ECE 530 – Analysis Techniques for Large-Scale Electrical Systems

Prof. Hao Zhu

Dept. of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

haozhu@illinois.edu

10/14/2014

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Lecture 15: DC Power Flow Model;

Sensitivity Analysis

Announcements

• HW 5 is due October 21

• Midterm exam is October 23 in class; closed book and notes, but one 8.5 by 11 inch note sheet and simple calculators allowed–Test covers up to dc power flow (before sensitivity)

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Fast Decoupled Power Flow

• Fast decoupled power flow (FDPF) further approximates Jacobian to be independent of the voltage magnitudes/angles

• This means the Jacobian need only be built/inverted once per power flow solution

• FDPF uses the same mismatch equations as standard power flow (just scaled) so it should have same solution, and so it DPF

• The FDPF is widely used, though usually when we only need an approximate solution

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FDPF Approximations

ij

The FDPF makes the following approximations:

1. G 0

2. 1

3. sin 0 cos 1i

ij ij

V

4

i1

i

1

P ( cos sin )

Which can also be written as

P( cos sin )

n

i k ik ik ik ik Gi Dik

nGi Di

k ik ik ik iki ik

VV G B P P

P PV G B

V V

To see the impact on the real power equations recall

𝑝𝑖=¿

FDPF Approximations

• With the approximations for the diagonal term we get

for the off-diagonal terms (k≠i )

• Hence the Jacobian for the scaled real equations can be approximated as –B

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𝜕𝑝𝑖

𝜕𝜃𝑖

=∑𝑘≠ 𝑖

𝐵𝑖𝑘 cos𝜃𝑖𝑘≈∑𝑘≠ 𝑖

𝐵𝑖𝑘=−𝐵𝑖 𝑖

𝜕𝑝𝑖

𝜕𝜃𝑘

=−𝐵𝑖𝑘 cos𝜃𝑖𝑘≈ −𝐵𝑖𝑘

FPDF Approximations

• For the reactive power equations we also scale by Vi

• Similarly, the Jacobian off-diagonals

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i1

i

1

Q ( sin cos )

Q( sin cos )

n

i k ik ik ik ik Gi Dik

nGi Di

k ik ik ik iki ik

V V G B Q Q

Q QV G B

V V

𝑞𝑖=¿

𝜕𝑞𝑖

𝜕𝑉𝑘

=−𝐵𝑖𝑘 cos𝜃 𝑖𝑘≈ −𝐵𝑖𝑘

http://nptel.ac.in/courses/108107028/module2/lecture9/lecture9.pdf

FDPF Approximations

• And for the reactive power Jacobian diagonal

• As derived the real and reactive equations have a constant Jacobian equal to –B–Usually modifications are made to omit from the real power

matrix elements that affect reactive flow (like shunts) and from the reactive power matrix elements that affect real power flow, like phase shifters

–We’ll call the real power matrix B’ and the reactive B”

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𝜕𝑞𝑖

𝜕𝑉 𝑖

≈ −2𝐵𝑖𝑖−∑𝑘≠𝑖

𝐵𝑖𝑘=−𝐵𝑖𝑖

FDPF Approximations

• It is also common to flip the sign on the mismatch equation, by changing it from (summation – injection) to (injection – summation)–Other modifications on the B matrix have been presented in

the literature (D. Jajicic and A. Bose, “A Modification to the Fast Decoupled Power Flow for Networks with High R/X Ratios,” IEEE Transactions on Power Sys., May 1988)

• Hence we have

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( ) ( )( )( ) 1 1

( ) ( )

( ) ( )v vvv

v v P x Q x

θ B V BV V

FDPF Three Bus Example

Line Z = j0.07

Line Z = j0.05 Line Z = j0.1

One Two

200 MW 100 MVR

Three 1.000 pu

200 MW 100 MVR

Use the FDPF to solve the following three bus system

34.3 14.3 20

14.3 24.3 10

20 10 30bus j

Y

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FDPF Three Bus Example, cont’d

1

(0)(0)2 2

3 3

34.3 14.3 2024.3 10

14.3 24.3 1010 30

20 10 30

0.0477 0.0159

0.0159 0.0389

Iteratively solve, starting with an initial voltage guess

0 1

0 1

bus j

V

V

Y B

B

(1)2

3

0 0.0477 0.0159 2 0.1272

0 0.0159 0.0389 2 0.1091

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FDPF Three Bus Example, cont’d

