Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT


 Edwin Shields
 2 years ago
 Views:
Transcription
1 Chapter 4 ECOOMIC DISATCH 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 system load plus transmsson loss.
3 IUT OUTUT CURVE OF GEERATIG UIT ower 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 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 mn and max. Input C n Rs / h mn max n MW Output Fg. InputOutput curve of a generatng unt
5 If the nputoutput curve of unt s quadratc, we can wrte C α β γ Rs / h () A power plant may have several generator unts. If the nputoutput characterstc of dfferent generator unts are dentcal, then the generatng unts can be equally loaded. But generatng unts wll generally have dfferent nputoutput characterstc. Ths means that, for partcular nput cost, the generator power 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 ΔC for a small change n power Δ. Snce dc = d Δ ΔC we can wrte ΔC = dc d Δ
6 ΔC = dc d Δ 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 d. The dmenson of dc d s Rs / MWh. The unt that has the nput output relaton as C α β γ Rs / h () has ncremental cost (IC) as IC dc α β () d d Here α, β and γ 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 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 LAT LOAD BETWEE GEERATIG UITS I A LAT Varous generatng unts n a plant generally have dfferent nputoutput 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 Fg. 3 Two unts case Intally, IC > IC. Decrease the output power n unt by Δ and ncrease output power n unt by Δ. ow IC Δ > IC Δ. Thus there wll be more decrease n cost and less ncrease n cost brngng the total cost lesser. Ths change can be contnued untl IC = IC at whch the total cost wll be mnmum. Further reducton n and ncrease n wll n IC >IC callng for decrease n and ncrease n untl IC = IC. Thus the total cost wll be mnmum when the ICREMETAL COSTS ARE EQUAL.
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 n order to decrease the total cost of generaton, 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. Inputoutput curve of the unts are denoted as C ( ),C( )......, C ( ). Our problem s, for a gven load demand D, fnd the set of s whch mnmzes the cost functon C T = C ( ) C ( )..... C ( ). (3) subject to the constrants D (..... ) 0 (4) and mn =,,.., (5) max Omttng the nequalty constrants for the tme beng, the problem to be solved becomes Mnmze C C ( ) (6) T subject to D 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(,,......,,λ ) C ( ) λ ( ) (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 L 0 =,,.., and 0. λ For a plant wth 3 unts L ( 3,,, λ ) = C ( ) C ( ) C 3 (3 ) λ(d 3 ) ecessary condtons for a mnmum are L C + λ (  ) = 0 L C + λ (  ) = 0 L 3 C λ (  ) = 0 L λ D 3 0
14 Generalzng the above, the necessary condtons are C λ =,,., and D 0 (9) (0) Here C 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 dc d Usng eqn.() n eqn. (9) we have () dc d λ =,,., ()
15 Thus the soluton for the problem Mnmze CT C ( ) subject to D 0 s obtaned when the followng equatons are satsfed. dc d λ =,,., (3) and D 0 (4) The above two condtons gve + number of equatons whch are to be solved for the + number of varables λ,. Equaton (3),,....., 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 OWER BALACE EQUATIO.
