ECE 740. Optimal Power Flow

Transcription

1 ECE 740 Optimal Power Flow 1

2 Economic Dispatch A B C L Objective: minimize the cost of generation Constraints Equality constraint: load generation balance Inequality constraints: upper and lower limits on generating units output 2

3 Limitations of Economic Dispatch B 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 3

4 Example A - Network Limitation A B L A L B Maximum flow on each line: 100MW C A C B 50 $/MWh 100$/MWh P A MAX P A P B MAX P B 4

5 Acceptable ED Solution A 300 MW 100 MW 0 MW B L A = 100 MW 100 MW L B = 200MW The solution of this (trivial) economic dispatch is: The flows on the lines are below the limit The economic dispatch solution is acceptable 5

6 Unacceptable ED Solution A 500 MW 200 MW 0 MW B L A = 100 MW 200 MW L B = 400MW The solution of the economic dispatch is: The resulting flows exceed their limit The economic dispatch solution is not acceptable 6

7 Modified ED Solution A 300 MW 100 MW 200 MW B L A = 100 MW 100 MW L B = 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 7

8 Example B - Marginal (Incremental) Cost Below are graphs associated with this two bus system. The graph on left shows the marginal cost for each of the generators. The graph on the right shows the system supply curve, assuming the system is optimally dispatched Generator Power (MW) Total Area Generation (MW) Current generator operating points 8

9 Two Bus Economic Dispatch Example Total Hourly Cost : 8459 $/hr Area Lambda : Bus A Bus B MW MW MW AGC ON AGC ON MW 9

10 Two Bus with Unconstrained Line With no overloads the OPF matches the economic dispatch Total Hourly Cost : 8459 $/hr Area Lambda : Transmission line is not overloaded Bus A $/MWh Bus B $/MWh MW MW MW AGC ON AGC ON MW Marginal cost of supplying power to each bus (locational marginal costs) This would be price paid by load and paid to the generators. 10

11 Two Bus with Constrained Line Total Hourly Cost : 9513 $/hr Area Lambda : Bus A $/MWh Bus B $/MWh MW MW MW AGC ON AGC ON MW With the line loaded to its limit, additional load at Bus A must be supplied locally, causing the marginal costs to diverge. Similarly, prices paid by load and paid to generators will differ bus by bus. (In practice, some markets such as ERCOT charge zonal averaged price to load.) 11

12 Optimal Power Flow (OPF) OPF functionally combines the power flow with economic dispatch. Minimize cost function (operating cost) while taking into account realistic equality and inequality constraints. Equality constraints: bus real and reactive power balance generator voltage set points area MW interchange 12

13 OPF Inequality constraints: transmission line/transformer/interface flow limits generator MW limits generator reactive power capability curves bus voltage magnitudes. Available Controls: generator MW outputs transformer taps and phase shifters 13

14 OPF 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 14

15 Mathematical Formulation of the OPF Control variables: Active power output of the generating units Voltage at the generating units Position of the transformer taps Position of the phase shifter taps Status of the switched capacitors and reactors Vector of control variables: 15

16 Mathematical Formulation of the OPF 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: 16

17 Mathematical Formulation of the OPF 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: 17

18 Mathematical Formulation of the OPF Classical objective function: Minimize total generating cost: Equality constraints: Power balance at each node - power flow equations Compact expression: 18

19 Mathematical Formulation of the OPF Inequality constraints: Limits on the control variables: Operating limits on flows: Operating limits on voltages Compact expression: 19

20 Compact form of the OPF problem Subject to: 20

21 OPF Challenges Size of the problem in a large power system 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 21

22 Solving the OPF using gradient methods Build 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 22

23 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 23

24 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 24

25 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: 1. Linearize the problem around an operating point 2. Find the solution to this linearized optimization 3. Perform a full ac power flow at that solution to find the new operating point 4. Repeat 25

26 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) 26

27 Example C - 3 Bus Example Consider a three bus case (bus 1 is system slack), with all buses connected through 0.1 pu reactance lines, each with a 100 MVA limit. Let the generator marginal costs be: Bus 1: 10 $ / MWhr; Range = 0 to 400 MW, Bus 2: 12 $ / MWhr; Range = 0 to 400 MW, Bus 3: 20 $ / MWhr; Range = 0 to 400 MW, Assume a single 180 MW load at bus 3. 27

28 B3 with Line Limits NOT Enforced 60 MW 60 MW Bus 2 Bus $/MWh 0.0 MW $/MWh 0 MW 60 MW Total Cost 60 MW 1800 $/hr Bus 3 0 MW 120 MW 120% 120% 120 MW $/MWh 180 MW MW Line from Bus 1 to Bus 3 is overloaded; all buses have same marginal cost (but not allowed to dispatch to overload line!) 28

29 B3 with Line Limits Enforced 20 MW 20 MW Bus 2 Bus $/MWh 60.0 MW $/MWh 0 MW 80 MW Total Cost 80 MW 1920 $/hr Bus 3 0 MW 100 MW 100% 100% 100 MW $/MWh 180 MW MW LP OPF re-dispatches to remove violation. Bus marginal costs are now different. Prices will be different at each bus. 29

30 Verify Bus 3 Marginal Cost 19 MW 19 MW Bus 2 Bus $/MWh 62.0 MW $/MWh 0 MW 81 MW 81% 81% Total Cost 81 MW 1934 $/hr Bus 3 0 MW 100 MW 100% 100% 100 MW $/MWh 181 MW MW One additional MW of load at bus 3 raised total cost by 14 $/hr, as G2 went up by 2 MW and G1 went down by 1MW. 30

31 Why is Bus 3 LMP = $14 /MWh? All lines have equal impedance. Power flow in a simple network distributes inversely to impedance of path. For bus 1 to supply 1 MW to bus 3, 2/3 MW would take direct path from 1 to 3, while 1/3 MW would loop around from 1 to 2 to 3. Likewise, for bus 2 to supply 1 MW to bus 3, 2/3MW would go from 2 to 3, while 1/3 MW would go from 2 to 1to 3. 31

32 Why is Bus 3 LMP $ 14 / MWh? With the line from 1 to 3 limited, no additional power flows are allowed on it. To supply 1 more MW to bus 3 we need: Extra production of 1MW: P g1 + P g2 = 1 MW No more flow on line 1 to 3: 2/3 P g1 + 1/3 P g2 = 0; Solving requires we increase P g2 by 2 MW and decrease P g1 by 1 MW for a net increase of $14/h for the 1 MW increase. That is, the marginal cost of delivering power to bus 3 is $14/MWh. 32

33 Both Lines into Bus 3 Congested 0 MW 0 MW Bus 2 Bus $/MWh MW12.00 $/MWh 0 MW 100 MW 100% 100% 100 MW 100% 100% 100 MW Total Cost 100 MW 2280 $/hr Bus 3 4 MW $/MWh 204 MW MW For bus 3 loads above 200 MW, the load must be supplied locally. 33

34 Limiting Carbon Dioxide Emissions There is growing concern about the need to limit carbon dioxide emissions. The two main approaches are (1) a carbon tax, or (2) a cap-and-trade system (emissions trading) The tax approach involves setting a price and emitter of CO 2 pays based upon how much CO 2 is emitted. A cap-and-trade system limits emissions by requiring permits (allowances) to emit CO 2. The government sets the number of allowances, allocates them initially, and then private markets set their prices and allow trade. 34