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Quasi steady-state simulation diagnosis usingNewton method with optimal multiplier

Patricia Rousseaux Thierry Van Cutsem, Fellow, IEEE

Abstract This paper deals with the quasi steady-state approx-imation of the long-term dynamics. This fast time simulationmethod assumes that the short-term dynamics are stable andcan be replaced by their equilibrium equations. When the latterstop having a solution, the simulation undergoes a singularity.This paper proposes a method to identify which component(s)are responsible for the loss of equilibrium. The correspondingequations are identied using the Newton method with optimalmultiplier. The method has been validated with respect to fulltime simulation. Very good results are shown on the Nordic-32system, in cases where long-term voltage instability triggers lossof synchronism. The proposed method enhances time simulation

at very low computational cost and can also help correcting modeland/or operating point errors.

Index Terms Long-term dynamics, voltage stability, time-domain simulation, quasi steady-state approximation, singularity,Newton method, line search

I. INTRODUCTION

I N power system dynamic studies, the trend is to performnumerical simulations over longer periods of time, withmore detailed models, and for more operating conditions anddisturbances. However, power system dynamic models arelarge and involve very different time scales, which makes theirsimulation over long time intervals very demanding.

Time-scale simplication of the model is a natural responseto this complexity. The idea is not new. For instance, it un-derlies the quasi-sinusoidal (or phasor) approximation used inmost stability studies [1], where electromagnetic transients areneglected and the network is modeled by algebraic equations.

The idea is further exploited in the Quasi Steady-State(QSS) approximation of long-term dynamics, which consistsof replacing the short-term differential equations of generators,motors, compensators, etc. by the corresponding algebraicequilibrium equations. QSS simulation is well suited to com-putationally intensive tasks such as security limit determina-

tion, real-time applications or training simulators [2], [3].Replacing the short-term dynamics by their equilibrium

equations requires that dynamics to be stable. Now, in somedegraded situations, it may happen that these dynamics losetheir equilibrium point. This may correspond for instance tosynchronous generators going out of step (angle instability) orinduction motors stalling.

P. Rousseaux (p.rousseaux@ulg.ac.be) is with the Dept. of ElectricalEngineering and Computer Science (Monteore Institute) of the Univer-sity of Li ege, Sart Tilman B37, B-4000 Li` ege, Belgium. T. Van Cutsem(t.vancutsem@ulg.ac.be) is with the Belgian National Fund for ScienticResearch (FNRS) at the same department.

When this happens, the QSS model meets a singularity.In practice, the Newton iterations diverge, and the simulationcannot proceed.

If it results from a loss of equilibrium, the singularity isby itself an indication that the system is subject to short-term instability. Using the method described in this paper, theobjective will be to further identify which system componentsare responsible for this instability.

On the other hand, the singularity might result from anumerical problem, i.e. a lack of convergence of the Newton

iterations. In this case, the proposed method is expected tosolve the QSS equations by performing more careful Newtoniterations.

To this purpose a scaling factor is applied to the Newtoncorrections as soon as an increase of the sum of squaredmismatches is detected. The scaling factor is adjusted tominimize that sum, using a line search in the direction of the Newton correction. By so doing, the mismatches tend toconcentrate on the subset equations most responsible for theinfeasibility.

To the authors knowledge, this optimal multiplier techniquehas been mainly applied to standard load ow equations. Itwas initially proposed in [4] to deal with ill-conditioned load

ow problems. The diagnosis capability of the method wasexploited in [5] to rank contingencies according to their impacton voltages. In [6], [7] the method was used in unsolvable loadow cases related to voltage instability, rst as a solvabilitymeasure, then to identify minimal corrective actions to restorefeasibility. Further in-depth analytical investigations were re-ported in [8].

This paper, on the contrary, deals with short-term equilib-rium equations, allowing to validate the output of the methodwith respect to detailed time simulation.

