Genetic Algorithms in Power System Small Signal Stability ... - CiteSeerX

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Genetic Algorithms in Power System Small Signal Stability Analysis Zhao Yang Dong

Yuri V. Makarov

David J. Hill

Department of Electrical Engineering The University of Sydney NSW 2006, Australia Abstract

Power system small signal stability analysis aims to explore dierent small signal stability conditions and controls, namely, 1) exploring the power system security domains and boundaries in the space of power system parameters of interest, including load

ow feasibility, saddle node and Hopf bifurcation ones, 2) nding the maximum and minimum damping conditions, and 3) determining control actions to provide and increase small signal stability. These problems are presented in the paper as dierent modi cations of a general optimization problem, and each of them has multiple minima and maxima. The usual optimization procedures converge to a minimum/maximum depending on the initial guesses of variables and numerical methods used. In the considered problems, all the extreme points are of interest. Additionally, there are diculties with nding the derivatives of the objective functions with respect to parameters. Numerical computations of derivatives in traditional optimization procedures are time consuming. In the paper, we propose a new black box genetic technique for comprehensive small signal stability analysis, which can eectively cope with highly nonlinear objective functions with multiple minima and maxima and derivatives which can not be expressed analytically. 1 Introduction

In the open access environment, the power utilities sometimes are forced to work far away from their pre-designed conditions. In this situation, it is necessary to re-approach the problems related to power system security, stability and transfer capability [1]. On the other hand, several recent major system blackouts in dierent countries and voltage collapses require additional attempts in power system stability area. Several power system oscillatory instability problems occurred recently again put forward attention in the research area of small signal stability. In the space of power system parameters, the

small signal stability domain is restricted by complicated surfaces of dierent kinds. These surfaces can be load ow feasibility, aperiodic and oscillatory boundaries. The last two are often referred to as Saddle node and Hopf bifurcation boundaries. One of the most important tasks is to obtain the system security measure or, in other words, the adequate stability margin. There are many de nitions for the stability margin [2] -[11]. But the de nition based on the distances from the system's state point to the stability domain boundary in the space of power system controlled parameters seems to be more appropriate [12], [13]. Due to complexity of the boundary, it is quite dif cult to nd out the critical shortest distance which is used as stability margin. Additionally, their are normally several subcritical distances which are close to the critical one. So the security margin must be assessed in several critical and subcritical directions. One of the approaches developed recently is the analytical approach [12], [14] -[17]. 2 Small Signal Stability Analytical Approaches

In certain practical cases, it is necessary to analysis the maximum transfer capability in a certain loading direction. For example, the problem can consist in assessment of the maximum power transfer from a particular generator to a particular load. To locate the saddle node and Hopf bifurcations as well as the load ow feasibility boundary points in a given loading direction within one procedure, the following constrained optimization problem is proposed, see [18], 2 ) max=min (1) subject to f (x; y0 + 
y) = 0 (2) t 0 0 00 ~ J (x; y0 + 
y)l ? l + !l = 0 (3) J~t (x; y0 + 
y)l00 ? l00 ? !l0 = 0 (4) li0 ? 1 = 0 (5)

2

li00 = 0

(6)

where  is the real part of an eigenvalue of interest, y0 is the current operation point,  and
y de nes the distance and direction to security boundaries from y0 , f (x; y0 + 
y) = 0 represents the load

ow condition, and J~ is the system Jacobian with l = l0 + jl00 as its eigenvector corresponding to the eigenvalue  =  + j!. To consider the load ow constraint, this constrained optimization problem can be represented with Lagrange function form as  = 2 + f t (x; y0 + 
y) ) minx;; (7) = 2 + 0 ) minx;; (8) The problem has many solutions including the load

