E O C NO N MIC C D I D SP S A P T A C T H C H A N A D N D UN U I N T T CO C MMITM T EN E T

Transcription

1 Chapter 4 ECOOMIC DISPATCH AD UIT COMMITMET

2 ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the generatng unts n dfferent power plants. Economc dspatch control determnes the power output of each power plant, and power output of each generatng unt wthn a power plant, whch wll mnmze the overall cost of fuel needed to serve the system load. We study frst the most economcal dstrbuton of the output of a power plant between the generatng unts n that plant. The method we develop also apples to economc schedulng of plant outputs for a gven system load wthout consderng the transmsson loss. ext, we express the transmsson loss as a functon of output of the varous plants. Then, we determne how the output of each of the plants of a system s scheduled to acheve the total cost of generaton mnmum, smultaneously meetng the load on the system.

3 IPUT OUTPUT CURVE OF GEERATIG UIT Power plants consstng of several generatng unts are constructed nvestng huge amount of money. Fuel cost, staff salary, nterest and deprecaton charges and mantenance cost are some of the components of operatng cost. Fuel cost s the major porton of operatng cost and t can be controlled. Therefore, we shall consder the fuel cost alone for further consderaton.

4 To get dfferent output power, we need to vary the fuel nput. Fuel nput can be measured n Tonnes / hour or Mllons of Btu / hour. Knowng the cost of the fuel, n terms of Rs. / Tonne or Rs. / Mllons of Btu, nput to the generatng unt can be expressed as Rs / hour. Let C Rs / h be the nput cost to generate a power of P MW n unt. Fg. shows a typcal nput output curve of a generatng unt. For each generatng unt there shall be a mnmum and a maxmum power generated as P mn and P max. C n Rs / h P mn P max P n MW Fg. Input-Output curve of a generatng unt

5 If the nput-output curve of unt s quadratc, we can wrte C α P + β P + γ Rs / h () A power plant may have several generator unts. If the nput-output characterstc of dfferent generator unts are dentcal, then the generatng unts can be equally loaded. But generatng unts wll generally have dfferent nput-output characterstc. Ths means that, for partcular nput cost, the generator power P wll be dfferent for dfferent generatng unts n a plant. 3 ICREMETAL COST CURVE As we shall see, the crteron for dstrbuton of the load between any two unts s based on whether ncreasng the generaton of one unt, and decreasng the generaton of the other unt by the same amount results n an ncrease or decrease n total cost. Ths can be obtaned f we can calculate the change n nput cost C for a small change n power P. Snce dc dp C we can wrte C P dc P dp

6 C dc P dp Thus whle decdng the optmal schedulng, we are concerned wth dc, ICREMETAL COST (IC) whch s determned by the slopes of the nputoutput curves. Thus the ncremental cost curve s the plot of versus dc dp P. The dmenson of dc dp s Rs / MWh. The unt that has the nput output relaton as C α P + β P + γ Rs / h () has ncremental cost (IC) as IC α P + dp dc β () dp Here α, β andγ are constants.

7 A typcal plot of IC versus power output s shown n Fg.. IC n Rs / MWh Lnear approxmaton Actual ncremental cost Fg. Incremental cost curve P n MW

8 Ths fgure shows that ncremental cost s qute lnear wth respect to power output over an apprecable range. In analytcal work, the curve s usually approxmated by one or two straght lnes. The dashed lne n the fgure s a good representaton of the curve. We now have the background to understand the prncple of economc dspatch whch gudes dstrbuton of load among the generatng unts wthn a plant.

9 4 ECOOMICAL DIVISIO OF PLAT LOAD BETWEE GEERATIG UITS I A PLAT Varous generatng unts n a plant generally have dfferent nput-output characterstcs. Suppose that the total load n a plant s suppled by two unts and that the dvson of load between these unts s such that the ncremental cost of one unt s hgher than that of the other unt. ow suppose some of the load s transferred from the unt wth hgher ncremental cost to the unt wth lower ncremental cost. Reducng the load on the unt wth hgher ncremental cost wll result n greater reducton of cost than the ncrease n cost for addng the same amount of load to the unt wth lower ncremental cost. The transfer of load from one to other can be contnued wth a reducton of total cost untl the ncremental costs of the two unts are equal. Ths s llustrated through the characterstcs shown n Fg. 3

10 IC IC IC P P P Fg. 3 Two unts case Intally, IC > IC. Decrease the output power n unt by P and ncrease output power n unt by P. ow IC P > IC P. Thus there wll be more decrease n cost and less ncrease n cost brngng the total cost less. Ths change can be contnued untl IC IC at whch the total cost wll be mnmum.

