A New Method for Estimating
Maximum Power Transfer and Voltage
Stability Margins to Mitigate the Risk of
Voltage Collapse
Bernie Lesieutre
Dan Molzahn
University of Wisconsin-Madison
PSERC Webinar, October 15, 2013
Voltage Stability
Static:
- Loading, Power Flows
- Local Reactive Power Support
Dynamic:
- Component Controls
- Network Controls
2
OVERVIEW: Static Voltage Stability Margin
Voltage, V
Margin
Power
A common voltage stability margin measures the distance from
a post-contingency operating point to the “nose point” on a
power-voltage curve.
3
OVERVIEW: Static Voltage Stability Margin
Voltage, V
Margin (negative)
Power
It’s even possible that a normal power flow solution won’t exist,
post contingency.
4
ISSUES
Voltage, V
Margin
Power
1. We don’t know the values on the curve.
2. We don’t know whether a post-contingency operating point
exists!
5
Typical Approach
• Run post-contingency power flow. This may or
may not converge.
• If successful, determine nose point: use a
sequence of power flows with increasing load, or
a continuation power flow.
6
Our Approach
Cast as an optimization problem:
• Minimize the controlled voltages while a
solution exists. (Claim: a solution exists.)
• Exploit the quadratic nature of the power
flow equations to directly obtain
Traditional Voltage Stability Margin
even when the margin is negative.
7
Advantages
• Eliminates need for repeated solutions
(multiple power flows, continuation power flows)
• Often offers provably globally optimal results
• Works when the margin is negative, i.e. when
there isn’t a solution.
8
OVERVIEW: Static Voltage Stability Margin
Voltage, V
Margin
V0
Vopt
P0
Pmax
Power
Optimal Solution
9
Power Flow Equations: Review
(ID1+jIQ1)
(VD1+jVQ1)
Electrical Network
I = YV
Y = G+jB
(VDN+jVQN)
(IDN+jIQN)
Current “Injection” equations in Rectangular Coordinates:
10
Power Flow Equations: Review
(P1+jQ1)
(VD1+jVQ1)
Electrical Network
I = YV
Y = G+jB
(VDN+jVQN)
(PN+jQN)
Power “Injection” equations in Rectangular Coordinates:
11
Power Flow Equations
The power flow equations are quadratic in voltage variables:
For reference, power engineers almost always express
these equations in voltage polar coordinates:
12
Classical Power Flow Constraints
Bus Type
Specified
Calculated
PQ (load)
P, Q
V, δ
PV (generator)
P, V
Q, δ
V, δ = 0º
P, Q
Slack
The “controlled voltages” are the problem-specified
slack and generator voltage magnitudes.
Locking the controlled voltages in constant
proportion, and allowing them to scale by α, we claim
a power solution exists for any power flow injection
profile. (subject to the unimportant small print.)
13
Optimization Problem
• Modify the power flow formulation to
• Slack bus voltage magnitude unconstrained
• PV bus voltage magnitudes scale with slack bus
voltage
• Minimize slack bus voltage
14
Optimization Problem
This optimization problem can be solved many different
ways…
We’ve been using the convex relaxation formulation for
the power flow equations (Lavaei, Low) because we
really want (provably) the minimum solution.
• The problem has a feasible solution
• The optimization using the convex relaxation can be
solved for a global minimum (and hopefully a feasible
power flow solution).
15
Relaxed Problem Formulation
In Rectangular coordinates, define
Then, power flow equations can be written in the form
where
Which is a rank one matrix by construction.
16
Relaxed Problem Formulation
The convex relaxation is introduced by relaxing the rank of
W. With that, we pose the following convex optimization
problem:
17
Semidefinite Relaxation Example
MISDP
18
Semidefinite Relaxation Example
MISDP
19
Bonus result concerning solution
to the power flow equations
• The existence of a power flow solution requires
•
Necessary, but not sufficient, condition for existence
• Conversely, no solution exists if
•
Sufficient, but not necessary, condition for non-existence
20
Controlled Voltage Margin
• A controlled voltage margin to the solvability
boundary
• Upper bound (non-conservative)
•
No power flow solution exists for
• Increasing the slack bus voltage (with proportional
increases in PV bus voltages) by at least
is required
for solution.
21
Power Injection Margin
• Uniformly scaling all power injections scales
• Uniformly scale power injections until
• Corresponding gives a power flow voltage stability
margin in the direction of uniformly increasing power
injections at constant power factor.
