Natural Modes and Their Stability in Power Systems

8/8/2019 Natural Modes and Their Stability in Power Systems

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Table 2.1Nonzero Eigenvalues and Their Eigenvectors

of a Uniform System

Eigenvalues

Eigenvectors

03

- j /nRok/H1 0

0 0

-1 0

0 1

0 0

0 -1

- j d3nRok/H

-1 0

2 0

-1 0

0 -1

0 2

0 -1

In this uniform syste m, the eigenvectors of thelinearlized ystem re lso he atural odes. To

+ j m. et the initial conditions beshow this roperty, onsider heomplex air-

Then the trajectory of the system is f the form

To pr ove (2.5), we substitute it into (2.1) and (2.2)to reduce the dynamics governing 1 and A3 to

and the dynamics governing 2 to

I = 2nR0w2 , A2(0) = 0,

2H; = 02 , w ( 0 ) = 0.2

System (2.7) yields the trivial solution 62(t)=O,op(t)=O. System (2.6) represents a model of a singlemachine infinite bus system with angle 8 and speedV , and exhibits sinusoidal oscillations for nonzero oand vo. Sin ce A1(t) = -d3(t), 61 and 63 cross zero orreach a maximum or minimum at the same time instant.Thisype of exact n-phase or out-of-phaseoscillation is called a natural r normal mode.

Similarly, we can show that the oscillation

6l(t) = -62(t)/2 = 63(t),

q ( t ) = -o2(t)/2 = w3(t),

is a natural mode associated with the complex pair- j e.

We now use the uniform system to illustrate someproperties of natural modes.

A property of a natural mode s that its trajectoryis normal o he quipotential urfaces of thepotential energy function I]

For the uniform system with natural mode (2.5), (2.8)

becomes

Vp~ (6) -(k os61 + k COS 63) + 2k. 2.9)

The gradient f VPE is

k sin 1

-k sinsin 3

k sin 8

avpE(b)/a6 = (2.10)

- - - -

which is colinear to the trajectory &l(t)=-63(t)=8(t),62(t)=O. his normal" roperty s n ssumptionused in the potential energy boundary surface method[51.

Another property of a natural mode is that itsstability can be determined from the equivalent singlemachine nfinite b u s model (2.6 ) using he nergyfunction

v(e, V ) = VKE(V) + vpE(e) (2.11)

consisting of the kinetic energy

VKE(V) = (1/2) (2H)(2nRo)v2 (2. 12)

and the potential energy

vpE(e) = -k COS 8 + k. (2.13)

From the Lyapunov stability theor y, a trajectory ofsystem (2.6) is stable if its energy satisfies

V(B, u) < 2.2.14)

Thus this property reduces the stability analysis of(2.1), (2 .2) in an En-dimensional pace to a2-dimensional anifold. he tability of the ullsystem (2.1 1, (2.2) an hen eetermined byrecovering 6 , o in terms f e , u. For heuniform sy stem , a trajectory is stable if its energysatisfies

V(6,w) = V K E ( U ) + Vp~(6) 4k2.15)

where the kinetic energy is

and he otential nergy i s given n 2.9). hisnatural mode property offers additional justificationsto direct methods using simplified energy functionssuch as the single machine r group of machines energyfunctionpproach15,16] . Furthermore, hetrajectory of a atural mode, when lose o eingunsta ble, will approach an unstable equilibrium. Thusthe methods in [3,4,63 will provide good results whenthe appropriate unstable equilibrium energy s used.


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The previous two properties hold or natural modesof any conservative system and are not restricted tothose of the niform ystem 2.1)-(2.3) nly.Howev er, a special property of the natural modes ofthe uniform system is that the relation (2.5) holdsindependent of the energy (2.11 ) of the system. Thati s , the trajectory when plotted in the phase plane of6 yields he o-called traight odal ine [8].This special property does not extend to a generaln-machine syste m, which ill e llustrated n henext section.

Another special property of the uniform system isthatonzeroamping Di=D, i=1,2,3 , can eintroduced obtain amped atural odes. orexample, for the mode (2.5), 2. 1) and ( 2.2 ) reduce to

e = 2 n ~ , e ( o ) = eo,(2.17)

2Hu = -Dv - k sin e , ~ ( 0 ) v0 .

