Load Flow Study using Tellegen’s Theorem

1. Load Flow Study using Tellegen’s Theorem

2. Load Flow • The load-flow study is an important tool involving numerical analysis applied to a power system • It focuses on various forms of AC power (ie: reactive, real, and apparent) rather than voltage and current. • It analyses the power systems in normal steady-state operation. • Many software implementations perform other types of analysis, such as short-circuit fault analysis and economic analysis. • In particular, some programs use linear programming to find the optimal power flow, the conditions which give the lowest cost per kilowatt generated.

3. Importance of load-flow studies • Great importance of power flow or load-flow studies is in the planning the future expansion of power systems as well as in determining the best operation of existing systems. • The principal information obtained from the power flow study is the magnitude and phase angle of the voltage at each bus and the real and reactive power flowing in each line.

4. Classification Of Load Flows • Classical Load Flow Method and • Exclusive Load Flow Method

5. Load Flow methods are categorized as Classical load Flow Methods • Gauss – Siedal(GS) Method • Netwon- Raphson (NR) Method • Fast Decoupled Load Flow Method Exclusive load Flow Methods are categorized into • Forward Sweeping Method • Dist Flow Method

6. TYPES OF LOAD FLOWS Newton- Raphson Method • Newton's method can often converge remarkably quickly, especially if the iteration begins "sufficiently near" the desired root. • In NR method, the changes in real power (P) are very much influenced by the changes in load angle only and no influence due to the voltage magnitude changes. • Similarly the changes in reactive power are very much influenced by changes in voltage magnitudes and no change takes place due to load angle changes.

7. Fast Decoupled Load Flow • It is reliable and fastest method in obtaining convergence. • This method with branches of high (R/X) ratios, could not solve problems with regard to non- convergence and long execution time.

8. Gauss- Seidel load flow method • The iteration process begins with a flat voltage profile assumption to all the buses expect the slack bus. • The bus voltages are updated and the convergence check is made on updated voltages and the iteration process is continued till the tolerance value is reached.

9. Load Flow study methods of Distribution networks • Efficient load flow method is one of the most important and highly demanded software in the power industry • The analysis of a distribution network has become an important area of activity for present day power systems engineer. • Due to the high (R/X) ratios of the branches of the network made the researches to develop an exclusive Load flow methods. • The conservation of power principle at a node level was the main principle used in the load flow methods.

10. Principle Involved In Exclusive Load Flow Methods The principle says that, at any node the power fed into the node is equal to the sum of the power dissipated in the series branch connected to that node and the power fed to the load connected to that node.

11. Forward Sweeping Method • The bus and branch numbering is simple and direct. • The node voltages are computed iteratively. • Initially branch power flows are assumed to be zero. • The branch power flows are computed using the power flow equations and terminal node voltages are updated and in turn branch power losses are updated. • A tolerance check is made to the updated branches power losses and iteration process is continued till the convergence is obtained. • Since the iteration process is carried in the forward direction of the power flow, the method is named “Forward Sweeping Method”.

12. Dist Flow Method • The distribution network is reduced to a single branch network and by using the power flow equation, the real and reactive powers injected into the reduced network are determined in an iterative way. • A convergence check is applied to these powers injected into the reduced network and the iteration process is continued till the tolerance is met. • If the convergence is not met, a new equivalent network is determined with new parameters and the process is continued till the convergence is achieved. • The node voltages and branch power losses are computed. • The main advantage of this exclusive method is the efficiency achieved by avoiding repeated computations of node voltage magnitudes.

13. Thought for the development of Tellegen theorem The majority of classical load flow solution methods have problems, like • Non- convergence • High memory requirement • Large computation time

14. So, to overcome some of these problems • The exclusive load flow methods for distribution networks are studied. • Power conservation principle is used in all exclusive load flow methods • If the same principle is applied to distribution networks a better load flow can be developed. • Conservation principle, when it is used to find a new load flow solution could also be real and near to the practical solution. • The power conservation principle for the entire network is designated as theorem called Tellegen theorem.