ow feasibility, saddle node and Hopf bifurcations and minimum and maximum damping points [18]. The result depends on the initial guesses of variables and selected eigenvalue. Our goal is to determine the small signal stability boundary point which is closest to the point y0 . To get this point, it is necessary to solve the problem for dierent eigenvalues and initial values of the loading parameter  , which is a very time consuming task. In case of exploring the shortest distances, the problem becomes even more complicated. In principle, it is possible to use the problem represented by equations (1) - (6) to get the critical distance vector. Ideas similar to those put forward in [12] can be tried. For example, after obtaining of the closest instability point along the given direction
y, it is possible to analyze the angle between the loading and left eigenvector of the corresponding matrix [load ow Jacobian or state matrix] in this point. As this vector shows the normal direction with respect to the stability boundary, the angle computed can be used to rotate
y in the direction where the distance decrease. Corresponding to the general method proposed in [18] the optimization problem can be depicted by Fig. 1. Then in each direction de ned by input
y, with considering of the state variables shown in Fig. 1, the load ow constraint is computed rst, while the state variables, which are to be optimized, provide input value to set up the state matrix. The eigenvalue computation is then based on this matrix. In the algorithm, the real part of the critical eigenvalue(s) will be used to form the objective function . The value of the objective function f , as the output, will then provide information used to proceed the optimization procedure and the state variables are to be adjusted accordingly. When the optimization process converged, all characteristic points along the direction de ned by
y can be located. To located all these points in

Input ∆y

Φ’

Φ

State Matrix

x

Output f

Eigenvalues

τ

η

State Variables

Optimization procedure Rotation

Fig. 1. System Model Diagram for Small Signal Stability General Method Optimization

the whole plane, a loop is needed to rotate the direction and repeat the optimization procedure locating all characteristic points in the whole plane/space of interest. Generally, this plane/space is a cut set of the space spanned by all parameters of interest [19]. 3 Genetic Algorithms with Sharing Function Optimization Method

Genetic Algorithms(GAs) [20] are heuristic probabilistic optimization techniques inspired by natural evolution process. In genetic algorithms, the tness function is used instead of objective function as in the traditional optimization procedures. Each concrete value of variables to be optimized is called as an individual. Then a current number of individuals composes the generation. In the process of GAs, individuals with better tness survive and those with lower tness die o, so to nally locate individual with the best tness as the nal solution. They are capable of locating the global optimum of a tness function in a bounded search domain, provided a sucient population size is given. The GA sharing function method is able to locate the multiple local maxima as well. Genetic algorithms have been already eectively applied in complicated multidimensional optimization problems, which can be hardly solved by traditional optimization methods. Genetic algorithms, which mimic the natural evolution, usually contain the following steps. Firstly, produce the initial population; secondly, evaluate tnesses for all individuals in the current population; thirdly, perform such operations as crossover, reproduction and mutation depending on the existing generation tnesses, and form a new generation. Then the procedure is repeated till some termination criterion is met and the optimum is thus obtained [20].

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To compute multiple maxima of the tness function, the genetic algorithm sharing function method, which is put forward by Goldberg and Richardson, [20], [21], can be applied. The method decreases the tnesses for similar individuals by the \niche count", m0 (i). For each individual i, the \niche count" is computed as a sum of sharing function values between the individual and all individuals j in generation, see Eqn. (10). The similarity of individuals is evaluated by the distance, d(i; j ) from each other, see Eqn. (9). The resulting shared tness 0 is changed through dividing the original tness,  by the corresponding niche count, see Eqn. (11), [20] -[22].

d(i; j ) = d(xi ; xj ) n X sh[d(i; j )] m0 (i) = j =1

i) 0 (i) = Pn ( j =1 sh[d(i; j )]

(9) (10) (11)

The sharing function is de ned so that it ful lls,

8 < 0  sh(d)  1 sh(d) = : sh(0) = 0 limd!1 sh(d) = 0

(12)

For example the sharing function can have the form,

sh(d) =



1 ? ( d ) ; if d <  0; otherwise

(13)

where  is a constant, and  is the given sharing factor. By doing so, an individual receives its full tness value if it is the only one in its own niche, otherwise its shared tness decreases due to the number and closeness of the neighboring individuals. In this paper we apply the genetic algorithm with sharing to small signal stability analysis. This optimization problem is highly non-linear and some times, even non-dierential-able, which makes it very dicult to be solved by normal optimization methods. 4 Black Box System Model for Optimization

