Study of multisolution quadratic load ow problems

Study of multisolution quadratic load ow problems and applied Newton-Raphson like methods. Yuri V. Makarov

Ian A. Hiskens

David J. Hill

University of Newcastle, Australia

Abstract - A number of facts about quadratic load

ow problems y = f (x) = 0, x 2 Rnx, y 2 Rny is proved. The main results are the following [1].  If any point x belongs to a straight line connecting a pair of distinct solutions in the state space Rnx, the Newton-Raphson iterative process goes along this line.  If a loading process y( ) reaches a singular point of the problem, the corresponding trajectory of state variables x( ) in Rnxtends to the right eigenvector nullifying the Jacobian matrix at the singular point.  In any singular point of the quadratic problem, there are two solutions which merge at this point.  The maximum number of solutions on any straight line in state the space Rnx is two.  Along a straight line through two distinct solutions of a quadratic problem, this problem can be reduced to a single scalar quadratic equation which locates these solutions. In addition, a number of other properties is reported. New proofs of them are given.  There is a point of singularity in the middle of a straight line connecting a pair of distinct solutions in the state space Rnx [2, 3, 4].  A vector co-linear to a straight line connecting a pair of distinct solutions in Rn nulli es the Jacobian matrix at the point of xsingularity in the middle of the line [2, 3].

Sydney University, Australia general idea behind all of them is to apply corrections to each step of methods in such a way that iterative processes do not oscillate or diverge. Due to its nonlinearity, the load ow problem can have a number of distinct solutions. Studies of of the multiple solution load ow problem play an important role in determining of stability margins and proximity to a voltage collapse [2, 11]. In order to obtain multiple load ow solutions, Tamura et al have used a set of quadratic equations and the NR optimal multiplier method [7]. Iba et al used the Tamura's approach and some newly discovered convergence peculiarities of the NR method with optimal multipliers to nd a pair of closest multiple solutions if there is one [8]. It is observed from experimental results in [8] that if a point comes close to a line connecting a couple of distinct solutions, a further NR iterative process in rectangular form goes along this line. The next observation mentioned in [8] is that in vicinity of a singular point the NR method with the optimal multiplier gives a trajectory which tends to the straight line connecting a pair of closely located but distinct solutions. These features are eectively used by in [8] to locate multiple load ow solutions. But the authors asked for a theoretical background of the experimentally discovered properties. The present paper attempts to make a number of proofs which meet this demand and explain some properities of the quadratic load ow problem and applied NR method.

1

Introduction

Theorem 1

For quadratic mismatch functions y = f (x), a variation of x along a straight line through a pair of distinct solutions of the problem f (x) = 0 results in variation of the mismatch vector y along a straight line in Rny. Proof. Let x be a point on the straight line connecting two distinct solutions x1, x2 of the problem f (x) = 0, and it is expressed as x = x1 + (x2 ? x1) = x1 + 
x21; (1) where  - is a scalar parameter, and
x21 = x2 ? x1. For quadratic mismatch functions, 2 f (x1 + 
x21) = f (x1 )+ J (x1 )
x21 + 2 W (
x21); (2) f (x2 ) = f (x1 ) + J (x1)
x21 + 0:5W (
x21); (3) where J (x) is the Jacobian matrix evaluated at x, and 1 2 W (
x21) is the quadratic term of the Taylor series expansion (3). At solution points x1, x2 , f (x1 ) = f (x2 ) = 0: (4)

The Newton-Raphson (NR) method and a variety of its modi cations are most popular numeric techniques for solution of the load ow problems. There are two main forms of load ow equations, namely, the polar and rectangular forms of equations. Both of them have their advantages and disadvantages. The polar form of equations provides signi cant reduction of computations. For instance, the method [5] and its modi cations, which use P-Q decomposition of a load

ow problem, are widely used on practice. The rectangular form of load ow equations can be effectively used as well [2, 3, 6, 7, 8, and others]. The most important fact is that the mismatch power balance functions f (x) can be exactly expressed using linear and second order terms of the Taylor series expansion. It is well known that the NR method has very good quadratic convergence if initial estimates are close to a solution point. But if they are far from a solution point or the load ow problem is an ill-conditioned one, convergence of the NR method can be slow or does not exist at all. To overcome this problem, a number of numeric techniques were proposed [6, 9, 10, and many others]. The 1

