Summary The paper considers the problem of nding points of maximum loadability which are closest (in a local

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1 Calculation of Power System Critical Loading Conditions Ian A. Hiskens Yuri V. Makarov Department of Electrical and Computer Engineering The University of Newcastle, Callaghan, NSW, 8, Australia Summary The paper considers the problem of nding points of maximum loadability which are closest (in a local sense) to the power system operating point. This optimization problem leads to a set of equations which describe such critical points. Not all solutions of this set of equations are critical points. The paper therefore explores the nature and characteristics of solutions. A two stage algorithm is proposed for solving the critical point problem. The rst stage is simply to nd a point of maximum loadability which lies is a specied direction. The second stage uses a continuation method to move from that initial point to the desired critical point. The proposed algorithm is tested on an eight bus power system example. INTRODUCTION The usual approach to the assessment of power system security, in a quasi-static sense, is to assume some particular loading pattern, then determine the corresponding point of maximum loadability [,, ]. Nose curve ideas, i.e., P-V and/or Q-V curves, follow from this approach. However such a method of assessment can result in optimistic predictions of security. The system may be much more sensitive to load changes for some dierent loading pattern. This paper presents a method for determining the most critical loading condition, and the associated security margin. The power ow is central to the calculation of quasistatic security margins. It describes the steady state relationships between system voltages and parameters such as loads, voltage setpoints and network impedances. That description takes the form of a set of n nonlinear algebraic equations y + f(x) = () where y R n is the vector of specied independent parameters such as active and reactive powers of loads and generators or xed voltages, x R n is the state, consisting of nodal voltages. The vector function f(x) denes the sum of power ows or currents into each bus from the rest of the network. The power ow equations f(x) generally dene a mapping from state space to a subset of parameter space, i.e.,?f : R n! L where L R n. The region L denes the set of parameters for which power ow solutions exist. As parameters y vary, the solutions of () will also move in state space. Parameters y may move to a point where two solutions coalesce, with further variation of y resulting in the disappearance of that solution. Behaviour of that form is referred to as a saddle node bifurcation. It follows from the Implicit Function Theorem that at such bifurcation points det D x f = det J(x) = () i.e., J(x) is the Jacobian matrix of f(x). Further, we see that the region L must be bounded by surfaces of points satisfying (). We shall dene the boundary as = f(x; y) : x; y R n ; y + f(x) = ; det Jj (x;y) = g () The projection of onto state space and parameter space will be referred to as x ; y respectively. To ensure adequate security of a power system, it is important that the operating point be prevented from moving too close to. That is, operating points should always be (at least) some specied distance from. As parameters y correspond to physical quantities that can be measured and controlled, it is useful to consider this distance in terms of parameter space, i.e., d(y) = ky? y k () where y is the operating point, and y y, i.e., y is a point on the solution boundary. The shortest (or critical) distance min yy d(y) gives a measure of power system security in the most dangerous direction of loading. In addition, the critical vector (y? y) denes the optimal way of controlling the power system to maximise security. Its largest components indicate parameters which contribute most to the security conditions [, 5]. In this paper we address the issue of robustly nding the minimum distance to. This question has been investigated before, see for example [, 5]. We are proposing a continuation approach to nding the points on which are closest (in a local sense) to the operating point. We shall refer to these points as critical points. The paper is organised as follows. Section establishes the mathematical description of critical points. Properties of the solutions of the critical point problem are discussed in Section. Section proposes an algorithm for nding the critical points. An eight bus example is considered in Section 5. Conclusions are given in Section 6. FORMULATION OF THE PROBLEM One of the rst things to consider in the formulation of the optimization problem min d(y) (5) y y is that not all parameters are free to vary. Some parameters, such as power injected at buses which have no load or generation, must always be xed. We therefore adopt the following power ow formulation. Let m equations in () contain xed values of parameters y = y = const. The other (n? m) parameters y are free to vary. Then the system () can be rewritten as y + f (x) = (6) y + f (x) = (7)

