583

Proc. of Int. Conf. on Control, Communication and Power Engineering 2010

Optimal Power Flow Solution Using Cluster Technique K.Radha Rani EEE Department R.v.r & J.c.college of Engg Guntur,A.P(State), INDIA E-mail:korrapati_radharani@yahoo.com

J. Amarnath

Index Terms – Optimal Power Flow(OPF), Economic schedule, Clusters.

NOMENCLATURE

t

bSH X u

Bus voltage angle vector.

Load (PQ) bus voltage magnitude vector.

Unit active power output vector.

Generation (PV) bus voltage magnitude vector.

Transformer tap settings vector.

Bus shunt admittance vector.

System state vector.

System control vector.

I.

II.

OPTIMAL POWER FLOW (OPF) PROBLEM

INTRODUCTION

The optimal power flow problem is the minimum generation cost, by considering so many constraints like voltage limits for generator buses, load buses, transformer tap setting limits, phase shifting transformer limits etc so that the constrained economic dispatch is called optimal power flow problem. Optimization calculation also balances the entire power flow. The objective function can take different forms other than minimizing the generation cost. It is common to express OPF as a minimization of electrical losses in the transmission system, or to express it as the minimum shift of generation and other controls from an optimum operating point. The OPF problem can be formulated as a mathematical optimization problem as follows:

The large interconnection of the electric networks, the energy crisis in the world and continuous rise in prices, it is very essential to reduce the running charges of the electrical energy i.e. reduce the fuel consumption for meeting a particular load demand. The optimal operation, involved the consideration of economy of operation, system security, emissions at certain fossil-fuel plants, optimal releases of water at hydro generation, etc. All these considerations may make for conflicting requirements and usually a compromise has to be made for optimal system operation.

Since its introduction as network constrained economic dispatch by Carpentier [1] and its definition as optimal power flow (OPF) by Dommel and Tinney [2], the OPF problem has been the subject of intensive research.

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EEE Department VITS,Deshmukhi Hyderabad,A.P,INDIA

The OPF optimizes a power system operating objective function (such as the operating cost of thermal resources) while satisfying a set of system operating constraints, including constraints dictated by the electric network. OPF has been widely used in power system operation and planning [3]. After the electricity sector restructuring, OPF has been used to assess the spatial variation of electricity prices and as a congestion management and pricing tool [4].

In its most general formulation, the OPF is a nonlinear, nonconvex, large-scale, static optimization problem with both continuous and discrete control variables. Even in the absence of nonconvex unit operating cost functions, unit prohibited operating zones, and discrete control variables, the OPF problem is nonconvex due to the existence of the nonlinear (AC) power flow equality constraints. The presence of discrete control variables, such as switchable shunt devices, transformer tap positions, and phase shifters, further complicates the problem solution.

The literature on OPF is vast, and [5] presents the major contributions in this area. Mathematical programming approaches, such as nonlinear programming (NLP) [6]–[9], quadratic programming (QP) [10], [11], and linear programming (LP) [12]– [14],have been used for the solution of the OPF problem.

In this paper a novel technique data clustering is introduced to solve the OPF with considerably less time.

Abstract -This paper presents a novel approach to Optimal Power Flow Solution using Data clustering Technique. The objective of an Optimal Power Flow(OPF) algorithm is to find steady state operation point which minimizes generation cost, losses etc. or maximizes loadability etc. while maintaining an acceptable system performance in terms of limits on generators real and reactive powers, line flow limits, output of various compensating devices etc.

Traditionally, classical optimization methods were used to effectively solve OPF. For improving the performance of direct search technique a novel technique based on clustering is introduced. The objective of this method provides Optimal Power Flow values with minimized total cost. Formation of clusters is done for different total load values. The proposed method is considerably fast and provides feasible near optimal solutions. Numerical solutions have proved the effectiveness of the proposed method in solving Optimal Power Flow problems with in reasonable execution time.

θ UL PG UG

S.Kamakshaiah

EEE Department J.N.T.U college of Engg Hyderabad,A.P,INDIA

Proc. of Int. Conf. on Control, Communication and Power Engineering 2010

Minimize Subject to

f(x,u) g(x,u)=0 h(x,u)≤0 uЄU x= [θˆΤ ULT]T u= [PˆGT UGT tT bSHT]T

where

(1) (2) (3) (4) (5) (6)

if they are “close” according to a given distance (in this case geometrical distance).This is called distance-based clustering. An important component of a clustering algorithm is the distance measure between data points. If the components of the data instance vectors are all in the same physical units then it is possible that the simple Euclidean distance metric is sufficient to successfully group similar data instances. However, even in this case the Euclidean distance can sometimes be misleading.

