EE 369 POWER SYSTEM ANALYSIS

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					          EE 369
 POWER SYSTEM ANALYSIS

        Lecture 13
Newton-Raphson Power Flow
  Tom Overbye and Ross Baldick



                                 1
         Announcements
Read Chapter 6 and (fourth edition) Chapter
 11, concentrating on sections 11.4 and 11.5;
 (fifth edition) Chapter 12, concentrating on
 sections 12.4 and 12.5.
Homework 10 is fourth edition: 6.27, 6.35,
 6.36, 6.44, 6.46, 6.57, 11.24, 11.29; fifth
 edition 6.28, 6.36, 6.37, 6.45, 6.51, 6.57,
 12.24, 12.28; due November 3. Turn in to in
 box next to my office door.
                                           2
Using the Power Flow: Example 1
                                                                                                                                              A
                                                                                                                                                                                                                                                                                         SLA C K3 4 5
                                                                                                                                              MVA
                                                                                                                                                                                                                                                     A


                                                                                                                                                                                                                                                  MVA

                                                                                                                                                                                                                                                                                                                    2 1 8 MW
                                                                                                                                                                             1 .0 2 pu                                   RA Y 3 4 5
                                                                                                                                                                                                                                                                                                       sla ck        5 4 M var
                                                                                                                                                                                        A                      A                                                                   A


              1 .0 2 pu                                                                                                                                                              MVA                      MVA                                        SLA C K1 3 8              MVA
                                         T IM 3 4 5
                                                                                                                                                                             1 .0 1 pu                                           RA Y 1 3 8


 1 .0 0 pu
                      A


                      MVA

                                               T IM 1 3 8
                                                                                           3 3 MW
                                                                                                                      A


                                                                                                                      MVA
                                                                                                                                                                                                  A
                                                                                                                                                                                                                                                 A


                                                                                                                                                                                                                                                MVA
                                                                                                                                                                                                                                                                                             1 .0 3 pu                                                                     Using
                                                                                                                                                                                                                                                                         1 .0 2 pu
                                                                                           1 3 M var

                                                                                                                                                                                                                                                                                                                                                                           case
                                                                                                                                                                                                  MVA
                                     A
                                                     A
                                                                                                                1 6 .0 M var                        1 8 MW
                                                                                                                                                                                  1 .0 2 pu                              RA Y 6 9
                                   MVA
                                                   MVA                                                                                               5 M var                                                                                                                                                    3 7 MW
                                                                                                                                                                       A
                                                                                                                                                                                                                        1 7 MW                            A

          1 .0 2 pu                            T IM 6 9                                                          P A I6 9                                                                                                                                                                                       1 3 M var
                                                                            1 .0 1 pu                                                                                                                                    3 M var

                                                                                                                                                                                                                                                                                                                                                                           from
                                                                                                                                                                       MVA                                                                                MVA
                                                                    A

          2 3 MW                                                                                                                      1 .0 1 pu                GRO SS6 9                                 A
  A                                                              MVA
           7 M var                                                                                                                                                                                      MVA
                                                                                                                                                                                                                    FERNA 6 9
 MVA                                                                                                                        A
                                                                                                                                                                                                                                                                                             1 .0 1 pu                     WO LEN6 9
                                                                                                                                                                                                                                      2 1 MW
       M O RO 1 3 8
                                                                                                                                                                                                                                                                                                                                                                           Example
                                               A
                                                                                                                            MVA

                                               MVA
                                                                               H ISKY 6 9                                                                                                                                              7 M var
                                                                                                                                                                                                                                                                                                                                                  A
                               A
                                                                                                                                                                                                                                      4 .8 M var
             1 2 MW            MVA
                                                                                                                                                                                                                                                                                                                    A                             MVA

              5 M var                                2 0 MW                                                                                                                                                         1 .0 0 pu


        1 .0 0 pu
                                         H A NNA H 6 9
                                                      8 M var                                    A


