Approach for Online Short-term Hydro-Thermal Economic Scheduling

Transcription

1 Approach for Onlne Short-term Hydro-Thermal Economc Schedulng Rav Brahmadev, Tukaram Moger, Member, IEEE and D. Thukaram, Senor Member, IEEE Department of Electrcal Engneerng Indan Insttute of Scence, Bangalore Emal: {rav, tukaramoger, Abstract : Ths paper ntroduces a new algorthm for real power generaton schedulng n power systems, where the system consstng of hydro and thermal plants. The proposed algorthm s dvded nto two parts, hydro and thermal sub problems. Hydro sub problem s solved by Hydro-Thermal schedulng algorthm where the losses n the network are ntally not consdered. Thermal sub problem s solved by Closed Loop Lambda method n whch the detaled network power flow wth practcal constrants and losses are consdered. Ths algorthm s applcable for practcal power systems for both real and reactve power patterns wth hgh accuracy and fast convergence. The proposed method s tested on IEEE New England 39 bus equvalent system wth 7 thermal power plants and 3 hydro power plants. Keywords- hydro-thermal schedulng, closed loop lambda, economc load dspatch, water value, B-coeffcents, ncremental transmsson loss, penalty factor. I. INTRODUCTION RESENT day power system are large grd networks wth nter connectons to neghborng systems. Due to ts large geographcal area the sources of hydro and thermal generatons spread unevenly. Also n vew of deregulaton envronment the hydro thermal schedulng plays an mportant role for the secured and optmzed operaton of power system. In recent tmes many heurstc approaches are reported for Short-Term Hydro-Thermal Schedulng (STHTS) such as Genetc Algorthm (GA), Artfcal Neural Networks (ANN), Evolutonary rogrammng (E), Bacteral Foragng Algorthm (BFA), Modfed BFA etc [1]-[3]. In spte of the advantages gettng from the heurstc technques mentoned, they have drawbacks of non-consderaton of many practcal constrants. The classcal optmzaton technques work well wth proper handlng of practcal constrants because the classcal methods have strong mathematcal foundaton [1]. In most of these methods, the network s accounted wth approxmate loss coeffcents or B-coeffcents. The optmal schedule obtaned wth B-coeffcents s not perfect and may not be apprecable for practcal modern power systems. In ths paper, a new algorthm of optmal power generaton schedulng for the Hydro-Thermal power system s proposed. The optmal power generaton of thermal and hydro power plants s obtaned by mnmzng the operatng cost of thermal power plants n the presence of hydraulc and electrc constrants. Ths method s applcable for practcal power system wth varous loads and generaton patterns as the network s drectly nvolved n optmzaton problem rather than by usng approxmate B-coeffcents. The earler methods of Hydro- Thermal schedulng cannot be applcable for varous load and generaton patterns along wth varous reactve power demand patterns. The Hydro-Thermal schedulng for the system s done for short range perods such as one day wth a number of subntervals by utlzng the avalable volume of water for each hydro plant. Ths algorthm s dvded nto two parts. The frst part deals wth hydro sub problem, whch s done by usng conventonal Newton-Raphson method; ths part gves the optmal generaton settngs for the hydro power plants for all tme ntervals. The second part s Closed Loop Lambda (CLL) method for thermal power plants, where the cost coeffcents for hydro plants are assumed as zero. Organzaton of the paper s as follows. Secton II provdes the part 1, hydro sub problem of the proposed algorthm, n whch the conventonal Newton-Raphson method for STHTS s explaned. Secton III deals wth part, thermal sub problem, whch uses Closed Loop Lambda method. The combnaton of hydro and thermal sub problems s presented n Secton IV as a new optmzaton algorthm for STHTS. Secton V gves the smulaton results for IEEE New England 39 bus equvalent system. The conclusons are drawn n Secton VI. II. HYDRO SUB-ROBLEM Hydro sub problem s the conventonal STHTS, whch uses the Newton-Raphson method [4]-[5]. Ths paper focuses only for short-term perod of one day or 4 hours. Throughout the day the head of the hydro power plant s more or less constant. So the head of the power plant s taken as constant and the problem s referred as Fxed Head STHTS. In ths part, the network s assumed as lossless. A. Hydro lant Model The output of hydro plant depends on manly water dscharge rate (q) to the turbne and avalable net head (h) of the plant. The nput-output characterstc of a hydro-electrc plant s gven by usng Glmn-Krchmayer model [6]. Ths model relates the water dscharge rate (q) wth power generaton () and head (h) of a hydro plant. 1

