1 Introduction. Keywords: parallel pumping stations, decomposition, aggregation, dynamic programming, optimization, adjustable-blade

Transcription

1 Study of Optmal Operaton for Hua an Parallel Pumpng Statons wth Adustable-Blade Unts Based on Two Stages Decomposton-Dynamc Programmng Aggregaton Method Y Gong, Jln Cheng, Rentan Zhang,2, Lhua Zhang College of Hydraulc Scence and Engneerng, Yangzhou Unversty, Yangzhou, Chna; 2 Jangsu Surveyng and Desgn Insttute of Water Resources Co., Ltd., Yangzhou, Chna gongy_8@63.com, lcheng@yzu.edu.cn, r_zhang@yzu.edu.cn, lhzhang@yzu.edu.cn Abstract: Two-stage decomposton-dynamc programmng aggregaton method has been frst proposed and ntroduced to solve the mathematcal model of daly optmal operaton for parallel pumpng statons wth adustable-blade unts. Takng mnmal daly electrcty cost of sngle pump staton as obectve functon, the water quantty pumped by each staton as coordnated varable, by means of the type of the pump unts ths model s decomposed nto several frst-stage sub-model of daly optmal operaton wth adustable-blade for sngle pump staton. Then takng mnmal daly electrcty cost of sngle pump unt as obectve functon, the water quantty pumped by each unt as coordnated varable, the frst-stage sub-model s decomposed nto several second-stage sub-model of daly optmal operaton wth adustable-blade for sngle pump unt whch takes the blade angle as decson varable, the dscrete values of water quantty pumped by each unt as state varable, and s solved by means of dynamc programmng method. The constructed aggregaton model takes daly water quantty pumped by each pump unt as decson varable, the dscrete values of water quantty pumped by parallel staton group as state varable, and s also solved by dynamc programmng method. The aggregaton process replaces the tradtonal method of constructng equatons. Ths method has frst solved the optmal operaton ssues for mult-unts of parallel statons wth varous operaton modes, tme perod dvson and daly average head of each staton, and also provded theoretcal support for the study on optmal operaton of mult-stage pumpng statons. Takng Hua'an No., No.2, and No.4 parallel pumpng statons as a study case, a seres of optmzaton results have been obtaned. Keywords: parallel pumpng statons, decomposton, aggregaton, dynamc programmng, optmzaton, adustable-blade Introducton The parallel pumpng statons have a huge energy consumpton n operaton because of contanng a large number of pump unts, whch makes t necessary to develop the study of optmal operaton method for mult-unts of parallel pumpng statons. At present, the maor study methods of optmal operaton for parallel pumpng statons contan decomposton-coordnaton method and decomposton-aggregaton method. The former would brng tedous combnatons whle more pump unts exst. And the latter always establshes regresson equaton whle n aggregaton process, whch would affect the soluton precson of optmzaton model. Therefor, two-stage decomposton-dynamc programmng aggregaton method has been proposed and appled to the study. Takng No., No.2 and No.4 Hua,an Pumpng Staton as a study case, we dvde one day nto several tme perods accordng to the nlfluence of peak-valley electrctyty prce and the demands of not hgh frequency start-up/shut