16 It s to be remembered that we have not yet consdered the nequalty constrants gven by mn =,,.., max 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 EXAMLE The cost characterstc of two unts n a plant are: C = K Rs./h C = K Rs. / h where and 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 = Rs. / MW h IC = Rs. / MW h Usng the equal ncremental cost rule λ and λ λ 60 λ 0 Snce + = 6.5 we get e. λ [ ] 6.5.e..36 λ =
18 .36 λ = Ths gves λ = 0 Rs / MWh Knowng λ and λ Optmum load allocaton s MW and 00 MW When the unts are equally loaded, 8.5 MW and we have devated from the optmum value of = 6.5 MW and = 00 MW. Knowng C = K Rs./h C = 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 EXAMLE A power plant has three unts wth the followng cost characterstcs: C C C Rs / h Rs / h Rs/h where s 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 Rs / MWh 70 Rs / MWh 60 Rs / MWh Usng the equal ncremental cost rule = λ; = λ; = λ
20 = λ; = λ; = λ Case ) Total load = 30 MW Snce = 30 we have λ 5.0 λ 70.0 λ e. λ [ ] e..43 λ = Ths gves λ = RM / MWh Thus = ( ) /.0 = MW = ( ) /.0 = MW 3 = ( ) /.4 = MW All ' s le wthn maxmum and mnmum lmts. Therefore, economc schedulng s = MW; = MW; 3 = MW
21 Case ) Total load = 00 MW Snce = 00 we have λ [ ] 00.e..49 λ = Ths gves λ = 300 Rs / MWh Thus = ( ) /.0 = 85 MW = ( ) /.0 = 5 MW 3 = ( ) /.4 = 00 MW It s noted that mn. Therefore s set at the mn. value of 39 MW. Then MW. Ths power has to be scheduled between unts and 3. Therefore λ [ ] 6.e..749 λ = Ths gves λ = 86 Rs / MWh Thus = ( 865 ) /.0 = 7 MW 3 = ( ) /.4 = 90 MW and 3 are wthn the lmts. Therefore economc schedulng s 7MW; 39 MW; 3 90 MW
22 EXAMLE 3 Incremental cost of two unts n a plant are: IC Rs / MWh IC Rs / MWh where and 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. = 0 MW. Total mnmum load beng 50 MW, when = 0 MW, must be equal to 30 MW. Thus ntally = 0 MW, IC = 76 Rs / MWh, = 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 = ( 76 0 ) / 0.9 = 6. MW. Untl that pont s reached, 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 = ( λ  60 ) / 0.8 MW = ( λ  0 ) / 0.9 MW Then the load allocaton wll be as shown below. lant IC λ Load on unt Load on unt Total load Rs/MWh MW MW + MW
24 Load on unt reaches the maxmum value of 5 MW, when λ = (0.9 x 5) + 0 = 3.5 Rs / Mwh. When the plant IC, λ ncreases further, shall reman at 5 MW and the load on unt alone ncreases and ts value s computed as = ( λ  60 ) / 0.8 MW. Such load allocatons are shown below. lant IC λ Load on unt Load on unt Total load Rs / MWh MW MW + MW The results are shown n graphcal form n Fg. 4
25 / MW Total load D Fg. 4 Load allocaton for varous plant load
26 Alternatve way of explanaton for ths problem s shown n Fg. 5 IC Rs/MWh , 76 30, 47 6., , , 60 5, / 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 L B B B B B B B B B (5) where respectvely. are the powers suppled by the plants,,.,,,...,
28 L B B B B B B B B B (5) From eq.(5) L = n n n n n n n n n B B B = n n n B + n n n B +. + n n n B = n B n m m n m = n B n m m n m Thus L can be wrtten as L = n B n m m n m (6)
29 L = m n m B mn n (6) When the powers are n MW, the B mn coeffcents are of dmenson / MW. If powers are n perunt, then B mn coeffcents are also n perunt. 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 L B B B B B B B B = B B B B
30 Snce B mn coeffcent matrx s symmetrc, for two plant system L B B B (7) In later calculatons we need the Incremental Transmsson Loss ( ITL ), L. For two plant system L B B (8) L B B (9) Ths can be generalzed as L m n B mn n m =,,., (0)
31 6 ECOOMIC DIVISIO OF SYSTEM LOAD BETWEE VARIOUS LATS I THE OWER 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. Inputoutput characterstcs of the plants are denoted as C ( ),C ( ),......, C ( ). Our problem s for a gven system load demand D, fnd the set of plant generaton whch mnmzes the cost functon,,...., C T = C ( ) C ( )..... C ( ) () subject to the constrants D (..... ) 0 () L and mn I =,,., (3) max Inequalty constrants are omtted for the tme beng.