For the analytical approaches, the optimization problems is highly non-linear and some times, even non-dierential-able. It is known that the traditional optimization methods meet serious diculties with convergence while solving such problems. Besides the rest, the constraint sets in Eqn. (2) -(6) take account of only one eigenvalue during the optimization. To get the stability margin for all eigenvalues of interest, as well as the critical load ow feasibility conditions, it is necessary to vary the initial guesses

and repeat the optimization. Additionally, the functions in Eqn. (2) -(6) can have breaks due to dierent limitations applied to power system parameters. For example, the generator current limiters may cause sudden changes in the model, and consequently breaks in the constraint functions. This makes the analytical optimization problem even more complicated. In the genetic optimization procedures, those diculties can be overcome by using the black box power system model given in Fig. 2, described in the sequel. Unlike the model used in the analytical form of Black Box System Model

γ Load flow calculation

converged

not converged

State variable

State matrix

Eigenvalue

calculation

formation

calculation

Fitness Function

Φ

α=0

Fig. 2. Black Box System Model for Optimization

small signal stability problems, this black box has control parameters as inputs, and the tness function  as outputs. Inside the black box, we compute the load ow rst. If it converges, then the state variables and matrix are computed, and then the eigenvalues of the state matrix are obtained. Thereafter, the critical eigenvalue is chosen for analysis. The critical eigenvalue's real part is used to compute a particular value of the tness function. If the load ow does not not converge, which means that a load ow solution does not exist, we put the critical eigenvalue real part to zero. By such a way, the load ow feasibility points are treated in the same way as the saddle node and Hopf bifurcation points. The tness function  can be changed quite exibly depending on the concrete task to be solved, see the next section for explanation. To demonstrate the advantages of the black box model, let's consider the tasks in Abstract. To reveal all characteristic small signal stability points, such as maximum loadability, saddle and Hopf bifurcation and minimum and maximum damping points, along a given ray y0 + 
y in the space of power system control parameters, y, the general small stability problem Eqn. (1) -(6) can be used. If the above problem is solved by traditional optimization methods, the solution obtained depends on initial selection of the eigenvalue traced, and variables x. Moreover, even for one eigenvalue selected, it is not possible to get all the characteristic points in one optimization procedure. By applying the black box model and GA techniques all the problem characteristic points can be found within one optimization procedure. In this case, the input is the loading parameter, =  , and

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the tness function is 2 for maximization and 12 for maximization. To compute the function, the load ow is computed for a given value of  . If the load ow converges, then the state matrix and its eigenvalues are computed, an eigenvalue of interest is selected (for example, the critical eigenvalue with the minimum real part.), and used to get the tness function. The black box model has only one input and one output, and is used in the standard GA optimization. To nd out all the critical distances to the load

ow feasibility and bifurcation boundaries in the problem, the same black box system model can be used. In this case, the inputs are
y and  , and the tness function is increased when the distance decrease and the critical eigenvalue real part tends to zero. It is understood that the shape of power system small signal stability boundaries can be very complicated, and there exist many niches or in other words, local maxima/minima. In order to ensure GA to locate the multiple maxima of the tness function, and to avoid the noise induced by genetic draft, sucient population size should be considered. However, too large population size will result in slow convergence. Techniques for choosing the population size can be found in [22]. In our test systems, the population size in the range from 30 to 160 was selected. It has been discovered that this population size is sucient to locate the maxima in the space of power system variables. 5 The Optimum Operation Direction

By using the methods addressed here, it is possible to obtain both global and local optima, which we call them critical and subcritical solutions. Upon obtaining them, optimum operation direction can be decided in case these solutions are distances to small signal stability boundaries. Say, for example, two solutions, V~1 and V~2 are distances to the boundaries. Then, a vector V~3 de ned below in Eqn. (14) gives the optimum direction of operation.