2 down operaton, by whch to search the optmzaton beneft carred out by optmal operaton of mult-unts wth adustable-blade n parallel pumpng statons operaton. 2 Optmal daly operaton model and ts solvng method for mult-unts wth adustable-blade n parallel pumpng statons In order to be convenent for dscusson, a seres of defntons have been made whch are as follows: () Operaton mode Operaton wth fxed blade angle and constant speed: The pump unts are operatng wth ratng speed and the blade angle at the desgnng degree. Optmal operaton wth adustable-blade: The pump unts are operatng wth constant speed and adustng the blade angle n each tme perod accordng to operaton condtons n order to obtan the mnmal water pumpng cost. (2) Full load, 80% load, 60% load The operaton of pump unts lastng 24 duratve hours s called full load operaton. 80% load operaton and 60% load operaton respectvely represents the pump unts are operatng wth 80% and 60% of water quantty pumped by the unts whle they are operatng wth full load at fxed blade angle and constant speed. (3) Begnnng tme and the combnaton between length of tme perod and peak-valley electrctyty prce Consderng peak-valley electrctyty prce and the demand of avodng hgh frequency of start-up/shut down operaton, we choose begnnng tme at 7 :00 and dvde one day nto 9 perods. The tme length and the electrctyty prce n each of perod s shown n Table. Tab. Tme perod dvson and peak-valley electrcty prce of each tme perod Length Electrctyty Length Electrctyty Seral number Tme dvson Seral of tme prce/yuan Tme dvson of tme prce/yuan - number - perod/h kw - h perod/h kw - h Ⅰ 7:00~9: Ⅵ 07:00~09: Ⅱ 9:00~2: Ⅶ 09:00~: Ⅲ 2:00~23: Ⅷ :00~4: Ⅳ 23:00~03: Ⅸ 4:00~7: Ⅴ 03:00~07: Optmal model of daly operaton for mult-unts wth adustable-blade n parallel pumpng statons Takng mnmal daly electrcty cost of entre parallel pumpng group as obectve functon, the tme perod as stage varable, the blade angle of each pump unt n each tme perod as decson varable, the water quantty pumped n defnte tme perod and power of electromotor equpped n staton as the constrant condtons, the optmal mathematcal model of daly operaton for mult-unts wth adustable-blade n parallel pumpng statons has been construced as follows : Obectve functon: G = mn k = G k = mn( ρgq k ( θ ) H k k k = = = z, k mot, k n t, k ΔT P ) k k () Water quantty constrant: Q (θ ) ΔT W (2) k = = = k k e

3 Power constrant: N k ( θ k ) N k0 (3) Where G s the mnmal daly electrcty cost of the entre parallel pumpng group. G k s the daly electrcty cost of the k-th pumpng staton. s quantty of pumpng statons n the parallel pumpng group. s quantty of pump unts n each sngle staton. s the quantty of tme perods dvded n one day. ρ s water densty and g s acceleraton of gravty. H k and Q k (θ k ) whch s correspondng to the blade angleθ k respectvely represent the average daly head and flow of the -th pump unt n the k-th pumpng staton and n the -th tme perod. T k and P k respectvely represent the tme length and the peak-valley electrcty prce of the -th tme perod n the k-th pumpng staton. z,k (θ k),mot,k, nt,k respectvely represent the effcency of equpment, electromotor and transmsson of the -th pump unt n the k-th pumpng staton. Among them z,k s relatve to the flow and average head of the -th perod. mot,k could be regarded as constant when the load s over 60%, whle n large electromotor themot,k could be regarded as 94%. Also we consdered as thent,k value n drect ont unt. W e s the obectve water quantty pumped by the whole parallel pumpng group n one day. And N k (θk) s the actual electromotor power of the -th pump unt n the k-th pumpng staton and n the -th tme perod whle the pump unt s operatng under the blade angleθk, whch should be less than the ratng power of N k Two-stage decomposton-dynamc programmng aggregaton method 2.2. Large-scale two-stage decomposton Frst-stage decomposton Takng water quantty pumped by each pumpng staton as the coordnated varable, we decompose eq. ()~(3) nto frst-stage subsystems accordng to the type of pump unt wth the assumpton that the unts have the same type n the same pumpng staton. Then the optmal mathematcal model of daly operaton for mult-unts wth adustable-blade for sngle pumpng staton s obtaned whch s shown from eq. (4)~(6). Ths model takes mnmal daly electrcty cost of sngle pumpng staton as obectve functon, the blade angle of each pump unt n each tme perod as decson varable, the water quantty pumped n defnte tme perod and power of electromotor equpped n staton as the constrant condtons. Obectve functon: F = mn = F = mn( ρgq, ( θ ) ( θ ) H = = z, mot, nt, ΔT P ) (4) Water quantty constrant: = = Q (θ ) ΔT W k (5) Power constrant: N k ( θ ) N k0 (=,2,, ; =,2,,) (6) Where F s the mnmal daly electrcty cost of sngle pumpng staton. F s daly electrcty cost of the -th pump unt. W k s the obectve water quantty pumped by sngle pumpng staton n one day Meanngs of other varables could be obtaned by analogy accordng to eq. ()~(3) Second-stage decomposton Takng water quantty pumped by each pump unt as the coordnated varable, eq. (4)~(6) s decomposed nto second-stage subsystems accordng to the quantty of pump unt n one sngle

