Load Flow Study using Load Flow Study using Tellegen’s TheoremTellegen’s Theorem

Load FlowLoad 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.

Importance of load-flow Importance of load-flow studiesstudies

• 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.

Classification Of Load FlowsClassification Of Load Flows

•Classical Load Flow Method

and•Exclusive Load Flow

Method

Load Flow methods are Load Flow methods are categorized ascategorized as

Classical load Flow MethodsClassical load Flow Methods • Gauss – Siedal(GS) Method• Netwon- Raphson (NR) Method• Fast Decoupled Load Flow Method

Exclusive load Flow Methods are Exclusive load Flow Methods are categorized intocategorized into

• Forward Sweeping Method• Dist Flow Method

TYPES OF LOAD FLOWSTYPES OF LOAD FLOWS

Newton- Raphson MethodNewton- 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.

Fast Decoupled Load FlowFast 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.

Gauss- Seidel load flow Gauss- Seidel load flow methodmethod

• 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.

Load Flow study methods of Load Flow study methods of Distribution networksDistribution 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.

Principle Involved In Exclusive Principle Involved In Exclusive Load Flow MethodsLoad 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.

Forward Sweeping MethodForward 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”.

Dist Flow MethodDist 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.

Thought for the development Thought for the development of Tellegen theoremof Tellegen theorem

The majority of classical load flow solution methods have problems, like

• Non- convergence • High memory requirement • Large computation time

So, to overcome some of these So, to overcome some of these problemsproblems

• 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 theoremellegen theorem.

Why Tellegen’s Theorem?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.

The TheoremThe 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

The Theorem..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.

AlgorithmAlgorithm

• 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.

NumberNode voltage

Real Power Loss(KW)

Reactiv Power Loss(KVAR)

1 1 32.7106 13.6115

2 0.981 33.0682 14.0692

3 0.9598 18.0104 7.4944

4 0.9476 10.5739 4.3655

5 0.9399 35.2935 14.6862

6 0.912 22.6388 9.4204

7 0.8927 10.0685 4.1896

8 0.8832 15.4245 6.4184

9 0.8672 16.5198 6.8742

10 0.8481 9.4573 2.6736

11 0.8361 3.5163 0.9941

12 0.8309 7.5706 2.14

13 0.8177 4.7139 0.9773

14 0.8075 2.1479 0.552

15 0.8015 1.0375 0.4400

16 0.7973 0.6101 0.115

17 0.7936 0.1019 0.0288

18 0.7924 0.0000 0.0000

Load flow results of Tellegen theorem method: Load flow results of Tellegen theorem method: Voltage magnitude in pu , branch real reactive power lossVoltage magnitude in pu , branch real reactive power loss – – 18 bus , 440 V distribution network with main feeder.18 bus , 440 V distribution network with main feeder.

Number of iterations: 4Execution time: 2.65Memory Requirement: 3190Total Real power loss: 223.4637KWTotal Reactive power toss: 89.0503KVAR

ResultResultINPUT 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];

OutputOutputTellegen’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 = 89.05121kvarans =Columns 1 through 9 1.0000 0.9810 0.9598 0.9476 0.9399 0.9120 0.8927 0.8832 0.8672Columns 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.

ApplicationsApplications

• 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.

.

ApplicationsApplications....

• 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