(1)2

3

i

i i1

(2)2

3

1 0.0477 0.0159 1 0.9364

1 0.0159 0.0389 1 0.9455

P ( )( cos sin )

V V

0.1272 0.0477 0.0159

0.1091 0.0159 0.0389

nDi Gi

k ik ik ik ikk

V

V

P PV G B

x

(2)2

3

0.151 0.1361

0.107 0.1156

0.924

0.936

0.1384 0.9224Actual solution:

0.1171 0.9338

V

V

θ V

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FDPF Region of Convergence

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FDPF Cautions

• The FDPF works well as long as the previous approximations hold for the entire system

• With the movement towards modeling larger systems, with more of the lower voltage portions of the system represented (for which r/x ratios are higher) it is quite common for the FDPF to get stuck because small portions of the system are ill-behaved

• The FDPF is commonly used to provide an initial guess of the solution or for contingency analysis

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“DC” Power Flow

• The “DC” power flow makes the most severe approximations:–completely ignore reactive power

• This makes the power flow a linear set of equations, which can be solved directly (no convergence issue!)

• The term dc power flow actually dates from the time of the old network analyzers (going back into the 1930’s)

• Not to be confused with the inclusion of HVDC lines in the standard NPF

1θ B P

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P sign convention is generation is positive

DC Power Flow References

• A nice formulation is given in the Power Generation and Control book by Wood and Wollenberg (with the 3rd edition out)

• The August 2009 paper in IEEE Transactions on Power Systems, “DC Power Flow Revisited” (by Stott, Jardim and Alsac) provides good coverage

• T. J. Overbye, X. Cheng, and Y. Sun, “A comparison of the AC and DC power flow models for LMP Calculations,” in Proc. 37th Hawaii Int. Conf. System Sciences, 2004, compares the accuracy of the approach

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DC Power Flow Example

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Example from Power System Analysis and Design, by Glover, Sarma, Overbye, fifth edition

DC Power Flow in PowerWorld

• PowerWorld allows for easy switching between the dc and ac power flows

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slack

One

Two

ThreeFourFiveA

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.000 pu 1.000 pu

1.000 pu

1.000 pu

1.000 pu 0.000 Deg -4.125 Deg

-18.695 Deg

-1.997 Deg

0.524 Deg

360 MW

0 Mvar

520 MW

0 Mvar

800 MW 0 Mvar

80 MW 0 Mvar

Notice with the dc power flow all of the voltage magnitudes are 1.0 per unit.

To use the dc approachin PowerWorldselect Tools,Solve, DCPower Flow

Notice thereare no losses

PowerWorld case is Bus5_GSO solved with the dc power flow

Sensitivity Analysis

• System description and notations• Motivation for the sensitivity analysis• From sensitivity to distribution factors• Definitions of the various distribution factors• Analysis of the distribution factors• Distribution factor applications

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• We consider a system with N buses and L lines given by the set given by the set – Some authors designate the slack as bus zero; an

alternative approach, that is easier to implement in cases with multiple islands and hence slacks, is to allow any bus to be the slack, and just set its associated equations to trivial equations just stating that the slack bus voltage is constant

• We may denote the kth transmission line or transformer in the system, , as , ),

Notation

from node to node

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Notation, cont.

• We’ll denote the real power flowing on line from bus to bus as

• The vector of active power flows on the L lines is:

• The bus real and reactive power injection vectors are(note we use lower-case p/q injection in later analysis)

p

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1 2f [ , , , ] TLf f f

• The series admittance of line is , and we define the matrix

• We define the LN incidence matrix:

Notation, cont.

where the element j of vector ai is nonzero , whenever line is coincident with bus j. Hence matrix A is quite sparse, with two nonzeros per row

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𝐀=[𝐚1𝑇

𝐚2𝑇

⋮𝐚𝐿

𝑇 ]

Analysis Example: Available Transfer Capability

• The power system available transfer capability (ATC) is defined as the maximum additional MW that can be transferred between two specific areas, while meeting all the specified pre- and post-contingency system conditions

