ECE 530 – Analysis Techniques for Large-Scale Electrical Systems
Prof. Hao Zhu
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
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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