                                                                                                 MVA
                                                                                                                P ET E6 9
                                                                                                                                      5 8 MW
                                                                                                                                                                             DEM A R6 9                       A
                                                                                                                                                                                                                                         1 .0 0 pu
                                                                                                                                                                                                                                                                        A
                                                                                                                                                                                                                                                                                    BO B1 3 8
                                                                                                                                                                                                                                                                                                            MVA
                                                                                                                                                                                                                                                                                                                A
                                                                                                                                                                                                                                                                                                                    MVA


                                                                                                                                                                                                                                                                                                                                                                           6.13
                                                                                                                                                                                                              MVA                                                       MVA
                                                  5 1 MW                                                                              4 0 M var
                                                                                                                                                        4 5 MW
                                                  1 5 M var                                                                                                                                                               A
                                                                                                                                                                                                                                      1 .0 2 pu                                              BO B6 9
                                                                                                                                                        1 2 M var
                                                                                  2 9 .0 M var                                                                                                                            MVA
                                                                                                                                        UIUC 6 9                0 .9 9 pu
                                                         1 4 .3 M var
                                                                                                              1 .0 0 pu                                                                                 1 4 0 MW                                                                   5 6 MW
                                                                                                                                                        1 2 .8 M var                        A

                                                                                                                                                                                                         4 5 M var
                                                                                                                                  A
                                                                                                                                                                                            MVA                                                                                    1 3 M var                                         LY NN1 3 8
                                                                              A
                                                                                                                                                                                                                                                0 MW
                                                                                                                                MVA
                                                          A                                                                                                                                                                                     0 M var
                                                                              MVA
                                                                                                                                                                                                                                                                A
                                                          MVA                                                    5 8 MW                                            A
                                                                                                                                                                                                                                                                                                                          1 4 MW
                                     0 .9 9 7 pu                                                                                                                                                                                 BLT 1 3 8
                                                                                                                 3 6 M var                                      MVA                                1 .0 0 pu                                                   MVA                                                         4 M var
                                              0 .9 9 pu                                  A M A NDA 6 9                                                  A
                                                                                                                                                                                                                                A

                               A
                                                                                                                                                                                                                                                                                    SH IM KO 6 9                                        1 .0 2 pu
          H O M ER6 9                                                       3 3 MW
                                                                                                                                                                                                                                MVA
                                                                                                                                                        MVA
                                                                                                                                                                                                                                                                                               7 .4 M var
                                                                                                                                                                                                                                                                                                                                                          A
                               MVA
                                                                            1 0 M var                                                 1 .0 1 pu                                                                                                                         A

                                                                                                                                                                                                                                      BLT 6 9                                                                                                             MVA
                                                                                                                                  A                                                1 .0 1 pu                                                                            MVA

        1 5 MW                                       A                                                                          MVA
                                                                                                                                                                                                                                                                                                                1 5 MW
         3 M var                                                                                             H A LE6 9                                  A                                                                       1 0 6 MW                                                                         5 M var
                                                   MVA
               1 .0 0 pu                                                                                                                                                                                                         8 M var                  A

                                                                                                                                                        MVA

                                                                                                                                  3 6 MW
                                                                                                                                                                                                                                                         MVA
  A
                                                                        A                                                                                                     A
                                                                                                                                                                                                                                                                                  1 .0 1 pu
                                          6 0 MW                        MVA                                                       1 0 M var          7 .2 M var              MVA
                                                                                                                                                                                                                                                                                                  A
                                                                                                                                                                                                                                                                                                                                                  A
 MVA
                                          1 2 M var
                                                                                                              1 .0 0 pu                                                                 1 .0 0 pu                        P A T T EN6 9                                                                                                            MVA

               0 .0 M var                                                                              A
                                                                                                                                                                                                                                                                                                 MVA



                                                                                                                                                                                                                        4 5 MW                  1 4 MW                        RO GER6 9
                                                                                                       MVA
                                                                                                                                        1 .0 0 pu                       WEBER6 9                                         0 M var
                                                                                    LA UF6 9                                                                                                                                                      2 M var
               1 .0 2 pu
                                                                                                                    2 3 MW
                                                                                                                                           2 2 MW                            0 MW
                                                                A                    A
                                                                                                                     6 M var                                                                                        1 4 MW                                          A