• ATC impacts measurably the market outcomes and system reliability and, therefore, the ATC values impact the system and market behavior

• A useful reference on ATC is Available Transfer Capability Definitions and Determination from NERC, June 1996 (available freely online)

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ATC and Its Key Components

• Total transfer capability (TTC ) – Amount of real power that can be transmitted across

an interconnected transmission network in a reliable manner, including considering contingencies

• Transmission reliability margin (TRM)– Amount of TTC needed to deal with uncertainties in

system conditions; typically expressed as a percent of TTC

• Capacity benefit margin (CBM) – Amount of TTC needed by load serving entities to

ensure access to generation; typically expressed as a percent of TTC 23

ATC and Its Key Components

• Uncommitted transfer capability (UTC) UTC = TTC – existing transmission commitments

• Formal definition of ATC is ATC = UTC – CBM – TRM

• We focus on determining Um,n, the UTC from node m to node n

• Um,n is defined as the maximum additional MW that can be transferred from node m to node n without violating any limit in either the base case or in any post-contingency conditions

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UTC Evaluation

nm

t t

maxf

i j

nolimit violation

,

( )

. .

m n

j max

U = max t

s t

f f f

L

for the base case j = 0 and each contingency case

j = 1,2 … , J

( )0f f

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Conceptual Solution Algorithm

1. Solve the initial power flow, corresponding to the initial system dispatch (i.e., existing commitments); Initialize the change in transfer t(0) = 0, k =0; set step size d ; j is used to indicate either the base case (j=0) or a contingency, j = 1,2,3…J

2. Compute t(k+1) = t(k) + d3. Solve the power flow for the new t(k+1)

4. Check for limit violations:

if violation is found set U(j)m,n = t(k) and stop;

else set k = k+1, and goto 2

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Conceptual Solution Algorithm, cont.

• This algorithm is applied for the base case (j=0) and each specified contingency case, j=1,2,..J

• The final UTC, Um,n is then determined by

• This algorithm can be easily performed on parallel processors since each contingency evaluation is independent of the other

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( ), ,

jm n m n

0 j JU = min U

Five Bus Example: Base Case

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Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

42 MW

67 MW

100 MW

118 MW

1.040 pu

1.042 pu

A

MVA

A

MVA

A

MVA

1.042 pu

A

MVA

1.044 pu

33 MW

MW200

258 MW

MW118

260 MW

100 MW

MW100

A

MVA

One Two

Three

Four

Five

PowerWorld Case: B5_DistFact

Five Bus Network Data

( MW )

1 2 0 -6.25 150

1 3 0 -12.5 400

1 4 0 -12.5 150

2 3 0 -12.5 150

3 4 0 -12.5 150

4 5 0 -10 1,000

i j g b

maxf

3

1

2

4

5

6

Five Bus UTC

• We evaluate U2,3 using the previous procedure– Gradually increase generation at Bus 2 and load at

Bus 3• We consider the base case and a single

contingency with line 2 outaged (between 1 and 3): J = 1

• Simulation results show for the base case that

• And for the contingency that

• Hence

( ) (1)2,3 2,3 2,3, 240U min U U MW

( )2,3 450U MW

(1)2,3 24U MW

UTC: Base Case

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Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

55 MW

71 MW

100 MW

150 MW

1.040 pu

1.041 pu

A

MVA

A

MVA

1.041 pu

A

MVA

1.043 pu

29 MW

MW200

258 MW

MW163

305 MW

100 MW

MW100

A

MVA

One Two

Three

Four

Five

100%A

MVA

( )2,3 450U MW

UTC: Contingency Case

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Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

34 MW

92 MW

100 MW

150 MW

1.040 pu

1.036 pu

A

MVA

A

MVA

1.038 pu

A

MVA

1.040 pu

8 MW

MW200

258 MW

MW142

284 MW

100 MW

MW100

One Two

Three

Four

Five

100%A

MVA

(1)2,3 24U MW

Computational Considerations

• Obviously such a brute force approach can run into computational issues with large systems

• Consider the following situation:– 10 iterations for each case– 6,000 contingencies– 2 seconds to solve each power flow

• It will take over 33 hours to compute a single UTC for the specified transfer direction from m to n.

• Consequently, there is an urgent need to develop fast tools that can provide satisfactory approximations

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