EEL 6266 Power System Operation and Control. Chapter 3 Economic Dispatch Using Dynamic Programming

Transcription

1 EEL 6266 Power System Operaton and Control Chapter 3 Economc Dspatch Usng Dynamc Programmng

2 Pecewse Lnear Cost Functons Common practce many utltes prefer to represent ther generator cost functons as sngle- or multple-segment, lnear cost functons Typcal examples: F(P) F(P) P mn P max P mn P max df(p)/dp df(p)/dp P mn P max P mn P max 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 2

3 Pecewse Lnear Cost Functons Pecewse lnear cost functons can not be used wth gradent based optmzaton methods lke the lambda-teraton such methods wll always land on P mn or P max A table-based method resolves ths problem technque for all unts runnng, begn to rase the output of the unt wth the lowest ncremental cost segment f ths unt hts the rght-hand end of a segment or hts P max, fnd the unt wth the next lowest ncremental cost segment and begn to rase ts output eventually, the total of all unts outputs equals the total load the last unt s adjusted to have a generaton, whch s partally loaded for one segment 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 3

4 Dynamc Programmng A wde varety of control and dynamc optmzaton problems use dynamc programmng (DP) to fnd solutons can greatly reduce the computaton effort n fndng optmal trajectores or control polces DP applcatons have been developed for economc dspatch hydro-thermal economc-schedulng unt commtment methods are based on the calculus of varatons but, applcatons are not dffcult to mplement or program prncples are ntroduced by presentng examples of onedmensonal problems 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 4

5 Dynamc Programmng Example consder the cost of transportng a unt shpment from locaton A to locaton N there are many short paths that connect many stops along the way, whch offers numerous parallel routes from gettng from A to N each path has an assocated cost e.g., dstance and level of dffculty results n fuel costs the total cost s the sum of the path costs of the selected route from the orgnatng locaton to the termnatng locaton the problem s to fnd the mnmum cost route 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 5

6 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 6 Dynamc Programmng A C B D F E G I H J M L K N 1-D Dynamc Programmng Example

7 Dynamc Programmng There are varous stages traversed startng at A, the mnmum cost path to N s ACEILN startng at C, the least cost path to N s CEILN startng at E, the least cost path to N s EILN startng at I, the least cost path to N s ILN startng at L, the least cost path to N s LN Obtanng the optmal route the choce of the route s made n sequence Theory of optmalty the optmal sequence s called the optmal polcy any sub-sequence s called a sub-polcy the optmal polcy contans only optmal sub-polces 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 7

8 Dynamc Programmng Example (contnued) dvde up the feld of paths nto stages (I, II, III, IV, V) at the termnus of each stage, there s a set of nodes (stops), {X } at stage III, the stops are [{X 3 } = {H, I, J, K}] a set of costs can be found for crossng a stage, {V III (X 2, X 3 )} a cost s dependent on the startng and termnatng nodes of a stage, V III (E, H) = 3, V III (F, I) = 11 the mnmum cost for traversng from stage I to stage and arrve at some partcular node (stop), X, s defned as f I (X ) the mnmum costs from stage I to stage II for nodes {B, C, D} are: f I (B) = V I (A, B) = 5, f I (C) = V I (A, C) = 2, f I (D) = V I (A, D) = 3 the mnmum cost from stage I to stage III for node {E} s: f II (E) = mn [ f I (X 1 ) + V II (X 1, E) ] = mn[ , 2 + 8, 3 + nf. ] {X 1 } X 1 = B = C = D f II (E) = 10 va ACE 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 8

9 A Dynamc Programmng I II III IV V 2 B 5 C 2 D E 10 F 6 G H 13 I 12 J 11 K 13 L 15 M 18 1-D dynamc programmng example: cost at each node N , 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 9