11 The same reasonng can be extended to a plant wth more than two generatng unts also. In ths case, f any two unts have dfferent ncremental costs, then decrease the output power n unt havng hgher IC and ncrease the output power n unt havng lower IC. When ths process s contnued, a stage wll reach wheren ncremental costs of all the unts wll be equal. ow the total cost of generaton wll be mnmum. Thus the economcal dvson of load between unts wthn a plant s that all unts must operate at the same ncremental cost. ow we shall get the same result mathematcally.

12 Consder a plant havng number of generatng unts. Input-output curve of the unts are denoted as C (P ),C(P )...,C (P ). Our problem s, for a gven load demand P D, fnd the set of P s whch mnmzes the cost functon C T C (P ) + C (P ) +... C (P ). (3) + subject to the constrants P (P + P P ) 0 (4) D and P mn P P,,.., (5) max Omttng the nequalty constrants for the tme beng, the problem to be solved becomes Mnmze C C (P ) (6) T D subject to P P 0 Ths optmzng problem wth equalty constrant can be solved by the method of Lagrangan multplers. In ths method, the Lagrangan functon s formed by augmentng the equalty constrants to the objectve functon usng proper Lagrangan multplers. For ths case, Lagrangan functon s (7)

13 D L ( P,P,...,P,λ ) C (P ) + λ (P P ) (8) where λ s the Lagrangan multpler. ow ths Lagrangan functon has to be mnmzed wth no constrants on t. The necessary condtons for a mnmum are L P L 0,,.., and 0. λ For a plant wth 3 unts L ( P 3, P, P, λ ) C (P ) + C (P ) + C 3 (P3 ) + λ(pd P P P3 ) ecessary condtons for a mnmum are L P C P + λ ( - ) 0 L P C P + λ ( - ) 0 L P 3 C P λ ( - ) 0 L λ PD P P P3 0

14 Generalzng the above, the necessary condtons are C P λ 0,,., (9) D and P P 0 (0) Here C P s the change n producton cost n unt for a small change n generaton n unt. Snce change n generaton n unt wll affect the producton cost of ths unt ALOE, we can wrte C P dc dp Usng eqn.() n eqn. (9) we have () dc dp λ I,,., ()

15 Thus the soluton for the problem Mnmze C C (P ) subject to P P 0 T D s obtaned when the followng equatons are satsfed. dc dp λ,,., (3) D and P P 0 (4) The above two condtons gve + number of equatons whch are to be solved for the + number of varables λ, P. Equaton (3), P,..., P smply says that at the mnmum cost operatng pont, the ncremental cost for all the generatng unts must be equal. Ths condton s commonly known as EQUAL ICREMETAL COST RULE. Equaton (4) s known as POWER BALACE EQUATIO.

16 It s to be remembered that we have not yet consdered the nequalty constrants gven n eqn. (5). Fortunately, f the soluton obtaned wthout consderng the nequalty constrants satsfes the nequalty constrants also, then the obtaned soluton wll be optmum. If for one or more generator unts, the nequalty constrants are not satsfed, the optmum strategy s obtaned by keepng these generator unts n ther nearest lmts and makng the other generator unts to supply the remanng power as per equal ncremental cost rule.

17 EXAMPLE The cost characterstc of two unts n a plant are: C 0.4 P + 60 P + K Rs./h C 0.45 P + 0 P + K Rs. / h where P and P are power output n MW. Fnd the optmum load allocaton between the two unts, when the total load s 6.5 MW. What wll be the daly loss f the unts are loaded equally? SOLUTIO Incremental costs are: IC 0.8 P + 60 Rs. / MW h IC 0.9 P + 0 Rs. / MW h Usng the equal ncremental cost rule 0.8 P + 60 λ and 0.9 P + 0 λ λ 60 λ 0 Snce P + P 6.5 we get e. λ [ + ] e..36 λ

18 .36 λ Ths gves λ 0 Rs / MWh Knowng 0.8 P + 60 λ and 0.9 P + 0 λ Optmum load allocaton s P 6.5 MW and P 00 MW When the unts are equally loaded, P P 8.5 MW and we have devated from the optmum value of P 6.5 MW and P 00 MW. Knowng C 0.4 P + 60 P + K Rs./h C 0.45 P + 0 P + K Rs. / h Daly loss can also be computed by calculatng the total cost C T, whch s C + C, for the two schedules. Thus, Daly loss 4 x [ C ( 8.5,8.5) C (6.5,00) ] T 4 x [ K + K ( K + K )] Rs T

19 EXAMPLE A power plant has three unts wth the followng cost characterstcs: C C C 3 0.5P.0P 0.7P 3 + 5P + 70P + 60P Rs / h Rs / h Rs/h where s P are the generatng powers n MW. The maxmum and mnmum loads allowable on each unt are 50 and 39 MW. Fnd the economc schedulng for a total load of ) 30 MW ) 00 MW SOLUTIO Knowng the cost characterstcs, ncremental cost characterstcs are obtaned as IC IC IC 3.0P.0P.4P Rs / MWh + 70 Rs / MWh + 60 Rs / MWh Usng the equal ncremental cost rule.0 P + 5 λ;.0 P + 70 λ;.4 P λ