32 The problem to be solved becomes Mnmze C C ( ) (4) T subject to 0 D L (5) For ths case the Lagrangan functon s L(,,....,, λ) = C ( ) λ ( D ) (6) L Ths Lagrangan functon has to be mnmzed wth no constrant on t. The necessary condtons for a mnmum are L 0 =,,.., L and 0. λ
33 L(.,,...,, λ) = ( λ ( ) C ) L D (6) For a system wth plants L ( λ,, ) = ) λ( ) ( C ) ( C L D L C + λ ( L  ) = 0 L C + λ ( L  ) = 0 0 λ L L D
34 Generalzng ths, for system wth plants, the necessary condtons for a mnmum are C λ [ L ] 0 =,,.., (7) and 0 D L (8) As dscussed n earler case, we can wrte C dc d (9) Usng eqn. (9) n eqn. (7) we have dc d L λ λ =,,.., (30) and 0 D L (3)
35 dc d L λ λ =,,.., (30) and 0 D L These equatons can be solved for the plant generatons.,,...., (3) As shown n the next secton the value of λ n eqn. (30) s the ICREMETAL COST OF RECEIVED OWER. The eqns. descrbed n eqn. (30) are commonly known as COORDIATIO EQUATIOS as they lnk the ncremental cost of plant dc d, ncremental cost of receved power λ and the ncremental transmsson loss ( ITL ) L. Equaton (3) s the OWER BALACE EQUATIO.
36 Coordnaton equatons dc d L λ λ =,,.., (30) 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. to obtan the,,....,
37 7 ICREMETAL COST OF RECEIVED OWER The value of λ n the coordnaton equatons s the ncremental cost of receved power. Ths can be proved as follows: The coordnaton equatons C λ L λ can be wrtten as C λ [ L ] (33).e. Δ C Δ Δ L Δ λ.e. ΔC Δ Δ L λ (34) Snce D L D L.e. Δ D Δ Δ (35) L Usng eqn,(35) n eqn.(34) gves Δ C Δ D C = λ D (36) Thus λ s the ncremental cost of receved power.
38 8 EALTY FACTORS To have a better feel about the coordnaton equatons, let us rewrte the dc L same as λ [ ] =,,.., (37) d Thus [ L ] dc d λ =,,.., (38) The above equaton s often wrtten as dc L = λ =,,.., (39) d where, L whch s called the EALTY FACTOR of plant, s gven by L = L =,,.., (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 OTIMUM SCHEDULIG OF SYSTEM LOAD BETWEE LATS  SOLUTIO ROCEDURE 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 Knowng C α β γ.e. IC = α + β substtute the value of λ nto the coordnaton equatons dc d L λ λ =,,..,.e. ( α β ) λ Bmn n n λ =,,.., The above set of lnear smultaneous equatons are to be solved for the values ' s.
40 Step 3 Compute the transmsson loss L from L = [ ] [ B ] [ t ] where [ ] = [.... ] and [ B ] s the loss coeffcent matrx. Step 4 Compare wth D + 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 by about 5% Value of λ s decreased by about 5% f f D + L. D + L In the subsequent teratons, usng lnear nterpolaton, value of λ can be updated as
41 k k L D k k k k k k ] [ λ λ λ λ (4) 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. k k k λ k λ k λ D + L k
42 EXAMLE 4 Consder a power system wth two plants havng ncremental cost as IC.0 00 Rs / MWh IC.0 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 MWh. 50MW. Intal value of λ = (.0 x 50 ) + 00 = 50 Rs / Coordnaton equatons dc d λ L λ ( 0.00 ( ) 50 ) 50.e and = ; On solvng 00 = MW and = MW
43 L = MW = + = MW and MW D L Snce < D L, λ value should be ncreased. It s ncreased by 4 %. ew value of λ = 50 x.04 = 60 Rs / MWh. Coordnaton equatons: ( 0.00 ( ) ) e and =.48 ; On solvng 0 = MW and = MW
44 L = MW = MW ; MW D L D L Knowng two values of λ and the correspondng total generaton powers, new value of λ s computed as λ k λ k λ k k λ k k [ D k L k ] λ ( ) 63 Rs / MWh Wth ths new value of λ, coordnaton procedure has to be repeated. equatons are formed and the
45 The followng table shows the results obtaned. λ L + D + L Optmum schedule s 5.40MW MW For ths transmsson loss s MW
46 0 BASE OIT AD ARTICIATIO 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 D keeps changng. When load changes are small, t s possble to move from one optmal schedule to another usng ARTICIATIG FACTORS. We start wth a known optmal generaton schedule, 0, 0,, 0, for a partcular load D. Ths schedule s taken as BASE OIT and the correspondng ncremental cost s λ 0. Let there be a small ncrease n load of Δ D. To meet wth ths ncreased load, generatons are to be ncreased as Δ, Δ,., Δ. Correspondngly ncremental cost ncreases by Δλ.
