NEW ALGORITHMS FOR THE COMPUTATION OF

NEW ALGORITHMS FOR THE COMPUTATION OF VOLTAGE COLLAPSE MARGINS: COMPUTATIONAL EXPERIENCE José Luis Fernández, Ana Cortes Red Eléctrica de España Pº Conde de los Gaitanes, 177 La Moraleja 28109 Madrid Fax: +34-1-650 45 42 Phone: +34-1-650 20 12 Abstract - This paper presents the results obtained by two new algorithms for the margin computation to the voltage collapse point. The voltage collapse point is considered to be equivalent to the maximum loading point. One of the proposed algorithms is a fast algorithm which yields an approximate margin with a low computational cost. The other one yields a more accurate margin with greater computational effort. WRITE SOMETHING ABOUT RESULTS. 1-INTRODUCTION

Julián Barquín, Tomás Gómez Instituto de Investigación Tecnológica Universidad Pontificia Comillas C/ Fernando el Católico, 63 Dup. 28015 Madrid Fax: +34-1-549 87 31 Phone: +34-1-544 90 88 included in the real time sequence of the system control computer. -

A fast computation margin procedure that could be launched after the occurrence of any contingency and whose results could be used to take decisions such as tap changers blocking or shedding of predetermined loads in an automatic way. This fast computation procedure could also be used to evaluate the effect of contingencies in voltage stability adding it to a contingency analysis program.

Due to the growing difficulties to construct new transmission lines because of environmental and economic constraints, power systems are being operated at increasing levels of load. As a consequence, normal operating points are getting closer to power system stability limits and it becomes necessary to evaluate the margins to that limits and also the control actions that can move the power system away from too risky operating conditions [1].

The remainder of this paper is organised as follows. The theoretical basis of the algorithms are presented in section 2. The fast algorithm is explained in section 3, and the exact algorithm ( which can be seen as the fast algorithm iterated ) in section 4. In section 5 WRITE SOMETHING ABOUT RESULTS.

In this paper two new algorithms for the computation of the margin to the voltage collapse are presented. One of them is a fast algorithm to estimate the margin from the working point of the system. The computing time of this algorithm is lower than that of a load flow. The other algorithm allows the exact calculation of the margin. This requires more computing time, approximately 4 times that of a load flow.

The theoretical basis of the algorithms are explained at length in [2,4]. The algorithm is based upon the analysis of the power-flow equations, which are written in a compact form as

Both algorithms are based on the search of the maximum loading point of the power flow equations. This point is mathematically equivalent to a saddle-node bifurcation of the system equations [2,3] The two different algorithms proposed respond to two different needs:

u are the power system variables, such as voltage magnitudes and angles at the load buses, and reactive power outputs and angles at generators not operating in their reactive power limits.

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An exact or very precise computation margin procedure that takes into account exciter current limits or reactive power limits of the generators is needed for use in real time control as an security enhancement tool for operator assistance. Recommended preventive control actions when they are needed are also a required output of this procedure. The computation of this precise margin should be done taking the results of the state estimator and should be

2-THEORETICAL BASIS

G (u , p) = 0 (1) where:

p are the operating parameters, that is, the quantities which define the current operating point, In this paper, only the active and reactive loads will be considered. The method can also consider voltage dependent load models. The linearization of equations (1) around an operating point yields Gu ∆u + Gp ∆p = 0 (2)

being Gu the power flow jacobian matrix.

limits and only requires a complete LU factorisation of the power flow jacobian matrix.

If Gu is non-singular, this formula provides a new operating point for any parameter increment, if small enough. However, in the maximum loading point -saddle-node bifurcation point- Gu becomes singular, equation (2) cannot be solved and no further load increment is possible. The singularity of this matrix implies the nullification of the smallest singular value of the pair (E, G u ), being E an arbitrary matrix. The singular values s and singular vectors w and v are defined by sEETw = Gu v sETE v = Gw

(3)

The main steps of the algorithm are: 1.

0 = wTGuu(v o v)x2+wTGp∆p (4) = ax2 + bT ∆p (5) = ax2 + µ (6) being wTGuu (v o v) a way to write the quadratic terms. This formula is known as the normal form of the saddle-node bifurcation. µ is the margin to the maximum loading point which can be shared among the different loads increments ∆p according the sensitivity vector b.

u

vector of system

p

vector of system

variables G (, ) = 0 parameters 2.