10 Dynamc Programmng at each stage, the mnmum cost should be recorded for all the termnus nodes (stops) use the mnmum cost of the termnus of the prevous stage dentfy the mnmum cost path for each of the termnatng nodes of the current stage (X 1 ) f I (X 1 ) path (X 2 ) f II (X 2 ) path (X 3 ) f III (X 3 )path (X 4 ) f IV (X 4 )path (X 5 ) f V (X 5 ) path B 5 A E 10 AC H 13 ACE L 15 ACEI N 19 ACEIL C 2 A F 6 AC I 12 ACE M 18 ADGK D 3 A G 9 AD J 11 ACF K 13 ADG 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 10

11 Economc dspatch Dynamc Programmng when the heat-rate curves exhbt nonconvex characterstcs t s not possble to use an equal H(P) ncremental cost method multple values of MW output exst for a gven value of ncremental cost dynamc programmng fnds optmal dspatch under such crcumstances the DP soluton s accomplshed as an allocaton problem dh(p)/dp the approach generates a set of outputs for an entre set of load values P mn P mn P max P max A nonconvex heat rate curve and ts correspondng ncremental heat rate curve 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 11

12 Dynamc Programmng Example consder a three-generator system servng a 310 MW demand the generator I/O characterstcs are not smooth nor convex Power Levels (MW) Costs ($/hour) P 1 = P 2 = P 3 F 1 F 2 F the demand does not ft the data exactly, nterpolate s needed between the avalable closest values, 300 and 325 MW 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 12

13 Dynamc Programmng Example the mnmum cost functon for schedulng unts 1 and 2: ( D) = F ( D P ) F ( ) f P2 let P 2 cover ts allowable range for demands of 100 to 350 MW D F 1 (D) P 2 = (MW) f 2 P 2 * (MW) ($/h) F 2 (P 2 )= ($/h) ($/h) (MW) , 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 13

14 Dynamc Programmng Example the mnmum cost functon for schedulng unts 1, 2 and 3: ( D) = f ( D P ) F ( ) f P3 let P 3 cover ts allowable range for the demand D f 2 (D) P 3 = (MW) f 3 P 3 * (MW) ($/h) F 3 (P 3 ) = ($/h) ($/h) (MW) , 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 14

15 Example the results show: Dynamc Programmng D Cost P 1 P 2 P generator #2 s the margnal unt t pcks up all of the addtonal demand ncrease between 300 MW and 325 MW P 1 = 50 MW, P 2 = 110 MW, P 3 = 150 MW, and P total = 310 MW the cost s easly determned usng nterpolaton F2 (110) = = ( ) F 1 = $ 810, F 2 = $1478, F 3 = $ 1998, and F total = $ , 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 15

16 Ramp Rate Constrants Generators are usually under automatc generaton control (AGC) a small change n load and a new dspatch causes the AGC to change the outputs of approprate unts generators must be able to move to the new generaton value wthn a short perod of tme large steam unts have a prescrbed maxmum rate lmt, P/ t (MW per mnute) the AGC must allocate the change n generaton to other unts, so that the load change can be accommodated quckly enough the new dspatch may be at the most economc values, but the control acton may not be acceptable f the ramp rate for any of the unts are volated 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 16

17 Ramp Rate Constrants To produce an acceptable dspatch to the control system, the ramp rate lmts are added to the economc dspatch formulaton requres a short-range load forecast to determne the most lkely load and load-rampng requrements of the unts system load s gven to be suppled at tme ncrements t = 1 t max wth loadng levels of P t load the N generators on-lne supply the load at each tme ncrement N t P = =1 P t load each unt must obey a rate lmt such that P t+ 1 = P P max t + P P P max 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 17

18 Ramp Rate Constrants The unts are scheduled to mnmze the cost to delver the power over the tme perod F constrants N = 1 and P total P t+ 1 t = P = = P max t P t max t= 1 t load N = 1 + P P F ( t P ) t = 1t P max max the optmzaton problem can be solved wth dynamc programmng 2002, 2004 Florda State Unversty EEL 6266 Power System Operaton and Control 18