made over limited periods of time. However, since te proposed metod does not ave an indication of emerging oscillatory instability, it sould take place somewat systematically after potentially dangerous events suc as faults, line or generator outages, etc. Of course, in terms of computational effort te above remedies will affect te overall performance of te proposed approac and may bring it close to a non-simplified simulation. V. CONCLUSION On te premise tat computational efficiency may be more important tan ig accuracy for some applications, tis paper demonstrates te possibility to perform simplified but fast time-domain simulations of a detailed dynamic model of te power system. Numerical approximation of te system response is tus preferred to simplification of te model, typical of te QSS approximation. Te proposed integration sceme allows neglecting te oscillatory dynamics wic are fast compared to te step size (suc as inter-macine rotor oscillations) wile concentrating on te average evolution, wit te possibility to track fast canges suc as loss of syncronism and motor stalling. Tis is obtained by exploiting te stiff decay property of te backward Euler metod, wic also allows a simplified andling of discontinuities. A variable step strategy as been proposed wic consists of integrating wit a maximum step unless convergence difficulties of te Newton iterations are met, corresponding to fast canges. Ten, te strategy focuses on te magnitude of te Newton mismatces, until te step size can be brougt back to its large default value. Oscillatory instability is likely to remain outside te range of penomena tat can be simulated by te proposed approac. Systems wit ligtly damped electromecanical modes sould tus be considered wit care. Tis is part of te compromise between accuracy and computational efficiency. Future work will be devoted to: (i) detection of emerging oscillatory instability (e.g. troug tracking of dominant system eigenvalues), (ii) alternative stiff-decay integration scemes, (iii) large-scale tests to assess te gain in computing times. REFERENCES [] B.Dembart,A.M.Eirsman,E.G.Cate,M.A.EptonandH.Dommel, Power System Dynamic Analysis - Pase I. Final Report EPRI EL-484, July 977 [2] B. Stott, Power systems dynamic response calculations, Proc. of te IEEE, Vol. 67, Feb. 979 [3] P. 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Lefebvre, Combined detailed and quasi steady-state time simulations for large-disturbance analysis, Electrical Power and Energy Systems, Vol. 28, 26, pp Autorized licensed use limited to: Tierry Van Cutsem. Downloaded on November 6, 29 at 3:9 from IEEE Xplore. Restrictions apply.