The magnitude of oscillation will decrease over time,but the magnitude and phase relation (2.5) will remainunchanged.

3. Series Solution of Natural Modes

There are no eneral esults n he umber fnatural modes that may exist in a conservative systemwith order larger than three , and on the computationof natural modes. For small perturbations about thestable equil ibriu m, the number of natural modes for ann-machine power system is expected to be (n-1). Forlarge erturbations, dditional atural odes ayarise in systems with certain parameters 18,173. Inthis se ction , we first discuss the characteristics ofnatural odes n ower syste ms, and he mpact ofnonconservative orces uch s njections, ransferconductances, and damping, on natural modes. Then wepropose a series expansion method for computing thenatural modes.

a. o injec tions , no transfer onductance nd odamping

In the case where i = O , Di=O, gij=O, i,j=l,Z, . . , ,we have D = O and table qui brium tw=O. Since the estoring orces re symmek?;:;about 6 =beq= 0 and bout w=O, an oscillationmay e ymmetrical oth bout 6 = 0 and w=O. Thesesymmetry conditions are necessary for the existence fnatural modes. The effects of the loss of thesesymmetry conditions are discussed next.

b. Nonzero injections, no transfer conductance and nodamping

The difference between this case and case a isthat he onzero njections PifO result in nonzero

equilibrium ngles deq#O. The nonzero quilibriummeans hat he estoring orces re no longersymmetrical bout d = d e q , although heymmetryabout o=O is preserved. n his ituation, t ispossible o ind ode of oscillation n which ,starting romith onzero w ( O ) , theachineangles ould each ither aximum or a inimumsimultaneously,ndhenrosssimultaneously. Howev er, beyond hequilibrium

crossing, the n-phase or out-of-phaseoscillations will no longer hold. Thus in this case,only the first half cycle of oscillation resembles anatural mode.

&eq

e. Nonzero indections, nonzero transfer conductancesand no dampinp,

With nonzero injections Pi#O and nonzero transferconductances gijtO , the symmetries about =Aeq and 0=0are both lost. The best approximation of a naturalmode that can be achieved is for the machine an gles,starting from with onzero w ( O ) , to reacheither a maximum or a minimum simultaneously. Beyondthat point, the in-phase or out-of-phase oscillatorybehavior will no longer old. However, to etect

first swing stability, this quarter cycle natural modelike oscillation s adequate.

d. Nonzero dampi ng

Only n he niform ystem ill atural odebehavior be preserved in the presence of damping. Ingeneral, the existence of damping will result in theloss of sy me tr y of system ynamics bout &=Aeqand 0=0, so that natural modes no longer exist. Asin ase c , the est ossible pproximation of anatural mode that can be obtained for a system withnonzero damping s a quarter cycle.

We now propose a method for computing the naturalmode like behavior of a power system for the firstquarter cycle of a machine swing. While the naturalmodes of a niform ystem re elatively traightforward to compu te, those of a eneral -machinesystem ayot eomputed xactly. romhecharacteristics of a natural mode, the angles 6 andspeeds o of an n-machine system (2.11, (2.2) can beexpressedseriesxpansionsithectorcoefficients VI , v2 , . . , v,

m

where i s atablequilibrium, nd e, vare the angle and speed of a single machine infinitebus system

6 = ~ V R , e ( o ) = eo

(3.2)

2Hv = -k sin e -Dv, ~ ( 0 ) v0 .

The even power term coefficients vzP, p=1,2, . . , ,are not needed if gij=O and Di=O, i,j=1,2, . . , ,because of symmetry. In contrast to a uniform syste m,the oefficients p's nd the parameters nddepend on the system energ y, that is , they depend onthe agnitude of oscillation. urthermore, ithnonzero igher rder oefficients vp, p22, thenatural mode oscillations, when plotted in the phaseplane of 6 , are urves ather han traight odallines. Thus the manifolds formed by natural modes aresurfaces, instead of straight lines as is the case ofa uniform system.