where Ai , J (0) are (n  n) constant matrices of Jacobian coecients, xi 2 x. Using (12), the equality (11) can be rewritten as 2J (x0 )
x21 = 0; (13) where x0 = (x1 + x2)=2. As
x21 6= 0, the vector
x21 is the right eigenvector corresponding to a zero eigenvalue of the Jacobian matrix. Moreover, for all x 6= 0 which are co-linear vectors with respect to
x21, we get J (x0)x = 0: Comments. The rst part of the theorem was proved in [4]. Both the rst and the second parts were proved in [2, 3]. The present way of proof seems to be more simple and compact compared to the mentioned works. An interesting conclusion follows from Theorems 1 and 2. Variations of x along a straight line connecting a couple of distinct solutions are actually motions of x along the right eigenvector nullifying J (x) in the middle of the line.

So, from (3),

0:5W (
x21) = ?J (x1 )
x21: (5) Using (5), the equation (2) transforms to f (x) = f (x1 + 
x21) = (1 ? )J (x1 )
x21 = ; (6) where = (1 ? );  = J (x1 )
x21: Thus the mismatch function y = f (x1 + 
x21) varies along the straight line  in Rny. Comments. The proved fact was mentioned in [2]. Theorem 1 has an interesting practical application. It gives a basis for nding of multiple solutions of a quadratic problem f (x) = 0. Really, (6) can be transformed to f (x1 +
x) + ( ? 1)J (x1)
x = 0; (7) where x1 is a known solution;
x is unknown increment of state variables;  is unknown scalar parameter. Except the trivial case
x = 0, the last equation corresponds to a dierent solution x2 = x1 + ?1
x; jj < 1: The system (7) has n equations and n + 1 unknown variables, and it is necessary to add an additional equation in (7). For instance, it can be rt
x ? 1 = 0; (8) where r is a nonzero vector. By varying of r and substitution of newly discovered solutions instead of x1 in (7), (8), it is possible to get all solutions of a quadratic problem.

2

3

If any point x is on a straight line connecting two distinct solutions of the quadratic problem f (x) = 0, the NewtonRaphson iterative process follows this line. Proof. If any point xi is on the line connecting two distinct solutions, it can be described in the form (1). The middle point of
x21 is x0 = 21 (x1 + x2). The quadratic mismatch function can be expressed as f (xi ) = f (x0 ) + J (x0 )(xi ? x0 ) + 0:5W (xi ? x0): (14) To express the last term of (14) in terms of Jacobian matrices, we write f (
x) = f (0) + J (0)
x + 0:5W (
x); (15) f (0) = f (
x) ? J (
x)
x + 0:5W (
x): (16) By summing of the last two equalities, W (
x) = [J (
x) ? J (0)]
x: So, 0:5W (xi ? x0) = 0:5[J (xi ? x0) ? J (0)](xi ? x0): On the other hand, taking into account quadratic nonlinearity of f (x) and (12), J (xi ? x0) = J (xi ) ? J (x0) + J (0): Therefore, in (14) we have 0:5W (xi ? x0) = 0:5[J (xi) ? J (x0)](xi ? x0): Noting (13), J (x0 )(xi ? x0) = 0: From (14), we get f (xi ) = f (x0 ) + 0:5J (xi)(xi ? x0 ): (17) For the Newton-Raphson method, we have the following expression for the correction vector
xi f (xi ) + J (xi )
xi = 0: (18)

Theorem 2

For a quadratic problem f (x) = 0, there is a point of singularity in the middle of a straight line connecting a pair of distinct solutions in Rnx, and a vector co-linear to this line nulli es the Jacobian matrix evaluated in the middle point. Proof. Let x1, x2 are two distinct solutions of a quadratic problem f (x) = 0. A line connecting these solutions can be de ned as (1), and due to quadratic nonlinearity, f (x1 ) = f (x2 ) ? J (x2 )
x21 + 0:5W (?
x21); (9) f (x2 ) = f (x1 ) + J (x1)
x21 + 0:5W (
x21): (10) It is clear that W (?
x) = W (
x): From (9), (10), and (4), [J (x1 ) + J (x2 )]
x21 = 0: (11) For a quadratic function f (x), the Jacobian matrix contains elements which are linear functions of x. So, it can be represented as

J (x) =

n X i=1

Ai xi + J (0);