2 Considering the revised system description (6),(7), it is shown in [6] that solutions of (5) are a subset of the solutions of the constrained optimization problem ext x ky + f (x)k (8) y + f (x) = (9) where `ext' denotes extrema of the cost function (8). The optimization is subject to nonlinear constraints (9). The cost function (8) denes the square of the distance between the points y and y, with both points belonging to the constraint hyperplane y = y = const. Using a Lagrangian multiplier approach, it is further shown in [6] that solutions of (8),(9) are given by the set of equations?s + y + f (x) = y + f (x) = () J t (x)s + J t (x) = The last equation in () can be rewritten as J t (x)s = () where J t = [J t J t] and s = [s t t ] t. If s 6=, the Jacobian matrix J(x) is singular, and the vector s is a left eigenvector corresponding to a zero eigenvalue. Therefore, considering the optimization problem (8),(9), and the condition () for s 6=, we can conclude that critical points, i.e., points on that are minimal distance (locally) from the operating point y, satisfy the system (). SOLUTIONS OF THE CRITICAL POINT PROBLEM The system () can be considered as an extended power ow problem where the usual power ow equations (the rst two equations in ()) are supplemented by the singularity condition (). State space, i.e., the space of unknown variables, is now extended to include the additional variables s. So the variables are (x; s ) R n.. Trivial and Nontrivial Solutions There are two kinds of solutions to the critical point problem (): Trivial solutions corresponding to the condition s =. Those solutions are actually the solutions of the usual power ow problem (). They are global minima (zeros) of the distance function (). All trivial solutions coincide in parameter space. Nontrivial solutions conforming to the condition s 6=. Those solutions belong to the singular margin given by (). Solution of () can result in either trivial or nontrivial solutions, depending on initial estimates of the variables and the numerical solution technique used for solving the problem. A technique which produces nontrivial solutions is proposed in Section.. Distance and the Left Eigenvector Consider nontrivial solutions of (). It can be seen from the rst equation of (), and (6) that at critical points, y? y = y? s + f (x) =?s () Therefore, because y = y, the compontent?s of the vector?s is the distance vector y? y. Further, it is known that the vector s is normal to the singular hypersurface y [5]. Dene the intersection of y and the y = y hyperplane as y;y. It follows that s is normal to the hypersurface y;y. So the vector y? y is orthogonal to y;y. This conrms that nontrivial solutions of () correspond to local minima, maxima or saddles of the distance from the point y to y;y. The minimum over the distances associated with all the nontrivial solution points characterizes the global \level" of power system security.. A raphical Illustration A graphical illustration of a nontrivial solution point of () is given by Figure. The solution point must lie somewhere on the intersection of the singular margin y and the constraint hyperplane y = y = const. The left eigenvector s at the nontrivial solution point y is perpendicular to the singular boundary, and its component s coincides with the vector from the singular point y to the operating point y. The component (the Lagrange multiplier vector) of s is orthogonal to the constraint hyperplane. det J(x) = - s λ - s - λ y o y y o y = const Figure : raphical illustration of a nontrivial solution point.. A Bus Example Consider the bus system of Figure. The values P, P are considered as free parameters y in (6). The voltage magnitudes V = pu and V = pu are taken as xed parameters y in (7). The optimization problem (8) therefore transforms to ext ; [(P? :) + (P + :6) ] () Figure shows the plane of state variables ;, and Figure the plane of free parameters P ; P, with parameters V ; V xed. Singular margins x (dashed lines) and contours of the cost function () are plotted on the plane ; in Figure. Solutions of the optimization problem () are also shown. Points A,A are minima. They correspond to trivial solutions of the critical point problem (). A is the `normal' operating point. Points marked B and C are nontrivial solutions of (), with B B being saddle points of (), and C,C maxima. All nontrivial solutions of () lie on the singular margin x. y