20 .0 P + 5 λ;.0 P + 70 λ;.4 P λ Case ) Total load 30 MW Snce P + P + P 3 30 we have λ 5 λ 70 λ e. λ [ + + ] e..43 λ Ths gves λ RM / MWh Thus P ( ) / MW P ( ) / MW P 3 ( ) / MW All P ' s le wthn maxmum and mnmum lmts. Therefore, economc schedulng s P MW; P MW; P MW

21 Case ) Total load 00 MW Snce P + P + P 3 00 we have λ [ + + ] e..49 λ Ths gves λ 300 Rs / MWh Thus P ( ) /.0 85 MW P ( ) /.0 5 MW P 3 ( ) /.4 00 MW It s noted that P < P mn. Therefore P s set at the mn. value of 39 MW. Then P 3 + P MW. Ths power has to be scheduled between unts and 3. Therefore 5 60 λ [ + ] e..749 λ Ths gves λ 86 Rs / MWh Thus P ( 86-5 ) /.0 7 MW P 3 ( ) /.4 90 MW P and P 3 are wthn the lmts. Therefore economc schedulng s P 7 MW; P 39 MW; P3 90 MW

22 EXAMPLE 3 Incremental cost of two unts n a plant are: IC 0.8 P + 60 Rs / MWh IC 0.9 P + 0 Rs / MWh where P and P are power output n MW. Assume that both the unts are operatng at all tmes. Total load vares from 50 to 50 MW and the mnmum and maxmum loads on each unt are 0 and 5 MW respectvely. Fnd the ncremental cost and optmal allocaton of loads between the unts for varous total loads and furnsh the results n a graphcal form. SOLUTIO For lower loads, IC of unt s hgher and hence t s loaded to mnmum value.e. P 0 MW. Total mnmum load beng 50 MW, when P 0 MW, P must be equal to 30 MW. Thus ntally P 0 MW, IC 76 Rs / MWh, P 30 MW and IC 47 Rs / MWh. As the load ncreased from 50 MW, load on unt wll be ncreased untl ts IC.e. IC reaches a value of 76 Rs / MWh. When IC 76 Rs / MWh load on unt s P ( 76 0 ) / MW. Untl that pont s reached, P shall reman at 0 MW and the plant IC,.e. λ s determned by unt.

23 When the plant IC, λ s ncreased beyond 76 Rs / MWh, unt loads are calculated as P ( λ - 60 ) / 0.8 MW P ( λ - 0 ) / 0.9 MW Then the load allocaton wll be as shown below. Plant IC λ Rs/MWh Load on unt P MW Load on unt P MW Total load P + P MW

24 Load on unt reaches the maxmum value of 5 MW, when λ (0.9 x 5) Rs / Mwh. When the plant IC, λ ncreases further, P shall reman at 5 MW and the load on unt alone ncreases and ts value s computed as P ( λ - 0 ) / 0.9 MW. Such load allocatons are shown below. Plant IC λ Load on unt Load on unt Total load Rs / MWh P MW P MW P + P MW The results are shown n graphcal form n Fg. 4

25 P / P MW P P Fg. 4 Load allocaton for varous plant load Total load P D

26 Alternatve way of explanaton for ths problem s shown n Fg IC Rs/MWh , 76 30, 47 6., , 36 P 6.5 P 00 5, 60 5, P / P MW Fg. 5 Load allocaton between two unts

27 5 TRASMISSIO LOSS Generally, n a power system, several plants are stuated at dfferent places. They are nterconnected by long transmsson lnes. The entre system load along wth transmsson loss shall be met by the power plants n the system. Transmsson loss depends on ) lne parameters ) bus voltages and ) power flow. Determnaton of transmsson loss requres complex computatons. However, wth reasonable approxmatons, for a power system wth number of power plants, transmsson loss can be represented as P L [ P P P ] B B B B B B B B B P P P (5) where respectvely. P are the powers suppled by the plants,,.,, P,..., P

28 The coeffcent matrx [ B ] s called as LOSS COEFFICIET MATRIX or B mn COEFFICIET MATRIX or B COEFFICIET MATRIX. Ths matrx s symmetrc. From eq.(5) P L [ P P ] P n n n B B B n n n Pn P n P n P B n Pn n + P B n Pn n +. + P B n Pn n P Bn Pn n + P B n Pn n + + P B n Pn n Thus P L can be wrtten as Pm Bmn Pn m n P L m n P m B mn Pn (6)