Computer the jacobian matrices G u and G p and the LU factors of Gu.

3.

Choose a direction d for the increments of the parameters.

The matrix E is a 0-1 matrix which purpose is to focus on a pre-determined geographical area and/or class of variables (voltage magnitudes or angles). Around the singular point (s = 0), it is possible to expand the power-flow equations, so that ∆u = vx [3,4,5], to get

Take the steady-state solution of the current operating point as departure point.

∆p = a d 4.

Choose a 0-1 filter matrix E. The objective of this matrix is to select the variables and equations that may be relevant at the collapse point. Special techniques have been developed to select this matrix E and they are an important part of the algorithm.

5.

Compute the minimum singular value, s0 , and singular vectors v and w of the pair of matrices (E, Gu) as in [6].

6.

Compute the coefficient a of the normal form using: a= wTGuu(vov)≈wT[G(u+ev)+G(u-ev)-2G(u)] with e a suitable small number.

In an operating point close to the addle-node bifurcation, the singular value of jacobian of the normal form is so = = 2ax (7)

7.

µ =8.

Therefore

9.

3-THE FAST ALGORITHM The fast algorithm is the application of the above formulae to estimate the margin to the maximum loading point. It takes into account the system

Estimate the parameters increment that lead to the voltage collapse using: µ = wTGp∆p = wTGp d a

µ = -ax2 =(8) As so can be computed from the singular analysis of the pair (E, Gu), and the "a" coefficient trough a numerical derivative computation, µ can be obtained.

Compute the margin µ of the normal form using:

Estimate the values of the variables in the collapse point. Algorithms based on the normal form have been developed to this end.

10. Check if there are reactive power outputs out of operating limits. If there are, then some of them are considered to be new PQ buses (to select which ones the criteria given in reference [7] is used). Otherwise, exit.

11. A new jacobian in the initial operating point is formed by adding the columns and rows corresponding to the new PQ buses. The reactive power outputs are added to the vector p , which dimension is increased. The associated new component in ∆p is set in such a way that these generators reach their limits.

5-RESULTS TO BE WRITTEN REFERENCES [1]

"Voltage Stability of Power Systems: Concepts, Analytical Tools, and Industry Experience". Published by IEEE, 1990. IEEE Catalogue Number 90TH0358-2-PWR.

[2]

"Inestabilidad de Tensiones: Criterios de Seguridad". PhD disertation, Julián Barquín, Universidad Pontificia Comillas, Madrid 1993.

[3]

Barquín, J., T. Gómez and F.L. Pagola "Voltage collapse and the saddle-node bifurcation". IIT Working Paper 92-028, July 1992 (available from the authors).

[4]

Barquín, J, T. Gómez and F.L. Pagola "Margin computations to the voltage instability". IIT Working Paper 92-031, July 1992 (available from the authors).

[5]

F.D. Galiana and Z.C. Zeng (1992, August) "Analysis of the load flow behaviour near a jacobian singularity". IEEE Transactions on Power Systems 7(3).

[6]

Löf, P.-A., T. Smed, G. Andersson, D.J. Hill (1992, February). "Fast Calculation of a Voltage Instability Index). IEEE Transactions on Power Systems 7 (1).

12. Refactorize the new Gu. 13. Go to step 5. 4-THE EXACT ALGORITHM The exact algorithm is, essentially the local algorithm iterated. Its main steps are: 1. Take the steady-state solution of the current operating point as departure point. 2. Choose a direction d for the increments of the parameters. ∆p = a d 3. Choose a 0-1 filter matrix E. The objective of this matrix is to select the variables and equations that may be relevant at the collapse point. 4. Computer the jacobian matrices G u and G p and the LU factors of Gu. 5. Apply the steps 5 to 9 of the fast algorithm 6. Actualise the parameters and variables according the previous results. 7.

Check if there are reactive power outputs out of operating limits. If there are, then some of them are considered to be new PQ buses (to select which ones the criteria given in reference [7] is used).

8.

The reactive power outputs of the new PQ buses are added to the vector p , which dimension is increased. The associated new component in p is actualised in such a way that these generators reach their limits.

9. Check if the minimum singular value and mismatch is small enough. If so finish. If not, go to step 4.

[7] Van Cutsem, T. (1991, February)."A Method to Compute Reactive Power Margins with Respect to Voltage Collapse". IEEE Transactions on Power Systems 6(1).