V~3 = ?1(k1 V~1 + k2 V~2 ) V~3 = ?1( jjV1 jj V~1 + jjV2 jj V~2 ) 1 2

(14) (15)

where 1 , 2 , k1 and k2 are weighting factors revealing the in uence of V~1 and V~2 . For instance, k1 and k2 can be de ned as proportional to the reciprocal of jjV~1 jj and jjV~2 jj respectively, as shown in Eqn. (15), to reveal the dierent in uences of critical and subcritical operation conditions.

Eo

~

0

yo

(−θ o- π/2)

V δ

Em ym (−θ m- π/2)

δm

~

C Pd+jQd

Fig. 3. The single-machine in nite-bus power system model

6 Power System Model for Small Signal Stability Analysis

A well known single machine in nite bus system is studied here with GAs and sharing function method to locate its critical and subcritical distances to instability. The model consists of four dierential equations, which cover both generator and load dynamics [23], see Fig. 3. The mathematical model of the system is the following: m_ = ! (16) M !_ = ?dm ! + Pm + +Em ym V sin( ? m ? m ) + +Em2 ym sinm (17) 2 2 _ Kqw  = ?Kqv2 V ? Kq vV + +E00 y00 V cos( + 00 ) + +Em ym V cos( ? m + m ) ? ?(y00 cos0 + ym cosm )V 2 ? ?Q0 ? Q1 (18) k4 V_ = Kpw Kqv2 V 2 + (Kpw Kqv ? Kqw Kpv )V + q 2 + K 2 [?E 0 y 0 V cos( +  ? h) ? + Kqw 0 pw 0 0 ?Em ym V cos( ? m + m ? h) + +(y00 cos(0 ? h) + ym cos(m ? h))V 2 ] ? ?Kqw (P0 + P1 ) + Kpw (Q0 + Q1 ) (19) qw where k4 = TKqw Kpv and h = tan?1 ( KKpw ). Parameters of the system are the following [23]: Kpw = 0:4, Kpv = 0:3, Kqw = ?0:03, Kqv = ?2:8, Kqv2 = 2:1, T = 8:5, P0 = 0:6, Q0 = 1:3; P1 and Q1 are taken zero at the initial operating point. Network and generator values are: y0 = 20:0, q0 = ?5:0, E0 = 1:0, C = 12:0, y00 = 8:0, 00 = ?12:0, E00 = 2:5, ym = 5:0, m = ?5:0, Em = 1:0, Pm = 1:0, M = 0:3, m = 0:05. All parameters are given in per unit except for angles, which are in degrees. The active and reactive loads are featured by the following equations: Pd = P0 + P1 + Kpw  + Kpv (V + T V_ ) (20)

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Acknowledgment

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This work was sponsored by the ERDC/ESAA Research Program Contract PN2420/94120. Z. Y. Dong's work was supported by Sydney University Electrical Engineering Postgraduate Scholarship. The authors would like to thank Mr. Haining Liu of Tianjin University for his helpful discussion on Genetic Algorithms.

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Reactive Load Power, p.u.

