Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow 48582 - P OWER S YSTEM A NALYSIS AND D ESIGN L ECTURE 6 - L OAD F LOW A NALYSIS D R . G ERMANE X A THANASIUS School of Electrical, Mechanical and Mechatronic Systems U NIVERSITY OF T ECHNOLOGY S YDNEY
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Lecture Outline Introduction 1 Nodal admittance matrix 2 NR method 3 Application of NR method for load flow analysis 4 Elements of J 1 Elements of J 2 Elements of J 3 Elements of J 4 Fast decoupled load flow 5
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Introduction Load flow analysis refers to the steady state analysis of the 1 power system with reference to voltages and angles at different buses and real and reactive power flow across different buses in an interconnected grid system. Load flow forms an important element in the power system 2 analysis. Load flow results give voltage magnitude and angles along 3 with real and reactive power flow across different buses in the system. The load flow results are essential for planning, load 4 scheduling and power system control.
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Introduction To start with the load flow analysis, the power system 1 network is represented by nodal admittance matrix. The power flow equations are nonlinear so analytical 2 solution is difficult. We use numerical iterative techniques to solve these 3 equations. Two methods of load flow solutions are available, 4 Gauss-Seidel method and Newton-Raphson (NR) method. NR method has better convergence and more suitable for 5 ill conditioned network situations when compared with Gauss-Seidel method. NR method is the most widely used method so we restrict 6 our discussion to NR method for load flow analysis.
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Admittance Matrix Consider a power system network consisting of two generators and four buses as shown in Figure 1. The network impedances can be represented using admittances and generators by constant current sources as shown in Figure 2. 1 2 12 10 13 23 34 Figure: Four bus two machine power system
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Admittance Matrix 10 20 12 2 1 13 23 34 Figure: Four bus two machine power system with admittance representation
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Admittance Matrix We can write nodal current equations for bus junctions of the network as, I 1 = y 10 V 1 + y 12 ( V 1 − V 2 ) + y 13 ( V 1 − V 3 ) I 2 = y 20 V 2 + y 12 ( V 2 − V 1 ) + y 23 ( V 2 − V 3 ) 0 = y 13 ( V 3 − V 1 ) + y 23 ( V 3 − V 2 ) + y 34 ( V 3 − V 4 ) 0 = y 34 ( V 4 − V 3 ) (1)
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Admittance Matrix Equation (1) can be rearranged as, I 1 = ( y 10 + y 12 + y 13 ) V 1 − y 12 V 2 − y 13 V 3 I 2 = − y 12 V 1 + ( y 20 + y 12 + y 23 ) V 2 − y 23 V 3 0 = − y 13 V 1 − y 23 V 2 ( y 13 + y 23 + y 34 ) V 3 − y 34 V 4 + = − y 34 V 3 + y 34 V 4 0
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Admittance Matrix In general we can write, = Y 11 V 1 + Y 12 V 2 + Y 13 V 3 + Y 14 V 4 I 1 I 2 = Y 21 V 1 + Y 22 V 2 + Y 23 V 3 + Y 24 V 4 I 3 = Y 31 V 1 + Y 32 V 2 + Y 33 V 3 + Y 34 V 4 I 4 = Y 41 V 1 + Y 42 V 2 + Y 43 V 3 + Y 44 V 4 (2) where Y 11 = y 10 + y 12 + y 13 Y 22 = y 20 + y 12 + y 23 Y 33 = y 13 + y 23 + y 34 Y 44 = y 34 Y 12 = Y 21 = − y 12 Y 13 = Y 31 = − y 13 Y 23 = Y 32 = − y 23 Y 34 = Y 43 = − y 34
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Admittance Matrix In (2), the admittance values will be zero if there is no connection between the particular busses. We can write (2) in a generalised form for system consisting of n buses as, I 1 Y 11 Y 12 Y 1 n V 1 . . . I 2 Y 21 Y 22 . . . Y 2 n V 2 = (3) . . . . . . . . . . . . . . . . . . I n Y n 1 Y n 2 Y nn V n . . .