4 staton. Then the optmal mathematcal model of daly operaton for sngle pump unt wth adustable-blade s obtaned whch s shown from eq. (7)~(9). Ths model takes mnmal daly electrcty cost of sngle pump unt as obectve functon, the blade angle of each pump unt n each tme perod as decson varable, the water quantty pumped n defnte tme perod and power of electromotor equpped n staton as the constrant condtons. The blade angle s chosen at nteger degree n order to be convenent for practcal operaton. Obectve functon: M ρgq = (θ )H, mn F = ΔT P (=,2, ) (7) = z, mot, nt, Water quantty constrant: Q (θ ) ΔT W (=,2,,;=,2, ) (8) = Power constrant: N θ ) N (=,2,,;=,2, ) (9) ( 0 Where M s the mnmal daly electrcty cost of sngle pump unt. F s daly electrcty cost of the -th pump unt. W s the obectve water quantty pumped by sngle pump unt n one day. Meanngs of other varables could be obtaned by analogy accordng to eq. ()~(3) Optmzaton of second-stage subsystem Eq. (7)~(9) are typcal one-dmenson dynamc programmng model whose stage varable s (=, 2,,) and decson varable s the blade angle θ. Also we could know from eq. (8) that the water quantty pumped n each tme perod s the state varableλ. Makng use of dynamc programmng method to solve ths model, a seres of F values correspondng to obectve water quantty W could be obtaned. The solvng detals of ths model have been shown as follows: Stage : M ( ) ( θ ) ρgq H λ (0) = mn ΔT P z mot nt The stage varable λ s dscrete wthn ts feasble regon: λ = 0 2, L, W, W, W. The decson varable θ s dscrete wthn ts feasble regon for example: -4, -3, -2, -, 0, +, +2, +3, +4. Also the condton Q ( θ ) ΔT λ should be satsfed. Accordng to the performance curve of pump devce, the flow and equpment effcency z correspondng to each θ and H could be obtaned. Stage : g ( ) ( θ ) ρgq H λ = mn[ ΔT P + g ( λ )] () z mot nt λ and θ are dscrete n the same way as above. Also Q ( θ ) ΔT λ should be satsfed. Accordng to eq. (8), the state transton equaton s as follows: λ Q ΔT (=2,3,,-) (2) = λ ( θ ) Stage : g ρgq ( θ ) H λ ) = mn[ ΔT P + g ( λ )] (3) ( z mot nt Where λ = W, W ;θ + = -4, -3, -2, -, 0, +, +2, +3, +4

5 = λ Q ΔT The state transton equaton: ( ) λ θ ( λ W ) (4) Wth the assumpton that pump statons are contaned n one parallel pumpng group and the pump unts contaned n each pumpng staton have the same type wth no performance dfference whle the pump unts n dfferent statons have dfferent types, each pumpng staton has one knd of performance curve of pump unt. Each pump unt has a blade angle correspondng to the maxmal flow wthn the power constrant under the average head of each tme perod. After takng a defnte water quantty step to dsperse the total water quantty W,max whch corresponds to the maxmal blade angle of all tme perods, the optmal mathematcal model of daly operaton for sngle pump unt wth adustable-blade could be appled to calculate the mnmal daly cost of sngle pump unt F,m (m =, 2,,max) whch respectvely corresponds to each water quantty W,m. Wth the fact that the pump unts n one staton have the same type wth no performance dfference, each pumpng staton only needs one optmal soluton. Therefor, groups of optmal soluton should be done n one parallel pumpng group, after whch W km ~F km (W km ) relatonshp could be obtaned Dynamc programmng aggregaton of large-scale system After a seres of W km ~F km (W km ) relatonshps are obtaned by means of the second-stage submodel solutons(k=,2,,;=,2,,;m=,2,,max), eq. ()~(3) could be transformed nto the followng aggregaton model. Obectve functon: G = mn F (5) k ( W k ) k = = Water quantty constrant: (6) W k W e k = = Power constrant: N k ( θ k ) N (7) k0 Takng pump statons as a suppostonal staton wth AZ pump unts (AZ= ), eq. (5) ~ (7) are also one-dmenson dynamc programmng model whose stage varable s n (n=, 2,, AZ), the decson varable W n s the daly water quantty pumped by each unt, and the state varable λ s the dscrete value of water quantty pumped by all unts. Applyng dynamc programmng method to solve the model above, the mnmal daly electrcty cost of entre parallel pumpng group correspondng to the obectve water quantty W e could be obtaned, by whch the optmal water quantty of each pump unt W * n (n=,2,,az)could be obtaned. The solvng detals of ths model are smlar to Chapter After gettng a seres of W n * values(n=,2,,az), by means of the results of solvng the second-stage subsystem whch s the optmal mathematcal model of daly operaton for sngle pump unt wth adustable-blade, we could get a seres of optmal operaton schemes of each pump unt whch s the optmal blade angle θ * n (=,2,,;n=,2,,AZ) n each tme perod correspondng to each W n *(n=,2,,az). 2.3 Analyss of optmal operaton for mult-unts wth adustable-blade n Hua an parallel pumpng statons 2.3. Basc nformatons of Hua an parallel pumpng statons The basc nformaton of No., No.2 and No.4 Hua an Pumpng Statons whch are the second stage statons n the Eastern Route of the South-to-North Water Transfer Proect s shown n Table 2. Durng