3 Delta, rad P - - P=..+j P = -.6 V = V = δ = var δ = var A.+j V = δ =.6+j Figure : Simple bus power system. B B B A Delta, rad Figure : The, plane for the bus example C B B P Figure : The P, P plane for the bus example. Figure shows the sections of the singular margin y plotted in the plane of free parameters P ; P. The solution space L has a number of `layers', with each layer restricted by a section of the singular margin. These layers reect the non-uniqueness of solutions of the power ow equations. Each layer eectively generates a pair of power ow solutions. For example, there is a single layer at the point A. Consequently, there are two distinct solutions of the power ow problem, A,A. They are shown in Figure. C A B B C B C All solutions of () except A are shown in Figure. (A coincides with A, so is not marked.) This gure also shows vectors from the operating point A to the solutions of (). Each of these vectors is normal to the singular margin y. (The apparent absence of orthogonality of the vectors with respect to the singular margin in Figure is caused by a dierence in the horizontal and vertical scales of the gure.) Figure shows clearly that B B are saddle points of the optimization problem (). From Figure, we see that they satisfy (5) locally. On the other hand, C,C are local maxima of (). They are also points that locally satisfy max yy d(y). Depending on initial guesses of variables, and the solution technique, any of the points identied in Figures and could be obtained as a solution of (). As seen in the example though, only some of those points are of interest to us. Thus, the problem is to nd saddle points on the boundary of the security region. This problem is quite dierent to usual optimization problems where minima or maxima are desired. An appropriate technique is discussed in Section. AN ALORITHM FOR FINDIN CRITI- CAL POINTS The set of equations () that describe critical points also have solutions that are not of interest, e.g., trivial solutions where s =. Therefore, an algorithm for nding critical points must consist of two parts, () a way of obtaining a good estimate of the unknown state variables x; s; in the vicinity of the critical point, and () a numerical technique that will converge reliably from that initial estimate to the critical point. Such an algorithm is described in this section.. Stage : Obtaining a ood Initial Estimate The rst stage of the critical point algorithm must produce a good estimate of state variables x; s; in the vicinity of the critical point. To achieve this, it is necessary to have some idea of the direction in parameter space from the operating point to the desired critical point. In practice this requirement does not restrict the usefulness of the method, as power system operators and planners will usually have a good idea of the way in which parameters of their system, such as loads, vary. Let the estimated loading direction be y. Recall y = y. The initial estimate of the critical point can be taken as the point on the solution boundary in the direction y from the operating point. That point is given by y + y + f (x) = () y + f (x) = (5) J t (x)s = (6) s t s = (7) where is the loading parameter in the specied direction y, and ky k =. An alternative formulation of (6),(7) uses the right eigenvector to achieve the singularity condition, rather than the left eigenvector s. Many techniques have been proposed for solving this problem, for example [,, ]. In some cases direct methods have been used, whilst others have applied continuation methods [7] to obtain the solution.. Stage : Motion Along the Singular Margin Stage one of the algorithm provided us with a point on in the vicinity of the desired critical point. Let that

4 point be x ; ; s = [st t ] t. We now wish to move from that point to the critical point. Consider the equations ( y + s )? s + y + f (x) = (8) y + f (x) = (9) J t (x)s = () The initial point x ; ; s is a solution of (8)-() when =. But when =, the problem is exactly that of (). So the critical point is a solution when =. Therefore, as is varied from to, the solution of (8)-() is distorted from the initial point given by stage one, to the critical point. Notice that because of (), all points along that path lie on. Also, (9) ensures that the path lies on the y = y hyperplane. In solving this continuation problem, it is helpful to scale s so that ksk = at the initial point =. This scaled s will still satisfy (). It is shown in [6] that as is varied from to, the distance ky? yk always reduces. This is an important property as it ensures that the stage two algorithm will never converge to maxima of (). Further, the algorithm will always nd a loading condition which is more critical, i.e., which corresponds to a lower security margin, than the direction specied at stage one. Many numerical techniques exist for solving this continuation problem [, 6, 7]. The solution method of [6] was used for the example given in Section 5. 5 AN 8 BUS EXAMPLE The algorithm of Section was tested on the eight bus example shown in Figure 5. Operating point values of generation and load parameters are given in Table. The system contains three generators with xed terminal voltage, and four nonzero loads. Bus 6 is the slack bus. The nominal voltage of all buses is kv. Line parameters and the operating point voltage prole are given in [6]. 5 5 L 5 L I-st critical point Table Critical Points II-nd critical point Bus Volt- Vector Eigen- Volt- Vector Eigenp age y? y vector age y? y vector kv MW MW kv MW MW Re/Im or kv or kv Re/Im or kv or kv Table Initial directions and convergence of the method Initial loading directions Convergence results Experi- P P P 5 Number of Solution ment MW MW MW iterations... I... I... I -... II s.p s.p I I 9... I s.p s.p s.p II II s.p I I I s.p s.p II II II I II L L 7 8 Figure 5: Eight bus test power system. Table Bus parameters for the 8-bus system Bus eneration, voltage Load no. Active Fixed Active Reactive p power voltage power power MW kv MW L The aim of the example was to determine the closest points on the power ow solution boundary (the critical points), if the real power injections at buses,, and 5 were free parameters, i.e., allowed to vary from their operating point values. Two critical points were obtained using the algorithm of Section. Details of these points are given in Table. The length of the vector y? y is 9.MW for the rst critical point, and 99.7MW for the second critical point. In both cases the angle between y? y (columns, 6 in Table ) and s, which is formed from the elements of the left eigenvector s (columns, 7 in Table ) that correspond to free parameters, is equal to 8deg. In obtaining these solutions, a number of dierent loading directions y were used in the rst stage of the critical point algorithm. These loading directions are given in Table (columns to ). The loading directions gave dierent (stage one) points on the solution boundary. Each of these points was used as the starting point for the second stage of the critical point algorithm. Table (columns 5, 6) shows convergence results for the second stage when the solution technique of [6] was used. Column 6 indicates which of the critical point was converged to, or whether a singular point (s.p.) was encountered. Figure 6 shows trajectories of the second stage solution