Theorem 3

(12) 2

From (6), and

f (xi ) = (1 ? )J (x1 )
x21; f (xi ) (if  6= 0; 1): J (x1)
x21 = (1 ? )

The rst case gives us the original solution point x = x . The second case corresponds to other solutions x 6= x (19) on the straight line directed by
x. But as it is clear from (24), the function (24) can not be negative. Thus

a2 + b + c  0; and in the case (b) it is possible to have only one additional At the point  = 21 we have xi = x0 , and it follows from solution except x , but not two or more. So, on the line we get one original root x = x , and we can have only one (19) and (20) that additional root corresponding to the condition (b). f ( x 1 i) (21) f (x0 ) = 4 (1 ? ) : (20)

5

Theorem 5

By substitution of (21) into (17), f (xi ) + 1 J (x )(x ? x ): f (xi ) = 41 (1 (22) ? ) 2 i i 0 Multiplying (22) by J ?1 (xi ) and taking into account (18),
xi = ? 21 44(1(1? ?)?) 1 (xi ? x0): (23) The equation (23) shows that the Newton-Raphson corrector vector
xi belongs to the straight line directed by the vector (xi ? x0), i.e. the iterative process goes along the line connecting solutions x1 and x2. Comments. The proved fact was experimentally discovered in the [8], where the authors asked for some theoretical proof of the phenomena.

For a straight line connecting two distinct solutions in Rnx, the initial system of quadratic equations f (x) = 0 can be reduced to a single scalar quadratic equation which locates these solutions. Proof. Let x1 and x2 are two unknown distinct solutions of a quadratic problem f (x) = 0. Suppose we have a point x and a direction
x which de ne a line connecting the pair of solutions. The mismatch function calculated along this line is f () = f (x + 
x); (25) where  is a scalar parameter. Let us take any xed value  =  and de ne  = f ( ) = const 6= 0: (26) Having (25) and (26), consider the following equation

4

t f () = 0: (27) It follows from Theorem 1 that  and f () are co-linear vectors, and (27) is true only if f () = 0. Using (26) and the Taylor series expansion f () = f (x ) + J (x )
x + 0:52W (
x); we get the scalar quadratic equation a2 + b + c = 0; (28) where a = 0:5tW (
x ); b = t J (x )
x ; c = t f (x ): According to conditions of the theorem, (28) is to have a pair of distinct real roots 1, 2, and we can de ne solutions of f (x) = 0 as x1 = x + 1
x ; x2 = x + 2
x :

Theorem 4

The maximumnumber of solutions of a quadratic equation f (x) = 0 on each straight line in the state space Rnx is two. Proof. Let us take the function () = f t (x + 
x)f (x + 
x): (24) For a quadratic mismatch function f (x), f (x + 
x) = f (x) + J (x)
x + 0:52W (
x): So, () = kf (x) + J (x)
x + 0:52W (
x)k2 = = kf (x)k2 + kJ (x)
xk2 + k0:52W (
x)k2 + + 2f t (x)J (x)
x + 3 W t (x)J (x)
x + + 2 f t (x)W (
x): The function () equals to zero if and only if f (x + 
x) = 0. At a solution point x = x , f (x ) = 0, and the function (24) is  () = 0:254kW (
x)k2 + 3 W t (
x)J (x)
x + +2 kJ (x)
xk2 = (a2 + b + c)2 : For any xed direction
x 6= 0,  () equals to zero in the two following cases (a)  = 0; (b) a2 + b + c = 0.

6

Theorem 6

If any loading process y( ) ends at a singular point x0 of the problem y( ) + f (x) = 0, where f (x) is a quadratic function of x and is a scalar loading parameter, there are two distinct solutions x1, x2 merging at the singular point, and the trajectories x1 ( ), x2( ) tend to the right 3