                            2 0 MW                                                                                                         1 5 M var                         0 M var
                                                                MVA                  MVA                                                                                                                             3 M var                                    MVA
                            3 0 M var
                                                                                                                                                                                                                                                                         1 .0 2 pu                                           JO 1 3 8          JO 3 4 5
                                                                                    LA UF1 3 8                                                                                                                       1 .0 2 pu             SA V O Y 6 9                                  4 2 MW
                                         1 .0 0 pu
                                                                                                                                                                                                                                                                                          2 M var
                                                                                                        1 .0 1 pu            BUC KY 1 3 8                                                                                                  A



                                                                                            A                                                                                                                                             MVA                                                                                            A

                                                                                                                                                                                                                                                                                                                                                                1 5 0 MW
                                                                                           MVA                                                                                                          1 .0 1 pu                          SA V O Y 1 3 8                                                                                MVA
                                                                                                                                                               A                                                                                                                         A
                                                                                                                                                                                                                                                                                                                                                                 0 M var
                                                                                                                                                               MVA                                                                                                                     MVA


                                                                                                                                                                                                                                                                                                                                                                1 5 0 MW
                                                                                                                                                                                    A
                                                                                                                                                                                                                                                                                                                                                                 0 M var
                                                                                                                                                                                    MVA
                                                                                                                                                                                                                                                                                                                                                1 .0 3 pu
                                                                                                                                                                                                                                                                                                                             1 .0 2 pu   A


                                                                                                                                                                                                                                                                                                                                         MVA
                                                                                                                                                                                                                                                                                                                                                                                     3
      Dishonest Newton-Raphson
Since most of the time in the Newton-Raphson
 iteration is spent dealing with the Jacobian, one
 way to speed up the iterations is to only calculate
 (and factorize) the Jacobian occasionally:
  – known as the “Dishonest” Newton-Raphson or
    Shamanskii method,
  – an extreme example is to only calculate the Jacobian
    for the first iteration, which is called the chord method.
       Honest:        x(v1)  x(v ) - J (x(v ) )-1f (x( v ) )
       Dishonest: x(v1)  x(v ) - J (x(0) )-1f (x( v ) )
       Both require f (x(v ) )   for a solution                4
            Dishonest Newton-Raphson
                     Example
Use the Dishonest Newton-Raphson (chord method)
to solve f ( x )  0, where:
f ( x)  x - 2   2

                                  1
x   (v)
               df ( x (0) )  f ( x ( v ) )
                 dx
                              
                               
x ( v )       1  (( x ( v ) )2 - 2)
                 2 x (0) 
                         
               (v)  1 
x ( v 1)    x                   (( x ( v ) )2 - 2)
                        2 x (0) 
                                
                                                        5
Dishonest N-R Example, cont’d
x   ( v 1)
               x   (v)      1  (( x ( v ) )2 - 2)
                            (0) 
                            2x 
Guess x (0)  1. Iteratively solving we get
             x ( v ) (honest)       x ( v ) (dishonest)
0                   1                      1             We pay a price
1                1.5                       1.5           in increased
2                1.41667                   1.375         iterations, but
                                                         with decreased
3                1.41422                   1.429         computation
4                1.41422                   1.408         per iteration
                                                                      6
      Two Bus Dishonest ROC
Slide shows the region of convergence for different initial
guesses for the 2 bus case using the Dishonest N-R
                                                 Red region
                                                 converges
                                                 to the high
                                                 voltage
                                                 solution,
                                                 while the
                                                 yellow region
                                                 converges
                                                 to the low
                                                 voltage
                                                 solution




                                                            7
Honest N-R Region of Convergence




                             Maximum
                                of 15
                             iterations




                                    8
    Decoupled Power Flow
The “completely” Dishonest Newton-
 Raphson (chord) is not usually used for
 power flow analysis. However several
 approximations of the Jacobian matrix
 are used that result in a similar
 approximation.
One common method is the decoupled
 power flow. In this approach
 approximations are used to decouple the
 real and reactive power equations.        9
       Decoupled Power Flow
           Formulation
General form of the power flow problem
   P ( v )   P   (v) 
                     