29 P L m n P m B mn Pn (6) When the powers are n MW, the B mn coeffcents are of dmenson / MW. If powers are n per-unt, then B mn coeffcents are also n per-unt. Loss coeffcent matrx of a power system shall be determned before hand and made avalable for economc dspatch. For a two plant system, the expresson for the transmsson loss s P L B B P P [ P P ] B B B P + B P [ P P ] B P + B P P B P P B P P B P B

30 Snce B mn coeffcent matrx s symmetrc, for two plant system P P B + P P B + P B (7) L In later calculatons we need the Incremental Transmsson Loss ( ITL ), P L P. For two plant system P L P B P + B P (8) P L P B P + B P (9) Ths can be generalzed as P P L m n B mn P n m,,., (0)

31 6 ECOOMIC DIVISIO OF SYSTEM LOAD BETWEE VARIOUS PLATS I THE POWER SYSTEM It s to be noted that dfferent plants n a power system wll have dfferent cost characterstcs. Consder a power system havng number of plants. Input-output characterstcs of the plants are denoted as C (P ),C (P ),...,C (P ). Our problem s for a gven system load demand P D, fnd the set of plant generaton P whch mnmzes the cost functon,p,..., P C T C (P ) + C (P ) +... C (P ) () + subject to the constrants P + P (P + P P ) 0 () D L and P mn P P I,,., (3) max Inequalty constrants are omtted for the tme beng.

32 The problem to be solved becomes Mnmze C C ( P ) (4) T subject to PD + PL P 0 (5) For ths case the Lagrangan functon s L( P,P,..., P, λ) ) C (P ) + λ ( P D + P L P ) (6) Ths Lagrangan functon has to be mnmzed wth no constrant on t. The necessary condtons for a mnmum are L P 0,,.., L and 0. λ

33 L( P,P,..., P, λ) C (P ) + λ ( P D + P P ) (6) L For a system wth plants L ( P, P, λ ) C (P ) + C (P ) + λ(pd + PL P P ) L C P P PL + λ ( P - ) 0 L P C P + λ ( P L P - ) 0 L λ PD + PL P P 0

34 Generalzng ths, for system wth plants, the necessary condtons for a mnmum are C P + λ [ P L P ] 0,,.., (7) and PD + PL P 0 (8) As dscussed n earler case, we can wrte C C P dc dp (9) Usng eqn. (9) n eqn. (7) we have dc dp PL + λ λ,,.., (30) P and PD + PL P 0 (3)

35 dc dp PL + λ λ,,.., (30) P and PD + PL P 0 (3) These equatons can be solved for the plant generatons P.,P,..., P As shown n the next secton the value of λ n eqn. (30) s the ICREMETAL COST OF RECEIVED POWER. The eqns. descrbed n eqn. (30) are commonly known as COORDIATIO EQUATIOS as they lnk the ncremental cost of plant dc dp, ncremental cost of receved power λ and the ncremental transmsson loss ( ITL ) P L P. Equaton (3) s the POWER BALACE EQUATIO.

36 Coordnaton equatons dc dp PL + λ λ,,.., (30) P can also be wrtten as IC + λ ITL λ,,.., (3) The number of coordnate equatons together wth the power balance equaton are to be solved for the plant loads economc schedule. P to obtan the,p,..., P

37 7 ICREMETAL COST OF RECEIVED POWER The value of λ n the coordnaton equatons s the ncremental cost of receved power. Ths can be proved as follows: The coordnaton equatons can be wrtten as C P λ [ P L P ] (33).e. C P P L P λ.e. C P P L λ (34) Snce P D P P D L P PL P P.e. P D P P (35) L Usng eqn,(35) n eqn.(34) gves C P D λ (36) Thus λ s the ncremental cost of receved power.

38 8 PEALTY FACTORS To have a better feel about the coordnaton equatons, let us rewrte the dc PL same as λ [ ],,.., (37) dp P Thus [ PL P ] dc dp λ,,.., (38) The above equaton s often wrtten as dc L λ,,.., (39) dp where L L, whch s called the PEALTY FACTOR of plant, s gven by P L P,,.., (40) The results of eqn. (39) means that mnmum fuel cost s obtaned when the ncremental cost of each plant multpled by ts penalty factor s the same for the plants n the power system.

