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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

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

48582 - POWER SYSTEM ANALYSIS AND DESIGN LECTURE 6 - LOAD FLOW ANALYSIS

  • DR. GERMANE X ATHANASIUS

School of Electrical, Mechanical and Mechatronic Systems UNIVERSITY OF TECHNOLOGY SYDNEY

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Lecture Outline

1

Introduction

2

Nodal admittance matrix

3

NR method

4

Application of NR method for load flow analysis

Elements of J1 Elements of J2 Elements of J3 Elements of J4

5

Fast decoupled load flow

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Load Flow Introduction

1

Load flow analysis refers to the steady state analysis of the 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.

2

Load flow forms an important element in the power system analysis.

3

Load flow results give voltage magnitude and angles along with real and reactive power flow across different buses in the system.

4

The load flow results are essential for planning, load scheduling and power system control.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Load Flow Introduction

1

To start with the load flow analysis, the power system network is represented by nodal admittance matrix.

2

The power flow equations are nonlinear so analytical solution is difficult.

3

We use numerical iterative techniques to solve these equations.

4

Two methods of load flow solutions are available, Gauss-Seidel method and Newton-Raphson (NR) method.

5

NR method has better convergence and more suitable for ill conditioned network situations when compared with Gauss-Seidel method.

6

NR method is the most widely used method so we restrict

  • ur discussion to NR method for load flow analysis.
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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 10 12 13 23 34

Figure: Four bus two machine power system

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Load Flow Admittance Matrix

1 10 12 13 23 34 2 20

Figure: Four bus two machine power system with admittance representation

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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, I1 = y10V1 + y12 (V1 − V2) + y13 (V1 − V3) I2 = y20V2 + y12 (V2 − V1) + y23 (V2 − V3) = y13 (V3 − V1) + y23 (V3 − V2) + y34 (V3 − V4) = y34 (V4 − V3) (1)

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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, I1 = (y10 + y12 + y13) V1 − y12V2 − y13V3 I2 = −y12V1 + (y20 + y12 + y23) V2 − y23V3 = −y13V1 − y23V2 (y13 + y23 + y34) V3 − y34V4 = −y34V3 + y34V4

+

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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, I1 = Y11V1 + Y12V2 + Y13V3 + Y14V4 I2 = Y21V1 + Y22V2 + Y23V3 + Y24V4 I3 = Y31V1 + Y32V2 + Y33V3 + Y34V4 I4 = Y41V1 + Y42V2 + Y43V3 + Y44V4 (2) where

Y11 = y10 + y12 + y13 Y22 = y20 + y12 + y23 Y33 = y13 + y23 + y34 Y44 = y34 Y12 = Y21 = −y12 Y13 = Y31 = −y13 Y23 = Y32 = −y23 Y34 = Y43 = −y34

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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,      I1 I2 . . . In      =      Y11 Y12 . . . Y1n Y21 Y22 . . . Y2n . . . . . . . . . . . . Yn1 Yn2 . . . Ynn           V1 V2 . . . Vn      (3)

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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, Ibus = YbusVbus (4) where Ybus is known as the bus admittance matrix of the system. The diagonal elements of Ybus 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 Ybus are known as transfer or mutual admittances.

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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

  • f the system assuming π model for the transmission lines.
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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: f1 (x1, x2, · · · , xn) = a1 f2 (x1, x2, · · · , xn) = a2 . . . fn (x1, x2, · · · , xn) = an (5) We shall make an initial estimate for the unknowns as x0

1, x0 2, · · · , x0

  • n. The corrections need to be added to the

estimates to get the correct solution be ∆x0

1, ∆x0 2, · · · , ∆x0 n.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

NR method Now we can write, f1

  • x0

1 + ∆x0 1, x0 2 + ∆x0 2, · · · , x0 n + ∆x0 n

  • =

a1 f2

  • x0

1 + ∆x0 1, x0 2 + ∆x0 2, · · · , x0 n + ∆x0 n

  • =

a2 . . . fn

  • x0

1 + ∆x0 1, x0 2 + ∆x0 2, · · · , x0 n + ∆x0 n

  • =

an (6)