   θ         V        θ( v )   P (x( v ) ) 
                                                  f (x( v ) )
   Q ( v )   Q (v)         (v)
                          V   Q(x ) 
                                            (v)
                                               
   θ
              V    
where
          P2 (x( v ) )  PD 2  PG 2 
                                     
P(x )  
    (v)
                                      
          P (x(v ) )  P  P 
          n               Dn     Gn                             10
             Decoupling Approximation
                                   P ( v )     Q ( v )
Usually the off-diagonal matrices,          and
                                   V           θ
are small. Therefore we approximate them as zero:
   P ( v )        
              0 
                       θ ( v )    P( x ( v ) ) 
    θ
                                               f (x( v ) )
            Q    V 
                (v)
                      
                             (v)
                                    Q( x ) 
                                    
                                            (v)
                                                    
     0
            V    
Then the problem can be decoupled
                     ( v ) 1                                   ( v ) 1
               P                             (v)       Q      
θ   (v)
                      P( x ( v ) ),   V                     Q( x ( v ) )
               θ                                       V             11
  Off-diagonal Jacobian Terms
Justification for Jacobian approximations:
   1. Usually r         x, therefore Gij        Bij
    2. Usually  ij is small so sin  ij  0
Therefore
    Pi
           Vi  Gij cos ij  Bij sin  ij       0
    Vj
    Qi
             Vi V j  Gij cos ij  Bij sin  ij   0
    θ j

                                                           12
Decoupled N-R Region of
     Convergence




                          13
      Fast Decoupled Power Flow
By further approximating the Jacobian we obtain
 a typically reasonable approximation that is
 independent of the voltage magnitudes/angles.
This means the Jacobian need only be built and
 factorized once.
This approach is known as the fast decoupled
 power flow (FDPF)
FDPF uses the same mismatch equations as
 standard power flow so it should have same
 solution if it converges
The FDPF is widely used, particularly when we
 only need an approximate solution.           14
           FDPF Approximations
The FDPF makes the following approximations:
    1.      Gij  0
    2.      Vi           1 (for some occurrences),
    3.      sin ij  0          cos ij  1
Then: θ( v )  B 1diag{| V |( v ) }1 P( x ( v ) ),
                        B 1diag{| V |( v ) }1 Q( x ( v ) )
                 (v)
         V
Where B is just the imaginary part of the Ybus  G  jB,
except the slack bus row/column are omitted.
Sometimes approximate diag{| V |( v ) } by identity.             15
      FDPF Three Bus Example
Use the FDPF to solve the following three bus system
                     Line Z = j0.07



One                                    Two

                                                     200 MW
                                                     100 MVR
           Line Z = j0.05      Line Z = j0.1



               Three                  1.000 pu


                                 200 MW
                                 100 MVR               34.3 14.3 20 
                                           Ybus    j  14.3 24.3 10 
                                                                      
                                                       20
                                                              10  30 
                                                                       
                                                                    16
FDPF Three Bus Example, cont’d
          34.3 14.3        20 
Ybus  j  14.3 24.3 10   B   24.3 10 
                                             10     30 
                                                          
          20
                    10     30 
                                
  1        0.0477 0.0159 
B        
            0.0159 0.0389   
Iteratively solve, starting with an initial voltage guess
                                       (0)
 2 
         (0)
                0           V 2              1
                        V               1
 3            0            3               
 2 
         (1)
                 0   0.0477 0.0159   2   0.1272 
                                  2    0.1091
 3             0   0.0159 0.0389                
                                                               17
FDPF Three Bus Example, cont’d
         (1)
V 2           1  0.0477 0.0159  1 0.9364 
 V   1   0.0159 0.0389  1   0.9455
 3                                         
Pi ( x ) n                                  P P
           Vk (Gik cosik  Bik sin ik )  Di Gi
   Vi      k 1                                 Vi

 2 (2)  0.1272   0.0477 0.0159  0.151  0.1361
    0.1091   0.0159 0.0389  0.107    0.1156
 3                                                 
         (2)
V 2            0.924 
V                   
 3              0.936
                      0.1384      0.9224 
Actual solution: θ               V
                      0.1171      0.9338
                                             
