1. Lecture 15 Economic Dispatch Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS

2. 1 Announcements Homework 7 is 6.46, 6.49, 6.52, 11.19, 11.21, 11.27; due date is October 30 Potential spring courses: ECE 431 and ECE 398RES (Renewable Electric Energy Systems) Be reading Chapter 11, concentrating on sections 11.4 and 11.5

3. 2 Power System Economic Operation Power system loads are cyclical. Therefore the installed generation capacity is usually much greater than the current load. This allows options on how to meet the current load Generation costs can vary widely, with different technologies balancing – the capital costs necessary to build the generator – the costs to actually produce electric power – for example, nuclear and some hydro have high capital costs and low operating costs. Natural gas generators have low capital costs, and higher operating costs

4. 3 “DC” Power Flow The “DC” power flow makes the most severe approximations: – completely ignore reactive power, assume all the voltages are always 1.0 per unit, ignore line conductance This makes the power flow a linear set of equations, which can be solved directly

5. 4 Power System Control A major problem with power system operation is the limited capacity of the transmission system – lines/transformers have limits (usually thermal) – no direct way of controlling flow down a transmission line (e.g., there are no valves to close to limit flow) – open transmission system access associated with industry restructuring is stressing the system in new ways We need to indirectly control transmission line flow by changing the generator outputs

6. 5 DC Power Flow Example

7. 6 DC Power Flow 5 Bus Example Notice with the dc power flow all of the voltage magnitudes are 1 per unit.

8. 7 Indirect Transmission Line Control What we would like to determine is how a change in generation at bus k affects the power flow on a line from bus i to bus j. The assumption is that the change in generation is absorbed by the slack bus

9. 8 Power Flow Simulation - Before One way to determine the impact of a generator change is to compare a before/after power flow. For example below is a three bus case with an overload

10. 9 Power Flow Simulation - After Increasing the generation at bus 3 by 95 MW (and hence decreasing it at bus 1 by a corresponding amount), results in a 31.3 drop in the MW flow on the line from bus 1 to 2.

11. 10 Analytic Calculation of Sensitivities Calculating control sensitivities by repeat power flow solutions is tedious and would require many power flow solutions. An alternative approach is to analytically calculate these values

12. 11 Analytic Sensitivities

13. 12 Three Bus Sensitivity Example

14. 13 Balancing Authority Areas An balancing authority area (use to be called operating areas) has traditionally represented the portion of the interconnected electric grid operated by a single utility Transmission lines that join two areas are known as tie-lines. The net power out of an area is the sum of the flow on its tie-lines. The flow out of an area is equal to total gen - total load - total losses = tie-flow

15. 14 Area Control Error (ACE) The area control error (ace) is the difference between the actual flow out of an area and the scheduled flow, plus a frequency component Ideally the ACE should always be zero. Because the load is constantly changing, each utility must constantly change its generation to “chase” the ACE.

16. 15 Automatic Generation Control Most utilities use automatic generation control (AGC) to automatically change their generation to keep their ACE close to zero. Usually the utility control center calculates ACE based upon tie-line flows; then the AGC module sends control signals out to the generators every couple seconds.

17. 16 Power Transactions Power transactions are contracts between generators and loads to do power transactions. Contracts can be for any amount of time at any price for any amount of power. Scheduled power transactions are implemented by modifying the value of P sched used in the ACE calculation

18. 17 PTDFs Power transfer distribution factors (PTDFs) show the linear impact of a transfer of power. PTDFs calculated using the fast decoupled power flow B matrix

19. 18 Nine Bus PTDF Example Figure shows initial flows for a nine bus power system

20. 19 Nine Bus PTDF Example, cont'd Figure now shows percentage PTDF flows from A to I

21. 20 Nine Bus PTDF Example, cont'd Figure now shows percentage PTDF flows from G to F

22. 21 WE to TVA PTDFs

23. 22 Line Outage Distribution Factors (LODFS) LODFs are used to approximate the change in the flow on one line caused by the outage of a second line – typically they are only used to determine the change in the MW flow – LODFs are used extensively in real-time operations – LODFs are state-independent but do dependent on the assumed network topology

24. 23 Flowgates The real-time loading of the power grid is accessed via “flowgates” A flowgate “flow” is the real power flow on one or more transmission element for either base case conditions or a single contingency – contingent flows are determined using LODFs Flowgates are used as proxies for other types of limits, such as voltage or stability limits Flowgates are calculated using a spreadsheet

25. 24 NERC Regional Reliability Councils NERC is the North American Electric Reliability Council

26. 25 NERC Reliability Coordinators

27. 26 Electric Fuel Prices Source: EIA Electric Power Annual, 2006 (October 2007)

28. 27 Natural Gas Prices: 1990’s to 2008

29. 28 Coal Prices: 2005 to Present There are four main types of coal: bituminous, subbituminous, lignite, and anthracite. Heat values range from a low of 8 Mbtu per ton to a high of 31 Mbtu per ton. For Illinois coal price per Mbtu is now about $3.8/Mbtu.

