03-Economic Dispatch 1. EE570 Energy Utilization & Conservation Professor Henry Louie

Transcription

1 03-Economic Dispatch 1 EE570 Energy Utilization & Conservation Professor Henry Louie 1

2 Topics Generator Curves Economic Dispatch (ED) Formulation ED (No Generator Limits, No Losses) ED (No Losses) ED Example 2

3 Introduction How should the real power output of a fleet of generators be determined? What factors are important in making the decision?

4 Economic Dispatch Generic problem: for an m generator system, what should P 1,, P m be set to? Several choices: minimize losses, minimize costs, etc. We focus on cost minimization known as Economic Dispatch (ED) This is an optimization problem Note: we will use $ as a unit of cost, but the results are generalizable to Kwacha 4

5 Optimization Problems Minimize or maximize an objective function Cost Profit CO 2 emissions Subject to constraints Generator output limits Power line thermal or stability limits CO 2 limits KCL, KVL, Ohm s Law, conservation of energy, etc 5

6 Economic Dispatch What factors influence the total cost of operation? fuel labor maintenance etc For simplicity, we will only analyze fuel costs We also assume that we have access to the fuel-cost curves for each generator Note: we are concerned with three-phase power, but the superscript will be suppressed 6

7 H i (P G ) (MBtu/MWh) C i (P G ) ($/hr) Fuel-Cost & Heat-Rate Curves Fuel cost curve Heat-rate curve determined by field testing heat energy supplied by burning fuel supplies efficiency information 34-39% MBtu/MWh P G (MW) P G (MW) 7

8 H i (P G ) (MBtu/MWh) Heat-Rate Curve Point x: 1200 MBtu to produce 100 MWh of electrical energy Point y: 1600 MBtu to produce 200 MWh of electrical energy At which point does the generator have the highest efficiency? At the minimum point of the heat-rate curve, the generator is most efficient 12 8 x 100 P G (MW) y 200 8

9 H i (P G ) (MBtu/MWh) Heat-Rate Curve Modern fossil-fuel power plants: minimum heat-rate is approx MBtu/MWh What efficiency does this correspond to? Approx. 1,055 joules/btu 39% 12 8 x 100 P G (MW) y 200 9

10 Input-Output Curve Heat input energy rate: F(P i Gi) PGiH(P i Gi) H i : MBtu/MWh P Gi : three-phase power, MW F i : heat input energy rate, MBtu/hr Plot of F i (P Gi ) is the input-output curve If the cost of fuel is K dollars/mbtu then: C i(p Gi) KF(P i Gi) fuel cost to supply per hour to supply P Gi MW of electricity 10

11 Example Compute the heat input energy rate (Mbtu/hr) and cost ($/hour) of a generator producing 100 MW of power given: Heat rate = 10 MBtu/MWh Fuel: $3/MBtu 11

12 Example Compute the heat input energy rate (Mbtu/hr) and cost ($/hour) of a generator producing 100 MW of power given: Heat rate = 10 MBtu/MWh Fuel: $3/MBtu F = 100 x 10 = 1,000 MBtu/hr C = 3 x 1,000 = $3,000/hr F(P i Gi) PGiH(P i Gi) C i(p Gi) KF(P i Gi) 12

13 Fuel Cost Curves Fuel cost curves can be approximated by 2 C P P P dollars/hr G G G (quadratic form) Nearly linear 5000 cost ($ / hr) power (MW) 13

14 Example Assume a 50-MW gas-fired generator has the following properties: 25 % of rating: MBtu/MWh 40 % of rating: MBtu/MWh 100 % of rating: MBtu/MWh Assume cost of gas is $5/MBtu 2 Find C(P G ) in the form of C P P P G G G 14

15 Example We are given the heat rate, H, at three points 25 % of rating: MBtu/MWh 40 % of rating: MBtu/MWh 100 % of rating: MBtu/MWh We need to relate C(P G ) and H(P G ) C(P ) KF(P ) KP H(P ) P P G 2 G G G G G G PG KH(P G) P H(P ) G KP K K G P G K: dollars/mbtu H: Mbtu/MWh F: MBtu/hr P: MW 15

16 Example Three unknowns, and three points. we can solve 12.5 MW: MBtu/MWh 20 MW: MBtu/MWh 50 MW: MBtu/MWh H P G P KP K K G G 3 equations, 3 unknowns 16

17 Example Solution: C P P P 2 G G G C P P 0.018P 2 G G G Find the fuel cost when fully loaded (50MW) $ / hr for 50 MW 5.85 cents/hr for 1 kw C

18 Example Find the fuel cost at 40% loaded, 25% loaded

19 Example Find the fuel cost at 40% loaded, 25% loaded C $ / hr for 20 MW 6.47 cents/kwh $891.1 / hr for 12.5 MW C cents/kwh

20 General Problem Formulation Now that we have discussed how costs are computed we can formulate the general economic dispatch problem We want to minimize the total cost while observing all the power flow equations, constraints on generators, line flow and voltage magnitude by selecting generator real power and voltage magnitudes 20