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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 ith equation we get, fi

  • x0

1 + ∆x0 1, x0 2 + ∆x0 2, · · · , x0 n + ∆x0 n

  • = fi
  • x0

1, x0 2, · · · , x0 n

  • +

∂fi ∂x1

  • ∆x0

1 +

∂fi ∂x2

  • ∆x0

2 + · · · +

∂fi ∂xn

  • ∆x0

n +

  • higher order terms of ∆x0

1, ∆x0 2, · · · , ∆x0 n

  • = ai

(7)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

NR method If we assume ∆x0s be small then we higher powers of ∆x0s can be neglected. If we arrange the expanded equations in matrix form we get,

     a1 − f1

  • x0

1, x0 2, · · · , x0 n

  • a2 − f2
  • x0

1, x0 2, · · · , x0 n

  • .

. . an − fn

  • x0

1, x0 2, · · · , x0 n

    =        

  • ∂f1

∂x1

  • ∂f1

∂x2

  • 0 · · ·
  • ∂f1

∂xn

  • ∂f2

∂x1

  • ∂f2

∂x2

  • 0 · · ·
  • ∂f2

∂xn

  • .

. .

  • ∂fn

∂x1

  • ∂fn

∂x2

  • 0 · · ·
  • ∂fn

∂xn

            ∆x0

1

∆x0

2

. . . ∆x0

n

     (8)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

NR method Equation (8) for jth iteration can be written as, ∆Aj = Jj∆X j ∆X j =

  • Jj−1

∆Aj (9) where Aj =         a1 − f1

  • xj

1, xj 2, · · · , xj n

  • a2 − f2
  • xj

1, xj 2, · · · , xj n

  • .

. . an − fn

  • xj

1, xj 2, · · · , xj n

       , ∆X j =       ∆xj

1

∆xj

2

. . . ∆xj

n

     

∆

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

NR method and Jj =         

  • ∂f1

∂x1

  • j
  • ∂f1

∂x2

  • j · · ·
  • ∂f1

∂xn

  • j
  • ∂f2

∂x1

  • j
  • ∂f2

∂x2

  • j · · ·
  • ∂f2

∂xn

  • j

. . .

  • ∂fn

∂x1

  • j
  • ∂fn

∂x2

  • j · · ·
  • ∂fn

∂xn

  • j

         and J is known as Jacobian Matrix. The next estimate is given by X j+1 = X j + ∆X j (10) The iterations are continued until ∆X falls below the tolerance value.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Application of NR method for load flow analysis Let us consider a power system with n buses. For the load flow analysis the buses in the system are designated under following categories:

1

Slack bus or Swing bus or Reference bus: One bus in the system is selected as slack bus for which the voltage magnitude (¯ V) and angle (δ) are specified. Real and reactive power are not specified for this bus, any power gap between total generation and load will be included in this bus.

2

Generator bus or P − V bus: Real power (P) and voltage magnitude (¯ V) are specified for this bus. Reactive power and angle are not known for this bus.

3

Load bus or P − Q bus: Real (P) and reactive power (Q) are specified for this bus. Voltage magnitude and angle are not known for this bus.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Application of NR method for load flow analysis Using (2), we can write the current at the ith node as, Ii =

n

  • j=1

YijVj (11) Equation (11) can be represented in polar form as, Ii =

n

  • j=1

|Yij|¯ Vj ∠(θij + δj) (12) where θ is the angle of the complex impedance and δ is the voltage angle. Voltage magnitude is indicated with a bar sign.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Application of NR method for load flow analysis The complex power at the ith node is given by, S∗ = Pi − Qi = V ∗

i Ii

Pi − Qi = ¯ Vi∠ − δi

n

  • j=1

|Yij|¯ Vj ∠(θij + δj) (13) Pi =

n

  • j=1

¯ Vi ¯ Vj|Yij| cos

  • θij − δi + δj
  • (14)