                                                         18
FDPF Region of Convergence




                             19
             “DC” Power Flow
The “DC” power flow makes the most severe
 approximations:
  – completely ignore reactive power, assume all the
    voltages are always 1.0 per unit, ignore line
    conductance
This makes the power flow a linear set of
 equations, which can be solved directly:

                       θ  B 1 P

                                                       20
DC Power Flow Example




                     P




                            21
    DC Power Flow 5 Bus Example

                       One                Five                                     Four             Three
                                  A                                                          A




    360 MW                                                                                                       520 MW
                                  MVA                                                        MVA
                                                             A




     0 Mvar
                                                             MVA


              slack
                                                                                                                  0 Mvar



                      1.000 pu           1.000 pu      A            A
                                                                                 1.000 pu                      80 MW
                      0.000 Deg         -4.125 Deg     MVA          MVA
                                                                                -1.997 Deg                     0 Mvar
                                                                                                   1.000 pu
                                                                                                   0.524 Deg




                                               1.000 pu                   Two
                                             -18.695 Deg

                                                                   800 MW
                                                                    0 Mvar




Notice with the dc power flow all of the voltage magnitudes are
1 per unit.
                                                                                                                           22
          Power System Control
A major problem with power system operation is
 the limited capacity of the transmission system
  – lines/transformers have limits (usually thermal)
  – no direct way of controlling flow down a
    transmission line (e.g., there are no low cost valves
    to close to limit flow, except “on” and “off”)
  – open transmission system access associated with
    industry restructuring is stressing the system in new
    ways
We need to indirectly control transmission line
 flow by changing the generator outputs.               23
Indirect Transmission Line Control
 What we would like to determine is how a change in
 generation at bus k affects the power flow on a line
 from bus i to bus j.
                                   The assumption is
                                   that the change
                                   in generation is
                                   absorbed by the
                                   slack bus




                                                   24
   Power Flow Simulation - Before
•One way to determine the impact of a generator
change is to compare a before/after power flow.
•For example below is a three bus case with an
overload.
                                     131.9 MW

                                     124%

            One                                   Two


      200.0 MW                                              200 MW
                                68.1 MW         68.1 MW
      71.0 MVR                                              100 MVR


       Z for all lines = j0.1
                                 Three           1.000 pu

                                                0 MW
                                                64 MVR                25
    Power Flow Simulation - After
•Increasing the generation at bus 3 by 95 MW
(and hence decreasing generation at the slack
bus 1 by a corresponding amount), results in a
31.3 MW drop in the MW flow on the line from
bus 1 to 2.                            101.6 MW

                                       100%

            One                                     Two


       105.0 MW                                               200 MW
                               3.4 MW             98.4 MW
       64.3 MVR                                               100 MVR

                                                     92%
       Z for all lines = j0.1
       Limit for all lines = 150 MVA
                                                   1.000 pu
                                 Three
                                                  95 MW
                                                  64 MVR
                                                                        26
Analytic Calculation of Sensitivities
  Calculating control sensitivities by repeated
   power flow solutions is tedious and would
   require many power flow solutions.
  An alternative approach is to analytically
   calculate these values
   The power flow from bus i to bus j is
              Vi V j                        i   j
      Pij              sin( i   j ) 
               X ij                           X ij
                i   j                                   ij
   So Pij                           We just need to get
                       X ij                                 PGk
                                                               27
        Analytic Sensitivities
From the fast decoupled power flow we know
   θ  B 1P (x)
So to get the change in θ due to a change of
generation at bus k , just set P( x) equal to
all zeros except a minus one at position k .
            0
             
             
   P       1  Bus k
            0
             
             
                                               28
 Three Bus Sensitivity Example
For the previous three bus case with Zline  j 0.1
          20 10 10 
Ybus  j  10 20 10   B   20 10 
                                 10 20 
                                          
          10 10 20 
                        
Hence for a change of generation at bus 3
                            1
     2       20 10   0   0.0333
                       1   0.0667 
    3          10 20                  
                 0.0667  0
Then P3 to 1               0.667 pu
                     0.1
P3 to 2  0.333 pu     P 2 to 1  0.333 pu         29

				
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