39 9 OPTIMUM SCHEDULIG OF SYSTEM LOAD BETWEE PLATS - SOLUTIO PROCEDURE To determne the optmum schedulng of system load between plants, the data requred are ) system load, ) ncremental cost characterstcs of the plants and ) loss coeffcent matrx. The teratve soluton procedure s: Step For the frst teraton, choose sutable ntal value of λ. Whle fndng ths, one way s to assume that the transmsson losses are zero and the plants are loaded equally. Step Substtute the value of λ nto the coordnaton equatons dc dp PL + λ λ,,.., P.e. n ( α P + β ) + λ Bmn Pn λ,,.., The above set of lnear smultaneous equatons are to be solved for the values P ' recommended. s by some effcent means. Gauss Sedel method s

40 Step 3 Compute the transmsson loss P L from P L [ P ] [ B ] [ P t ] where [ P ] [ Step 4 P ] and [ B ] s the loss coeffcent matrx. P...P Compare P wth P D + P L to check the power balance. If the power balance s satsfed wthn a specfed tolerance, then the present soluton s the optmal soluton; otherwse update the value of λ. Frst tme updatng can be done judcously. Value of λ s ncreased or decreased by about 5% dependng on P < P D + P L or P > P D + P L In the subsequent teratons, usng lnear nterpolaton, value of λ can be updated as

41 k λ + λ k k λ P k P k P D + P L k λ λ k k k + k k k λ λ + [PD + PL P ] (4) k k P P Here k-, k and k+ are the prevous teraton count, present teratve count and the next teraton count respectvely. Step 5 Return to Step and contnue the calculatons of Steps, 3 and 4 untl the power balance equaton s satsfed wth desred accuracy. The above procedure s now llustrated through an example.

42 EXAMPLE 4 Consder a power system wth two plants havng ncremental cost as IC.0 P + 00 Rs / MWh IC.0 P + 50 Rs / MWh Loss coeffcent matrx s gven by B Fnd the optmum schedulng for a system load of 00 MW. SOLUTIO Assume that there s no transmsson loss and the plants are loaded equally. Then P 50MW. Intal value of λ (.0 x 50 ) Rs / MWh. Coordnaton equatons:.0 P.0 P ( 0.00P ( 0.00P P P ) 50 ) 50.e..5 P 0.5 P 50.e. P P 0.5P +. P 00 P P These equatons are solved by Gauss Sedel method.

43 P P P P Iteraton count P P P L [ ] MW [ ] P + P MW P + P MW D L Snce P + P < P D + PL, λ value should be ncreased. It s ncreased by 4 %. ew value of λ 50 x Rs / MWh.

44 Coordnaton equatons:.0.0 P P ( 0.00P ( 0.00P P P ) ) e..5 P 0.6 P 60.e. P P 0.6 P +.48 P 0 P P These equatons are solved by Gauss Sedel method. Iteraton count P P P L [ ] [ ] MW P P MW ; P + P MW D L

45 P + + P PD PL Knowng two values of λ and the correspondng total generaton powers, new value of λ s computed as λ 60 + ( ) 63 Rs / MWh Wth ths new value of λ, coordnaton equatons are formed and the procedure has to be repeated. The followng table shows the results obtaned. λ P P P L P + P P D + P L Optmum schedule s P P 5.40 MW MW For ths transmsson loss s MW

46 0 BASE POIT AD PARTICIPATIO FACTORS The system load wll keep changng n a cyclc manner. It wll be hgher durng day tme and early evenng when ndustral loads are hgh. However durng nght and early mornng the system load wll be much less. The optmal generatng schedulng need to be solved for dfferent load condtons because load demand P D keeps changng. When load changes are small, t s possble to move from one optmal schedule to another usng PARTICIPATIG FACTORS. We start wth a known optmal generaton schedule, P 0, P 0,, P 0, for a partcular load P D. Ths schedule s taken as BASE POIT and the correspondng ncremental cost s λ 0. Let there be a small ncrease n load of P D. To meet wth ths ncreased load, generatons are to be ncreased as P, P,., P. Correspondngly ncremental cost ncreases by λ.

47 Knowng that for th unt, C α P + β P + γ, ncremental cost s IC α P + β λ (4) Small change n ncremental cost and correspondng change n generaton are related as λ α (43) P Thus P λ α for,,, (44) Change n load should be equal to the total change n generatons. Therefore P D P λ αα (45) From the above two equatons P P D α α k for,,, (46) The rato P s known as the PARTICIPATIO FACTOR of generator I, P D represented as k. Once all the k s, are calculated from eq.(46), the change n generatons are gven by P k P D for,,, (47)

48 EXAMPLE 5 Incremental cost of three unts n a plant are: IC 0.8 P + 60 Rs / MWh; IC 0.9 P + 0 Rs / MWh; and IC 3.5 P Rs / MWh where P, P and P 3 are power output n MW. Fnd the optmum load allocaton when the total load s 4.5 MW. Usng Partcpatng Factors, determne the optmum schedulng when the load ncreases to 60 MW. Soluton Usng the equal ncremental cost rule 0.8 P + 60 λ ; 0.9 P + 0 λ ;.5 P λ λ 60 λ 0 λ 0 Snce P + P + P we get e λ [ + + ] e. 3.6 λ Ths gves λ 0 Rs / MWh Optmum load allocaton s P 6.5 MW ; P 00 MW ; P 3 80 MW