10 5 0 −5 −10 −15

References

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[1] M. Ilic, F. Galiana, L. Fink, A. Bose, P. Mallet and H. Othman, \Transmission Capacity in Power Networks", Proc. of the /em 12th Power System Computation Conference, Dresden, August 19-23, 1996, pp. 5{21. [2] \Proposed Terms & De nitions for Power System Stability", Task Force on Term & De nitions, System Dynamic Performance Subcommittee, Power System Engineering Committee, IEEE Trans. on Power Apparatus and Systems, Vol. PAS101, No. 7, July 1982, pp. 1894 {1898. [3] S. Greene, I. Dobson and F. L. Alvarado, \Sensitivity of the Loading Margin to Voltage Collapse with Respect to Arbitrary Parameters", IEEE PES Winter Power Meeting, 96 WM 278-2-PWRS. [4] J. Barquin, T. Gomez and F. Luis Pagola, \Estimating the Loading Limit Margin Taking into Account Voltage Collapse Areas", IEEE PES Winter Power Meeting, Paper 95 WM 183-4 PWRS, 1995. [5] A. Berizzi, P. Bresesti, P. Marannino, G. P. Granelli and M. Montagna, \System-area Operating Margin Assessment and Security Enhancement Against Voltage Collapse", IEEE PES Summer Power Meeting, Paper 95 SM 584-3 PWRS. [6] A. C. Zambroni and V. H. Quintana, \New Techniques of Network Partitioning for Voltage Collapse Margin Calculations", IEE Proceedings of Part C., Generations, Transmission, and Distribution, Vol. 141, pp. 630-6, Nov. 1994. [7] I. Dobson and L. Lu, \New Methods for Computing a Closest Saddle Node Bifurcation and Worst Case Load Power Margin for Voltage Collapse", IEEE Trans. on Power Systems, Vol. 8, No. 3, pp. 905 {913, August 1993. [8] N. Flatabo, O. B. Fosso, R. Ognedal, T. Carlsen, and K. R. Heggland, \Method for Calculation of Margins to Voltage Instability Applied on the Norwegian System for Maintaining Required Security Levels," IEEE Trans. on Power Systems, Vol. 8, pp. 920{928, Aug. 1993. [9] O. B. Fosso, N. Flatabo, T. Carlsen. O. Gjerde, and M. Jostad, \Margins to Voltage Instability Calcu-

−25 −30

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−10

0 10 Active Load Power, p.u.

20

30

Fig. 4. The Closest Stability Boundary Points

Qd = Q0 + Q1 + Kqw  + Kqv V + Kqv2 V 2 (21) The system (16) -(19) depends on four state variables , m , !, V . Their values at the initial load

ow point are the following:  = 2:75, m = 11:37, ! = 0, and V = 1:79. Note that the initial point is not a physical solution as the voltage V is too high as Q1 is zero. To show the results in the parameter plane, which have been obtained formerly by the authors in [18], [24], the boundaries are plotted in solid line in Fig. 4. In the same gure, critical and subcritical stability points by Genetic Algorithm with sharing function method are plotted in . it evident that this method with the black box model can locate the desired results. They can be used to nd out optimum operation directions P by Eqn. (14) is the second item is changed to ni=2 ki V~i instead of k2 V~2 . 7 Conclusion

A black box power system model is given in this paper for GA based small signal stability analysis. With appropriate system model the method provided here can locate global as well as local optimum solution in the form of critical and subcritical distances to to instability/oscillation.They can be used to make decision on control actions. The black box model together with GA are capable of solving most of the small stability problems. To make the techniques more practical and ecient, further research on fast genetic algorithms, GAs with sharing function method and simpli ed eigenvalue computation algorithms are required.

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lated for Normal and Outage Conditions", CIGRE, paper 38-209, 1992. [10] Y. Sekine, K. Takahashi, Y. Ichida, Y. Ohura, and N. Tsuchimori, "Method of Analysis and Assessment on Power System Voltage Phenomena, and Improvements Including Control Strategies for Greater Voltage Stability Margins", CIGRE, paper 38-206, 1992. [11] T. V. Cutsem, \A Method to Compute Reactive Power Margins with Respect to Voltage Collapse," IEEE Trans. on Power Systems, Vol. 6, pp. 145{ 156, Feb. 1991. [12] I. Dobson, \Computing a Closest Bifurcation in Stability Multidimensional Parameter Space", Journal of Nonlinear Science, Vol. 3., No. 3., pp. 307{327, 1993. [13] Y. V. Makarov, D. J. Hill and J. V. Milanovic, \Eect of Load Uncertainty on Small Disturbance Stability Margins in Open-Access Power Systems", Proc. Hawaii International Conference on System Sciences HICSS-30, Kihei, Maui, Hawaii, January 7-10, 1997. [14] Y. V. Makarov, V. A. Maslennikov, and D. J. Hill, \Calculation of Oscillatory Stability Margins in the Space of Power System Controlled Parameters", Proc. of the International Symposium on Electric Power Engineering Stockholm Power Tech: Power Systems, Stockholm, Sweden, 18-22 June, 1995, pp. 416-422 [15] I. Dobson and L. Lu, \Computing an Optimum Direction in Control Space to Avoid Saddle Node Bifurcation and Voltage Collapse in Electric Power Systems", IEE Trans. on Automatic Control, Vol. 37, No. 10, Oct. 1992, pp.1616 {1620. [16] I. Dobson, \An Iterative Method to Compute a Closest Saddle Node or Hopf Bifurcation Instability in Multidimensional Parameter Space", Proc. of the IEEE International Symposium on Circuits & Systems, San Diego, CA, May 1992, pp. 2513 { 2516. [17] I. Dobson, \Observations on Geometry of Saddle Node Bifurcation and Voltage Collapse in Electrical Power Systems", IEEE Trans. on Circuits and Systems - I: Fundamental Theory and Applications, Vol. 39, No. 3, March 1992, pp. 240 {243. [18] Y. V. Makarov, Z. Y. Dong and D. J. Hill, \A General Method for Power System Small Signal Stability Analysis", Proc. the 1997 Power Industry Computer Application conference, PICA'97, May 11-16, Columbus, Ohio, pp. 280 {286. (to appear in IEEE Trans. on Power Systems) [19] Y. V. Makarov, D.J . Hill, and Z. Y. Dong, \Computation of Bifurcation Boundaries for Power sys-

