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Optimal Power Flow Daniel Kirschen© 2011 D. Kirschen and the University of Washington
Economic dispatch Objective: minimize the cost of generationL Objective: minimize the cost of generation Constraints Equality constraint: load generation balance Inequality constraints: upper and lower limits on generating units output © 2011 D. Kirschen and the University of Washington
Limitations of economic dispatchB C Network A D 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 © 2011 D. Kirschen and the University of Washington
Example of network limitationB A LA LB Maximum flow on each line: 100MW 50 $/MWh 100$/MWh PA PB PAMAX PBMAX © 2011 D. Kirschen and the University of Washington
Acceptable ED solution100 MW 0 MW 300 MW B A LA = 100 MW LB = 200MW The solution of this (trivial) economic dispatch is: The flows on the lines are below the limit The economic dispatch solution is acceptable © 2011 D. Kirschen and the University of Washington
Unacceptable ED solution200 MW 0 MW 500 MW B A LA = 100 MW LB = 400MW The solution of the economic dispatch is: The resulting flows exceed their limit The economic dispatch solution is not acceptable © 2011 D. Kirschen and the University of Washington
Modified ED solution B A100 MW 200 MW 300 MW B A 100 MW LA = 100 MW LB = 400MW 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 following inequality constraint: However, adding inequality constraints for each problem is not practical in more complex situations We need a more general approach © 2011 D. Kirschen and the University of Washington
Optimal Power Flow (OPF) - OverviewOptimization 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 © 2011 D. Kirschen and the University of Washington
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: © 2011 D. Kirschen and the University of Washington
Mathematical formulation of the OPF (2)State variables Describe the response of the system to changes in the control variables Magnitude of voltage at each bus Except generator busses, which are control variables Angle of voltage at each bus Except slack bus Vector of state variables: © 2011 D. Kirschen and the University of Washington
Mathematical formulation of the OPF (3)Parameters Known characteristics of the system Assumed constant Network topology Network parameters (R, X, B, flow and voltage limits) Generator cost functions Generator limits … Vector of parameters: © 2011 D. Kirschen and the University of Washington
Mathematical formulation of the OPF (4)Classical objective function: Minimize total generating cost: Many other objective functions are possible: Minimize changes in controls: Minimize system losses … © 2011 D. Kirschen and the University of Washington
Mathematical formulation of the OPF (5)Equality constraints: Power balance at each node - power flow equations Compact expression: © 2011 D. Kirschen and the University of Washington
Mathematical formulation of the OPF (6)Inequality constraints: Limits on the control variables: Operating limits on flows: Operating limits on voltages Compact expression: © 2011 D. Kirschen and the University of Washington
Compact form of the OPF problemSubject to: © 2011 D. Kirschen and the University of Washington
OPF Challenges Size of the problem Problem is non-linear1000’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 © 2011 D. Kirschen and the University of Washington
Solving the OPF using gradient methodsBuild the Lagrangian function The gradient of the Lagrangian indicates the direction of steepest ascent: Move in the opposite direction to the point with the largest gradient Repeat until © 2011 D. Kirschen and the University of Washington
Problems with gradient methodsSlow 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 © 2011 D. Kirschen and the University of Washington
Solving the OPF using interior point methodBest 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 © 2011 D. Kirschen and the University of Washington
Linearizing the OPF problemUse 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 © 2011 D. Kirschen and the University of Washington
Sequential LP OPF Consequence of linear approximationThe 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 © 2011 D. Kirschen and the University of Washington
Advantages and disadvantagesAdvantages 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) © 2011 D. Kirschen and the University of Washington
DC Power Flow Approximation
Power Flow Equations Set of non-linear simultaneous equationsNeed a simple linear relation for fast and intuitive analysis dc power flow provides such a relation but requires a number of approximations
Neglect Reactive Power
Neglect Resistance of the Branches
Assume All Voltage Magnitudes = 1.0 p.u.