•
indicates that no solution exists for the original
power flow problem
22
Power Injection Margin
The power injection margin answers the question
For a given voltage profile, by what factor can we
change our power injections (uniformly at all
buses) while still potentially having a solution?
Answer:
23
Examples
• IEEE 14-Bus System
• IEEE 118-Bus System
• Tested many other systems and loadings
24
Controlled Voltage Margin
IEEE 14-Bus Continuation Trace: Bus 5
1.1
1
0.9
0.8
V5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Injection Multiplier
25
Power Injection Margin
IEEE 14-Bus Continuation Trace: Bus 5
1.1
1
0.9
0.8
V5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Injection Multiplier
26
IEEE 14 Bus System
Injection
Multiplier
1.000
2.000
3.000
4.000
4.010
4.020
4.030
4.040
4.050
4.055
4.056
4.057
4.058
4.059
4.060
4.061
4.062
4.063
4.064
4.065
5.000
NewtonRaphson
Converged?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
No
dim(null(A)
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
0.5261
0.7440
0.9112
1.0522
1.0535
1.0548
1.0561
1.0575
1.0588
1.0594
1.0595
1.0597
1.0598
1.0599
1.0601
1.0602
1.0603
1.0605
1.0606
1.0607
1.1764
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
27
Voltage Margin
28
Voltage and Power Injection Margins
IEEE 14-Bus Continuation Trace: Bus 5
1.2
1
V5
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Injection Multiplier
29
Controlled Voltage Margin
IEEE 118-Bus Continuation Trace: Bus 44
1
V44
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
Injection Multiplier
30
Power Injection Margin
IEEE 118-Bus Continuation Trace: Bus 44
1
V44
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
Injection Multiplier
31
IEEE 118 Bus System
Injection
Multiplier
NewtonRaphson
Converged?
1.00
1.50
2.00
2.50
3.00
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
4.00
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
No
No
No
No
No
No
(lower bound)
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
1.035
0.5724
0.7010
0.8095
0.9050
0.9914
1.0159
1.0175
1.0191
1.0207
1.0223
1.0239
1.0255
1.0271
1.0287
1.0303
1.0319
1.0335
1.0351
1.0366
1.0382
1.1448
dim(null(A)
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
32
Voltage Margin
IEEE 118 Bus System
33
Voltage and Power Injection Margins
IEEE 118-Bus Continuation Trace: Bus 44
1.2
1
V44
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Injection Multiplier
34
Alternate Power Injection Profiles
Power injection margin is in the direction of a uniform,
constant-power-factor injection profile
We can alternatively specify any profile that is a linear
function of powers and squared voltages
‒ However, insolvability condition
is not necessarily valid
35
Reactive Power Limits
Previous work models generators
as ideal voltage sources
‒ Limit-induced bifurcations
Two approaches to
modeling these limits:
Reactive Power versus Voltage Magnitude Characteristic
Q
max
Reactive Power
Detailed models limit
reactive outputs
Q
min
‒ Mixed-integer semidefinite
programming
0
∗
V
Voltage Magnitude
‒ Infeasibility certificates using sum
of squares programming
36
MISDP Formulation
Model reactive power limits using binary variables
Reactive Power versus Voltage Magnitude Characteristic
Reactive Power
Qmax
Qmin
0
V∗
Voltage Magnitude
Insolvability
37
MISDP Formulation
Model reactive power limits using binary variables
Reactive Power versus Voltage Magnitude Characteristic
Reactive Power
Qmax
Qmin
0
V∗
Voltage Magnitude
Insolvability
38
MISDP Formulation
Model reactive power limits using binary variables
Reactive Power versus Voltage Magnitude Characteristic
Reactive Power
Qmax
Qmin
0
V∗
Voltage Magnitude
Insolvability
39
MISDP Formulation
Model reactive power limits using binary variables
Reactive Power versus Voltage Magnitude Characteristic
Reactive Power
Qmax
Qmin
0
V∗
Voltage Magnitude
Insolvability
40
Reactive Power Limits Results
14-Bus System Power vs. Voltage
1.025
1.02
1.015
V5
1.01
1.005
1
0.995
P-V Curve
ηmax
0.99
0.985
Infeasibility Certificate Found
1
1.1
1.2
1.3
1.4
PQ Bus Injection Multiplier
41
Conclusions
• Cast the problem of computing voltage stability margins as
an optimization problem – to minimize the slack bus voltage.