2 For j th hydro power plant, h q j K j j (1) where K s proportonalty constant; ψ (hj) s a quadratc functon n terms of h; ϕ(j) s a quadratc functon n terms of ; As the head of the plant s constant for short perods, ψ(hj) s constant and the model s represented only n terms of water dscharge and power output. Therefore, for j th hydro plant n k th sub nterval of tme the water dscharge rate s gven as, q jk x jjk y j jk j 3 / z m hr () where xj, yj and zj are water dscharge coeffcents; Each hydro plant s constraned by the amount of water avalable for scheduled perod of tme (Tsch). For j th hydro plant, t s gven as, T sch 0 q j() t dt Vj (3) where Vj s the amount/volume of water avalable n m 3 ; B. Thermal lant Model The nput-output characterstcs of a thermal power plant are gven as, ( k ) k k Rs./hr (4) F a b c where, parameters a, b, c are cost coeffcents of th thermal power plant; k s the power generaton by th thermal plant n k th tme nterval. C. roblem Formulaton: The man objectve of optmal power generaton schedulng s to mnmze the operatng cost of generators. Let the system has N thermal plants, M hydro plants and the number of tme ntervals are T. the hydro power plants has neglgble operatng cost. The objectve functon s defned as, mn T N k k k11 J t F ( ( t)) (5) where J s the total operatng cost of system; Electrcal constrants on objectve functon are, ower balance equaton (neglectng losses), NM k dk (6) 1 Unt constrants: mn k,=1,, N+M (7) where k s the power output of th plant (thermal/hydro) n k th nterval; Dk s the power demand n k th nterval; mn s mnmum power generaton for th plant; max s maxmum power generaton for th plant; Hydraulc constrants: Water avalablty constrants as descrbed n (3), can be represented n dscrete form as, T tkq jk Vj, for j 1,... M (8) k1 where, T s total number of subntervals t k s the tme duraton n hours for k th nterval; The Lagrangan functon s formed usng above objectve functon and set of constrants s gven as, T N M L( k, k, v j ) J k dk k k1 1 M T v j tk q jk V j j1 k1 where J s objectve functon; λ k and ν j are Lagrangan coeffcents also known as the margnal cost of electrcty for each plant; λ k represents the cost of electrcty produced by Thermal power plants durng k th sub nterval n Rs/MWhr; ν j water value or water converson factor n Rs./Mm 3 ; These Lagrangan coeffcents represent the mnmum cost of supplyng an addtonal unt of power over the system load. The Kaursh-Kuhn-Tucker condtons are obtaned from Lagrangan functon L as, L L L 0, 0, 0 v k k j (9) (10) These partal dervatves gve the set of equatons. These equatons are to be solved by Newton-Raphson method. D. A Bref Computatonal Algorthm Step 1: Read system data such as number of thermal and hydro plants (N and M), number of tme ntervals (T) and tme lapsed n each nterval (t), water dscharge coeffcents (x, y, z) of hydro plants, cost coeffcents (a, b, c) of thermal plants, ower demand pattern ( D ) and avalable volume of water (V) for each hydro plant. Step : Guess the ntal generatons ( k ) and Lagrangan coeffcents (λ k, v j ) for each nterval. Step 3: Start teraton count, r =1. Step 4: Form Jacoban matrx by applyng Newton-Raphson method to (10). Step 5: Fnd the changes n power generaton (Δ k ) and change n Lagrangan coeffcents (Δλ k, Δv j ) for each plant n each nterval. Step 6: Update the power generaton values and Lagrange coeffcents. Step 7: Check the lmts of generaton at each generators. If any generator s volatng lmts set the generaton to the volated lmt and dsallow that generator nto schedulng problem partcpaton.