Assume all angles are small
Why is it called dc power flow?Reactance plays the role of resistance in dc circuit Voltage angle plays the role of dc voltage Power plays the role of dc current
Example of LPOPF Solving the full non-linear OPF problem by hand is too difficult, even for small systems We will solve linearized 3-bus examples by hand © 2011 D. Kirschen and the University of Washington
Example Economic dispatch: 1 2 3 B A 450 MW 10 $/MWh PA PAMAX=390MWPB PBMAX= 150 MW Economic dispatch: © 2011 D. Kirschen and the University of Washington
Flows resulting from the economic dispatch1 2 3 B A 450 MW 390MW 60MW Fmax = 200MW Fmax = 260MW Assume that all the lines have the same reactance Do these injection result in acceptable flows? © 2011 D. Kirschen and the University of Washington
Calculating the flows using superpositionBecause we assume a linear model, superposition is applicable 1 60 MW 2 3 450 MW 390 MW 1 2 3 390 MW 60 MW 1 2 3 © 2011 D. Kirschen and the University of Washington
Calculating the flows using superposition (1)2 3 390 MW FA FB FA = 2 x FB because the path has twice the reactance of the path 1-3 FA + FB = 390 MW FA = 260 MW FB = 130 MW © 2011 D. Kirschen and the University of Washington
Calculating the flows using superposition (2)60 MW FD FC 1 2 3 60 MW FC = 2 x FD because the path has twice the reactance of the path 2-3 FC + FD = 60 MW FC = 40 MW FD = 20 MW © 2011 D. Kirschen and the University of Washington
Calculating the flows using superposition (3)1 2 3 390 MW 60 MW 260 MW 130 MW 20 MW 40 MW 1 60 MW 2 3 450 MW 390 MW 110 MW 280 MW 170 MW Fmax = 260 MW © 2011 D. Kirschen and the University of Washington
Correcting unacceptable flows390 MW 60 MW 280 MW 1 2 Must reduce flow F1-3 by 20 MW 3 450 MW 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 © 2011 D. Kirschen and the University of Washington
Check the solution using superposition1 2 3 330 MW 120 MW 220 MW 110 MW 40 MW 80 MW 1 120 MW 2 3 450 MW 330 MW 70 MW 260 MW 190 MW Fmax = 260 MW © 2011 D. Kirschen and the University of Washington
Comments (1) The OPF solution is more expensive than the ED solutionCED = 10 x x 60 = $5,100 COPF = 10 x x 120 = $5,700 The difference is the cost of security Csecurity = COPF - CED = $600 The constraint on the line flow is satisfied exactly Reducing the flow below the limit would cost more © 2011 D. Kirschen and the University of Washington
Comments (2) We have used an “ad hoc” method to solve this problemIn practice, there are systematic techniques for calculating the sensitivities of line flows to injections These techniques are used to generate constraint equations that are added to the optimization problem © 2011 D. Kirschen and the University of Washington
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 © 2011 D. Kirschen and the University of Washington
Example: base case solution of OPF1 2 3 B A 450 MW 330MW 120MW 70 MW 260 MW 190 MW © 2011 D. Kirschen and the University of Washington
Example: contingency case1 2 3 B A 450 MW 330MW 120MW 0 MW 330 MW 120 MW Unacceptable because overload of line 1-3 could lead to a cascade trip and a system collapse © 2011 D. Kirschen and the University of Washington
Formulation of the Security Constrained OPFSubject to: Power flow equations for the base case Operating limits for the base case Power flow equations for contingency k Operating limits for contingency k Subscript 0 indicates value of variables in the base case Subscript k indicates value of variables for contingency k © 2011 D. Kirschen and the University of Washington
Preventive security formulationsubject to: This formulation implements preventive security because the control variables are not allowed to change after the contingency has occurred: © 2011 D. Kirschen and the University of Washington
Corrective security formulationsubject to: 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 control variables in the base case © 2011 D. Kirschen and the University of Washington
Size of the SCOPF problemExample - European transmission network: 13,000 busses 13,000 voltage constraints 20,000 branches 20,000 flow constraints N-1 security 20,000 contingencies In theory, we must consider 20,000 x (13, ,000) = 660 million inequality constraints… However: Not all contingencies create limit violations Some contingencies have only a local effect © 2011 D. Kirschen and the University of Washington
Limitations of N-1 criterionNot 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 © 2011 D. Kirschen and the University of Washington
Probabilistic security analysisGoal: 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 © 2011 D. Kirschen and the University of Washington
New Formulations of the Optimal Power Flow Problem
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© 2010 D. Kirschen and The University of Manchester1 New Formulations of the Optimal Power Flow Problem Prof. Daniel Kirschen The University of Manchester.
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