II. QSS APPROXIMATION : PRINCIPLE AND SINGULARITIES

A. Principle of the QSS approximation

In stability studies, the general model of a power systemtakes on the Differential-Algebraic (DA) form:

0 = g(x , y , z) (1)x = f (x , y , z) (2)

z(t+k ) = h (x , y , z(t

k )) (3)

For an N -bus system, the 2N algebraic equations (1) relateto the network and involve the vector y of bus voltages.

The differential equations (2), involving the vector x of continuous states, relate to a wide variety of phenomena andcontrols including:

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the short-term dynamics of generators, turbines, gover-nors, Automatic Voltage Regulators (AVRs), Static VarCompensators, induction motors, HVDC links, etc.

the long-term dynamics of secondary frequency and volt-age control, load self-restoration, etc.

The discrete-time equations (3) capture discrete events thatstem from:

controllers acting on shunt compensation, generator set-points, Load Tap Changers (LTCs), etc. [1], [9] equipment protections such as OverExcitation Limiters

(OELs), etc. [1], [9] system protection schemes acting on loads and/or gener-

ators.The corresponding (shunt susceptance, transformer ratio, etc.)variables are grouped into z, which undergoes step changesfrom z (t k ) to z (t

+k ) at some times tk , most often dictated by

the whole system dynamics.As indicated previously, the QSS approximation of long-

term dynamics consists of representing faster phenomena bytheir equilibrium conditions instead of their full dynamics. The

correspondingly simplied model takes on the form:

0 = g(x 1 , x 2 , y , z) (4)0 = f 1 (x 1 , x 2 , y , z) (5)

x 2 = f 2 (x 1 , x 2 , y , z) (6)z(t+k ) = h (x 1 , x 2 , y , z (t

k )) (7)

in which x (resp. f ) has been decomposed into x1 and x 2(resp. f 1 and f 2 ).

Three algebraic equations (5) can be used to describe eachgenerator, accounting for saturation, AVR voltage droop andgovernor speed droop effects [2], [3]. These equations aredetailed in Appendix A.

System frequency is a component of x 1 , if its dynamics areneglected. This additional variable is balanced by the equationsetting the voltage phase angle to zero at a reference bus [2].

In principle, (6) relates to long-term dynamics, and hencex 2 are the slow state variables. However, if frequencydynamics are taken into account, it may be required foraccuracy purposes to retain some time constants in the order of a second in the (hydro) turbine and speed governor models, asreported in [10]. The corresponding differential equations areincluded in (6). Assuming the same speed for all generatorsand accounting for their inertia, the rate of change of frequencyprovides an additional equation (6). Frequency is then acomponent of x2 .

B. Singularities of the QSS approximation

While the assumption that the short-term dynamics arestable holds in most practical long-term stability studies [3],[2], [10], [11], it is invalidated in the following situations:

1) following a large disturbance, the system has a short-term equilibrium point but is not attracted by the latter.This is typical of transient (angle) instability, whichtakes place over a few seconds after the disturbance;

2) as x 2 and z evolve according to the long-term dynamics,the short-term equilibrium becomes oscillatory unstable;

3) as x 2 and z evolve according to the long-term dynamics,the short-term equilibrium disappears.

The rst two cases cannot be detected by the QSS simu-lation. Coupling with detailed time simulation has been pro-posed in [11] to deal with the rst case. The second case wouldrequire additional eigenvalue analysis. This paper focuses onthe third case.

When the short-term equilibrium point disappears, the QSSequations can no longer be solved; the QSS model meets asingularity. Note that as the latter is approached, the time de-coupling assumption is less and less valid and the short-termdynamics stop being faster [12]. Hence, before reaching asingularity, the QSS model (4-7) is expected to somewhatdepart from the reference model (1-3).

QSS singularities may be encountered:1) following discrete transitions (7) such as LTC move-

ments or OEL activations. In this case, equations (4,5)cannot be solved for the updated value z(t+k ) and thecurrent (unchanged) value of x2 ;

2) immediately after an external disturbance, for which no

post-disturbance short-term equilibrium exists. This caseis similar to the previous one (with other parameters thanz undergoing a discontinuity);

3) due to a change in the continuous states x2 , making itimpossible at some time t to proceed to time t + h,where h is the simulation time step. If a partitionedscheme is used to integrate the DA model (4-7), a valueis assumed (iteratively) for x 2(t+ h) and the singularityoccurs in the course of solving (4,5) for the so updatedvalue of x 2 and the current (unchanged) value of z. If a simultaneous scheme is used, the equations that can-not be solved include additional relationships stemmingfrom the discretization of (6).