6 the optmzaton process, one standby unt contaned n No.4 Hua an Pumpng Staton s not consdered. Table 2. Basc nformaton of No., No.2 and No.4 Hua an Pumpng Statons Pumpng staton Type of pump unt Unt quantty Impeller dameter /mm Rated speed /r mn - Match motor power /kw Rated blade angle/ Range of adustableblade No. Axal-flow pump ~+4 No.2 Axal-flow pump ~+4 No.4 Axal-flow pump ~+4 The upstream and downstream rvers of Hua an parallel pumpng statons have a bg enough cubage, whch makes the daly head has a small change scope. Therefor, wth the assumpton that the daly average head has a constant value, and wthn the feasble doman of parallel pumpng statons, we dsperse t nto 6 average daly heads whch are 3.3m, 3.53m, 3.93m, 4.3m, 4.53m and 4.93m. In each of daly average head, full load, 80% load and 60% load of water quantty correspondng to operaton wth fxed blade angle and constant speed are consdered as the optmal obectve water quantty. Takng use of two-stage decomposton-dynamc programmng aggregaton method, we could get the electrcty cost per 0 4 m 3 water quantty correspondng to mnmal daly electrcty cost of entre parallel pumpng group under each daly average head and operaton load Optmzaton results of optmal operaton model for mult-unts wth adustable-blade n Hua an parallel pumpng statons Makng use of the method above, the optmal operaton scheme of No., No.2 and No.4 Hua an Pumpng Staton under each daly average head and operaton load could be obtaned. Takng the daly average head of 4.3m, 80% load for example, the optmal operaton scheme s shown n Tab. 3 whose electrcty cost per 0 4 m 3 water quantty s yuan/0 4 m 3. Fg. shows the electrcty cost per 0 4 m 3 water quantty of optmzaton under each operaton load. And Fg.2 shows the optmal water quantty allocaton among all pmup unts respectvely under full load, 80% load and 60% load whle the daly average head s 4.3m. Cost per unt/yuan 0 4 m Daly average head/m full load 80% load 60% load Water quantty/0 4 m Number of pump unt Fg. Unt cost of water pumpng under Fg 2. Optmal water quantty allocaton the optmal operaton wth among unts under dfferent loads adustable -blade and daly average head of 4.3m full load 80% load 60% load