tems: A New
-plane Method", to be submitted to IEEE Trans. on Circuits and Systems. [20] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, AddisonWesley Publishing Co. Inc., 1989. [21] D. E. Goldberg and J. Richardson, \Genetic Algorithms with Sharing for Multimodal Function Optimization", Genetic Algorithms and Their Applications: Proc. of the 2nd international Conference on GAs, July 28-31, 1987, MIT, pp. 41{49. [22] S. W. Mahfoud, \Population Size and Genetic Drift in Fitness Sharing", L. D. Whitile and M. D. Vose edt. Foundations of Genetic Algorithms  3, Morgan Kaufmann Publishing, Inc. 1995, pp. 185{ 223. [23] H. -D. Chiang, I. Dobson, et al. \On Voltage Collapse in Electric Power Systems". IEEE Trans. Power Systems, Vol. 5, No. 2, May 1990. [24] Z. Y. Dong, Y. V. Makarov and D. J. Hill, \Computing the Aperiodic and Oscillatory Small Signal Stability Boundaries in the Modern Power Grids", Proc. Hawaii International Conference on System Sciences HICSS-30, Kihei, Maui, Hawaii, January 7-10, 1997. Zhao Yang Dong was born in China, 1971. He received his BSEE degree as rst class honor in July, 1993. Since 1994, he had been studying in Tianjin University (Peiyang University), Tianjin, China for his MSEE degree. Now, he is continuing his study as a postgraduate research student in the Department of Electrical Engineering, the University of Sydney, Australia. His research interests include power system analysis and control, electric machine and drive systems areas. Yuri V. Makarov received the M.Sc. in Computer Engineering (1979), and the Ph.D. in Electrical Networks and Systems (1984) from the St. Petersburg State Technical University (former Leningrad Polytechnic Institute), Russia. He is an Associate Professor at Department of Power Systems and Networks at the same University. Now he is conducting his research work at the Department of Electrical Engineering in the University of Sydney, Australia. His research interests are mainly in the eld of power system analysis, stability and control with emphasis on mathematical aspects and numerical methods. David J. Hill (M'76, SM'91, F'93) received his B.E. and B.Sc. degrees from the University of Queensland, Australia in 1972 and 1974 respectively. In 1976 he received his Ph.D. degree in Electrical Engineering from the University of Newcastle, Australia. He currently occupies the Chair in Electrical Engineering at the University of Sydney. Previous appointments include research positions at the University of California, Berkeley and the Department of Automatic Control, Lund Institute of Technology, Sweden. From 1982 to 1993 he held various academic positions at the University of Newcastle. His research interests are mainly in nonlinear systems and control, stability theory and power system dynamics and security. His resent applied work consists of various projects in power system stabilization and power plant control carried out in collaboration with utilities in Australia and Sweden.