Without loss of generality, we assume that a partitionedscheme is used. Hence, the equations to be handled are (4,5).In a g-generator system, there are 2N + 3 g + 1 such equations(one of them setting the phase angle at a reference bus). Tokeep the notation simple, they are rewritten in compact formas:

(u ) = 0 with = gf 1

and u = yx 1 (8)

III . N EWTON METHOD WITH OPTIMAL MULTIPLIER

According to the well-known Newton method, equation (8)is solved through the following iterations:

u k+1 = u k + u k k = 0 , 1, 2, . . . (9)

with the correction u k given by:

u k = 1u (u k ) (u k ) (10)

where u (u k ) stands for the Jacobian matrix of withrespect u at the k-th iteration. The above iterations convergerapidly provided the initial guess u o is sufciently close tothe solution. In case of ill-conditioned systems, however, theprocedure may fail to converge. On the other hand, in case of a singularity, the system (8) has no solution and the Newtoniterations (9) diverge.

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0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0 10 20 30 40 50 60 70 80

t(s)

V(pu)

t(s)

V(pu) voltage at bus 4032voltage at bus 4044voltage at bus 4047

Fig. 2. Case 1. Time evolution of voltages - QSS simulation

B. Case 1

We simulate the tripping, at t = 1 s, of the 400-kVtransmission line between buses 4032 and 4044.

The long-term evolution of voltages provided by QSSsimulation is shown in Fig. 2. The system evolves under theeffect of LTCs and OELs. For already mentioned reasons,many machines have their eld currents limited: g7 at t = 22 ,g14 at t = 23 , g12 at t = 27 , g15 at t = 46 , g16 at t = 47 ,and g6 at t = 73 s.

The QSS simulation meets a singularity at t = 79 s. At thistime step, following the operation of four LTCs, Eqs. (4,5) canno longer be solved.

The results of the optimal multiplier method in terms of nalmismatches are shown in Table I. The tolerances used to stopthe Newton iterations are 0.5 MW (resp. Mvar) for active (resp.reactive) power equations and 0.001 pu for voltage equations.

Table I only shows the mismatches above these tolerances,with the corresponding equations (some of them are found inAppendix A).

TABLE I

CAS E 1. F INAL MISMATCHES AT THE SINGULARITY POINT

eqn. group eqn. type system component nal mismatch(4) active current bus 1042 -0.7 MW/pu(5) (16) generator g6 0.7 MW

Clearly the method points out generator g6 as being re-sponsible for the singularity, with a mismatch on the activepower balance equation of this generator and the active current

balance equation of the bus where it is connected.To check the validity of this diagnosis a time simulation

was run with the detailed model (1-3) from which the QSSmodel was derived.

The corresponding voltage evolutions are shown in Fig. 3.As can be seen, over the time interval of the QSS simulation,both models yield very similar voltage evolutions (except of course the stable electromechanical oscillation neglected inthe QSS model). The detailed simulation, however, showsa nal collapse that corresponds to instability of the short-term dynamics. This is conrmed by the rotor angle curvesof Fig. 4, relative to generators in the central area (all angles

0 20 40 60 80 1000.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

V (pu)

t (s)

voltage at bus 4032

voltage at bus 4044

voltage at bus 4047

Fig. 3. Case 1. Time evolution of voltages - detailed simulation

0 20 40 60 80 1005

0

5

10

15

20

25

t (s)

g6

g7, g14, g15, g16

(rad)

Fig. 4. Case 1. Time evolution of rotor angles - detailed simulation

are referred to that of generator g20). The curves clearly showthat generator g6 is going out step, dragging some other nearbygenerators. This nicely corroborates the results of Table I.

