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represented by A*, that st il l a llows a stable operation of thesystem. At this point the voltage stability index S Ie is equalto zero. It is shown in [6] that A* c an b e computed directly as

{ r e - xQe + J[ r 2 + x2 P; + ~ ] }

A = v 2 . (2)s [ xP e - rQe 2]

Fo r other operating points A I 1), Eqs. (1) and (2) areused by replacing P e and Qe respectively by APe and AQe

It was also shown in [6] that it is possible to obtain atwo-bus equiva lent sys tem from any radial DS, by keepingthe substat ion bus s a nd s ome particular load bus k. Theimpedance of the equivalent l ine that connects bus s to k isgiven by

(3)

where Z sk is the equivalent impedance between the substationbus s and the bus k f is the set of branches of t he pa th tha tconnects bus s to bus k Jk is the current injected at bus k andZi and J i are the impedance and the current through branchi.

In method I, A* is i tera tively obtained for any radial DS.The procedure can be summarized as follows.

(a) Compute the volt age st abi li ty index for all load busesusing (1).

(b) Obtain the two-bus equiva lent sys tem containing thesubstat ion and the weakest bus (which has the smalles tvoltage stability index), using (3).

(c) Compute the max imum load factor for the equiva lent

system using (2).(d) Per fo rm s teps a - c unt il the max imum load factor

converges to *.Forthe sake of illustrating the performance of method I, Fig.

2 shows the evolution of the i tera tive process for the 33-busdistribution system [5].

real value. Fig. 2 shows an example of such a situation,where the real maximum loading is 370.77 and methodI converges to 361.36 , with a final error of 2.54 .Even though these errors are small for most cases , there

is no control over them. In other words, there maybe cases, for ce rt ain condi tions, where thi s e rro r cou ldbe unacceptable. The accuracy of met hod I c ould beimproved if there were some control over the final error.

• The pene tra tion of dis tr ibuted generat ion has been increasing significantly [7]. For DSs containing independent generators with vol tage regulat ion, method I mayprovide load factors larger than the real value, resulting ininfeasible operating points and divergence of the conventionalload flow methods. This aspect was not consideredin [6]. An example of such a s itua tion will be shownlater. The robustness of method I could be improved if theinfeasibility situations were dealt with in an appropriate

manner.This pa pe r has the purpos e of making a cont ribut ion to

solving the two points discussed above.

B Load.flow method with step size optimization LFSSO

In general terms, the load flow method with step sizeoptimization (LFSSO) consists of controlling the step size ofthe Newton s iterative method by updating the voltages usinga scalar, usual ly referred to as the opt ima l multiplier. Thi sidea was initially propos ed in [8], where the nodal powersand the voltages were represented in rectangular coordinates.The LFSSO in whi ch the voltages are repres ented in pol arcoordinates, being easily incorporated to production grade loadflow programs, was proposed in [9]. Recently, the efficiencyof LFSSO was val idated in [10].

The basic equation of LFSSO at iteration n is

[D. ) ] n

1 [ D. ) ] n . - 1 n [ D.P ] n

1

D.V D.V +M J D.Q 4)

380 . . . . . r . . r . .

370

~ 6

where D. ) and V are the voltage phase angle and magnitudecorrections, D.P and D.Q are the real and react ive power mismatches, and J is the Jacobian matrix. Th e optimal multiplierJ is added to the convent ional load flow formulation, beingcomputed to minimize a quadratic function based on the powermismatches. The derivation of M is shown in [9], where it is

clear that second or der informat ion is us ed to improve theperformance of the load flow method.

Fo r well-condit ioned systems, Mis close to one, and doesnot affect the iterative process significantly. For ill-conditionedsystems, M assumes different values to minimize the illcondi tioning effects , providing the correct solut ion for theproblem. For infeas ible sys tems (those for which the re areno solut ions for the load flow equa tions ) Mtends to zero ina few iterations, indicating that the power mismatches cannotbe decreased fur ther and the best possible solut ion has beenreached. Also, Overbye [11] showed that, for infeasible cases,LFSSO leads to a point on the feasibility boundary rather thanto simply diverge. These features are of particular interest for

voltage stability.

10

.

330

Bu 350 0

o

340

1

- Maximum loadability- - - Method I

320 L---L_--I.---=:::i::::===::L:::=:::::==:t:===::::::::t:===:t====:::L==::J1 5 6 8

IterationFig. 2. Method I - 33-bus distribution system.

After exhaustive simulations using method I for several DSs,the following comments can be made.