49 Partcpaton Factors are: k k k Change n load P D MW Change n generatons are: P x MW P x MW P x MW Thus optmum schedule s: P MW; P MW; P MW

50 Example 6 A power plant has two unts wth the followng cost characterstcs: C 0.6 P + 00 P Rs / hour C. P + 50 P Rs / hour where P and P are the generatng powers n MW. The daly load cycle s as follows: 6:00 A.M. to 6:00 P.M. 50 MW 6:00 P.M. to 6:00 A.M. 50 MW The cost of takng ether unt off the lne and returnng to servce after hours s Rs Consderng 4 hour perod from 6:00 A.M. one mornng to 6:00 A.M. the next mornng

51 a. Would t be economcal to keep both unts n servce for ths 4 hour perod or remove one unt from servce for hour perod from 6:00 P.M. one evenng to 6:00 A.M. the next mornng? b. Compute the economc schedule for the peak load and off peak load condtons. c. Calculate the optmum operatng cost per day. d. If operatng one unt durng off peak load s decded, up to what cost of takng one unt off and returnng to servce after hours, ths decson s acceptable?

52 Soluton To meet the peak load of 50 MW, both the unts are to be operated. However, durng 6:00 pm to 6:00 am, load s 50 MW and there s a choce ) both the unts are operatng ) one unt (ether or to be decded) s operatng ) When both the unts are operatng IC. P + 00 Rs / MWh IC.4 P + 50 Rs / MWh Usng equal IC rule λ λ ;.5 λ and λ Therefore P MW; P MW Then C T Rs / h For hour perod, cost of operaton Rs

53 ) If unt s operatng, C Ι P Rs / h If unt s operatng, C Ι P Rs / h Between unts and, t s economcal to operate unt. If only one unt s operatng durng off-peak perod, cost towards takng out and connectng t back also must be taken. Therefore, for off-peak perod (wth unt alone operatng) cost of operaton (3000 x ) Rs Between the two choces () and (), choce () s cheaper. Therefore, durng 6:00 pm to 6:00 am, t s better to operate unt alone.

54 b. Durng the peak perod, P D 50 MW. Wth equal IC rule λ λ ;.5 λ and λ Therefore P 86. MW; P MW Thus, economc schedule s Durng 6:00 am to 6:00 pm P 86. MW; P MW Durng 6:00 pm to 6:00 am P 0; P 50 MW c. Cost of operaton [ C Ι P C Ι P ] for peak perod x Rs Cost of operaton for off-peak perod Rs 6000 Therefore, optmal operatng cost per day Rs

55 d. If both the unts are operatng durng off-peak perod, cost of operaton Rs 7833 If unt alone s operatng durng Rs (3000 x ) + x off-peak perod, cost of operaton Rs x For Crtcal value of x: x 7833 x Rs 5833 Therefore, untl the cost of takng one unt off and returnng t to servce after hours, s less than Rs 5833, operatng unt alone durng the off-peak perod s acceptable.

56 UIT COMMITMET Economc dspatch gves the optmum schedule correspondng to one partcular load on the system. The total load n the power system vares throughout the day and reaches dfferent peak value from one day to another. Dfferent combnaton of generators, are to be connected n the system to meet the varyng load. When the load ncreases, the utlty has to decde n advance the sequence n whch the generator unts are to be brought n. Smlarly, when the load decrease, the operatng engneer need to know n advance the sequence n whch the generatng unts are to be shut down. The problem of fndng the order n whch the unts are to be brought n and the order n whch the unts are to be shut down over a perod of tme, say one day, so the total operatng cost nvolved on that day s mnmum. Ths s known as Unt Commtment (UC) problem. Thus UC problem s economc dspatch over a day. The perod consdered may a week, month or a year.

57 But why s the problem n the operaton of electrc power system? Why not just smply commt enough unts to cover the maxmum system load and leave them runnng? ote that to commt means a generatng unt s to be turned on ; that s, brng the unt up to speed, synchronze t to the system and make t to delver power to the network. Commt enough unts and leave them on lne s one soluton. However, t s qute expensve to run too many generatng unts. A great deal of money can be saved by turnng unts off (decommtng them) when they are not needed. Example 7 The followng are data pertanng to three unts n a plant. Unt : Mn. 50 MW; Max. 600 MW C P P Rs / h Unt : Mn. 00 MW; Max. 400 MW C P P Rs / h Unt 3: Mn. 50 MW; Max. 00 MW C P P 3 Rs / h What unt or combnaton of unts should be used to supply a load of 550 MW most economcally?

58 Soluton To solve ths problem, smply try all combnaton of three unts. Some combnatons wll be nfeasble f the sum of all maxmum MW for the unts commtted s less than the load or f the sum of all mnmum MW for the unts commtted s greater than the load. For each feasble combnaton, unts wll be dspatched usng equal ncremental cost rule studed earler. The results are presented n the Table below.