Figure shown below illustrates this with an example of the width and height measurements of an object, despite of both the measurements being taken in the same physical units. As Fig. 2 shows, different scalings can lead to different clustering.

The equality constraints (2) are the nonlinear power flow equations. The inequality constraints (3) are the functional operating constraints, such as • branch flow limits (MVA, MW or A); • load bus voltage magnitude limits; • generator reactive capabilities; • slack bus active power output limits.

Constraints (4) define the feasibility region of the problem control variables such as • unit active power output limits; • generation bus voltage magnitude limits; • transformer-tap setting limits (discrete values); • bus shunt admittance limits (continuous or discrete control).

III. CLUSTER TECHNIQUE Clustering is the unsupervised classification of patterns (observations, data items or feature vectors) into groups (clusters). Cluster analysis is the organization of a collection of patterns (usually represented as a vector of measurements, or a point in a multidimensional space) into clusters based on similarity. It is important to understand the difference between clustering (unsupervised classification) and discriminant analysis (supervised classification). In supervised classification, we are provided with a collection of labeled (pre classified) patterns; the problem is to label a newly encountered, yet unlabeled, pattern. In the case of clustering, the problem is to group a given collection of unlabeled patterns into meaningful clusters. A cluster is therefore a collection of objects which are “similar” between them and are “dissimilar” to the objects belonging to other clusters. This can be shown with a simple graphical example:

Figure 1. Formation of clusters

In this case 4 clusters are easily identified, into which the data can be divided; the similarity criterion is distance: two or more objects belong to the same cluster

Figure 2. Different clusters with scalings

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Proc. of Int. Conf. on Control, Communication and Power Engineering 2010

Another kind of clustering is conceptual clustering: two or more objects belong to the same cluster if this one defines a concept common to all that objects. In other words, objects are grouped according to their fit to descriptive concepts, not according to simple similarity measures. However, this is not only a graphic issue, the problem arises from the mathematical formula used to combine the distances between the single components of the data feature vectors into a unique distance measure that can be used for clustering purposes. Different formulas leads to different clusterings. Again, domain knowledge must be used to guide the formulation of a suitable distance measure for each particular application

these data points is then compared and the best solution with respect to minimum cost of generation is chosen.

The flow chart for the proposed method is presented in Fig. 3 and the algorithm is presented below.

B. ALGORITHM FOR ECONOMIC SCHEDULE PROBLEM USING CLUSTER TECHNIQUE 1. Start 2.Take bus data, line data, megawatt limits, and cost coefficients.

3. Calculate the system minimum and maximum generations.

The minimum generation of the system is the minimum generation amongst all units and maximum generation is the sum of maximum generations of all units.

4. Set load point count i = 1.Set PLoad1 = minimum generation of the system 5. Execute economic dispatch program based on Lagrangian approach for the load by considering losses.

6. Take solution of step-5 as initial approximate solution for the load.

7. Increment load count i = i + 1 and pLOAD (i ) = pLOAD (i −1) + ΔpL Where ΔPL =step increase

IV. OPTIMAL POWER FLOW SOLUTION USING CLUSTER TECHNIQUE A. Cluster Technique The cluster Technique proposed in this paper is explained in 5 stages.

Stage-1: Obtain Initial Approximate Economic Scheduling Data

For a given system load, execute economic scheduling problem by Lagrangian method. This solution is assumed an approximate optimal solution due to non consideration of some or many constraints. It can be assumed the exact optimal solution shall be at the vicinity of this approximate solution. Lagrangian solution is taken as the initial scheduled data.

in load ΔPL 8. For the considered load take the scheduled data as the cluster 9. Repeat step-5 to step-8 for all the load points.

10. Input the data for the system load.

11. Choose the cluster for the system load.

12. Generate population of data points by providing upper and lower limits for all the generating buses in scheduled data. The limits are provided for all the generating buses except for the slack bus. This population consists of individual generating buses of all possible combination of power generation within the boundary of error limits.

One data point in this population provides information regarding the active power generation at all the buses except slack bus.

13. Set data point count for the population j=1.

14. Run load flow studies.

15.Obtain total cost of generation for the result of step- 14.

16. Increment j=j+1.

17. Repeat step-14 to step-16 for all the data points in the population.

18. Compare the total cost of each data point in the population and obtain minimum out of all.

19.Print results.

Stage-2: Formation of Clusters

The scheduled data by Lagrangian method for a given load shall form a cluster. This cluster consists of data points regarding the system load and allocation of this total load amongst various units. It can be said the number of clusters contained are exactly equal to the load points.