21 General Problem Formulation Minimize total cost C T ($/hr) m generators committed (on-line) n buses S Di given subject to: m T i Gi i1 min C C (P ) P P P P i 1,2,,m min max Gi Gi Gi ij min i i i max Gi max V V V i 1,2,,m,,n P P Optimization Problem all lines power flow equations satisfied 21

22 General Problem Formulation See text page 404 for further comments (to be provided) Economic dispatch problem is a nonlinear optimization problem Nonlinear programming is beyond the scope of this class We will make some problem alterations 22

23 Assumptions Losses ignored Weak dependence on voltage magnitude and reactive power demand Therefore, we expect the voltage magnitudes and reactive power demand to have little effect on the flow of real power We then make the approximation that all voltages are equal to 1.0 p.u. and formulate the problem entirely based on real power generation and flows 23

24 Classical Economic Dispatch Simplified ED Problem min C ( P) P T subject to m Gi D Di i1 i1 n P P P conservation of power P P P i 1,2,,m min max Gi Gi Gi if we ignore generator limits the problem becomes very simple 24

25 Classical Economic Dispatch Incremental costs (IC) Derivative of fuel cost curve dc P IC IC i i dp Gi Gi 2P i i i Gi What are the units? dollars/mwh Linear, monotonically increasing 25

26 C 2 (P G2 ) C 1 (P G1 ) Optimal Dispatch Rule Consider two generators operating at different incremental costs to serve a 150 MW load If we decrease P G1 by 10 MW, we decrease the cost by a significant amount (large slope) P G2 increases by 10, but the cost only mildly increases (small slope) slope = IC 1 P G1 (MW) slope = IC 2 P G2 (MW) 26

27 Optimal Dispatch Rule Using the problem as formulated on the previous slide, if we have no losses and no generator limits then: Operating the generator at the same incremental cost is the optimal solution note 1: this rule only gives an optimality condition, not a solution (but the problem has been greatly reduced) note 2: it is implicitly assumed that there are no constraints preventing the generators from operating at the same incremental cost 27

28 Example Consider two generators with cost curves: C (P ) P 0.01P 2 1 G1 G1 G1 C (P ) P 0.003P 2 2 G2 G2 G2 Let the total demand be 700 MW Find P G1, P G2 that serves the load and minimizes the cost 28

29 Example C P P 0.01P 2 1 G1 G1 G1 C P P 0.003P 2 2 G2 G2 G2 Find the incremental cost curves: 29

30 Example C P P 0.01P 2 1 G1 G1 G1 C P P 0.003P 2 2 G2 G2 G2 Find the incremental cost curves: dc1 IC P dp 1 G1 G1 dc2 IC P dp 2 G2 G2 add the power balance constraint P P 700 G1 G2 30

31 Example We have three equations, and three unknowns; solve dc1 IC P dp 1 G1 G1 dc2 IC P dp 2 G2 G2 P P 700 G1 G P P G1 P 700 P P P G1 G1 G2 G2 84.6MW 615.4MW G2 31

32 Example Incremental costs are dc1 IC $46.69 / MWh dp G1 dc2 IC $46.69 / MWh dp G2 Total costs are: T 2 2 C $34,877 Average cost is: $34,877/700 MW = $49.82/MWh 32

33 Incremental Costs Example Another way to view it IC P G (MW) IC 2 33

34 Optimal Dispatch Rule Proof Using the simplified formulation we need to find the values of P Gi that minimize CT C 1(P G1) C 2(P G2) C m(p Gm) P P P P G1 G2 Gm D P P P P P Gm D G1 G2 Gm1 We have m-1 independent variables CT C1 PG1 C2 PG2 Cm PD PG1 PG2 PGm 1 The problem is now unconstrained 34

35 Optimal Dispatch Rule Proof Remember how to solve an unconstrained minimization problem? We set the partial derivative wrt all the other variables equal to zero C C (P ) C (P ) C (P P P P ) T 1 G1 2 G2 m D G1 G2 Gm1 C T dci dc P m P P dp P Gm Gi Gi Gm Gi dci dcm 0 P dp Gi Gm for i = 1, 2,, m-1 35

36 Optimal Dispatch Rule Proof Note that we have assumed convexity Assuming that the cost curves are monotonically increasing, we expect only one solution to this problem Therefore, equal ICs is a necessary and sufficient condition to determine for optimality 36

37 Solution Via Lagrange Multipliers It is often easier to use Lagrange Multipliers to minimize or maximize a constrained function See Appendix 3 for more details We rewrite the cost function with an augmented cost function l m C C P P i1 T T Gi D l is the Lagrange Multiplier 37

38 Solution Via Lagrange Multipliers Optimal point: stationary points of C T wrt l and all the P Gi variables l m C C P P i1 T T Gi D C P C T Gi T l 0 i 1,2,,m 0 note: this point satisfies the constraint 38