59 Unt Mn Max Unt Unt Unt 3 Mn. Gen Max. Gen P P P 3 Total cost Off Off Off 0 0 Infeasble On Off Off Off On Off Infeasble Off Off On Infeasble On On Off Off On On On Off On On On On ote that the least expensve way of meetng the load s not wth all the three unts runnng, or any combnaton nvolvng two unts. Rather t s economcal to run unt one alone.

60 Example 8 Daly load curve to be met by a plant havng three unts s shown below. 00 MW 500 MW noon 4 pm 8 pm am 6 am noon Data pertanng to the three unts are the same n prevous example. Startng from the load of 00 MW, takng steps of 50 MW fnd the shutdown rule.

61 Soluton For each load startng from 00 MW to 500 MW n steps of 50 MW, we smply use a brute-force technque wheren all combnatons of unts wll be tred as n prevous example. The results obtaned are shown below. Load Optmum combnaton Unt Unt Unt 3 00 On On On 50 On On On 00 On On On 050 On On On 000 On On Off 950 On On Off 900 On On Off 850 On On Off 800 On On Off 750 On On Off 700 On On Off 650 On On Off 600 On Off Off 550 On Off Off 500 On Off Off

62 The shut-down rule s qute smple. When load s above 000 MW, run all three unts; between 000 MW and 600 MW, run unts and ; below 600 MW, run only unt. It s to be noted that, n ths brute-force technque, neglectng the number of nfeasble solutons, we need to solve the economc dspatch problem ( ) M where s the number of unts and M s the load steps. Ths number wll be too large for practcal case. UC problem become more complcated when we need to consder power system havng several plants each plant havng several generatng unts. So far, we have only obeyed one smple constrant: Enough unts wll be connected to supply the load. There are several other constrants to be satsfed n practcal UC problem.

63 COSTRAITS O UC PROBLEM Some of the constrants that are to be met wth whle solvng UC problem are lsted below.. Spnnng reserve: There may be sudden ncrease n load, more than what was predcted. Further there may be a stuaton that the generatng unt may have to be shut down because of fault n generator or any of ts auxlares. Some system capacty has to be kept as spnnng reserve ) to meet an unexpected ncrease n demand and ) to ensure power supply n the event of any generatng unt sufferng a forced outage.. Mnmum up tme: When a thermal unt s brought n, t cannot be turned off mmedately. Once t s commtted, t has to be n the system for a specfed mnmum up tme. 3. Mnmum down tme: When a thermal unt s decommtted, t cannot be turned on mmedately. It has to reman decommtted for a specfed mnmum down tme.

64 4. Crew constrant: A plant always has two or more generatng unts. It may not be possble to turn on more than one generatng unt at the same tme due to non-avalablty of operatng personnel. 5. Transton cost: Whenever the status of one unt s changed some transton cost s nvolved and ths has to be taken nto account. 6. Hydro constrants: Most of the systems have hydroelectrc unts also. The operaton of hydro unts, depend on the avalablty of water. Moreover, hydro-projects are multpurpose projects. Irrgaton requrements also determne the operaton of hydro plants.

65 7. uclear constrant: If a nuclear plant s part of the system, another constrant s added. A nuclear plant has to be operated as a base load plant only. 8. Must run unt: Sometme t s a must to run one or two unts from the consderaton of voltage support and system stablty. 9. Fuel supply constrant: Some plants cannot be operated due to defcent fuel supply. 0. Transmsson lne lmtaton: Reserve must be spread around the power system to avod transmsson system lmtaton, often called bottlng of reserves.

66 PRIORITY- LIST METHOD In ths method the full load average producton cost of each unt s calculated frst. Usng ths, prorty lst s prepared. Full load average producton of a unt Producton cost correspondng Full load to full load Example 9 The followng are data pertanng to three unts n a plant. Unt : Max. 600 MW C P P Rs / h Unt : Max. 400 MW C P P Rs / h Unt 3: Max. 00 MW Obtan the prorty lst C P P 3 Rs / h

67 Soluton Full load average producton of a unt x x Full load average producton of a unt x x Full load average producton of a unt x x A strct prorty order for these unts, based on the average producton cost, would order them as follows: Unt Rs. / h Max. MW

68 The commtment scheme would (gnorng mn. up / down tme, start up costs etc.) smply use the followng combnatons. Combnaton Load P D MW P D < 00 MW MW P D < 000 MW P D < 400 MW ote that such a scheme would not gve the same shut down sequence descrbed n Example 7 wheren unt was shut down at 600 MW leavng unt. Wth the prorty lst scheme both unts would be held on untl load reached 400 MW, then unt would be dropped.