Stage-3: Error limits for Scheduled Data

Due to non consideration of some or more constraints in the scheduled problem by the analytical Lagrangian method, the initially scheduled data shall have some error. Now error limits are provided for the initial scheduled data in the upper and lower regions. The exact solution shall be definitely in the region bounded by these limits. Now the aim is to find the exact solution.

While providing lower and upper error limits, if for any generator the minimum or maximum limits are violated, then the generation of that particular unit is set to either pimin or pimax as the case may be.

Stage-4: Generation of Limited Number of Population

With in the bounded error limits, a limited population is generated. The exact optimal solution shall be the part in generated population. This method avoids consideration randomly generated large population and restrict to a limited population.

Stage-5 Final Economic Scheduling Solution

For all the data points generated for the population in stage-4, run the load flow study and obtain individual units of generation. For this individual generation obtain total cost of generation .The total cost generation of all 288 © 2009 ACEEE

Proc. of Int. Conf. on Control, Communication and Power Engineering 2010 CLUSTER MATRIX :

FLOWCHART: GEN1

GEN2

50.0000 69.0703 51.1836 51.9433 83.6367 42.2739 94.9871 36.8969 102.6057 35.3556 109.0749 35.2172 115.0939 35.6979 120.9077 36.4936 126.6195 37.4673 132.2789 38.5501 137.9120 39.7042 143.5340 40.9074 149.1541 42.1463 154.4146 43.3371 158.2977 44.1986 162.1696 45.0708 165.9752 45.9489 169.4915 46.7492 173.0142 47.5551 176.5485 48.3660 180.0994 49.1815 183.3574 49.9439 186.6487 50.6914 189.9671 51.4420 193.3172 52.1956 196.7034 52.9522 200.0000 57.1847 200.0000 58.3527 200.0000 60.8887 200.0000 63.4154 200.0000 65.9314

GEN3

GEN4

GEN5

GEN6

LOAD

26.2703 22.1618 19.3321 17.6414 17.1280 17.0560 17.1827 17.4125 17.7015 18.0276 18.3784 18.7465 19.1275 19.5322 19.8020 20.0776 20.4138 20.6729 20.9361 21.2030 21.4736 21.7965 22.0524 22.3120 22.5753 22.8425 22.8828 23.0354 23.8223 24.6139 25.4108

35.0000 21.6681 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.3934 12.5405 14.7062 16.6205 18.5761 20.5299 22.4772 24.4139 26.1625 27.8908 29.5945 31.2692 32.9107 31.8159 35.0000 35.0000 35.0000 35.0000

24.0626 11.5864 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.2684 10.9344 11.6056 12.2821 12.9641 13.6637 14.3034 14.9490 15.6009 16.2598 16.4433 17.9004 19.8899 21.8957 23.9195

23.2624 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.0000 12.4626 13.0772 13.6980 14.3255 14.9603 15.8523 17.0175 18.9212 20.8360 22.7632

14.1700 28.3400 42.5100 56.6800 70.8500 85.0200 99.1900 113.3600 127.5300 141.7000 155.8700 170.0400 184.2100 198.3800 212.5500 226.7200 240.8900 255.0600 269.2300 283.4000 297.5700 311.7400 325.9100 340.0800 354.2500 368.4200 382.5900 396.7600 410.9300 425.1000 439.2700

3. Input the system load. Given load value is 283.4MW 4. Now, choose the cluster belonging to the given system load 5. Fix the error limits and generate populations.

6. Extract the minimum cost data point from the generated population

TABLE I TEST RESULTS OF IEEE-30 BUS POWER SYSTEM BY CLUSTER TECHNIQUE METHOD

Figure 3

V. CASE STUDIES AND RESULTS The proposed cluster technique solution of the OPF is evaluated for the IEEE 30-bus system [6]. The test data of IEEE-30 bus power system is taken from [6].

Following the sequential steps enlightened in the algorithm the following results have been obtained.

1.Approximate economic scheduling for the different system loads is obtained by Lagrangian method.

2. The clusters are formed based on the results obtained

by Lagrangian methods.

BUS NO

BUS

POWER

VOLTAGE

GENERATION FOR EACH GENERATOR

1

1

1.06

176.044

468.306

2

2

1.043

48.866

127.303

3

5

1.01

21.203

49.3009

4

8

1.01

22.4772

77.2442

5

11

1.082

12.2821

40.6177

6

13

1.071

12

39.6

292.872

802.372

TOTAL

289 Š 2009 ACEEE

GEN NO

COST OF GENERATION FOR EACH GENERATOR

Proc. of Int. Conf. on Control, Communication and Power Engineering 2010 [7] R. R. Shoults and D. T. Sun, “Optimal power flow based on P-Q decomposition,” IEEE Trans. Power Apparat. Syst., vol.