47 Knowng that for th unt, C = α + β + γ, ncremental cost s IC = α + β = λ (4) Small change n ncremental cost and correspondng change n generaton are related as Δλ = α (43) Δ Thus Δ = Δλ α for =,,, (44) Total change n generatons s equal to the change n load. Therefore Δ = Δ D.e. Δ D = Δλ α (45) From the above two equatons Δ Δ D α α = k for =,,, (46) The rato Δ s known as the ARTICIATIO FACTOR of generator, Δ D represented as k. Once all the k s, are calculated from eq.(46), the change n generatons are gven by Δ = k Δ D for =,,, (47)
48 EXAMLE 5 Incremental cost of three unts n a plant are: IC = Rs / MWh; IC = Rs / MWh; and IC 3 = Rs / MWh where, and 3 are power output n MW. Fnd the optmum load allocaton when the total load s 4.5 MW. Usng artcpatng Factors, determne the optmum schedulng when the load ncreases to 50 MW. Soluton Usng the equal ncremental cost rule λ ; λ ; = λ λ 60 λ 0 λ 0 Snce = 4.5 we get e λ [ ] 4.5.e. 3.6 λ = Ths gves λ = 0 Rs / MWh Optmum load allocaton s MW ; 00 MW ; 3 80 MW
49 artcpaton Factors are: k = k = k 3 = = 3.6 = = = = 3.6 = 0.53 Change n load Δ D = = 7.5 MW Change n generatons are: Δ = x 7.5 =.9655 MW Δ = x 7.5 =.6363 MW Δ 3 = 0.53 x 7.5 =.898 MW Thus optmum schedule s: = MW; = MW; 3 = MW
50 Example 6 A power plant has two unts wth the followng cost characterstcs: C = Rs / hour C = Rs / hour where and are the generatng powers n MW. The daly load cycle s as follows: 6:00 A.M. to 6:00.M. 50 MW 6:00.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 Maxmum generaton of each unt s 00 MW. 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.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? e. If the cost of takng one unt off and returnng to servce after hours exceeds the value calculated n d, what must be done durng off peak perod?
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 = Rs / MWh IC = Rs / MWh λ Usng equal IC rule λ 50 = 50;.5 λ = and λ = Therefore = MW; = MW Then C T = C ( = ) + C ( = ) = Rs / h For hour perod, cost of operaton = Rs
53 ) Cost of operaton for hours = Rs ) If unt s operatng, C Ι = 50 = 3500 Rs / h If unt s operatng, C Ι = 50 = 3000 Rs / h Between unts and, t s economcal to operate unt. If only one unt s operatng durng offpeak perod, cost towards takng out and connectng t back also must be taken. Therefore, for offpeak 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, D = 50 MW. Wth equal IC rule λ Therefore = 86. MW; Thus, economc schedule s: λ 50 = 50;.5 λ = and λ = = MW Durng 6:00 am to 6:00 pm = 86. MW; = MW Durng 6:00 pm to 6:00 am = 0; = 50 MW c. Cost of operaton = [ C Ι = C Ι = ] for peak perod = x = Rs Cost of operaton for offpeak perod = Rs 6000 Therefore, optmal operatng cost per day = Rs
55 d. If both the unts are operatng durng offpeak perod, cost of operaton = Rs 7833 If unt alone s operatng durng = Rs (3000 x ) + x offpeak 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 offpeak perod s acceptable. e. If the cost of takng one unt off and returnng t to servce exceeds Rs.5833, then both the unts are to be operated all through the day.
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 decreases, 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, 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 ths 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 when the load s not large enough. As seen n prevous example, 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 = Rs / h Unt : Mn. = 00 MW; Max. = 400 MW C = Rs / h Unt 3: Mn. = 50 MW; Max. = 00 MW C 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 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 bruteforce 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 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 The shutdown rule s qute smple. When load s above 000 MW, run all three unts; more than 600 MW and less than 000 MW, run unts and ; below 600 MW, run only unt.