To solve for tne oefficients vp's nd theparameters H and k , we substitute the expansions (3.1)and (3. 2) into 2.1), 2.2) oatch he imederivatives of the state variables and the restoringforces longuarter ycle fhe (e , U )trajectory. In general, the solution depends on the

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energy V(B, u) in the system (3.2). For V(B, u ) small,(3.2) is close to a linear system, implying that V Ii s approximately n igenvector. hus ne way offinding heatural odes i s toomputeheeigenvectors and use hem to initiate he olutionprocess. Since it is impossible o olve for allv Is , a practical approach i s to solve for vp's oftge leading terms in (3.1) by matching the expansionsin (2.11, (2.2) at selected points on the ( 8 , U )trajectory. If r vp's are oe solved, rxnnonlinear lgebraic quations re needed. Then hecoeff icien ts vp's and the parameters H and k resolved from these load flow like equations. No timesimulation i s required. However, an iterativetechnique such as Newton's method is needed to obtainthe olution. or arge power systems, he pproachin [18] can be used to retain the load buses so thatsparsity i s maintained.

4. Example

Let us use a three machine system to illustratenatural odes for case here beq#0. The ystemparameters are

H1 = 0.5, H2 = 1.0, H3 = 1.5,k12 = 0.15, k13 = 0.07, k23 = 0.2,PI = -0.120, P3 = 0.0403, P3 = 0.0797, (4.1)

Theysteminearized at 6=beq hasoscillatory ode f 1.72 Hz. The igenvectorsthis mode are given y

where

( 4

anfor

2)

V1 = [-0.6937 0.4349 -0.05873IT. (4 .3 )

Using his ode as the tarting point, a ewtoniterative echnique as used to solve or VI , v3and v5 of the expansio n (3.1), that is , a 5th orderapproximation f his atural ode, at differentlevels of energy. The results are shown in Table 4.1.

In able 4.1, 8 is the maximum an gle chlevedby the equivalent single machine infinite bus system(3.21, and increases with system energy levels. Theequivalent inertia H is fixed at 0.5 for convenience.Note that v1 varies nonlinearly as the energylevel. AS em goes to zero, J R o k / uapproaches 1.72 Hz and v1 approaches the eigenvector

4 5 . i - 6 2 e q30.

1 5 .

0

-15.

-30.

2 2 5 ,

1 5 0 .

h

75 .wWw0

v oLLI-Ia

4-75.

-150.

0 0 . 6

T I M ES E C O N D S )

Figure 4.1 Natural Mod e fo r 8,=60

1. 2

1

1I

/

I /

0 0 . 6

T I M ES E C O N D S )

Figure 4.2 Natural Mode for 8,=170

1 . 2

(4.3). he results for +,,=180 are not shownbecause he teration did not converge. A higherorder xpansion eems to eeeded. he system

em=17Oo re hown in Figu:&6e(l. 1 and 4.2. Notesimulations tarting at for 8,=60 and

the natural mode behavior for the first half cycle.As pointed out earlier in the case b discussion, this

Table 4.1

Series Solution of Natural Mode

System Energy

1705020000'00m(maximum 8 )

0.00201.00150.00104.000645.000316.0000883.00000270

H 0.5.5.3.5.5.5.5

k 0.381 0.303.261.243.239.248.267

-0.661 0.1790.07180.2210.3340.4550.580

'0.510 0.501.560.576.577.566.539

-0.120 -0.3940.3500.3100.2730.226 -0.166

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behavior o onger holds after he Ae q crossing.Forhe 8,=170 si mu la ti on , oteheelative"flatness" o f the peaks , signify ing that the syste m isclose to instability. In this simulation, even thoughthis system i s stable for the first half cycle, itsubsequently goes unstable, that is , it is first swingstable but "back swing" unstable.

5. Conclusions

In this paper we have discussed natural modes andtheir characteristics in ower system electromechanicalmodel. We have shown that for a system oscillating ina natural mode, its stability can be determine d froman quivalent ingle achine nfinite bus system.practical ethod for obtaining series expansio nsolution f atural odes as een roposed. hismethod of determin ing natural mode stability requiressolving several sets of load flow like equations. Nosimulation is necessary. heethod has beendemonstrated on a small system.

ACKNOWLEDGEMENT

This work i s supported by the U.S. Depar tment ofEnergy under Contract DE-AC01-84CE76249.

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Natural Modes and Their Stability in Power Systems