15. Why Tellegen’s Theorem? • Based solely on network topology and kirchoff’s laws • Is a power conservation theorem. • States that vectors of flows and forces are orthogonal. • Simple to code, less storage & execution time.

16. The Theorem • Consider an arbitrary lumped network whose graph G has b branches and nt nodes. • Suppose that to each branch of the graph we assign arbitrarily a branch potential difference Wk and a branch current Fk for k= 1 ,2 … b and suppose that they are measured with respect to arbitrarily picked associated reference directions. • If the branch potential differences W1 , W2 , W3 , … , Wb satisfy all the constraints imposed by KVL and if the branch currents F1 , F2 , F3 … Fb satisfy all the contraints imposed by KCL, then

17. The Theorem.. • The Tellegen theorem is extremely general. • It is valid for any lumped network that contains any elements, linear or nonlinear, passive or active, time-varying or time-invariant. The generality follows from the fact that Tellegen's theorem depends only on the two Kirchhoff laws.

18. Algorithm • Make an initial guess of all unknown voltage magnitudes and angles. It is common to use a "flat start" in which all voltage angles are set to zero and all voltage magnitudes are set to 1.0 p.u. • solve the power balance equations using the most recent voltage angle and magnitude values. • linearize the system around the most recent voltage angle and magnitude values • solve for the change in voltage angle and magnitude • update the voltage magnitude and angles • check the stopping conditions, if met then terminate, else go to step 2.

19. Load flow results of Tellegen theorem method: Voltage magnitude in pu , branch real reactive power loss – 18 bus , 440 V distribution network with main feeder. Number of iterations: 4 Execution time: 2.65 Memory Requirement: 3190 Total Real power loss: 223.4637KW Total Reactive power toss: 89.0503KVAR

20. Result INPUT CASE FILE • % this function gives data for r, x, pl and ql,nb,epslon,vr of 18 bus system • function [r,x,pl,ql,nb,epslon,vr] = case18bus • r =[1.8216 2.2270 1.3662 0.9180 3.6432 2.7324 1.4573 2.7324 3.6432 2.7520 1.3760 4.1280 4.1280 3.0272 2.7520 4.1280 2.7520 0.000] • x =[0.7580 0.9475 0.5685 0.3790 1.5160 1.1370 0.6064 1.1370 1.5160 0.7780 0.3890 1.1670 0.8558 0.7780 1.1670 0.7780 0.7780 0.0000] • pl =[0.00 140.00 80.0 80.0 100.0 80.0 90.0 90.0 80.0 90.0 80.0 80.0 90.0 70.0 70.0 70.0 60.0 60.0] • ql =[0.00 90.0 50.0 60.0 60.0 50.0 40.0 40.0 50.0 50.0 50.0 40.0 50.0 40.0 40.0 40.0 30.0 30.0] • nb=[18]; • epslon=[0.0001]; • vr=[440];

21. Output Tellegen’s Therom results are as follows: • The number of iterations are = 10 • The total load = 1410.00000kw • Total reactive load = 810.00000kvar • Total real power losses = 223.46564kw • Total reactive power losses = 89.05121kvar ans = Columns 1 through 9 1.0000 0.9810 0.9598 0.9476 0.9399 0.9120 0.8927 0.8832 0.8672 Columns 10 through 18 0.8481 0.8361 0.8309 0.8177 0.8075 0.8015 0.7973 0.7936 0.7924 The above mentioned array indicate the voltage at different nodes.

22. Applications • The classical application area for network theory and Tellegen's theorem is electrical circuit theory. It is mainly in use to design filters in signal processing applications. • A more recent application of Tellegen's theorem is in the area of chemical and biological processes. • The assumptions for electrical circuits (Kirchhoff laws) are generalized for dynamic systems obeying the laws of irreversible thermodynamics. .

23. Applications.. • Topology and structure of reaction networks (reaction mechanisms, metabolic networks) can be analyzed using the Tellegen theorem. • Another application of Tellegen's theorem is to determine stability and optimality of complex process systems such as chemical plants or oil production systems. • The Tellegen theorem can be formulated for process systems using process nodes, terminals, flow connections and allowing sinks and sources for production or destruction of extensive quantities