Qi = −

n

  • j=1

¯ Vi ¯ Vj|Yij| sin

  • θij − δi + δj
  • (15)
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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Application of NR method for load flow analysis In (14) Pi = f(¯ V, δ) and in (15) Qi = f(¯ V, δ), also for a n bus system we will have 2n nonlinear equations with 2n unknowns. We can use NR method to solve for these unknowns. Now using (8) and (9) we write, for the lth estimate             ∆Pl

1

. . . ∆Pl

n

− − − ∆Ql

1

. . . ∆Ql

n

            =                

∂P1 ∂δ1 l

· · ·

∂P1 ∂δn l

  • ∂P1

∂ ¯ V1 l

· · ·

∂P1 ∂ ¯ Vn l

. . . . . .

∂Pn ∂δ1 l

· · ·

∂Pn ∂δn l

  • ∂Pn

∂ ¯ V1 l

· · ·

∂Pn ∂ ¯ Vn l

− − − − − − − − − − − − −−

∂Q1 ∂δ1 l

· · ·

∂Q1 ∂δn l

  • ∂Q1

∂ ¯ V1 l

· · ·

∂Q1 ∂ ¯ Vn l

. . . . . .

∂Qn ∂δ1 l

· · ·

∂Qn ∂δn l

  • ∂Qn

∂ ¯ V1 l

· · ·

∂Qn ∂ ¯ Vn l

                            ∆δl

1

. . . ∆δl

n

− − − ∆ ¯ V l

1

. . . ∆ ¯ V l

n

            (16)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Application of NR method for load flow analysis This can be written in short as, ∆P ∆Q

  • =
  • J1

J2 J3 J4 ∆δ ∆ ¯ V

  • (17)

1

For the slack bus voltage and angle are specified so ∆δ1 and ∆ ¯ V1 will be zero. Therefore the first and n + 1th row and first and n + 1th column can be removed from the equation set (16).

2

Similarly for P − V buses ∆ ¯ V will be zero, so the rows and columns involving ∆Qm and ∆ ¯ V can be eliminated.

3

If we have m P − V buses, then the size of J1 will be (n − 1) × (n − 1), J2 will be (n − 1) × (n − m − 1),J3 will be (n − m − 1) × (n − 1) and J4 will be (n − m − 1) × (n − m − 1).

4

The overall size of the Jacobian matrix will be (2n − m − 2) × (2n − m − 2).

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Elements of J1 Diagonal elements: ∂Pi ∂δi =

  • j=i

¯ Vi ¯ Vj|Yij| sin

  • θij − δi + δj
  • (18)

Off-diagonal elements: ∂Pi ∂δj = −¯ Vi ¯ Vj|Yij| sin

  • θij − δi + δj
  • j = i

(19)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Elements of J2 Diagonal elements: ∂Pi ∂ ¯ Vi = 2 ¯ Vi|Yij| cos θij +

  • j=i

¯ Vj|Yij| cos

  • θij − δi + δj
  • (20)

Off-diagonal elements: ∂Pi ∂ ¯ Vj = ¯ Vi|Yij| cos

  • θij − δi + δj
  • j = i

(21)

ii ii

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Elements of J3 Diagonal elements: ∂Qi ∂δi =

  • j=i

¯ Vi ¯ Vj|Yij| cos

  • θij − δi + δj
  • (22)

Off-diagonal elements: ∂Qi ∂δj = −¯ Vi ¯ Vj|Yij| cos

  • θij − δi + δj
  • j = i

(23)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Elements of J4 Diagonal elements: ∂Pi ∂ ¯ Vi = −2 ¯ Vi|Yij| sin θij −

  • j=i

¯ Vj|Yij| sin

  • θij − δi + δj
  • (24)

Off-diagonal elements: ∂Qi ∂ ¯ Vj = −¯ Vi|Yij| sin

  • θij − δi + δj
  • j = i

(25)

Qi

ii ii

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Application of NR method for load flow analysis Once the Jacobian matrix is computed, ∆δi and ∆ ¯ Vi can be

  • btained. The next estimate will be given by,

δl+1

i

= δl

i + ∆δl i

¯ V l+1

i

= ¯ V l

i + ∆¯

V l

i

(26)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Application of NR method for load flow analysis The load flow analysis procedure can be summarised as follows:

1

Form the admittance matrix of the system.