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Load Flow Admittance Matrix Equation (3) can be represented as, I bus = Y bus V bus (4) where Y bus is known as the bus admittance matrix of the system. The diagonal elements of Y bus matrix is known as driving point admittance or self admittance and it is the sum of all admittances connected to the particular bus. Off diagonal elements of Y bus are known as transfer or mutual admittances .
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow Problem 1 Consider a 2 generator 4 bus power system shown in Figure 3. The pu impedance data of the network is given in Figure 3 at 100 MVA, 220 kV base. Bus 1 and 4 operate at 220 kV and have load connected to them. Formulate the admittance matrix of the system assuming π model for the transmission lines.
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow NR method NR method offers solution to a set of nonlinear equations through iterative procedure. Let us consider a set of n equations with n unknowns as given below: f 1 ( x 1 , x 2 , · · · , x n ) = a 1 f 2 ( x 1 , x 2 , · · · , x n ) = a 2 . . . f n ( x 1 , x 2 , · · · , x n ) = a n (5) We shall make an initial estimate for the unknowns as x 0 1 , x 0 2 , · · · , x 0 n . The corrections need to be added to the estimates to get the correct solution be ∆ x 0 1 , ∆ x 0 2 , · · · , ∆ x 0 n .
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow NR method Now we can write, � � x 0 1 + ∆ x 0 1 , x 0 2 + ∆ x 0 2 , · · · , x 0 n + ∆ x 0 = f 1 a 1 n � � x 0 1 + ∆ x 0 1 , x 0 2 + ∆ x 0 2 , · · · , x 0 n + ∆ x 0 f 2 = a 2 n . . . � � x 0 1 + ∆ x 0 1 , x 0 2 + ∆ x 0 2 , · · · , x 0 n + ∆ x 0 = f n a n (6) n
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow NR method Each of the functions in (6) can be expanded using Taylor’s series of expansion. If we expand the i th equation we get, � � � � x 0 1 + ∆ x 0 1 , x 0 2 + ∆ x 0 2 , · · · , x 0 n + ∆ x 0 x 0 1 , x 0 2 , · · · , x 0 f i = f i + n n � ∂ f i � ∂ f i � ∂ f i � � � ∆ x 0 ∆ x 0 ∆ x 0 1 + 2 + · · · + n + ∂ x 1 ∂ x 2 ∂ x n 0 0 0 � � higher order terms of ∆ x 0 1 , ∆ x 0 2 , · · · , ∆ x 0 = a i n (7)
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow NR method If we assume ∆ x 0 s be small then we higher powers of ∆ x 0 s can be neglected. If we arrange the expanded equations in matrix form we get, � � � � � � ∂ f 1 ∂ f 1 ∂ f 1 0 · · · x 0 1 , x 0 2 , · · · , x 0 ∆ x 0 � � ∂ x 1 ∂ x 2 ∂ x n a 1 − f 1 0 0 n 1 � � � � � � ∂ f 2 ∂ f 2 ∂ f 2 x 0 1 , x 0 2 , · · · , x 0 ∆ x 0 � � 0 · · · a 2 − f 2 n 2 ∂ x 1 ∂ x 2 ∂ x n 0 0 = (8) . . . . . . . . . x 0 1 , x 0 2 , · · · , x 0 ∆ x 0 � � a n − f n � � � � � � ∂ f n ∂ f n ∂ f n n n 0 · · · ∂ x 1 ∂ x 2 ∂ x n 0 0
Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow NR method Equation (8) for j th iteration can be written as, ∆ A j J j ∆ X j = J j � − 1 � ∆ X j ∆ A j = (9) where � � x j 1 , x j 2 , · · · , x j a 1 − f 1 ∆ x j n 1 � � x j 1 , x j 2 , · · · , x j ∆ x j a 2 − f 2 n A j = , ∆ X j = 2 . . . ∆ . . . ∆ x j � � x j 1 , x j 2 , · · · , x j a n − f n n n
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