69 Most prorty schemes are bult around a smple shut down algorthm that mght operate as follows: At each hour when the load s droppng, determne whether droppng the next unt on the prorty lst wll leave suffcent generaton to supply the load plus spnnng reserve requrements. If not, contnue operatng as s; f yes, go to next step. Determne the number of hours, H, before the unt wll be needed agan assumng the load s ncreasng some hours later. If H s less than the mnmum shut down tme for that unt, keep the commtment as t s and go to last step; f not, go to next step. Calculate the two costs. The frst s the sum of the hourly producton costs for the next H hours wth the unt up. Then recalculate the same sum for the unt down and add the start up cost. If there s suffcent savng from shuttng down the unt, t should be shut down; otherwse keep t on. Repeat the entre procedure for the next unt on the prorty lst. If t s also dropped, go to the next unt and so forth.

70 Questons on Economc Dspatch and Unt Commtment. What do you understand by Economc Dspatch problem?. For a power plant havng generator unts, derve the equal ncremental cost rule. 3. The cost characterstcs of three unts n a power plant are gven by C 0.5 P C 0.6 P C 3.0 P P Rs / hour + 60 P Rs / hour + 00 P Rs / hour where P, P and P 3 are generatng powers n MW. Maxmum and mnmum loads on each unt are 5 MW and 0 MW respectvely. Obtan the economc dspatch when the total load s 60 MW. What wll be the loss per hour f the unts are operated wth equal loadng?

71 4. The ncremental cost of two unts n a power statons are: dc dp dc dp 0.3 P + 70 Rs / hour 0.4 P + 50 Rs / hour a) Assumng contnuous runnng wth a load of 50 MW, calculate the savng per hour obtaned by usng most economcal dvson of load between the unts as compared to loadng each equally. The maxmum and mnmum operatonal loadngs of both the unts are 5 and 0 MW respectvely. b. What wll be the savng f the operatng lmts are 80 and 0 MW?

72 5. A power plant has two unts wth the followng cost characterstcs: C 0.6 P + 00 P Rs / hour C. P + 50 P Rs / hour where P and P are the generatng powers n MW. The daly load cycle s as follows: 6:00 A.M. to 6:00 P.M. 50 MW 6:00 P.M. to 6:00 A.M. 50 MW The cost of takng ether unt off the lne and returnng to servce after hours s Rs Consderng 4 hour perod from 6:00 A.M. one mornng to 6:00 A.M. the next mornng a. Would t be economcal to keep both unts n servce for ths 4 hour perod or remove one unt from servce for hour perod from 6:00 P.M. one evenng to 6:00 A.M. the next mornng? b. Compute the economc schedule for the peak load and off peak load condtons. c. Calculate the optmum operatng cost per day. d. If operatng one unt durng off peak load s decded, up to what cost of takng one unt off and returnng to servce after hours, ths decson s acceptable?

73 6. What do you understand by Loss coeffcents? 7. The transmsson loss coeffcents B mn, expressed n MW - of a power system network havng three plants are gven by B Three plants supply powers of 00 MW, 00 MW and 300 MW respectvely nto the network. Calculate the transmsson loss and the ncremental transmsson losses of the plants. 8. Derve the coordnaton equaton for the power system havng number of power plants.

74 9. The fuel nput data for a three plant system are: f 0.0 P +.7 P Mllons of BTU / hour f 0.0 P +.4 P Mllons of BTU / hour f P P Mllons of BTU / hour where P s are the generaton powers n MW. The fuel cost of the plants are Rs 50, Rs 30 and Rs 40 per Mllon of BTU for the plants, and 3 respectvely. The loss coeffcent matrx expressed n MW - s gven by B The load on the system s 60 MW. Compute the power dspatch for λ 0 Rs / MWh. Calculate the transmsson loss. Also determne the power dspatch wth the revsed value of λ takng 0 % change n ts value. Estmate the next new value of λ.

75 0. What are Partcpatng Factors? Derve the expresson for Partcpatng Factors.. Incremental cost of three unts n a plant are: IC IC IC 3. P.5 P.6 P 3 + Rs / MWh + 5 Rs / MWh + 4 Rs /MWh where P, P and P 3 are power output n MW. Fnd the optmum load allocaton when the total load s 85 MW. Usng Partcpatng Factors, determne the optmum schedulng when the load decreases to 75 MW.. What s Unt Commtment problem? Dstngush between Economc Dspatch and Unt Commtment problems.

76 3. Dscuss the constrans on Unt Commtment problem. 4. Explan what s Prorty Lst method. ASWERS MW MW 9.45 MW Rs Rs.39 Rs It s economcal to operate unt alone durng the off peak perod. 86. MW MW 0 50 MW Rs Rs MW MW 5.88 MW 8.5 MW MW MW MW Rs / MWh. 3.5 MW; 30.0 MW;.5 MW MW; MW; MW