eigenvector r corresponding to a zero eigenvalue of the References Jacobian matrix J (x0). [1] Y.V. Makarov, and I.A. Hiskens, Solution characterProof. Let after execution of the loading process istics of the quadratic power ow problem, Technical Report no EE9377, Department of Electrical and y( ) + f (x) = 0; = var (29) Computer Engineering, University of Newcastle, Australia, May 1994. we get a singular point x0 corresponding to the value 0 of the loading parameter . At the singular point, [2] Y. Tamura, K. Sakamoto, and Y. Tayama, "Voltage ! instability proximity index (VIPI) based on multiple dy load ow solutions in ill-conditioned power systems", d =
yd : dy = d Proc. of the 27th Conference on Decision and Con = 0 trol, Austin, Texas, December 1988. [3] Y. Tamura, Y. Nakanishi, and S. Iwamoto, "On the The implicit function theorem gives multiple solution structure, singular point and existence condition of the multiple load ow solutions", J (x0)dx + dy = J (x0 )dx +
yd = 0: (30) Proc. of the IEEE PES Winter Meeting, N.Y., February 1980. t By multiplying (30) by s , where s is the left eigenvector of J (x0 ) corresponding to a zero eigenvalue, we get [4] V.I. Idelchik, and A.I. Lazebnik, "An analytical research of solution existence and uniqueness of electrical power system load ow equations", Izvestia AN st J (x0)dx + st
yd = st
yd = 0: SSSR. Energetika i Transport, no. 2, 1972, pp. 18-24 (in Russian). For the general case of loading [5] B. Stott, and O. Alsac, "Fast decoupled load ow", st
y 6= 0; IEEE Trans. on Power App. and Syst., vol. PAS-93, no. 3, May-June 1974, pp. 859-869. and, therefore, d = 0. This means that the loading parameter reaches its extremal value 0 at the singular [6] S. Iwamoto, and Y. Tamura, Y, "A load ow calculation method for ill-conditioned power systems", point x0. Alternatively, when st
y = 0, the loading traIEEE Trans. on Power App. and Syst., vol. PAS-100, jectory y( ) tends to the tangent hyperplane to the sinno. 4, April 1981, pp. 1736-1743. gular margin of the problem f (x) = 0 at the point x0 . It follows from the fact that s is an orthogonal vector with [7] Y. Tamura, K. Iba, and S. Iwamoto, "A method for respect to the singular margin in the space of mismatches. nding multiple load- ow solutions for general power Using (30), we have systems", Proc. of the IEEE PES Winter Meeting, N.Y., February 1980. J (x0 )dx = 0: (31) [8] K. Iba, H. Suzuki, M. Egawa, and T. Watanabe, "A method for nding a pair of multiple load ow soSo, the increment dx has the same direction as the right lutions in bulk power systems", Proc. of the IEEE eigenvector r of the Jacobian matrix corresponding to its Power Industry Computer Application Conference, zero eigenvalue, and the last part of Theorem 6 has been Seattle, Washington, May 1989. proved. On the other hand, having (31), [9] V.A. Matveev, "A method of numerical solution of sets of nonlinear equations", Jurnal Vychislitelnoi !0 Matematiki i Matematicheskoi Fiziki, vol. 4, no. 6, ?
y
= J (x0)
x + 0:5W (
x)
?! 0:5W (
x); 1964, pp, 983-994 (in Russian). and [10] A. M. Kontorovich, Y. V. Makarov, and A. A. W (
x) = W (?
x): Tarakanov, "Improved methods for load ow anal ysis", Acta Polytechnica, Prace CVUT v Praze, vol. Consequently, for a small increment
at the point x0 we 5/III, no. 1, 1983, pp. 121-125. have two increments of x of opposite signs directed along r. Therefore, the rst part of the Theorem 6 has been [11] Y. Tamura, Y, H. Mori, and S. Iwamoto, "Reproved. lationship between voltage instability and multiple Comments. The theorem is proven in [12]. It explains load ow solutions in electric power systems", IEEE the experimental fact obtained in [8]. It was observed that Trans. on Power App. and Syst., vol. PAS-102, no. in vicinity of a singular point the NR method with optimal 5, May 1983. multiplier gives a trajectory which tends to the straight line connecting a pair of closely located load ow solutions [12] Y.V. Makarov, and I.A. Hiskens, "A continuation x1, x2. Actually, the loading trajectory y( ) in (29) can method approach to nding the closest saddle node be represented as the convergence trajectory of the NR bifurcation point", Proc. NSF/ECC Workshop on method. If it comes close to the singular margin, it tends Bulk Power System Voltage Phenomena III, Davos, to the right eigenvector r which nulli es the Jacobian maSwitzerland, August 1994; published by ECC Inc., trix in the middle point between closely located solutions. Fairfax, Virginia. Theorem 3 says that further iterative process goes along the line directed by the vector (x1; x2), and that line is co-linear to r. So, Theorems 3,6 explain the phenomena. 4