04-01-2013, 09:59 AM
Optimal Power Flow
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Economic dispatch
Objective:
minimize the cost of generation
Constraints
Equality constraint: load generation balance
Inequality constraints: upper and lower limits on generating units output
Limitations of economic dispatch
Generating units and loads are not all connected to the same bus
The economic dispatch may result in unacceptable flows or voltages in the network
Modified ED solution
In this simple case, the solution of the economic dispatch can be modified easily to produce acceptable flows.
This could be done mathematically by adding the followinginequality constraint:
However, adding inequality constraints for each problem is not practical in more complex situations
We need a more general approach
Optimal Power Flow (OPF) - Overview
Optimization problem
Classical objective function
Minimize the cost of generation
Equality constraints
Power balance at each node - power flow equations
Inequality constraints
Network operating limits (line flows, voltages)
Limits on control variables
Mathematical formulation of the OPF (1)
Decision variables (control variables)
Active power output of the generating units
Voltage at the generating units
Position of the transformer taps
Position of the phase shifter (quad booster) taps
Status of the switched capacitors and reactors
Control of power electronics (HVDC, FACTS)
Amount of load disconnected
Vector of control variables:
OPF Challenges
Size of the problem
1000’s of lines, hundreds of controls
Which inequality constraints are binding?
Problem is non-linear
Problem is non-convex
Some of the variables are discrete
Position of transformer and phase shifter taps
Status of switched capacitors or reactors
Problems with gradient methods
Slow convergence
Objective function and constraints must be differentiable
Difficulties in handling inequality constraints
Binding inequality constraints change as the solution progresses
Difficult to enforce the complementary slackness conditions
Solving the OPF using interior point method
Best technique when a full AC solution is needed
Handle inequality constraints using barrier functions
Start from a point in the “interior” of the solution space
Efficient solution engines are available
Linearizing the OPF problem
Use the power of linear programming
Objective function
Use linear or piecewise linear cost functions
Equality constraints
Use dc power flow instead of ac power flow
Inequality constraints
dc power flow provides linear relations between injections (control variables) and MW line flows
Sequential LP OPF
Consequence of linear approximation
The solution may be somewhat sub-optimal
The constraints may not be respected exactly
Need to iterate the solution of the linearized problem
Algorithm:
Linearize the problem around an operating point
Find the solution to this linearized optimization
Perform a full ac power flow at that solution to find the new operating point
Repeat
Advantages and disadvantages
Advantages of LPOPF method
Convergence of linear optimization is guaranteed
Fast
Reliable optimization engines are available
Used to calculate nodal prices in electricity markets
Disadvantages
Need to iterate the linearization
“Reactive power” aspects (VAr flows, voltages) are much harder to linearize than the “active power aspects” (MW flows)
Correcting unacceptable flows
Must use a combination of reducing the injection at bus 1 and increasing the injection at bus 2 to keep the load/generation balance
Decreasing the injection at 1 by 3 MW reduces F1-3 by 2 MW
Increasing the injection at 2 by 3 MW increases F1-3 by 1 MW
A combination of a 3 MW decrease at 1 and 3 MW increase at 2 decreases F1-3 by 1 MW
To achieve a 20 MW reduction in F1-3 we need to shift 60 MW of injection from bus 1 to bus 2
Security Constrained OPF (SCOPF)
Conventional OPF only guarantees that the operating constraints are satisfied under normal operating conditions
All lines in service
This does not guarantee security
Must consider N-1 contingencies
Corrective security formulation
This formulation implements corrective security because the control variables are allowed to change after the contingency has occurred
The last equation limits the changes that can take place to what can be achieved in a reasonable amount of time
The objective function considers only the value of the controlvariables in the base case
Limitations of N-1 criterion
Not all contingencies have the same probability
Long lines vs. short lines
Good weather vs. bad weather
Not all contingencies have the same consequences
Local undervoltage vs. edge of stability limit
N-2 conditions are not always “not credible”
Non-independent events
Does not ensure a consistent level of risk
Risk = probability x consequences
Probabilistic security analysis
Goal: operate the system at a given risk level
Challenges
Probabilities of non-independent events
“Electrical” failures compounded by IT failures
Estimating the consequences
What portion of the system would be blacked out?
What preventive measures should be taken?
Vast number of possibilities