PAS-101, pp. 397–405, Feb. 1982.

[8] M. H. Bottero, F. D. Galiana, and A. R. Fahmideh-Vojdani, “Economic dispatch using the reduced Hessian,” IEEE Trans.

Power Apparat. Syst., vol. PAS-101, pp. 3679–3688, Oct. 1982.

[9] J. A. Momoh, “A generalized quadratic-based model for optimal power flow,” IEEE Trans. Syst., Man, Cybern, vol.

SMC-16, 1986.

[10] G. F. Reid and L. Hasdorf, “Economic dispatch using quadratic programming,” IEEE Trans. Power Apparat. Syst., vol. PAS-92, pp. 2015–2023, 1973.

[11] R. C. Burchett, H. H. Happ, and K. A. Wirgau, “Largescale optimal power flow,” IEEE Trans. Power Apparat. Syst., vol. PAS-101, pp. 3722–3732, Oct. 1982.

[12] B. Stott and E. Hobson, “Power system security control calculation using linear programming,” IEEE Trans. Power Apparat. Syst., pt. I and II, vol. PAS-97, pp. 1713–1731 [13] B. Stott and J. L. Marinho, “Linear programming for power system network security applications,” IEEE Trans. Power Apparat. Syst., vol. PAS-98, pp. 837–848, May/June 1979.

[14] R. Mota-Palomino and V. H. Quintana, “A penalty function-linear Programming method for solving power system constrained economic operation problems,” IEEE Trans. Power Apparat. Syst., vol. PAS-103, pp. 1414–1442, June 1984.

[15] Anastasios G. Bakirtzis and Pandel N. Biskas, “Optimal Power Flow by Enhanced Genetic Algorithm”, IEEE Transactions On Power Systems, Vol. 17, No. 2, May 2002.

[16] L. Chen, H. Suzuki, and K. Katou, “Mean field theory for optimal power flow,” IEEE Trans. Power Syst., vol. 12, pp.

1481–1486, Nov. 1997.

[17] L. Chen, S. Matoba, H. Inabe, and T. Okabe, “Surrogate constraint method for optimal power flow,” IEEE Trans. Power Syst., vol. 13, pp. 1084–1089, Aug. 1998.

[18] L. L. Lai, J. T. Ma, R. Yokoyama, and M. Zhao, “Improved genetic algorithms for optimal power flow under both normal and contingent operation states,” Elec. Power Energy Syst., vol.

19, no. 5, pp. 287–292, 1997.

[19] T. Numnonda and U. D. Annakkage, “Optimal power dispatch in multinode electricity market using genetic algorithm,” Elec. Power Syst. Res., vol. 49, pp. 211–220, 1999.

VI. CONCLUSION In this paper a novel approach to Optimal Power Flow Solution by using Data- clustering Technique is presented. The unit active powers, voltage magnitudes, and reactive power limits are taken as continuous controlled variables. The proposed method is considerably fast and provides feasible optimal solutions.

Numerical solutions have proved the effectiveness of the proposed method in solving Optimal Power Flow problems with in reasonable execution time. The execution time is 15seconds nearly, where the execution time in reference [15] is nearly 85seconds with more constraints. This method is simple as compared to conventional methods like interior point method, gradient method, Newton method, Lambda iteration method. The cluster technique approach is more suitable for on-line applications.

VII. REFERENCES [1] J. Carpentier, “Contibution a.’l’etude du dispatching economique,” Bull. Soc. Francaise Elect, vol. 3, pp. 431–447, Aug. 1962.

[2] H.W. Dommel and W. F. Tinney, “Optimal power flow solutions,” IEEE Trans. Power Apparat. Syst., vol. PAS-87, pp. 1866–1876, Oct. 1968.

[3] J. A. Momoh, R. J. Koessler, M. S. Bond, B. Stott, D. Sun, A Papalexopoulos, and P. Ristanovic, “Challenges to optimal power flow,” IEEE Trans. Power Syst., vol. 12, pp. 444–455, Feb. 1997.

[4] R. D. Christie, B. F. Wallenberg, and I. Wangensteen, “Transmission management in the deregulated environment,” Proc. IEEE, vol. 88, pp. 170–195, Feb. 2000.

[5] J. A. Momoh, M. E. El-Hawary, and R. Adapa, “A review of selected optimal power flow literature to 1993,” IEEE Trans.

Power Syst., pt. I and II, vol. 14, pp. 96–111, Feb. 1999.

[6] O. Alsac and B. Stott, “Optimal load flow with steady state security,” IEEE Trans. Power Apparat. Syst., vol. PAS-93, pp.

745–751, May/June 1974.

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