5 process. Each trajectory starts from the point on obtained from stage one for the dierent loading directions. The labels of these starting points correspond to the loading directions given in Table. P I P 7 9 II P 6 Figure 6: Trajectories of the stage two iterative process in the space of free parameters. Distance, MW Singular points Solution II Solution I Iterations Figure 7: Distance changes during the stage two iterative process. The following observations were made about the stage two solution process: The iterative solution process always moved along the solution space boundary, i.e., one eigenvalue of the power ow Jacobian J(x) was always zero. The distance d(y) = ky? y k steadily decreased as the iterative process moved from the initial point on given by stage one, to the nal point (either of the critical points, or a singular point). This behaviour is shown in Figure 7. In most cases solutions were obtained after -8 iterations. More iterations were required (-), and singular points were encountered, when inappropriate initial loading directions y were chosen. Convergence to trivial solutions or maxima never occured. 6 CONCLUSIONS The minimum distance from an operating point to the power ow solution space boundary, i.e., to points of maximum loadability, gives a measure of the security of a power system. Points which (locally) provide this minimum distance satisfy a constrained optimization problem. The optimization problem leads to a set of equations which describe such critical points. Not all solutions of this set of equations are critical points however. Trivial solutions correspond to solutions of the usual power ow problem. Other nontrivial solutions describe extrema of the optimization problem that are not of interest. Care must therefore be taken to ensure that algorithms for nding critical points do in fact nd the correct type of points. A two stage algorithm can be used to nd critical points. Because there may be many critical points, it is necessary to provide an estimate of the direction in parameter space of the desired critical point. The rst stage of the algorithm nds a point on the solution space boundary which lies in that specied direction. The second stage uses a continuation method to move along the boundary from that initial point to the desired critical point. The distance from the operating point to the boundary point always decreases along the continuation path. The proposed algorithm converges reliably to desired critical points under normal conditions. However, if a very bad estimate of the direction of the critical point is used, singularity of the Jacobian of the critical point equations may occur. 7 ACKNOWLEDEMENT This work was sponsored in part by an Australian Electricity Supply Industry Research Board project grant \Voltage Collapse Analysis and Control". 8 REFERENCES [] A.M. Kontorovich and A.V. Krukov, Stability limit load ows of power systems (Fundamentals of the theory and computational methods), Publishing House of the Irkutsk University, Irkutsk, 985 (in Russian). [] C.A. Ca~nizares and F.L. Alvarado, \Computational experience with the point of collapse method on very large AC/DC systems", Proc. NSF/ECC Workshop on Bulk Power System Voltage Phenomena II, Deep Creek Lake, MD, August 99; published by ECC Inc., Fairfax, Virginia. [] I.A. Hiskens and R.J. Davy, \A technique for exploring the power ow solution space boundary", Technical Report EE97, Department of Electrical and Computer Engineering, The University of Newcastle, Australia, May 99. [] A.M. Kontorovich, A.V. Krukov, M.K. Lukina, Y.V. Makarov, V.E. Saktoev and R.. Khulukshinov, Methods of stability indices computations for complicated power systems, Publishing House of the Irkutsk University, Irkutsk, 988 (in Russian). [5] I. Dobson and L. Lu, \New methods for computing a closest saddle node bifurcations and worst case load power margin for voltage collapse", IEEE Transactions on Power Systems, Vol. 8, No., August 99, pp [6] Y.V. Makarov and I.A. Hiskens, \A continuation method approach to nding the closest saddle node bifurcation point", Proc. NSF/ECC Workshop on Bulk Power System Voltage Phenomena III, Davos, Switzerland, August 99; published by ECC Inc., Fairfax, Virginia. [7] R. Seydel, From Equilibrium to Chaos, Elsevier Science Publishing Co., New York, 988.

Properties of quadratic equations and their application to power system analysis

Electrical Power and Energy Systems 22 (2000) 313 323 www.elsevier.com/locate/ijepes Properties of quadratic equations and their application to power system analysis Y.V. Makarov a,1, D.J. Hill b, *, I.A.