Power Flow Solution

Power Flow Analysis

In computer application in power system analysis

Purpose of Load Flow Calculations 

Network planning tasks      



Network operation  



Determination of equipment loading Identification of weak points Impact of load increase Investigation of peak / low load and generation conditions Voltage control, reactive power compensation Security of supply (n-1 criterion) and reliability Loss reduction Investigation of network configurations during maintenance

Initial state for  

Stability calculations Motor start 2

Results of load flow calculation 

Load currents  



Node voltages 



magnitude and angle Equipment loading, overloading

magnitude and angle

Powers   

Active and reactive power balance Active and reactive power of generators Losses 3

Modeling for Load Flow 

Modeling mathematically as voltage or power source Slack bus - voltage (magnitude and angle) fixed, real and reactive power variable PU-/PV-bus - voltage (magnitude) and real power fixed, reactive power variable (normal operation mode of generator) PQ-bus - real and reactive power fixed, voltage (magnitude and angle) variable 4

Importance of Slack Generator 

Task of slack generator (swing bus)  

Fixing of voltage angle Balance of power difference between loads and infeed

5

Fundamentals of load flow calculation  • 

infeeds and loads, buses, branches description of network topology, i.e. solving load flow calculation Node

Load Infeed ~

Branch

6

Description of infeeds 

Slack feed:

voltage fixed δ fixed P, Q variable 1 slack needed in each network to balance powers



PU-feed: voltage fixed real power fixed Q, δ variable



PQ-feed: real power fixed reactive power fixed U, δ variable 7

Description of loads 

PQ-load:

real and reactive power fixed description by P,Q P, cos phi S, cos phi I, cos phi...

8

Description of branches 

Impedance

ZAB=RAB+ jXAB

or 

Admittance

Y AB

1 1 = = Z AB RAB + jX AB

9

Description of network topology

− Y 12 − Y 13 0 − Y 15  Y 12 + Y 13 + Y 15   − Y Y + Y − Y 0 0 21 21 23 23    Y = − Y 31 − Y 32 Y 31 + Y 32 + Y 34 − Y 34 0   0 0 − Y Y + Y − Y 43 43 45 12    − Y 51 0 0 − Y 54 Y 51 + Y 54  2

Y ii − Sum of all admittance s connected to node i Y ik − Negative admittance

3 I2

I12 Infeed ~

U1

Load

Y12 1

Branch

between node i and node k 5

4

10

Properties of admittance matrix 

large matrix



elements are complex numbers



sparse (for large networks only few elements nonzero)



diagonal elements positive



non-diagonal elements zero or negative 11

Load flow problem [I] = [ Y ] ⋅ [U] [ Y ] − admittance matrix [U] − matrix of node voltages [I] − matrix of node currents (signed sum of all currents at node)

non-linear problem for non-impedance loads (typical)

12

Load flow problem power at nodes

Pi + jQi = 3U i ⋅ I

* i

power at nodes, expressed as matrix equation

p + jq = 3 ⋅ diag ( u ) ⋅ i = 3 ⋅ diag ( u ) ⋅ Y ⋅ u *

*

*

13

Solving technique   

Guauss-Seidel method Newton-Raphson method Fast Decoupled method

14

Gauss-Seidel technique 1 0.9 0.8 0.7

f(x)

0.6 0.5 0.4 0.3 0.2 0.1 0

f1(x)=x       f2(x)=exp(­x)

Root  0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Similar to the fixed-point iteration method 15

Convergence of Fixed-point iteration y

y

y1 = x

y1 = x y 2 = g( x) •

•

x2

y

x1

x0

y 2 = g(x )

x1 x 2

x

x0

x

y y 2 = g( x ) y1 = x

y 2 = g( x )

•

x 0 x1 x 2

y1 = x

•

x

x1

x0

x 16

Newton-Raphson method f(x i )  0 f '(x i )  x i  x i 1

f(x)

df(xS2) dxS2

xL1

xS1

xS3

xS2

xL2

f(x i ) x i 1  x i  f '(x i ) x

17

Limits of load flow calculation 

Iteration boundary  high accuracy (ε small) vs. high calculation time



Load model  assumption of constant power for PQ-loads only valid near rated voltage  for low voltages load assumption too high -> voltage collapse



Possible reasons for non-convergence  load too high (PQ-load instead of Z-load)  reactive power problem -> voltage collapse  long lines  slack bus badly positioned



Steady state solution might not be reachable because of stability problems 18

Principle procedure of load flow calculation by iteration Start

Ui = U r

Start values for node voltages

δi = 0

Start values for deviations

∆U i = ∆δ i = 0

U i = U i + ∆U i

Adjustment of node voltages

δ i = δ i + ∆δ i

Calculation of node power

p + jq = 3 ⋅ diag( u ) ⋅ i = 3 ⋅ diag( u ) ⋅ Y ⋅ u *

*

Comparison with allowed divergence

Calculation of ΔUi and Δδi

*

no

Pi − Pnom < ε Qi − Qnom < ε

yes End

19

Gauss-Seidel method







Calculation continues with the new values of voltage for new iteration The process is repeated until the difference in voltage between the consecutive iterations is small enough Converges slowly 20

Gauss-Seidel acceleration factor 

Correction in voltage is multiplied by the constant ω



Selection of the multiplier depends on the network to be analyzed; 1.6 being a common value

21

Newton-Raphson method  

f(x) = 0 Initial guess x0



Find ∆x1 such that f(x0 + ∆x1) = 0



Taylor series: f (x0) + f ’(x0)∆x1 = 0

22

Newton-Raphson method 

The process is repeated with the value x1 = x0 + ∆x1

23

Newton-Raphson method 

Power equations for load nodes



Alternative representation of power equations

24

Newton-Raphson method 







Initially guess for voltage magnitude and angle Corresponding Pi and Qi to guessed voltage are calculated Compare with initial data of P and Q to get mismatch ∆ Pi and ∆ Qi Repeat until mismatches are small enough

25

Newton-Raphson method 

Selection of initial values Ui0 and δi0



Calculation of mismatches (actual-calculated)



Form linearization of node equations

26

Newton-Raphson method 



 

Determine inverse Jacobian matrix and solve the corrections for angles and voltages Substitute new values to voltages and angles and calculate the new partial derivative matrix Calculate the new power mismatches If the mismatches > given tolerance, repeat the process until the tolerance is small enough 27

The elements of Jacobian matrix

28

Newton-Raphson method – branch flow 

Power flow in branch is calculated by Iij = Yij(Vi – Vj)

And 

Sij = ViI*ij

Loss in branch is calculated by SL = Sij - Sji 29

Decoupled load flow (DLF)

In a power transmission network, JB and JC can be assumed zero Therefore, construction of the Jacobian and finding its inverse become easier

30

Fast decoupled load flow





The Jacobian matrix replaced by real constant matrix has to be constructed and inverted only once These accelerated (approximate) methods nevertheless give accurate results, because the calculated powers are always compared with the real ones 31

Possibilities to reach The following tips that may help to achieve convergence. It convergence should remembered that changes to the network may have to 

be reversed again and plausibility of results must be checked.  

 

   





change PQ-loads to Z-loads (impedance load conversion) change PU-generator to PQ-generator, relax operating limits of generators set starting points change method of calculation (current iteration, NewtonRaphson) disconnect long lines divide network in independent sub-networks try different positions of slack depending on network structure insert reactive power (capacitive or inductive) increase number of iterations and change accuracy requirements set tap changer to variable setting 32