• Calculated voltage stability margins – power injection/flows,
and controlled voltages.
• Tested with numerical examples
Advantages:
• Eliminates repeated solution (multiple power flows,
continuation power flows)
• Often offers provably globally optimal results
• Works when the margin is negative, i.e. when there isn’t a
solution.
42
Related Publications
[1]
B.C. Lesieutre, D.K. Molzahn, A.R. Borden, and C.L. DeMarco, “Examining the Limits of the
Application of Semidefinite Programming to Power Flow Problems,” 49th Annual Allerton
Conference on Communication, Control, and Computing (Allerton), 2011, pp.1492-1499, 28-30
Sept. 2011.
[2]
D.K. Molzahn, J.T. Holzer, and B.C. Lesieutre, and C.L. DeMarco, “Implementation of a LargeScale Optimal Power Flow Solver Based on Semidefinite Programming,” To appear in IEEE
Transactions on Power Systems.
[3]
D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, “An Approximate Method for Modeling ZIP
Loads in a Semidefinite Relaxation of the OPF Problem,” submitted to IEEE Transactions on
Power Systems, Letters.
[4]
D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, "A Sufficient Condition for Global Optimality of
Solutions to the Optimal Power Flow Problem,” to appear in IEEE Transactions on Power
Systems, Letters.
[5]
D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, “A Sufficient Condition for Power Flow
Insolvability with Applications to Voltage Stability Margins,” IEEE Transactions on Power
Systems, Vol 28, No. 3, pp. 2592-2601.
[6]
D.K. Molzahn, V. Dawar, B.C. Lesieutre, and C.L. DeMarco, “Sufficient Conditions for Power
Flow Insolvability Considering Reactive Power Limited Generators with Applications to Voltage
Stability Margins,” presented at Bulk Power System Dynamics and Control - IX. Optimization,
Security and Control of the Emerging Power Grid, 2013 IREP Symposium, 25-30 Aug. 2013.
D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, “Investigation of Non-Zero Duality Gap
Solutions to a Semidefinite Relaxation of the Power Flow Equations,” To be presented at the
ConclusionHawaii International Conference on System Sciences, January 2014.
[7]
43
Questions?
44
Extra Slides: Feasibility
45
Feasibility
For lossless systems:
1. Show a solution exists for zero power injections at
PQ buses and zero active power injection at PV
buses, for α = 1.
2. Use implicit function theorem to argue that
perturbations to zero power injections solutions also
exist. Specifically choose one in the direction of
desired power injection profile.
3. Exploit the quadratic nature of power flow equations
to scale voltages and power to match injection profile.
46
Feasibility: Zero Power Injection
Solution
Easy: Construct a solution.
• Open Circuit PQ buses for zero power, zero
current injection.
• Use Ward-type reduction to eliminate PQ buses.
(not really necessary, but clean) small print
• Choose uniform angle solution for all buses.
• Directly use reactive power flow equation to
calculated the reactive power injections at
generator buses.
0
0
0
47
Note: Zero Power Injection Solutions for
Lossy Systems
• Not all systems have a zero-power injection solution
• Ability to choose θ such that PPV = 0 depends on
•
Ratio of VPV to Vslack
•
Ratio of g to b
•
Systems with small resistances and small voltage magnitude
differences are expected to have a zero power injection solution
48
Nearby Solutions
A nearby non-zero solution exists
provided the Jacobian is nonsingular. For a connected
lossless system at the zero-power injection solution,
the appropriate Jacobian is nonsingular, generically.
small print
49
Feasible Solution
Exploit the quadratic nature of power flow
equations to scale voltages to match desired
power profile:
50
Extra Slides: Infeasibility Certificates
51
Infeasibility Certificates
Guarantee that a system of polynomial is infeasible
Positivstellensatz Theorem
If
such that
then the system of polynomials has no solution
52
Power Flow in Polynomial Form
Power Injection and Voltage Magnitude Polynomials
Reactive Power versus Voltage Magnitude Characteristic
Qmax
Reactive Power
Reactive Power Limit Polynomials
Qmin
0
V∗
Voltage Magnitude
53
Power Flow Infeasibility Certificates
Find a sum of squares polynomial of the form
such that
by finding polynomials
and sum of squares polynomials
Then the power flow equations have no solution.
54