2

Specify the real power and voltage for P − V bus and real and reactive power for P − Q bus. Select the slack bus and specify the voltage and angle for the slack bus. Generally the angle for slack bus is selected as zero.

3

Initialise unknown δ’s and voltages.

4

Find Pi and Qi using (14) and (15) and find the deviations ∆Pi and ∆Qi from the specified.

5

Compute the Jacobian matrix using (18) to (26).

6

Find ∆δ and ∆ ¯ V using (16). Update δ and ¯ V using (22).

7

Repeat the above procedure until ∆Pi and ∆Qi falls below the preset tolerance value.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Problem 2 Consider a 2 generator power system feeding a load through transmission network as shown in Figure 4. The impedance data of the network is given in pu values on 100 MVA base. Using NR method find the bus voltages after two iterations. The scheduled power and nominal bus pu bus voltages are indicated in Figure 4.

Figure: 3 Generator 3 Bus power system.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Fast decoupled load flow

1

In a power system active power flow mainly depends on the angle difference between sending end and receiving end voltages.

2

Similarly the reactive power flow on voltage magnitudes.

3

Using these facts the dependance of real power on voltage magnitude and the reactive power flow on the voltage angles can be neglected.

4

This assumption will help to decouple ∆P and ∆Q equations.

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Fast decoupled load flow We can rewrite (17) as, ∆P ∆Q

  • =
  • J1

J4 ∆δ ∆ ¯ V

  • (27)

which can be written as, ∆P = J1∆δ = ∂P ∂δ

  • ∆δ

∆Q = J4∆ ¯ V = ∂Q ∂ ¯ V

  • ∆ ¯

V (28)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Fast decoupled load flow Now the diagonal elements of J1 given by (18) can be written as, ∂Pi ∂δi =

n

  • j=1

¯ Vi ¯ Vj|Yij| sin

  • θij − δi + δj
  • −V¯i

2|Yii|sinθ

= −Qi − ¯ V 2

i Bii

(29) where Qi is given by (15) and Bii = Imaginary part of Yii. Generally in the power system Bii ≫ Qi, so Qi can be neglected and ¯ V 2

i

≈ ¯

  • Vi. With these assumptions the equation

(29) becomes, ∂Pi ∂δi = −¯ ViBii (30)

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Fast decoupled load flow To get the off diagonal elements of J1 we assume, θij − δi + δj ≈ θij and write equation (19) as, ∂Pi ∂δj = −¯ Vi ¯ VjBij (31) If we assume ¯ Vj ≈ 1 equation (31) becomes, ∂Pi ∂δj = −¯ ViBij (32) Using equations (30) and (32) we can write, ∆P ¯ Vi = −BP∆δ (33) ∆δ = −B−1

P

∆P ¯ Vi (34) where BP is the imaginary part of the system admittance matrix of size (n − 1).

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Fast decoupled load flow With similar assumptions we can get the diagonal elements of J4 as ∂Qi ∂ ¯ Vi = −¯ ViBii (35) and off diagonal elements as, ∂Qi ∂ ¯ Vj = −¯ ViBij (36) Combining (35) and (36) we can write, ∆Q ¯ Vi = −BQ∆ ¯ Vi (37) ∆ ¯ Vi = −B−1

Q

∆ ¯ Vi (38) where BQ is the imaginary part of the system admittance matrix of size (n − 1 − m).

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Introduction Nodal admittance matrix NR method Application of NR method for load flow analysis Fast decoupled load flow

Fast decoupled load flow

1

The updates for the iterative procedures are given by (34) and (38). BP and BQ are constant matrices and need to be computed once only.

2

Compared with NR-method, fast decoupled method will take more iterations for solution but will be fast and suitable for online control solutions.