# ECE 476 POWER SYSTEM ANALYSIS Lecture 17 Economic

- Slides: 45

ECE 476 POWER SYSTEM ANALYSIS Lecture 17 Economic Dispatch and Optimal Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering

Announcements l l Homework 8 is 11. 19, 11. 21, 11. 26, 11. 27, due on Thursday Homework 9 is 7. 1, 7. 17, 7. 20, 7. 24, 7. 27 – l l you do not need to turn it in, but should do it before the exam Second exam is Tuesday Nov 13 in class Design Project 2 from the book (page 345 to 348) was due on Nov 15, but I have given you an extension to Nov 29. The Nov 29 date is firm! 1

Power System Economic Operation l l 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 2

Thermal versus Hydro Generation l l The two main types of generating units are thermal and hydro For hydro the fuel (water) is free but there may be many constraints on operation – – – l l 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, looking at short-term optimization 3

Generator types l 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 4

Generator Cost Curves l Generator costs are typically represented by up to four different curves – – l 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 = 1 x 106 Btu 1 MBtu = 0. 29 MWh 5

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 6

Incremental (Marginal) cost Curve l l Plots the incremental $/MWh as a function of MW. Found by differentiating the cost curve 7

Economic Dispatch: Formulation l 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 8

Unconstrained Minimization l l l 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: 9

Minimization with Equality Constraint l l 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 10

Economic Dispatch Lagrangian 11

Minimization with Equality Constraint l l 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 12

Economic Dispatch Lagrangian 13

Economic Dispatch Example 14

Economic Dispatch Example, cont’d 15

Lambda-Iteration Solution Method l l The direct solution only works well if the incremental cost curves are linear and no generators are at their limits A more general method is known as the lambdaiteration – – the method requires that there be a unique mapping between a value of lambda and each generator’s MW output the method then starts with values of lambda below and above the optimal value, and then iteratively brackets the optimal value 16

Lambda-Iteration Algorithm 17

Lambda-Iteration: Graphical View In the graph shown below for each value of lambda there is a unique PGi for each generator. This relationship is the PGi( ) function. 18

Lambda-Iteration Example 19

Lambda-Iteration Example, cont’d 20

Lambda-Iteration Example, cont’d 21

Lambda-Iteration Example, cont’d 22

Lambda-Iteration Solution Method l l The direct solution only works well if the incremental cost curves are linear and no generators are at their limits A more general method is known as the lambdaiteration – – the method requires that there be a unique mapping between a value of lambda and each generator’s MW output the method then starts with values of lambda below and above the optimal value, and then iteratively brackets the optimal value 23

Generator MW Limits l l l Generators have limits on the minimum and maximum amount of power they can produce Often times the minimum limit is not zero. This represents a limit on the generator’s operation with the desired fuel type Because of varying system economics usually many generators in a system are operated at their maximum MW limits. 24

Lambda-Iteration with Gen Limits 25

Lambda-Iteration Gen Limit Example 26

Lambda-Iteration Limit Example, cont’d 27

Lambda-Iteration Limit Example, cont’d 28

Lambda-Iteration Limit Example, cont’d 29

Thirty Bus ED Example Case is economically dispatched without considering the incremental impact of the system losses 30

Back of Envelope Values l Often times incremental costs can be approximated by a constant value: – – $/MWhr = fuelcost * heatrate + variable O&M Typical heatrate for a coal plant is 10, modern combustion turbine is 10, combined cycle plant is 7 to 8, older combustion turbine 15. Fuel costs ($/MBtu) are about 1 to 1. 5 for coal, 12 for natural gas, 0. 5 for nuclear, probably 13 to 15 for fuel oil. Hydro costs tend to be quite low, but are fuel (water) constrained 31

Aside: Cost of Electricity Generation All values are in 2003 dollars. Nuclear costs do not include decommissioning costs, which are < 0. 1 cents/k. Wh Source: California Energy Commission: http: //www. energy. ca. gov/electricity/comparative_costs-v 1. html 32

Natural Gas Prices Over the Years (adjusted for inflation) Source: US FERC, http: //www. ferc. gov/market-oversight/mkt-gas/overview/2007/ngas-ovr-hh-pr. pdf 33

Professor Chapman’s Solar House l Our own Professor Chapman recently installed a 2870 W solar system for his new house – Details at http: //www. patrickchapman. com/solar. htm 34

Inclusion of Transmission Losses l l l The losses on the transmission system are a function of the generation dispatch. In general, using generators closer to the load results in lower losses This impact on losses should be included when doing the economic dispatch Losses can be included by slightly rewriting the Lagrangian: 35

Impact of Transmission Losses 36

Impact of Transmission Losses The penalty factor at the slack bus is always unity! 37

Impact of Transmission Losses 38

Calculation of Penalty Factors 39

Two Bus Penalty Factor Example 40

Thirty Bus ED Example Because of the penalty factors the generator incremental costs are no longer identical. 41

Area Supply Curve The area supply curve shows the cost to produce the next MW of electricity, assuming area is economically dispatched Supply curve for thirty bus system 42

Eastern US Supply Curve The y-axis units are $/MWh 43

Economic Dispatch - Summary l l Economic dispatch determines the best way to minimize the current generator operating costs The lambda-iteration method is a good approach for solving the economic dispatch problem – – l l generator limits are easily handled penalty factors are used to consider the impact of losses Economic dispatch is not concerned with determining which units to turn on/off (this is the unit commitment problem) Economic dispatch ignores the transmission system limitations 44

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