39 Solution Via Lagrange Multipliers applied to our problem C C P P l m T T PG Gi D i1 dc dp m i1 T Gi l i 1,2,,m P P 0 Gi D l is the system incremental cost 39

40 Incremental Costs Solution Via Lagrange Multipliers IC 1 IC 2 IC 3 l P G1 P G2 P G3 40

41 Solution Via Lagrange Multipliers If the cost curves are quadratic then the incremental cost curves are linear and the problem can be readily solved If the cost curves are nonlinear, then an iterative process is used 1) pick an initial value of l 2) find the corresponding P G1 (l), P G2 (l), P G3 (l) 3) if the generation is less than load, then increase l and repeat step 2. if the generation is greater than the load, decrease l and repeat step 2. If generation and load balance then stop. 41

42 Generator Limits Included Formulation so far as ignored generator constraints, we now will add them P subject to m min C T P P P P Gi D Di i1 i1 n min max P P P i 1,2,,m Gi Gi Gi conservation of power 42

43 Incremental Costs Generator Limits Included Now we add limits to our generator output At the current operating point, there is no problem IC 1 IC 2 IC 3 l 1 P G1 P G2 P G3 43

44 Incremental Costs Generator Limits Included What if the system load increases? IC 1 IC 2 l 2 IC 3 l 1 P G1 P G2 P G3 44

45 Incremental Costs Generator Limits Included What if the system load increases further? Operate Gen 3 at P G3, gen 1 and gen 2 should have equal l IC 1 IC 2 l 3 l 2 IC 3 l 1 P G1 P G2 P G3 45

46 Optimal Dispatch Rule (No losses) Operate all generators that are not at their limits at equal l Procedure: pick a l such that all generators operate at the same incremental costs and within their constraints If the generation is not equal to the load at this l, then adjust l as in the unconstrained case Repeat this process until the load is met, or a generator reaches is limit If a generator reaches its limit, then fix its output to the limit and continue to adjust the other values 46

47 Optimal Dispatch Rule (No losses) Note 1: it usually helpful to draw graphs of the incremental curves Note 2: this rule applies to committed generators only (already on-line) Note 3: this rule can be interpreted as trying to operate the system as closely as possible to equal incremental costs 47

48 Example Assume there are two generators with cost functions: 2 C1 PG P G1 0.01P G1 2 C P P 0.003P 2 G2 G2 G2 Let the generator limits be: 50MW P 200MW G1 50MW P 600MW G2 if P D = 700, find the optimal dispatch 48

49 Example Start with an initial l guess l = 45.5 C P P 0.01P 2 1 G1 G1 G1 C P P 0.003P 2 2 G2 G2 G2 IC P P 25 1 G1 G1 IC P P G2 G2 Is this feasible? No (P G1 <50MW) Is the demand met? No, so increase l We don t fix P G1 at its minimum since we need to increase l 49

50 Example Increase l to IC P P G1 G1 IC P P G2 G2 Is the demand met? yes Are the generators within their limits? No (P G2 > 600) fix P G2 at 600, P G1 = =

51 Incremental Costs Example IC 1 IC P G (MW) 51

52 Line Losses Considered Line losses can reasonably be neglected if all the generators are located geographically close to one another However, this may not be the case. we then need to account for transmission loses Generators closer to the loads will tend to have lower losses than those far away 52

53 Line Losses Considered Let P L be the total line losses We then have P P (P ) P P,,P L L G L G2 Gm If we assume that each P Di is fixed, then: m m P P P L Gi Di i1 i1 assuming bus 1 is the slack bus 53

54 Line Losses Considered G min C T P G m P subject to P P P,,P P 0 Gi L G2 Gm Di i1 i1 min max P P P i 1,2,,m Gi Gi Gi n 54

55 Line Losses Considered Once again, we will use Lagrange multipliers (for now, ignore generator limits) m m n CT Ci PGi PGi PL P G2,,PGm PDi 0 l i1 i1 i1 The stationary points are: dc T dl dc dp m PGi PL PD 0 i1 T 1 G1 dc l 0 (slack bus) dp G1 dc T dci PL l 1 0 i 2,...,m dpgi dpgi PGi 55

56 Line Losses Considered rewrite dct dci PL l 1 0 i 2,...,m dpgi dpgi PGi as 1 P 1 P L Gi dc dp i Gi l i 2,...,m now define: 56 L L i i 2,...,m PL 1 P Gi

57 Line Losses Considered L i is called the penalty factor of generator i We can rewrite the necessary conditions for the optimal solution as dc dc dc L L L l 1 2 m 1 2 m dpg1 dpg2 dpgm Recall that the incremental costs are: We can now formulate the optimal dispatch rule with line losses considered and generator limits ignored dc dp i Gi 57

58 Optimal Dispatch Rule (Line Losses Considered, No Generator Limit) Operate all generators so that L i x IC i = l for every generator Note: we no longer need to operate each generator at the same incremental cost 58

59 Optimal Dispatch Rule (Line Losses Considered) Operate all generators not at their limits so that L i x IC i = l 59