3 Step 8: Check for convergence. If convergence s acheved go to Step10 else go to Step 9 Step 9: Advance the teraton count by 1, r = r+1 and go to Step 4. Step 10: prnt the optmal generaton schedule for each nterval of tme, calculate total cost of generaton, cost of electrcty for thermal plant and water value for hydro plant (Lagrange coeffcents). III. CLOSED LOO -METHOD FOR THERMAL SUB- ROBLEM The closed loop λ-method [7]-[8] gves the optmal generaton schedule for thermal power plants whch has quadratc fuel cost functon. In the proposed hydro-thermal schedulng algorthm ths Closed Loop Lambda method s used to get optmal schedule for thermal plants. Ths method of schedulng uses penalty factors of generators, whch are calculated by power flow Jacoban method. The formulaton of method for one partcular tme nterval s as follows, Objectve functon, N T F (11) 1 mn F ( ) Subject to the condtons load loss (1) 1 and mn max (13) where, s total number of generators n system (N+M); N s number of thermal plants; load s total power load; loss s total power loss; FT s generaton cost for the whole system; F s generaton cost of th thermal generator for the specfed tme nterval; The augmented Lagrange functon consderng the constrants s gven by, T ( load loss ) (14) 1 F where, λ s called as the Lagrange multpler or slack varable; For the objectve functon to be mnmum, then, df loss (1 ) or d df L d, (15) where df / d :Incremental Cost (IC ) of th generator; loss / : Incremental Transmsson Loss (ITL) of th bus; L : penalty factor for th bus; The penalty factor for th bus a gven by, 1 L (16) (1 loss ) In a system, the total power loss s the sum of real power values at all the buses.e. NB loss (17) 1 where NB s number of buses n the system; In ths study, Bus 1 s taken as a reference bus. So the change n loss of (17) s gven by, d NB loss d1 d (18) And the transmsson loss can be descrbed as a functon of power generated at generator bus s gven by, 1,... (19) loss loss G loss G Usng the Newton-Raphson method, the power msmatch matrx can be wrtten as; d Jp Jpv d dq Jq J qv dv (0) The real power msmatch of the reference Bus 1 s gven by, 1 1 d 1 [ ] T d d V dq dq T [ ] 1 1 J 1 p Jpv V Jq Jqv,,...,, NB ITL of th generator s, (1) () dloss ITL 1 fo r,.... and (3) d From (15), the penalty factor, 1 L (4) For reference Bus 1, the ITL1 = 0, so L1=0 and γ1= -1. In optmal condton, the IC of th generator s gven by, df 1 ITL (5) d Form (5), the ncremental cost s, df b c d (6) From (5) and (6), the optmum generaton for th generator s gve as, ( ) b 1 b c L c (7) 3

4 For economc dspatch operaton, the total transmsson loss at optmum dspatch s, 0 loss ( G ) loss ( G ) 1 d loss ( G ) 1 G G 1 (8) The followng cases are studed by consderng the volaton and non-volaton condton of the generators; Case A: when no generator s volatng ts lmt, the system lambda s gven as 0 G b 1 L L c 1 1 cl (9) Case B: When some generators volate ther lmts, the generaton shftng can be done only on no-volatng generators. So G G load loss G dloss 1 1 v nv Here d loss s the change n loss due to change n the nonvolatng generators b load loss G G G L c v nv nv 1 1 cl nv (30) where, v: volatng condton; nv: non-volatng condton Knowng the values of λ, the optmum settngs of the plant s obtaned from (7). IV. ROOSED OTIMIZATION ALGORITHM FOR STHTS The new optmzaton algorthm has two parts, hydro sub problem dscussed n Secton II and thermal sub problem dscussed n Secton III. The sequence of steps for new approach s gven below; Step 1: Read the system data