30. 29 US Generation Mix (Energy) 2006 Gen TypeUS %Illinois %California% Coal49.047.6 1.0 Nuclear19.448.914.7 Hydro 7.1 0.122.2 Gas20.0 2.949.8 Petroleum 2.0 0.1 1.0 Renewable 2.5 0.4 11.8 (14.4 in 1990) Source: http://www.eia.doe.gov/cneaf/electricity/st_profiles/toc.html 2006 datahttp://www.eia.doe.gov/cneaf/electricity/st_profiles/toc.html Indiana is 94% coal, while Oregon is 71% hydro, Washington State is 76% hydro. Canada is about 60% hydro, France is also 80% nuclear, China is about 80% coal

31. 30 Thermal versus Hydro Generation The two main types of generating units are thermal and hydro, with wind rapidly growing For hydro the fuel (water) is free but there may be many constraints on operation – fixed amounts of water available – reservoir levels must be managed and coordinated – downstream flow rates for fish and navigation Hydro optimization is typically longer term (many months or years) In 476 we will concentrate on thermal units and some wind, looking at short-term optimization

32. 31 Generator types Traditionally utilities have had three broad groups of generators – baseload units: large coal/nuclear; always on at max. – midload units: smaller coal that cycle on/off daily – peaker units: combustion turbines used only for several hours during periods of high demand

33. 32 Block Diagram of Thermal Unit To optimize generation costs we need to develop cost relationships between net power out and operating costs. Between 2-6% of power is used within the generating plant; this is known as the auxiliary power

34. 33 Generator Cost Curves Generator costs are typically represented by up to four different curves – input/output (I/O) curve – fuel-cost curve – heat-rate curve – incremental cost curve For reference - 1 Btu (British thermal unit) = 1054 J - 1 MBtu = 1x10 6 Btu - 1 MBtu = 0.29 MWh

35. 34 I/O Curve The IO curve plots fuel input (in MBtu/hr) versus net MW output.

36. 35 Fuel-cost Curve The fuel-cost curve is the I/O curve scaled by fuel cost. A typical cost for coal is $ 1.70/Mbtu.

37. 36 Heat-rate Curve Plots the average number of MBtu/hr of fuel input needed per MW of output. Heat-rate curve is the I/O curve scaled by MW Best for most efficient units are around 9.0

38. 37 Incremental (Marginal) cost Curve Plots the incremental $/MWh as a function of MW. Found by differentiating the cost curve

39. 38 Mathematical Formulation of Costs Generator cost curves are usually not smooth. However the curves can usually be adequately approximated using piece-wise smooth, functions. Two representations predominate – quadratic or cubic functions – piecewise linear functions In 476 we'll assume a quadratic presentation

40. 39 Coal Usage Example A 500 MW (net) generator is 35% efficient. It is being supplied with Western grade coal, which costs $1.70 per MBtu and has 9000 Btu per pound. What is the coal usage in lbs/hr? What is the cost?

41. 40 Wasting Coal Example Assume a 100W lamp is left on by mistake for 8 hours, and that the electricity is supplied by the previous coal plant and that transmission/distribution losses are 20%. How much irreplaceable coal has he/she wasted?

42. 41 Incremental Cost Example

43. 42 Incremental Cost Example, cont'd

44. 43 Economic Dispatch: Formulation The goal of economic dispatch is to determine the generation dispatch that minimizes the instantaneous operating cost, subject to the constraint that total generation = total load + losses Initially we'll ignore generator limits and the losses

45. 44 Unconstrained Minimization This is a minimization problem with a single inequality constraint For an unconstrained minimization a necessary (but not sufficient) condition for a minimum is the gradient of the function must be zero, The gradient generalizes the first derivative for multi-variable problems:

46. 45 Minimization with Equality Constraint When the minimization is constrained with an equality constraint we can solve the problem using the method of Lagrange Multipliers Key idea is to modify a constrained minimization problem to be an unconstrained problem

47. 46 Lambda-Iteration Limit Example,cont’d