This angular instability corresponds to a loss of equilibriumof the system, caused by a lack of synchronizing torqueon the eld current limited generator g6 [2]. Under theeffect of constant excitation and sagging network voltages, themaximum electromagnetic torque of that generator decreasesprogressively until it becomes smaller than the mechanicaltorque imposed by its turbine, making steady-state synchro-nous operation impossible.

We note that QSS singularity occurs at t = 79 s whilesynchronism is lost near t = 95 s. This is explained by (i) theQSS assumption that short-term dynamics are innitely fast,which stops being true (see Section II.B); (ii) the time takenby the loss of equilibrium to reveal, owing to rotor inertia.

C. Case 2

The same disturbance is simulated but a longer thermaloverload capability is assumed for g6, allowing this machineto be overexcited over the whole simulation.

A singularity is met in the QSS simulation at t = 86 s,

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0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

.104

0

01

min

2

2 13 3 4 4

55

0

Fig. 6. Searching for the minimum of ( )

this choice, the number of -function evaluations is between8 and 15 (see Appendix B). This is conrmed by the effectivenumber of evaluations shown in Table V.

TABLE V

C AS E 1. ITERATIONS OF THE OPTI MAL MULTIPLIER METHOD

iter min ( min ) nb. of evaluations1 1.000 0.3449 10 3 82 0.617 0.1535 10 3 143 0.000 0.1531 10 3 8

V. C ONCLUSION

This paper has described the use of the optimal multipliertechnique in the Newton method for the diagnosis of QSStime simulation. The method is able to provide a diagnosis of singularities corresponding to loss of short-term equilibrium. Acomparison with detailed time simulation on cases where anglestability is lost has shown that the generators most responsiblefor the singularity are adequately and unambiguously identi-

ed. The diagnosis is obtained at a low computational costsince the method is used at the singularity point only andinvolves a few Jacobian updates and a moderate number of mismatch evaluations.

The next steps will consist in applying the method toQSS equations involving induction motors (to identify stallingconditions) and performing tests on a wider range of systems.

In practice, the identication of the components most in-volved in the singularity will allow the user to quickly check for possible data errors, and decide whether model adjustmentsare needed. In the context of Monte-Carlo simulations, whererandom operating points are generated, it may also helprejecting unrealistic scenarios.

A PPENDIX A . S YNCHRONOUS GENERATOR QSS MODEL

Each synchronous machine is characterized by three vari-ables of the type x1 :E q the emf proportional to eld currentE sq the corresponding emf behind saturated synchronous re-

actances the internal rotor (or load) angle [1].

which are involved in three equations of the type (5): the machine saturation relationship:

E q k(E q , E sq , ,V ) E sq = 0 k > 1 (14)

the voltage regulation relationship:

E q G (V o V ) = 0 (15)

where G is the open-loop steady-state gain of the AVRand V o its voltage setpoint;

the speed regulation relationship. Assuming the mechan-ical power P m entirely converted into active power P

and considering the steady-state governor effect yields:P P m = P (E q , E sq , ,V ) P o + g = 0 (16)

where P o is the power setpoint, the per unit frequencydeviation from nominal value, and g is a function of thepermanent speed droop and turbine rating [1].

With the armature resistance neglected, the active andreactive powers produced by the generator are given by:

P =E sq V X sd

sin + V 2

2 1X sq

1X sd

sin2 (17)

Q =E sq V X sd

cos V 2sin2

X sq+

cos2 X sd

(18)

where X sd and X sq are the saturated direct- and quadrature-

axis synchronous reactances, respectively [1], [2]. They relateto their unsaturated values X d and X q through:

X sd = X l + X d X l

k X sq = X l +

X q X lk

(19)

where X l is the leakage reactance and k is the saturationcoefcient involved in (14). According to a widely usedsaturation model:

k = 1 + m(V l )n m,n > 0 (20)

where V l is the magnitude of the voltage behind leakagereactance. The latter is obtained from the generator voltageV and current I through:

V l = V + jX l I (21)

We use network equations (4) expressed in terms of activeand reactive currents. Replacing in (17,18) X sd and X sq bytheir expressions (19) and k by the ratio E q /E sq , we obtain:

I P = P V

=E sq E q

X l E q + ( X d X l )E sqsin +

V E q2

[

1X lE q + ( X q X l )E sq

1

X l E q + ( X d X l )E sq]sin2

(22)I Q =

QV

=E sq E q

X lE q + ( X d X l )E sqcos

V E q[ sin2

X lE q + ( X q X l )E sq+

cos2 X lE q + ( X d X l )E sq

] (23)

The k coefcient is expressed in terms of the same variablesas follows:

k = 1 + m(V l )n = 1 + m (V + X l I Q )2 + ( X l I P )2n/ 2

in which I P and I Q have to be replaced by (22,23).

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a

4 3 5 2

cb

1 3 2

2 3 1

1

Fig. 7. Subdividing the search interval

A PPENDIX B . L INE SEARCH PROCEDURE

The procedure is analog to a binary search for ndingfunction roots. It starts with the evaluation of the function tobe minimized at three consecutive points 1 , 3 , 2 . It thenconsists of building smaller and smaller intervals [1 , 3 , 2]in which we are guaranteed to nd the minimum.

The procedure starts with [1 , 2 ] = [0, 1]. A new value 3is taken in the middle of the interval and (3) is computed.

Consider rst the situation depicted in Fig. 7.a, where:

(3 ) < (1 ) and (3 ) < (2 ) . (24)

We only know that the function has its minimum somewherebetween 1 and 2 . To further bracket the minimum, iscomputed at two other points 4 and 5 . The new smaller

interval including the minimum is formed by three consecutivevalues of , the middle one corresponding to the smallestvalue of computed so far. In other words, the new 3 is theelement of {3 , 4 , 5} yielding the smallest value. In theexample of Fig. 7.a, 3 does not change and the new intervalis [4 , 3 , 5].

Note that two function evaluations, as in Fig. 7.a, are notnecessarily needed. Indeed, if the rst point considered, say4 , brings a lower function value than the central point, i.e.(4 ) < (3), it is useless to try a second point, since bothnew points could not be lower than the central point under theassumption of a single minimum in the search interval.

Consider now the situation depicted in Figs. 7.b and 7.c,

where the three points do not satisfy (24), but rather make upa decreasing sequence. Clearly, either the minimum is = 1 ,

as in Fig. 7.b, or is somewhere in the interval [3 , 2], asin Fig. 7.c. When partitioning this interval, either a newdecreasing sequence is found, as in Fig. 7.b, or the valuessatisfy (24), as in Fig. 7.c, in which case we are brought back to the situation of Figs. 7.a. Increasing sequences are obviouslyhandled in the same way, with possible convergence towards = 0 .

The above procedure is repeated until the difference be-tween the upper and lower bounds of the interval is smallerthan a specied tolerance. In all cases, we place a new pointin the middle of the surrounding interval.

This technique is a variant of the golden section search,which considers unequal partitioning of the search interval.With respect to the latter, the above described approachguarantees that the interval is divided by two at each iteration,possibly at the price of a few additional function evaluations.

REFERENCES

[1] P. Kundur, Power System Stability and Control , New York, Mc Graw Hill,1994

[2] T. Van Cutsem, C. Vournas, Voltage Stability of Electric Power Systems ,Boston, Kluwer Academic Publishers, 1998

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[6] T.J. Overbye, A power ow solvability measure for unsolvable cases,IEEE Trans. on Power Systems, vol. 9, 1994, pp. 1359-1365

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[8] Y. Makarov, A. Kontorovic, D. Hill, I. Hiskens, Solution characteristicsof quadratic power ow problems, Proc. 12th Power System Computa-tion Conference, Dresden (Germany), Aug. 1996, pp. 460-467

[9] C.W. Taylor, Power System Voltage Stability , McGraw Hill, EPRI PowerSystem Engineering series, 1994

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[13] W.H. Press, S.A. Teukolsky, W. T. Vetterling, B.P. Flannery, Numericalrecipes in Fortran (2nd edition), Cambridge University Press, 1994

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