• For DSs without distributed generation, the iterative pro

cess always converges to a load factor sma ll er than the

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Substation

The behavior of LFSSO can be bet ter understood with thehelp of Fig. 3, where a two-bus example system is shown. Fig.3 also shows the parameter load) space for this system, wherepoints A, a nd C correspond respectively to the base case, an

infeasible point and the operating point provided by LFSSO incase point B were specified. The dashed line which connectspoints A and B corresponds to a pre-determined direct ion ofload increase that could be, for instance, related to maintaininga constant power factor the real and reactive powers increaseproportionally). is the boundary that separates the feasible nternal) and infeasible external) regions. Note that point Cprovided by LFSSO, in case point B were specified, is on thefeasibility boundary but not necessarily on the dashed line thatindicates the direction of the load increase. Experience showsthat point C is always within the close vicini ty of the dashedline.

Load

j0

A.-

Fig. 3. Example of two-bus system and its respective parameter load) space.

Incorporating LFSSO into method I provides the robustnessand efficiency discussed earlier in the Sec. II-A, as shown inthe next section.

C. roposed method

The proposed method compri ses three basic steps: loadincrease, load shedding and binary search.

In the first step, met hod I [6] is used to obtain the valuesof the load factor ,\ corresponding to the load increase. SinceLFSSO is very robust and deals properly with infeasible cases,the idea is to use i ts powerful character ist ics to increase theload very quickly, intentional ly seeking a load factor largerthan the real maximum loading in the infeasible region), butnot too far from it. In or der to achieve this goal, method I isslightly modified, by multiplying the computed load factor in 2) by an arbitrarily chosen scalar. In this paper, this scalar wasset to 1.1 in other words, the load increase will be 10 largerthan the one determined by method I). This step is carried outiteratively until an infeasible point is found, for which LFSSOdoes not converge (M tends to zero), but provides an operating

point on the feasibility boundary.

In the second step, the best solution given by LFSSO, whichis an operat ing point on the feasibil ity boundary, is used asa starting point for a load shedding procedure. Then, eachcomputed power on the boundary) is divided by its respective

base case power, providing a load factor very close to thereal maximum loading. However, a more representative valuefor the whole system is obtained by computing the averageof these computed values for all load buses. Three kinds ofvalues that are inadequate are eliminated from this averagecalculat ion, namely the values corresponding to computedpowers that are larger than the specif ied values in module) ,since they indicate that they are not on the feasibility boundarybut in the infeasible region; positive values, since they indicatethat the respective buses contain generation rather than load;and values for which the specified powers are equal to zero,since it results in division by zero. As the idea is to obtain avalue of the load factor in the feasible region, but very close

to the feasibility boundary, the computed average is multipliedby an arbitrary scalar in this paper this scalar is equal to0.9, corresponding to a 10 reduct ion) . This step is carriedout iteratively until a load factor within the feasible region isobtained.

In the third step, the smallest load factor corresponding toan infeasible point obtained in the first step) and the largestload factor cor responding to a feasible point obtained inthe second step) are used to star t a binary search procedure,which consists of computing a load factor halfway from thesetwo initial points. Using LFSSO, it is verified whether th isnew value is within o r without the feasible region. In casethis point is feasible, it replaces the old feasible value. f itis infeasible, it replaces the old infeasible value. Th is stepis carri ed ou t it eratively until the value of the load factorconverges according with a desired precision. In this paper, theconvergence criteria of the binary search process was based onthe difference between the last two load factors. The processis interrupted in case this difference becomes smaller than 1 .Note that there is now a control over the final error.

Af ter comput ing the maximum load factor, the voltagestability margin is easily computed as the difference from thisrecently obtained value and the load factor of the base case.The proposed method is described below.

a) Compute the vol tage stabil ity index for all load buses,using 1).

b) Obtain a two-bus equivalent system containing the sub

station and the weakest bus which has the smallestvoltage stability index), using 3).

c) Compute the maximum load factor of the equivalentsystem with 2) and multiply i t by a scalar to force theloading even further. In this paper, this scalar was set to1.1 loading forced 10 further).

d ) Perform steps a ) to c) unti l the new operating point isinfeasible LFSSO provides 0). Then, proceed tostep e).

e) From the power mismatches provided by LFSSO, decrease the load factor curtai l load), and multiply i t bya scalar to force the load ing to fall into the feasibil ityregion. In this paper, the scalar was set to 0.9. Repeat

this step unt il a feasible point is obtained.

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