such as lne data, bus data, generator operatng characterstcs, schedulng perod and specfed load pattern. Solve the hydro sub problem by followng the computatonal procedure shown n Secton II. At the end of ths step the results obtaned are: Approxmate generaton schedule for thermal plants Fnal generaton schedule for hydro plants. Water dscharge rate n each nterval. Water value for hydro plants. Step : Take the coeffcents of the quadratc cost characterstc of the hydro power plant are assumed as a=b=0 and c s taken as neglgbly small value. For these assumptons, the plant generaton wll ht ts maxmum lmt. That means only thermal plants are consdered for schedulng n further. Step 3: Set subnterval number, k=1 Step 4: Set upper lmt of power generaton for hydro power plant as scheduled generaton, whch s obtaned n Step 1 corresponds to subnterval k. Step 5: Apply the closed loop λ-method for the system for k th subnterval by followng the procedure presented n Secton III. In ths step the fnal generaton schedule and cost of generaton for each plant λ n Rs/MWhr s obtaned. Step 6: Fnd the total cost of generaton for the k th nterval tme n Rupees. Step 7: Increase subnterval count, k=k+1. If k > T (total number of subntervals) goto Step 8, else goto Step 4. Step 8: rnt the fnal generaton schedule and cost of generaton for each subnterval and cost of generaton. Sum up all the cost of generatons, whch gves the total optmal cost of generaton for the gven schedulng perod. V. SIMULATION RESULTS The proposed algorthm for Short-term Hydro-thermal Schedulng s appled for IEEE New England 39 bus equvalent system. Fgure 1. IEEE New England 39 bus equvalent system Ths system has 10 generators, 1 transformers and 34 transmsson lnes. The sngle lne dagram of the system s shown n Fg. 1. eak real and reactve power loads on the system are 5941MW and 131MVAR respectvely. 7 among the 10 generators are consdered as thermal and rest all consdered as hydro [9]. In [9] the system has hydro 4

5 plants on the same stream, but n ths paper the hydro plants are assumed to be on dfferent streams. Ths assumpton on the system s reasonable because the STHTS s done only for short duraton tmngs wth the avalable amount of water. The thermal power plants cost characterstcs are adopted from [10], n whch all plants are consdered as thermal but the selectve 7 thermal plants characterstcs are taken nto account. The adopted cost characterstcs are shown n Table I. TABLE I. THERMAL GENERATORS COST CHARACTERISTICS Generator Bus b (Rs/MWh) c (Rs/MW h) max (MW) G G G G G G G The cost characterstcs of the thermal plant as descrbed n (4), n whch coeffcent a represents the fxed cost of plant per hour. The fxed cost does not partcpate n the cost mnmzaton process, so a coeffcent s consdered as zero. Hydro power plant water dscharge characterstcs are adopted from the test systems presented n [11]. The sutable dscharge characterstcs for the present network are gven n Table II. TABLE II. Gen Bus WATER DISCHARGE COEFFICIENTS OF HYDRO LANTS x (m 3 /M W h) G G G y (m 3 /M Wh) z (m 3 /h) max (MW) Fgure. Daly load duraton curve Water Volume (Mm 3 ) The total scheduled perod one day (4 hours) s dvded nto 1 equal sub ntervals and each sub nterval conssts hrs duraton. Snce maxmum power capacty s avalable at G10, whch s located at Bus-39 s consdered as reference bus n ths study. The load demand patterns of real and reactve powers are shown n Table III and the typcal load duraton curve s shown Fg.. TABLE III. LOAD ATTERNS IN EACH SUB INTERVAL Sub nterval -load (MW) Q-load (MVAR) The algorthm was successfully appled to ths problem consderng all the possble constrants. The optmum power generaton schedule for hydro power plants s obtaned for the problem by applyng the procedure of hydro sub problem. The optmal hydro generaton schedule and water dscharge rate n each nterval are shown n Table IV and Table V respectvely. TABLE V. TABLE IV. OTIMAL HYDRO OWER SCHEDULE Interval G1(MW) G4(MW) G9(MW) WATER DISCHARGE RATE FOR EACH HYDRO LANT Interva l q1 (Mm 3 /h) q4 (Mm 3 /h) q9 (Mm 3 /h) Water values obtaned for the plants are gven as, v1 = Rs/m 3 ; v4 = Rs/m 3 ; v9 = 0.97 Rs/m 3 ; The power generaton schedule of hydro plants shown n Table IV wll acts as maxmum generatons respectvely for 5

6 further optmzaton process. The thermal sub problem s solved by followng Secton IV along wth Secton III (CLL method). The soluton of ths sub problem gves the optmal thermal generaton schedule, whch s presented n Table VI. TABLE VI. OTIMAL THERMAL GENERATION SCHEDULE ( MW) Sl G G3 G5 G6 G7 G8 G Fgure 3 gves the plot for optmal generatons of all generators wth respect to schedulng tme perods. The total runnng cost of the system s Rs Fgure 3. Optmal soluton for the problem VI. CONCLUSION In ths paper Hydro-Thermal schedulng s solved by usng a new approach whch ncludes the objectves of Hydrothermal schedulng and Closed Loop Lambda method. Ths approach has strong mathematcal foundaton. Ths approach takes both real and reactve power load patterns and lmts on generators nto account. Ths method wll be applcable to any practcal power system for onlne economc load dspatch. The results for IEEE New England 39 bus equvalent system llustrate the effectveness of ths algorthm. REFERENCES [1] I. A. Farhat and M. E. El-Hawary, Fxed-Head Hydro-Thermal Schedulng Usng a Modfed Bacteral Foragng Algorthm, IEEE Electrcal ower and Energy Conference, 010. [] H. Y. Yamn, Revew on methods of generaton schedulng n electrc power systems, Electrc ower System Research 69, 004, pp [3] R. W. Ferrero, Jorge F. Rvera and S.M.Shahdehpour, Effect of deregulaton on hydrothermal systems wth transmsson Constrants, Electrc ower System Research 38, 1997, pp [4] Abdul Halm, Abdul Rashd and Khald Mohamed Nor, An Effcent Method for Optmal Schedulng of Fxed Head Hydro and Thermal plants. IEEE Transactons on ower Systems, Vol.6,, May 1991, pp [5] D.. Kothar and J. S. Dhllon, ower System Optmzaton, rentce- Hall Inda, New Delh, 004. [6] M. E. El-Hawary and G. S. Chrstensen, Optmal Economc Operaton of Electrc ower Systems, Academc ress, New York,1979. [7] D. Thukaram, K. arthasarathy, B. S. R. Iyengar and T. S. Swamy, Fast Economc dspatch and allevaton of over-voltage under-voltage condtons n a large power system operaton, Natonal ower System Conference, Bombay, June [8] R. Ramanathan, Fast Economc Dspatch Based on The enalty Factors From Newtons Method, IEEE Transactons on AS, Vol. AS-104, 7, July [9] G. X. Luo, H. Habbollahzadeh and A. Semlyen, Short- Term Hydro- Thermal Dspatch Detaled model and Solutons, IEEE Transactons on ower Systems, Vol.4, 4, October 1989, pp [10] T. B. Nguyen and M. A. a, Dynamc Securty- Constraned Reschedulng of ower Systems Usng Trajectory Senstvtes, IEEE Transactons on ower Systems, Vol.18,, May 003,pp [11] J. S. Dhllon, S. C. art and D.. Kothar, Fuzzy decson makng n stochastc multobjectve Short term Hydro Thermal Schedulng, IEE roc.-gener. Transm. Dstrb.. Vol. 14,, March 00. [1] A. J. Wood and B. Wollenberg, ower Generaton, Operaton and Control, Wley, New York,