7 Pumpng staton No. No.2 No.4 Tab 3. Optmal operaton schemes of 80% load wth adustable-blade under daly average head of 4.3m consderng peak-valley electrcty prce Tme perod Unt number Unt Stop Stop Stop Stop -2-2 Unt 2 Stop Stop Stop Stop -2-2 Unt 3 Stop Stop Stop Stop -2-2 Unt 4 Stop Stop Stop Stop -2-2 Unt 5 Stop Stop Stop Stop -2-2 Unt 6 Stop Stop Stop Stop -2-2 Unt 7 Stop Stop Stop Stop -2-2 Unt 8 Stop Stop Stop Stop -2-2 Unt Stop Stop Unt Stop Stop Unt Stop Stop Stop Stop + + Unt 2 Stop Stop Stop Stop Unt 3 Stop Stop Stop Stop Dscusson of optmzaton results on optmal operaton of mult-unts wth adustable-blade n Hua an parallel pumpng statons Analyzng upon the fgures and tables obtaned from the optmzaton on mult-unts wth adustable-blade n Hua an parallel pumpng statons by two-stage decomposton-dynamc programmng aggregaton method amng to each daly average head and operaton load, followng results could be obtaned. () Average electrcty cost per 0 4 m 3 water quantty of all daly average heads correspondng to full load, 80% load and 60% load operaton are respectvely 94.60yuan/0 4 m 3, 78.98yuan/0 4 m 3 and 64.37yuan/0 4 m 3. (2) Optmzaton results show that shut-down perods always appear n the perod of hgh electrcty prce (0.978 yuan/kw h) and whle n operaton perod the hgh prce corresponds to small blade angle of pump unt and vce versa. In the meantme, t s necessary to ncrease the shut-down perods nstead of operatng at the mnus blade angle n order to save the electrcty cost. That means there s a preferental consderaton of controllng the number of operaton unts, after whch the adustable-blade measure would be taken. (3) Fg.2 shows that as a result of hgher unt performance of No.2 Hua an pumpng staton, the water quantty dstrbuted to No.2 staton s more that the others, whch reflects effcency prorty prncple. (4) As the two-stage decomposton-dynamc programmng aggregaton method frstly takes optmal operaton calculaton for sngle pump unt wth adustable-blade by means of dynamc programmng, after whch the general coordnaton of water quantty by means of the aggregaton model s taken, we could obtan optmal operaton mode under the dfferent blade angles of each pump unt n the same tme perod. Therefore, ths method s sutable for solvng the optmal daly operaton problems of parallel pumpng statons wth dfferent daly average heads, dfferent tme perod dvsons and dfferent adustable mode of pump unt n each pumpng staton.

8 3 Concluson Two-stage decomposton-dynamc programmng aggregaton method s frst put forward to solve the optmal mathematcal model of daly operaton for mult-unts wth adustable-blade n parallel pumpng statons, by whch the notable optmzaton results could be obtaned. Ths method has a general gudng sgnfcance for the optmzaton problems of complex nonlnear mathematcal models whch are smlar to eq. ()~(3), and could solve the optmal daly operaton problems of parallel pumpng statons wth dfferent daly average heads, dfferent tme perod dvsons and dfferent adustable mode of pump unt n each pumpng staton. Besdes, a set of optmal operaton schemes of parallel pumpng statons under dfferent daly average heads and operaton loads have been establshed by means of calculatng typcal parallel pumpng statons, whch could offer references for the optmal operaton of parallel pumpng statons wth small daly average head ampltude, and also make the study bass for the optmal operaton for mult-stage pumpng statons. References. Lu Jachun, Zhang Zxan, Zhang Mufe, et al: Determnaton of economcal operaton program at dranng pump staton, Dranage and Irrgaton Machnery. J. 24(6), (2006) 2. Farhad Ghassem-Tar, Eshagh Jahangr: Development of a hybrd dynamc programmng approach for solvng dscrete nonlnear knapsack problems, Scence Drect. J. 88, (2007) 3. Chen Shoulun, Ru Jun, Xu Qng, et al: Daly Optmal Operaton for Pumpng Statons. Hydroelectrc Energy. J. 2(3), (2003) 4. Zhu Huzhu, Xa Fuzhou: Applcaton of decomposton-aggregaton method of large-scale system on whole structural optmzaton of arch aqueduct. Journal of Hydraulc Engneerng. J. 0, --7 (995) 5. Shaaban.Hassan, Gruc.Lubomr: Decomposton-aggregaton method appled to a multmachne power system. Large Scale Systems. J. 0(2), (986) 6. Cheng Jln, Zhang Lhua, Zhang Rentan, et al: Study on optmal day-operaton of sngle adustable-blade pump unt. Journal of Hydraulc Engneerng. J. 4(4), (200) 7. Wang Dezh, Dong Zengchuan, Dng Shengxang: Research on aggregaton-decomposton-coordnaton model of feedng reservor group. Journal of Hoha Unversty (Natural Scences). J. 34(6), (2006) 8. L.Cooper, Mary W.Cooper: Introducton to Dynamc Programmng. Pergamon Press, New York (98)