63 The above shutdown rule s qute smple; but t fals to take the economy over a day. In a power plant wth unts, for each load step, (neglectng the number of nfeasble solutons) economc dspatch problem s to solved for ( ) tmes. Durng a day, f there are M load steps, (snce each combnaton n one load step can go wth each combnaton of another load step) to arrve at the economy over a day, n ths bruteforce technque, economc dspatch problem s to be solved for ( ) M. Ths number wll be too large for practcal case. UC problem become much more complcated when we need to consder power system havng several plants each plant havng several generatng unts and the system load to be served has several load steps. 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.
64 COSTRAITS O UC ROBLEM 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 one 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.
65 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 nonavalablty 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, hydroprojects are multpurpose projects. Irrgaton requrements also determne the operaton of hydro plants.
66 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.
67 RIORITY 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 roducton 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 = Rs / h Unt : Max. = 400 MW C = Rs / h Unt 3: Max. = 00 MW Obtan the prorty lst C 3 = Rs / h
68 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
69 The shutdown scheme would (gnorng mn. up / down tme, start up costs etc.) smply use the followng combnatons. Combnaton Load D MW D < 00 MW MW D < 000 MW 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.
70 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.
71 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 C = 0.6 C 3 = Rs / hour Rs / hour Rs / hour where, and 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?
72 4. The ncremental cost of two unts n a power statons are: dc d dc d = Rs / hour = 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?
73 5. A power plant has two unts wth the followng cost characterstcs: C = Rs / hour C = Rs / hour where and are the generatng powers n MW. The daly load cycle s as follows: 6:00 A.M. to 6:00.M. 50 MW 6:00.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.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?
74 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.
75 9. The fuel nput data for a three plant system are: f = Mllons of BTU / hour f = Mllons of BTU / hour f 3 = Mllons of BTU / hour where 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.
76 0. What are artcpatng Factors? Derve the expresson for artcpatng Factors.. Incremental cost of three unts n a plant are: IC IC IC Rs / MWh Rs / MWh Rs /MWh where, and 3 are power output n MW. Fnd the optmum load allocaton when the total load s 85 MW. Usng artcpatng 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.
77 3. Dscuss the constrans on Unt Commtment problem. 4. Explan what s rorty 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
Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationDynamic Constrained Economic/Emission Dispatch Scheduling Using Neural Network
Dynamc Constraned Economc/Emsson Dspatch Schedulng Usng Neural Network Fard BENHAMIDA 1, Rachd BELHACHEM 1 1 Department of Electrcal Engneerng, IRECOM Laboratory, Unversty of Djllal Labes, 220 00, Sd Bel
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationTime Value of Money Module
Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationFormulating & Solving Integer Problems Chapter 11 289
Formulatng & Solvng Integer Problems Chapter 11 289 The Optonal Stop TSP If we drop the requrement that every stop must be vsted, we then get the optonal stop TSP. Ths mght correspond to a ob sequencng
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationRELIABILITY, RISK AND AVAILABILITY ANLYSIS OF A CONTAINER GANTRY CRANE ABSTRACT
Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March RELIABILITY RIK AND AVAILABILITY ANLYI OF A CONTAINER GANTRY CRANE
More informationA GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION
A GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION Helena Vasconcelos INESC Porto hvasconcelos@nescportopt J N Fdalgo INESC Porto and FEUP jfdalgo@nescportopt
More informationEducational Software for Economic Load Dispatch for Power Network of Thermal Units Considering Transmission Losses and Spinning Reserve Power
Educatonal Software for Economc Load Dspatch for ower Network of Thermal Unts Consderng Transmsson Losses and Spnnng Reserve ower Mohammad T. Amel Saed Moslehpour Massoud ourhassan ower and Water Unversty
More informationSUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.
SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976761000
More informationFeasibility of Using Discriminate Pricing Schemes for Energy Trading in Smart Grid
Feasblty of Usng Dscrmnate Prcng Schemes for Energy Tradng n Smart Grd Wayes Tushar, Chau Yuen, Bo Cha, Davd B. Smth, and H. Vncent Poor Sngapore Unversty of Technology and Desgn, Sngapore 138682. Emal:
More informationResearch Article Enhanced TwoStep Method via Relaxed Order of αsatisfactory Degrees for Fuzzy Multiobjective Optimization
Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced TwoStep Method va Relaxed Order of αsatsfactory Degrees for Fuzzy
More informationActivity Scheduling for CostTime Investment Optimization in Project Management
PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta San Sebastán, September 8 th 10 th 010 Actvty Schedulng
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationAn MILP model for planning of batch plants operating in a campaignmode
An MILP model for plannng of batch plants operatng n a campagnmode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafeconcet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationLecture 7 March 20, 2002
MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationJoint Scheduling of Processing and Shuffle Phases in MapReduce Systems
Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, AlcatelLucent
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationPeriod and Deadline Selection for Schedulability in RealTime Systems
Perod and Deadlne Selecton for Schedulablty n RealTme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng
More information2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet
2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet CORE Voe du Roman Pays 34 B1348 LouvanlaNeuve, Belgum. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 Emal: corestatlbrary@uclouvan.be
More informationExhaustive Regression. An Exploration of RegressionBased Data Mining Techniques Using Super Computation
Exhaustve Regresson An Exploraton of RegressonBased Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The
More informationLogical Development Of Vogel s Approximation Method (LDVAM): An Approach To Find Basic Feasible Solution Of Transportation Problem
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77866 Logcal Development Of Vogel s Approxmaton Method (LD An Approach To Fnd Basc Feasble Soluton Of Transportaton
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationAnt Colony Optimization for Economic Generator Scheduling and Load Dispatch
Proceedngs of the th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lsbon, Portugal, June 118, 5 (pp17175) Ant Colony Optmzaton for Economc Generator Schedulng and Load Dspatch K. S. Swarup Abstract Feasblty
More informationChapter 7: Answers to Questions and Problems
19. Based on the nformaton contaned n Table 73 of the text, the food and apparel ndustres are most compettve and therefore probably represent the best match for the expertse of these managers. Chapter
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More informationResponse Coordination of Distributed Generation and Tap Changers for Voltage Support
Response Coordnaton of Dstrbuted Generaton and Tap Changers for Voltage Support An D.T. Le, Student Member, IEEE, K.M. Muttaq, Senor Member, IEEE, M. Negnevtsky, Member, IEEE,and G. Ledwch, Senor Member,
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationOptimal Bidding Strategies for Generation Companies in a DayAhead Electricity Market with Risk Management Taken into Account
Amercan J. of Engneerng and Appled Scences (): 86, 009 ISSN 94700 009 Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a DayAhead Electrcty Market wth Rsk Management Taken nto Account
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More informationFinancial Mathemetics
Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationFINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals
FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant
More informationMaintenance Scheduling by using the BiCriterion Algorithm of Preferential AntiPheromone
Leonardo ournal of Scences ISSN 5830233 Issue 2, anuaryune 2008 p. 4364 Mantenance Schedulng by usng the BCrteron Algorthm of Preferental AntPheromone Trantafyllos MYTAKIDIS and Arstds VLACHOS Department
More informationExamensarbete. Rotating Workforce Scheduling. Caroline Granfeldt
Examensarbete Rotatng Workforce Schedulng Carolne Granfeldt LTH  MAT  EX   2015 / 08   SE Rotatng Workforce Schedulng Optmerngslära, Lnköpngs Unverstet Carolne Granfeldt LTH  MAT  EX   2015
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationMathematics of Finance
CHAPTER 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty; Amortzaton Revew Exercses Extended Applcaton: Tme, Money, and Polynomals Buyng
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationFuzzy Set Approach To Asymmetrical Load Balancing In Distribution Networks
Fuzzy Set Approach To Asymmetrcal Load Balancng n Dstrbuton Networks Goran Majstrovc Energy nsttute Hrvoje Por Zagreb, Croata goran.majstrovc@ehp.hr Slavko Krajcar Faculty of electrcal engneerng and computng
More informationAPPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT
APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT Toshhko Oda (1), Kochro Iwaoka (2) (1), (2) Infrastructure Systems Busness Unt, Panasonc System Networks Co., Ltd. Saedocho
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationWeek 6 Market Failure due to Externalities
Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to
More informationUTILIZING MATPOWER IN OPTIMAL POWER FLOW
UTILIZING MATPOWER IN OPTIMAL POWER FLOW Tarje Krstansen Department of Electrcal Power Engneerng Norwegan Unversty of Scence and Technology Trondhem, Norway Tarje.Krstansen@elkraft.ntnu.no Abstract Ths
More informationGibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments)
Gbbs Free Energy and Chemcal Equlbrum (or how to predct chemcal reactons wthout dong experments) OCN 623 Chemcal Oceanography Readng: Frst half of Chapter 3, Snoeynk and Jenkns (1980) Introducton We want
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationA generalized hierarchical fair service curve algorithm for high network utilization and linksharing
Computer Networks 43 (2003) 669 694 www.elsever.com/locate/comnet A generalzed herarchcal far servce curve algorthm for hgh network utlzaton and lnksharng Khyun Pyun *, Junehwa Song, HeungKyu Lee Department
More informationMulticomponent Distillation
Multcomponent Dstllaton need more than one dstllaton tower, for n components, n1 fractonators are requred Specfcaton Lmtatons The followng are establshed at the begnnng 1. Temperature, pressure, composton,
More informationRate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Prioritybased scheduling. States of a process
Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? RealTme Systems Laboratory Department of Computer
More informationMethod for Production Planning and Inventory Control in Oil
Memors of the Faculty of Engneerng, Okayama Unversty, Vol.41, pp.2030, January, 2007 Method for Producton Plannng and Inventory Control n Ol Refnery TakujImamura,MasamKonshandJunIma Dvson of Electronc
More informationRetailers must constantly strive for excellence in operations; extremely narrow profit margins
Managng a Retaler s Shelf Space, Inventory, and Transportaton Gerard Cachon 300 SH/DH, The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana 90 cachon@wharton.upenn.edu http://opm.wharton.upenn.edu/cachon/
More informationAvailabilityBased Path Selection and Network Vulnerability Assessment
AvalabltyBased Path Selecton and Network Vulnerablty Assessment Song Yang, Stojan Trajanovsk and Fernando A. Kupers Delft Unversty of Technology, The Netherlands {S.Yang, S.Trajanovsk, F.A.Kupers}@tudelft.nl
More informationPrice Competition in an Oligopoly Market with Multiple IaaS Cloud Providers
Prce Competton n an Olgopoly Market wth Multple IaaS Cloud Provders Yuan Feng, Baochun L, Bo L Department of Computng, Hong Kong Polytechnc Unversty Department of Electrcal and Computer Engneerng, Unversty
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More information2. SYSTEM MODEL. the SLA (unlike the only other related mechanism [15] we can compare it is never able to meet the SLA).
Managng Server Energy and Operatonal Costs n Hostng Centers Yyu Chen Dept. of IE Penn State Unversty Unversty Park, PA 16802 yzc107@psu.edu Anand Svasubramanam Dept. of CSE Penn State Unversty Unversty
More informationVOLTAGE stability issue remains a major concern in
Impacts of Mert Order Based Dspatch on Transfer Capablty and Statc Voltage Stablty Cuong P. guyen, Student Member, IEEE, and Alexander J. Flueck, Member, IEEE Abstract In ths paper, the goal s to nvestgate
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationSurvey on Virtual Machine Placement Techniques in Cloud Computing Environment
Survey on Vrtual Machne Placement Technques n Cloud Computng Envronment Rajeev Kumar Gupta and R. K. Paterya Department of Computer Scence & Engneerng, MANIT, Bhopal, Inda ABSTRACT In tradtonal data center
More informationThe circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:
polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng
More informationINVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMAHDR NETWORKS
21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS
More informationFINANCIAL MATHEMATICS
3 LESSON FINANCIAL MATHEMATICS Annutes What s an annuty? The term annuty s used n fnancal mathematcs to refer to any termnatng sequence of regular fxed payments over a specfed perod of tme. Loans are usually
More informationData Broadcast on a MultiSystem Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819840 (2008) Data Broadcast on a MultSystem Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationCredit Limit Optimization (CLO) for Credit Cards
Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationPerformance Analysis of Energy Consumption of Smartphone Running Mobile Hotspot Application
Internatonal Journal of mart Grd and lean Energy Performance Analyss of Energy onsumpton of martphone Runnng Moble Hotspot Applcaton Yun on hung a chool of Electronc Engneerng, oongsl Unversty, 511 angdodong,
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationEnabling P2P Oneview Multiparty Video Conferencing
Enablng P2P Onevew Multparty Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract MultParty Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P
More information